chiral dynamics how s and why s
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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. - PowerPoint PPT PresentationTRANSCRIPT
Chiral DynamicsChiral DynamicsHowHowss and Why and Whyss
1st lecture: basic ideas
Martin Mojžiš, Comenius University23rd Students’ Workshop, Bosen, 3-8.IX.2006
effective theories
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
a a fundamentalfundamental
theorytheory
an effectivean effective theorytheory
certain circumstances:certain circumstances:the same resultsthe same results
valid inmuch wider range
calculationsconsiderably simpler
derivation
some examples
underlying theory effective theory
general relativity Newtonian gravity
kinetic theory hydrodynamics
electroweak SM Fermi theory
QCD ChPT
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin 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 reconcilethe principles of quantum mechanics with those of special relativity.
if possible at all, it has to be QFT
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
the most general relativistic Lagrangian the most general relativistic Lagrangian
should include all relativistic quantum physicsshould include all relativistic quantum physics
a simple example
(x)xdL L 4 2 3 4
5
21
2 2
21
2 213151 0 dmceeddcc 23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
a scalar field φ(x)
21
321
55
44
33
221
ee
ddd
ccccc(x)L
why do some constants vanish
• c1 φ redefinition of fields
• c5 φ5 renormalizability
• d1 μμφ 4-divergence
• d3 φμμφ linear combination: d2 + 4-div
• e1 μμννφ renormalizability
• e2 μμφννφ renormalizability23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
the most important constrains
exploited substantially
in EFT
relaxed completely
in EFT
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
non-renormalizable non-feasible
• 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
n
n)(eff LL
• effective field theory needs some organizing principle
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin 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
n
n)(eff LL
21
321
55
44
33
221eff
ee
ddd
cccccL (0)L(2)L(4)L
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin 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 Nth 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?
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
the answer for spinless massless particles
for any Feynman diagram the amplitude
is a homogeneous function of external momenta
pi pi Mfi Mfi
NL # of loops
NI # of internal lines
d # of derivatives (in the vertex)
Nd # of vertices with d derivatives
to do list
1. prove this
2. show, how this answers the question
d dIL dNNN 24 ω
(which will turn out to be relevant later on)
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
• 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
the proof for the tree diagrams
d dI dNN 2 ω
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
the proof for the loop diagrams
),kpf(...kd ) ,kf(p...kd jiji 14
14
ii pp
ii kk substitute
)k,pf(...kd ) ,kpf(...kd jiji 144
14
• amplitude amplitude d dIL dNNN 24 ω
• dimensional regularization does not spoil the picture ( ln 0 )
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
the consequences (the answer)
d dIL dNNN 24 ω1 VLI NNN
d dL dNN )2( 22
• bonus: a systematic order-by-order renormalization
• if for some reason (0) = (1) = 0then to an amplitude of order to an amplitude of order
only only (n) with with n = dn = d can contribute can contribute
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
order-by-order renormalization
d dL dNN )2( 22
• 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
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
21
321
55
44
33
221eff
ee
ddd
cccccL (0)L(2)L(4)L
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin 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
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
the symmetries of QCD
... qMDiqQCDL
s
d
u
q
000
000
000
000
s
d
u
m
m
m
M
igAD
various accidental approximate symmetriesevery relevance for EFT(ChPT is based on these symmetries)
fundamental SU(3)-color symmetryno relevance for EFT(since hadrons are color singlets)
23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
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
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 enough23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
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 QCD23rd Students’ Workshop, Bosen, 3-8.IX.2006 Martin Mojžiš, Comenius University
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