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