Resonance Chiral Theory towards a "Swiss knife" for low

Transcription

Resonance Chiral Theory
towards a "Swiss knife" for low-energy
particle interactions
Sergiy IVASHYN
[email protected]
Instytut Fizyki, Uniwersytet Śla̧ski, Katowice
and
A.I. Akhiezer Inst. for Theoretical Physics
NSC KIPT, Kharkov, Ukraine
12 May 2009 @ IF US
1
2
3
4
Introduction
Sample problems
ρ → e+ e−
ρ → π+π−
Pion and kaon form factors
Building the framework
Power counting
Symmetries and external fields
Conclusions
2 / 45
Universal tool for low-energy particle phenomenology
Lagrangian-based framework
light pseudoscalar mesons (π, K , η)
vector resonances (ρ, ω, φ, . . . )
axial-vector resonances (a1 , . . . )
scalar resonances (a0 , f0 , . . . )
strong interactions
electromagnetic interactions
weak interactions
Preliminaries
3 / 45
Strong interactions in effective theories
Hadrons at low (and intermediate) energies
use known symmetries
work in terms of hadronic fields
respect the phenomenology
Introduction
4 / 45
Strong interactions in effective theories
Hadrons at low (and intermediate) energies
use known symmetries
work in terms of hadronic fields
respect the phenomenology
VMD, vector meson dominance
sigma-models (LσM [ Schwinger 1957; Gell-Mann, Lévy 1960 ], NLσM)
unitarized chiral theory UχPT [ Oset; Oller nucl-th/0207086; Oller, Meissner (2000), (2001) ]
IAM, inverse amplitude method
[ Dobado, Pelaez Phys. Rev. D 56 (1997) ]
PKU dispersive representation
[ H.Q. Zheng, Z.Y. Zhou; see refs. in arXiv:0905.0528 ]
HMET
[ Jenkins, Manohar, Wise Phys. Rev. Lett. 75 (1995) ]
Heavy baryon χPT
[ Jenkins, Manohar Phys. Lett. B 255 (1991) ]
Resonance chiral theory RχT
Introduction
[ Ecker et al. Nucl. Phys. B 321 (1989); Phys. Lett. B 223 (1989) ]
4 / 45
What people do with RχT
direct application for given decays
Introduction
5 / 45
What people do with RχT
direct application for given decays
Beyond the scope of the seminar:
consistent SU(3) flavour breaking
renormalizable sectors
[ Cirigliano et al. JHEP 0306 (2003) ]
[ Rosell et al. JHEP 0512 (2005); Sanz Cillero ]
short-distance constraints
[ Jamin et al. Nucl. Phys. B 622 (2002) ]
resonance “saturation” of ChPT LEC’s
resonance “saturation” up to O(p6 )
[ Cirigliano et al. Nucl. Phys. B753 (2006) ]
SU(3) flavour symmetric case
resonance “saturation” at NLO in 1/N − c
[ Knecht, Nyffeler EPJ C 21 (2001) ]
[ Rosell et al. arXiv:0903.2440 ]
Tensor field vs. vector field for vector mesons
Introduction
5 / 45
To whom may it concern?
Agreed to apply RχT soon for given problem
How to start?
What exactly is needed?
Helpful tricks?
Not going to use RχT , but..
Use of the basic ideas?
General model-building?
What’s up? (“Swiss knife”, heh?)
Introduction
Outline of seminar
6 / 45
Target consumers of seminar
Common: had not worked out the model yet
Before starting, estimate how long it’s going to take
Expected achievements?
What is possible to do with this tool?
Introduction
Outline of seminar
7 / 45
Quick start guide
1
“Read in manual” what to take and what to do
Lagrangian terms
Introduction
Outline of seminar
8 / 45
Quick start guide
1
Lagrangian terms
2
Work out ( and compare with data, . . . )
Couplings?
Widths, form factors
Introduction
Outline of seminar
8 / 45
Quick start guide
1
Lagrangian terms
2
Couplings?
3
Understand why to take/do this-or-that
(Theoretical grounds of model building)
Introduction
Outline of seminar
8 / 45
Quick start guide
1
Lagrangian terms
2
Couplings?
3
Understand why to take/do this-or-that
(Theoretical grounds of model building)
4
Limits of applicability
5
Troubleshooting
In a “review" some steps would be hidden, and the order would differ
Introduction
Outline of seminar
8 / 45
Contents
1
2
3
4
Sample problems
Introduction
Sample problems
ρ → e+ e−
ρ → π+π−
Power counting
Conclusions
ρ → e+ e−
9 / 45
Leptonic electromagnetic decays of vector mesons
FV L = √ Vµν f+µν
2 2
[ Lagrangian — Ecker et al. Phys. Lett. B 223 (1989) ]
Sample problems
ρ → e+ e−
10 / 45
2 2
Vµν — tensor field for vector meson:
√
 0 √

+
∗+
ρ
K
ρ / 2 + ω/ 2
√
√
Vµν =  ρ−
−ρ0 / 2 + ω/ 2 K ∗0 
K ∗−
K̄ ∗0
φ µν
Sample problems
ρ → e+ e−
10 / 45
2 2
√
 0 √

+
∗+
ρ
K
ρ / 2 + ω/ 2
√
√
Vµν =  ρ−
−ρ0 / 2 + ω/ 2 K ∗0 
K ∗−
K̄ ∗0
φ µν
f+µν ⇒ eF µν (uQu † + u † Qu)
F µν — e.m. field strength tensor
Q = diag( 23 , − 13 , − 13 ) — quark charge matrix
√
Φ2
u = exp(iΦ/ 2F ) = 1 + i √Φ − 4F
2 + ···
2F
thus f+µν ⇒ 2QeF µν + O(Φ2 )
Sample problems
ρ → e+ e−
10 / 45
2 2
√
 0 √

+
∗+
ρ
K
ρ / 2 + ω/ 2
√
√
Vµν =  ρ−
−ρ0 / 2 + ω/ 2 K ∗0 
K ∗−
K̄ ∗0
φ µν
f+µν ⇒ eF µν (uQu † + u † Qu)
F µν — e.m. field strength tensor
√
Φ2
u = exp(iΦ/ 2F ) = 1 + i √Φ − 4F
2 + ···
2F
thus f+µν ⇒ 2QeF µν + O(Φ2 )
LγV
Sample problems
= eFV F µν
1 0
1
1
ρµν + ωµν − √ φµν
2
6
3 2
ρ → e+ e−
10 / 45
Vertex function: eFV [gνλ qµ − gνµ qλ ] × coeff
coeff
ρ
ω
φ
1
2
1
6
−1
√
3 2
normalization for tensor field: h0|vµν (0)|v , Qi = i
[Qµ Eν −Qν Eµ ]
MV−1
[ Ecker et al. Nucl. Phys. B 321 (1989) ]
Eµ — vector meson polarization vector; Eµ Q µ = 0
Γρ→e+ e− =
Sample problems
e4 FV2
12πMρ
ρ → e+ e−
11 / 45
Contents
1
2
3
4
Sample problems
Introduction
Sample problems
ρ → e+ e−
ρ → π+π−
Power counting
Conclusions
ρ → π+ π−
12 / 45
ρ → π+π−
iGV
L = √ hVµν u µ u ν i
2
Sample problems
ρ → π+ π−
13 / 45
ρ → π+π−
iGV
2
uµ
i[u † (∂µ − irµ )u − u(∂µ − ilµ )u † ]
√
√
2
2
∂µ Φ −
ieBµ [Φ, Q] + O(Φ3 )
⇒ −
F
F
(lµ , rµ ⇒ −eQBµ )
=
√
√

π 0 / 2 + η8 / 6
π+
K+
√
√
Φ =  π−
−π 0 / 2 + η8 / 6
K0 
√
−
0
K
K̄
−2η8 / 6

Sample problems
ρ → π+ π−
13 / 45
ρ → π+π−
iGV
2
keeping only O(Φ2 ) terms
and dropping the terms with photon field Bµ ,
LVPP
= iGV
1
fπ2
(2 ρ0µν ∂ µ π + ∂ ν π − )
√
1 0
(ρµν + ωµν − 2φµν )(∂ µ K + ∂ ν K − )
2
fK
√
1
+ 2 (−ρ0µν + ωµν − 2φµν )(∂ µ K 0 ∂ ν K̄ 0 ) .
fK
+
Sample problems
ρ → π+ π−
13 / 45
ρ → π+π−
Vertex function:
π ± (fP = fπ )
K ± (fP = fK )
K 0 (fP = fK )
GV
2fP2
ρ
2
1
−1
−+
lµ lλ − lµ+ lλ− × coeff
ω
0
1
1
Γtot,ρ (Q 2 ) =
Sample problems
φ
0
√
−√2
− 2
3/2
GV2 Mρ2
Q 2 − 4mπ2
4
2
48πfπ Q
ρ → π+ π−
14 / 45
Contents
1
2
3
4
Sample problems
Introduction
Sample problems
ρ → e+ e−
ρ → π+π−
Power counting
Conclusions
15 / 45
γ → π+π−
L=
F2
huµ u µ i
4
recall
uµ
Sample problems
i[u † (∂µ − irµ )u − u(∂µ − ilµ )u † ]
√
√
2
2
⇒ −
∂µ Φ −
F
F
(lµ , rµ ⇒ −eQBµ )
=
16 / 45
γ → π+π−
L=
F2
huµ u µ i
4
√
uµ
√
2
2
= −
∂µ Φ −
F
F
↔
LγPP
↔
= −ieBµ (π + ∂µ π − + K + ∂µ K − ),
↔
a ∂µ b ≡ a ∂µ b − b ∂µ a
Sample problems
16 / 45
γ → π+π−
Vertex function: i e (q + − q − )ν
coincides with the scalar QED result
Sample problems
17 / 45
Electromagnetic form factor
µ
hP1 (q1 )P2 (q2 )|Jem
(0)|0i ≡ (q1 − q2 )µ FP (Q 2 )
All possible intermediate vector resonances V = ρ0 , ω, φ, ... in
general contribute.
For real photons only the first term on the r.h.s. is non-zero.
Sample problems
18 / 45
µ
hP1 (q1 )P2 (q2 )|Jem
(0)|0i ≡ (q1 − q2 )µ FP (Q 2 )
π
(Q 2 ) = 1 −
Fem
FV GV 2
Q Dρ (Q 2 ),
fπ2
p
DV (Q 2 ) = [Q 2 − MV2 + ı Q 2 Γtot,V (Q 2 )]−1 .
Γtot,ρ (Q 2 ) =
Sample problems
GV2 Mρ2
2
2 3/2
Q
−
4m
π
48πfπ4 Q 2
18 / 45
µ
hP1 (q1 )P2 (q2 )|Jem
(0)|0i ≡ (q1 − q2 )µ FP (Q 2 )
In SU(3) symmetry limit; charged kaon
K+
Fem
(Q 2 )
FV GV 2
= 1−
Q
fK2
1
1
1
2
2
2
Dρ (Q ) + Dω (Q ) + Dφ (Q ) ,
2
6
3
For advanced approach see [Ivashyn, Korchin EPJ C 49 (2007)]
p
DV (Q 2 ) = [Q 2 − MV2 + ı Q 2 Γtot,V (Q 2 )]−1 .
Γtot,ρ (Q 2 ) =
Sample problems
GV2 Mρ2
2
2 3/2
Q
−
4m
π
48πfπ4 Q 2
18 / 45
µ
hK 0 (q1 )K̄ 0 (q2 )|Jem
(0)|0i ≡ (q1 − q2 )µ FP (Q 2 )
In SU(3) symmetry limit; neutral kaon
K0
Fem
(Q 2 )
=
FV GV 2
Q
fK2
1
1
1
2
2
2
Dρ (Q ) − Dω (Q ) − Dφ (Q ) ,
2
6
3
For advanced approach see [Ivashyn, Korchin EPJ C 49 (2007)]
p
DV (Q 2 ) = [Q 2 − MV2 + ı Q 2 Γtot,V (Q 2 )]−1 .
Γtot,ρ (Q 2 ) =
Sample problems
GV2 Mρ2
2
2 3/2
Q
−
4m
π
48πfπ4 Q 2
18 / 45
Constraints on the form factor behavior
Pion form factor:
π
Fem
(Q 2 ) = 1 −
FV GV
Q2
fπ2 Q 2 − Mρ2
1
(Q 2 → 0)
π (Q 2 ) = 1
limQ 2 →0 Fem
2
(Q 2 → ∞)
π (Q 2 ) = 0
limQ 2 →∞ Fem
π
Fem
(Q 2 ) ≈
Mρ2
FV GV
1−
−
+ ···
fπ2
Q2
Thus one gets a short-distance constraint for effective couplings:
FV GV = fπ2
Sample problems
19 / 45
Electromagnetic vertex for the off-shell pseudoscalar
Γµ (q + , q − ) = −ie(q + − q − )µ Fem (Q 2 )
Q = q+ + q−
Sample problems
20 / 45
Contents
1
2
3
4
Introduction
Sample problems
ρ → e+ e−
ρ → π+π−
Power counting
Conclusions
Power counting
21 / 45
Chiral perturbative expansion
Lagrangian is organized as series in the masses of light quarks mq
and derivatives ∂µ acting on the pseudoscalar NGB fields
Leff = L2 + L4 + L6 + ...,
where
L2n ∼ O(p2n ).
Expansion coefficients are called low energy constants
Power counting
22 / 45
Chiral perturbative expansion
Lagrangian is organized as series in the masses of light quarks mq
and derivatives ∂µ acting on the pseudoscalar NGB fields
Leff = L2 + L4 + L6 + ...,
where
L2n ∼ O(p2n ).
Expansion coefficients are called low energy constants
Generic RχT Lagrangian at O(p6 ):
0
0
R
RR
R
+ LRR
LRχT
= LNGB
+ LNGB
+ LNGB
+ LR
0
2 + L4 + L2
2
4
6
6
00
RχT Lagrangian at O(p4 ) of [ Ecker et al. Nucl. Phys. B 321 (1989); Phys. Lett. B 223 (1989) ]
LRχT
= LNGB
+ LR
2
2
4
Power counting
22 / 45
Power counting
[ Weinberg Physica 96 A (1979) ]
Chiral power counting scheme
LEC’s ∼ R ∼ O(1)
u ∼ R ∼ O(1)
∂µ ∼ lµ ∼ rµ ∼ uµ ∼ O(p)
mq ∼ O(p2 ) (because mq ∝ M 2 , where M is the NGB mass)
s, p ∼ O(p2 )
χ ∼ χ± ∼ O(p2 )
Power counting
23 / 45
Power counting
LEC’s ∼ R ∼ O(1)
u ∼ R ∼ O(1)
∂µ ∼ lµ ∼ rµ ∼ uµ ∼ O(p)
s, p ∼ O(p2 )
χ ∼ χ± ∼ O(p2 )
(Exercise 1)
LNGB
=
2
χ+ = u † χu † + uχu,
F2 µ
hu uµ + χ+ i
4
χ ≡ 2B0 (s + i p) ≈ 2B0 diag(mu , md , ms )
Power counting
23 / 45
Power counting
LEC’s ∼ R ∼ O(1)
u ∼ R ∼ O(1)
∂µ ∼ lµ ∼ rµ ∼ uµ ∼ O(p)
s, p ∼ O(p2 )
χ ∼ χ± ∼ O(p2 )
(Exercise 2)
2 2
f±µν = uFLµν u † ± u † FRµν u,
FRµν = ∂ µ r ν − ∂ ν r µ − i[r µ , r ν ],
FLµν = ∂ µ l ν − ∂ ν l µ − i[l µ , l ν ]
Power counting
23 / 45
Main RχT Lagrangian
The lowest-order resonance piece [ Ecker et al. Nucl. Phys. B 321 (1989); Phys. Lett. B 223 (1989) ]
R
LR
= LR
2
kin + Lint
LR
kin
*
M2
∇ Rλµ ∇ν R − R Rνµ R νµ
2
R=V ,A
E
1 X D ν
∇ R ∇ν R − MR2 R 2 ,
+
2
1 X
= −
2
λ
+
νµ
R=S,P
∇µ X = ∂µ X + [Γµ , X ],
Γµ =
1 †
{u (∂µ − irµ )u + u(∂µ − ilµ )u † }
2
(with antisymmetric tensor representation for spin-1 fields Vµν , Aµν )
Power counting
24 / 45
Main RχT Lagrangian
The lowest-order resonance piece [ Ecker et al. Nucl. Phys. B 321 (1989); Phys. Lett. B 223 (1989) ]
R
LR
= LR
2
kin + Lint
LR
int
=
iG
iF FV √ Vµν f+µν + √ V hVµν u µ u ν i + √A Aµν f−µν
2 2
2
2 2
µ
+cd hSuµ u i + cm hSχ+ i + idm hPχ− i ,
Power counting
24 / 45
Large-Nc
[ t’ Hooft Nucl. Phys. B 772 (1974) ]
Single flavour trace terms dominate.
Each additional flavour trace brings a suppression of order 1/Nc
Meson resonances, are the narrow states with equal masses
within the multiplet in large-Nc limit
Power counting
25 / 45
Contents
1
2
3
4
Introduction
Sample problems
ρ → e+ e−
ρ → π+π−
Power counting
Conclusions
26 / 45
Symmetries of Lagrangian
Fundamental theory: massless QCD
1 a µν a
L0QCD = − Gµν
G
+ i q̄L γ µ Dµ qL + i q̄R γ µ Dµ qR
4
Color structure
local color SU(3)c
a — gluon field Ga strength tensor
Gµν
µ
a
Dµ = ∂µ − igs λ2 Gµa — covariant derivative
Gell-Mann matrices λa (a = 1, ..., 8) in the color space
gs is the strong interaction constant.
27 / 45
1 a µν a
L0QCD = − Gµν
G
4
Chiral structure
The “right” and “left” quark spinor fields (chiral basis) ⇔ Dirac
spinor q
1
1
(1 + γ5 )q + (1 − γ5 )q ≡ qR + qL
2
2
Left and right massless fermions do not communicate with each
other
q=
27 / 45
1 a µν a
L0QCD = − Gµν
G
4
Flavour structure
quark fields q have got flavour indices: q = (u, d, s)T ⇔ q b
Lagrangian L0QCD has global SU(3)L × SU(3)R flavour symmetry
qL → UL qL ,
qR → UR qR ,
b
UR/L = exp(iλb θR/L
/2)
(in addition to global U(1)V )
The axial global U(1)A symmetry is broken on the quantum level
27 / 45
1 a µν a
L0QCD = − Gµν
G
4
Flavour structure
Important in building the effective theory
There is a flavour manifestation in QCD spectrum.
Meson, baryon fields can not be only flavour singlets; they may carry
flavour indices as well.
27 / 45
What’s “wrong” at this point?
We are about to have an effective theory at ≈ 1 GeV.
Mass pattern for the light quarks
Can not neglect it!
Masses are due to interaction with the scalar field (Higgs
mechanism)
s = diag(mu , md , ms )(1 + H/v ) = diag(mu , md , ms ) + . . .
( VEV of the scalar field v ≈ 246 GeV )
But... how to handle the chiral symmetry?
28 / 45
What’s “wrong” at this point?
We are about to have an effective theory at ≈ 1 GeV.
Important in building the effective theory
Quark masses can be related to the light pseudoscalar masses via the
quark condensate hq̄qi
[ GOR Phys. Rev. 175 (1968) ]
−
3Mη28
Mπ20,±
MK2 0
MK2 ±
hq̄qi
=
=
=
=
mu + md
md + ms
mu + ms
mu + md + 4ms
F2
hq̄qi ≈ (−240 ± 10 MeV)3 (at scale µ = 1 GeV)
But... how to handle the chiral symmetry?
28 / 45
External sources
Generic external vector, axial-vector, scalar and pseudoscalar fields:
vµ = vµb λb /2,
aµ = aµb λb /2,
s = sb λb ,
p = p b λb
b = 0, 1, . . . , 8
λb — Gell-Mann matrices in the flavour space of the light quarks
Properties of the external sources
hermitean: vµ† = vµ , aµ† = aµ , s† = s, p† = p
3 × 3 matrices in flavour space (mind λb )
vector and axial sources are traceless: hvµ i = haµ i = 0
scalar source s contains the quark mass matrix
29 / 45
QCD interaction with external fields
[ Gasser, Leutwyler Ann. Phys. 158 (1984); Nucl. Phys. B 250 (1985) ]
Lext. fields = q̄[γ µ (vµ + γ5 aµ ) − (s − ipγ5 )]q
= q̄L γ µ lµ qL + q̄R γ µ rµ qR − q̄R (s + ip)qL − q̄L (s − ip)qR
rµ = vµ + aµ ,
q = (u, d,
s)T
lµ = vµ − aµ — right and left vector fields
— quark fields
In general, the source term of QCD Lagrangian is not invariant under
chiral transformations!
30 / 45
External sources w.r.t. global chiral transformations
Let us require
(see spurion method)
rµ → UR rµ UR† ,
lµ → UL lµ UL† ,
s + ip → UR (s + ip) UL† ,
s − ip → UL (s − ip) UR† .
then, the Lagrangian is invariant!
30 / 45
External sources w.r.t. global local chiral transformations
One can require even more:
rµ → UR (x) rµ UR (x)† + iUR (x) ∂µ UR (x)† ,
lµ → UL (x) lµ UL (x)† + iUL (x) ∂µ UL (x)† ,
s + ip → UR (x) (s + ip) UL (x)† ,
s − ip → UL (x) (s − ip) UR (x)† .
which, also leave Lagrangian invariant!
(Exercise)
30 / 45
SU(3)L × SU(3)R ⇒ SU(3)L+R=V
Chiral symmetry SU(3)L × SU(3)R of Lagrangian is hidden
(spontaneously broken).
[ Nambu, Goldstone ]
G = SU(3)L × SU(3)R = SU(3)R−L × H
H = SU(3)L+R=V
The mesonic degrees of freedom transform under G in a
complicated way.
We’ve just introduced local chiral symmetry
⇒ non-trivial geometry on G/H, chiral covariant derivative, etc.
31 / 45
External sources in the RχT Lagrangian:
resonance part
R
LR
= LR
2
kin + Lint
LR
kin
*
M2
∇λ Rλµ ∇ν R νµ − R Rνµ R νµ
2
R=V ,A
E
1 X D ν
∇ R ∇ν R − MR2 R 2 ,
+
2
1 X
= −
2
+
R=S,P
∇µ X = ∂µ X + [Γµ , X ],
Γµ =
1 †
{u (∂µ − irµ )u + u(∂µ − ilµ )u † }
2
32 / 45
External sources in the RχT Lagrangian:
resonance part
R
LR
= LR
2
kin + Lint
LR
int
=
iG
iF FV √ Vµν f+µν + √ V hVµν u µ u ν i + √A Aµν f−µν
2 2
2
2 2
+cd hSuµ u µ i + cm hSχ+ i + idm hPχ− i ,
32 / 45
External sources in the RχT Lagrangian: NGB part
LNGB
=
2
F2 µ
hu uµ + χ+ i
4
uµ = i[u † (∂µ − irµ )u − u(∂µ − ilµ )u † ]
χ+ = u † χu † + uχu
χ ≡ 2B0 (s + i p) ≈ 2B0 diag(mu , md , ms )
33 / 45
Electromagnetic interactions in RχT
External vector field only is responsible for electromagnetic interactions
vµ = −eQBµ ,
aµ = 0
Bµ — electromagnetic field
√
e = 4πα
34 / 45
Weak interactions
[ Moussallam hep-ph/0407246 ][ Ecker Prog. Part. Nucl. Phys. 35 (1995) ]
g sin2 θW
QZµ ,
cos θW
g
1
lµ =
(Q sin2 θW + − Q)Zµ
cos θW
6


0
Vud Wµ+ Vus Wµ+
g
0
0 
− √  Vud Wµ−
2
−
Vus Wµ
0
0
rµ =
g = e/ sin θW ,
CKM matrix : Vud = 0.97418 ± 0.00027 (from nuclear β-decays) Vus = 0.2255 ± 0.0019 (from K decays),
θW is the weak mixing angle with sin2 θW ≈ 0.23 [ see PDG ].
for application in HMET see [ Bijnens et al. Phys. Lett. B 429 (1998) ]
35 / 45
Conclusions
Conclusions
Summary and conclusions
36 / 45
Summary
Thorough respect of fundamental symmetries
Symmetry breaking can be introduced
Number of parameters is small
Short-distance properties are satisfied
Interference phases are fixed
Conclusions
37 / 45
Conclusions
Strong
Weak
Electromagnetic
In most cases the application is straightforward
Predictive power is promising
In the implicit form, Lagrangian has a very good shape
Explicit notation is sometimes messy
Automatizing routine is in order
Conclusions
38 / 45
Spare parts
SPARES
Technical issues
39 / 45
Transformation of coset representative u(Φ)
The NGB fields Φ parametrize the coset space G/H
[ Coleman et al.; Callan et al. Phys. Rev. 177 (1969) ]
u(Φ) transforms under G in a non-linear way:
u(Φ) → u(Φ0 ) = UR (x) u(Φ) h(U, Φ)−1 = h(U, Φ) u(Φ) UL−1 (x)
under U(x) = (UR (x), UL (x)).
The explicit form of the compensating transformation
h(U, Φ) ∈ SU(3)V is usually not needed.
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Transformation of coset representative u(Φ)
The NGB fields Φ parametrize the coset space G/H
[ Coleman et al.; Callan et al. Phys. Rev. 177 (1969) ]
Usually the exponential
parametrization
√
√
u = exp(iΦ/ 2F ) = 1 + iΦ/ 2F − Φ2 /4F 2 + ... is used. Φ is the octet
of the pseudoscalar (J P = 0− ) NGB’s
√

 0 √
π+
K+
π / 2 + η8 / 6
√
√
Φ =  π−
−π 0 / 2 + η8 / 6
K0 
√
K−
K̄ 0
−2η8 / 6
LO ChPT corrections to F are in order;
ηη 0 mixing can be accounted for
[ Ivashyn, Korchin arXiv:0904.4823 ]
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Principal guidelines for model building I
Nambu-Goldstone bosons; via coset representative u(Φ),
external sources s, p, (vµ , aµ ) = (lµ , rµ ),
low-energy coefficients B0 , fπ , . . .
⇓
g
µν (em)
(f±
special Lorentz-tensors, X , obeying X → hXh†
= eF µν (em) (uQu † ± u † Qu), Q = diag(2/3, −1/3, −1/3))
g
f±µν → hf±µν h†
⇓
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Principal guidelines for model building II
combine X tensors to form Lorentz-invariant object and take the trace
of it in the flavour space
h· · · iflavour
⇓
× effective coupling constants FV , GV , cd , . . .
⇓
Lorentz-invariant, chiral-symmetric effective Lagrangian Leff , carrying
a given chiral power
(due to counting rules)
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Basic building blocks
L2 =
fπ2
huµ u µ + χ+ i
2
basis ("vielbein field")
uµ
=
i(u † (∂µ − irµ )u − u(∂µ − ilµ )u † )
g
uµ → huµ h†
mass terms via the external scalar source
χ+
=
u + χu + + uχu
χ
=
2B0 (s + ip) = 2B0 diag(mu , md , ms )
≈
diag(mπ2 , mπ2 , 2mK2 − mπ2 )
g
χ+ → hχ+ h†
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Transformation properties of resonances and massive
mesons
The other fields, massive mesons or resonances, transform under
chiral group as SU(3)V multiplets (octets)
R → R 0 = h R h†
or singlets
R → R0 = R
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More complicated building blocks
Covariant derivative
∇µ X
Γµ
g
= ∂µ X + [Γµ , X ]
∇µ X → h∇µ Xh†
1 †
{u (∂µ − irµ )u + u(∂µ − ilµ )u † } "connection"
=
2
(cf. uµ = i(u † (∂µ − irµ )u − u(∂µ − ilµ )u † )
other operators
Γµν
=
1
i
[uµ , uν ] − f+ µν
4
2
"vielbein field")
g
Γµν → hΓµν h†
more traces (suppressed with (1/Nc )n )
h· · · iflavour · · · h· · · iflavour
|
{z
}
n
It turns out, that the number of the building blocks is finite!
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Resonance Chiral Theory towards a "Swiss knife" for low

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