Download Resonances in chiral effective field theory Jambul Gegelia

Resonances in chiral effective field theory
Jambul Gegelia
Johannes Gutenberg-Universität Mainz
In collaboration with:
T. Bauer, D. Djukanovic, G. Japaridze,
S. Scherer und L. Tiator
PWA Workshop,
Washington, DC, 23-27 May, 2011
Outline
• Magnetic moment of the Roper
• Vector form factor of the pion
• Some aspects of complex-mass scheme
• Imaginary parts of loop integrals
• Perturbative unitarity in CMS
• Summary
Magnetic moment of the Roper
Effective Lagrangian relevant for the magnetic moment of the Roper
L = L0 + Lπ + LR + LN R + L∆R .
Here
L0 = N̄ (iD
/ − mN 0)N + R̄(iD
/ − mR0)R
−
3
Ψ̄µξ 2 [(iD
/−
m∆0
) g µν
− i (γ µDν
+ γ ν Dµ) + i γ µD
/ γν
+ m∆0
γ µγ ν
3
] ξ 2 Ψν .
N and R denote nucleon and Roper isospin doublets. Ψν are the Rarita3
Schwinger fields of the ∆ resonance, ξ 2 is the isospin-3/2 projector.
Covariant derivatives are defined as follows:
µ
DµH =
(s)
∂µ + Γµ − i vµ
¶
H,
(s)
(DµΨ)ν,i = ∂µΨν,i − 2 i ²ijk Γµ,k Ψν,j + ΓµΨν,i − i vµ Ψν,i ,
³
´i
1 h †
†
†
†
Γµ =
u ∂µu + u∂µu − i u vµu + uvµu
= τk Γµ,k .
2
Lowest-order Goldstone boson Lagrangian:
´
´
F 2M 2 ³ †
F2 ³
µ
†
=
Tr ∂µU ∂ U +
Tr U + U
4
4
´ i
F 2 h³
†
†
+ i
Tr ∂µU U + ∂µU U vµ + · · · .
2
v µ = −e τ23 Aµ; Pion fields are contained in U ; F is the pion-decay constant in the chiral limit; M 2 = 2B m̂, where B is related to the quark
condensate hq̄qi0 in the chiral limit.
(2)
Lπ
Leading order pion-Roper Lagrangian:
g
(1)
LR = R R̄γ µγ5uµR ,
2
where gR is an unknown coupling constant and
uµ = i
√
where u = U .
h
u† ∂
µ u − u∂µ
u†
−i
³
u†vµu − uvµu†
´i
,
Second and third order Roper Lagrangians:
"
(2)
LR
= R̄
c∗6
2
+ +
fµν
c∗7 (s)
vµν
2
#
σ µν R + · · · ,
µ
i ∗ h µ +i ν
(s)
(3)
LR =
d6R̄ D , fµν D R + h.c. + 2 i d∗7R̄ ∂ µvµν
2
where
(s)
vµν
(s)
¶
Dν R + h.c. + · · · ,
(s)
= ∂µvν − ∂ν vµ ,
e Aµ
(s)
,
vµ
= −
2
+ = uf u† + u† f u ,
fµν
µν
µν
fµν = ∂µvν − ∂ν vµ − i[vµ, vν ]
and c∗6, c∗7, d∗6 and d∗7 are unknown coupling constants.
Leading order interaction between the nucleon and the Roper:
g
(1)
LN R = N R R̄γ µγ5uµN + h.c.
2
with an unknown coupling constant gN R .
Leading-order interaction between the delta and the Roper:
(1)
L∆R
=
3
−g∆R Ψ̄µ ξ 2
(g µν + z̃ γ µγ ν ) uν R + h.c. ,
where g∆R is a coupling constant and we take z̃ = −1.
We apply the complex-mass scheme (CMS):
R. G. Stuart, in Z0 Physics, ed. J. Tran Thanh Van (Editions Frontiers,
Gif-sur-Yvette, 1990), p.41.
A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, Nucl. Phys. B560,
33 (1999).
A. Denner, S. Dittmaier, M. Roth and L. H. Wieders, Nucl. Phys. B724,
247 (2005).
Generalization of the on-mass-shell scheme to unstable particles.
Well suited for unstable particles in perturbation theory.
Bare parameters of the Lagrangian are split into complex (in general)
renormalized parameters and complex (in general) counterterms.
Renormalized masses are chosen as poles of dressed propagators in chiral
limit:
mR0 = zχ + δzχ ,
mN 0 = mχ + δm ,
m∆0 = z∆χ + δz∆χ .
(1)
zχ - complex pole of the Roper propagator in the chiral limit.
mχ - mass of the nucleon in the chiral limit.
z∆χ - pole of the delta propagator in the chiral limit.
Renormalized parameters zχ, m, and z∆χ are included in the propagators
and the counterterms are treated perturbatively.
Power counting:
Interaction vertex obtained from an O(q n) Lagrangian ∼ q n,
Pion propagator ∼ q −2,
Nucleon propagator ∼ q −1,
∆ propagator ∼ q −1,
Roper propagator ∼ q −1,
Loop integration ∼ q 4.
Within CMS, such a power counting is respected in the range of energies
close to the Roper mass.
Dressed propagator of the Roper
i
iSR (p) =
,
/
p − zχ − ΣR (p
/)
where −iΣR (p
/ ) denotes the self-energy of the Roper.
the pole of the dressed propagator SR is obtained by solving
z − zχ − ΣR (z) = 0 .
(2)
The pole mass and the width:
ΓR
z = mR − i
.
2
Dressed propagator has a pole only on the second Riemann sheet.
ΣR (zχ) = 0
on the second Riemann sheet.
ΣR (zχ) 6= 0
on the first Riemann sheet.
(3)
Roper propagator close to the pole
iSR (p) =
iZ
+ n.p. .
/
p −z
Physical quantities characterizing unstable particles have to be extracted
at pole positions using the complex-valued Z.
Up to order O(q 3), Z is obtained by calculating the Roper self-energy
diagrams shown in Figure.
1
1
(1)
1
1
(2)
1
1
(3)
Diagrams contributing in Roper form factors
Tree diagrams:
1
2
3
(T1)
(T2)
(T3)
Loop Diagrams:
2
1
1
1
1
1
1
1
1
1
(4)
(1)
(2)
(3)
2
1
1
1
1
1
1
1
1
1
(8)
(5)
(6)
(7)
1
1
1
(10)
(9)
2
1
1
1
1
1
1
1
1
1
(14)
2 = z 2:
=
p
Parameterize the renormalized vertex function for p2
i
f
√
"
#
√
i σ µν qν
i
µ
j
i
µ
2
Z w̄ (pf )Γ (pf , pi)w (pi) Z = w̄ (pf ) γ F1(q ) +
F2(q 2) wj (pi).
2 mN
F1(q 2) and F2(q 2) are complex valued functions.
Z× tree diagrams subtracts all power counting violating loop contributions in F1(q 2). We obtain F1(0) = (1 + τ3)/2.
Power counting violating loop contributions in magnetic form factor are
absorbed in the renormalization of c∗6 and c∗7.
Subtracted loop contributions satisfy the power counting.
Magnetic moment:
µR = F2(0).
F2(t) = mN [ G1(t) + τ3G2(t)].
Tree order result:
∗
Gtree
1 (0) = mN c7 ,
∗
Gtree
2 (0) = 2 mN c6 .
Loop contributions:
Gloop
(0)
1
=
−
Gloop
(0)
2
=
+
2
3gR
³ ´
2
2
³
´
[zχ − A0 zχ
2 π2
16F 2zχ M 2 − 4zχ
³
´
³
´
³
2
2
2
2
2
2
M − 3zχ B0 zχ , M , zχ ]M + M 2
2
gR
n
n
³
2
−A0 zχ
´
M2
2
− 2zχ
³
´
3M 2
³
´
+
2
2
2
2
16F zχ M − 4zχ π
h
³
´
³
´i
o
2
2
2
2
2
2
zχ M − 2 M − 4zχ B0 zχ , 0, zχ + · · · .
A0
³
M2
2
− 10zχ
´
´
···
A0
o
³
M2
´
Loop functions are given as
Z
(2π µ)4−n
dnk
m
2
2
2
2
A0 m
=
= −32π λ m − 2 m ln ,
2
2
2
iπ
k − m + i²
µ
4−n Z
³
´
(2π
µ)
dnk
2
2
2
h
ih
i
B0 p , m1, m2 =
2
2
2
iπ
k 2 − m2
1 + i² (p + k) − m2 + i²
µ
ω
2
= −32π λ + 2 ln
− 1 − 2F1 (1, 2; 3; ω )
m2
2
Ã
!
Ã
!
2
2
m2
m2
1
−
1+ 2
,
2 F1 1, 2; 3; 1 + 2
2
m1(ω − 1)
m1(ω − 1)
³
´
ω =
m2
1
− m2
2
+ p2
+
r³
m2
1
− m2
2
2m2
1
´2
2
+p
2
− 4m2
1p
,
where 2F1 (a, b; c; z ) is the standard hypergeometric function, µ is the
scale parameter of the dimensional regularization and
½
i¾
1
1 h
1
0 (1) + 1
λ=
−
ln(4π)
+
Γ
.
16 π 2 n − 4 2
Vector form factor of the pion
Effective Lagrangian relevant for the pion form factor calculation (up
to higher order terms)
L =
+
−
−
2
h
i
h
i
F
F2
†
µ
†
†
Tr DµU (D U ) +
Tr χU + U χ
4
4
h
i
2
M + cxTr χ+ /4
µ − iΓµ gρ − iΓ
Tr
gρ
)( µ
[(
µ )]
2
g
1
Tr [ρµν ρµν ] + i dxTr [ρµν Γµν ]
2
√
n
o
2
µν
fV Tr ρµν f+ ,
2
(4)
Here
Ã
!
iΦ(x)
,
F
DµA = ∂µA − irµA + iAlµ ,
χ+ = M 2(U † + U ) ,
´i
³
1 h †
†
†
†
,
Γµ =
u ∂µu + u∂µu − i u rµu + ulµu
2
µν
ρ
= ∂ µρν − ∂ ν ρµ − ig [ρµ, ρν ] ,
U (x) = u2(x) = exp
Γµν = ∂µΓν − ∂ν Γµ + [Γµ, Γν ],
µν
µν
µν
f± = uFL u† ± u†FR u ,
rµ = vµ + aµ ,
lµ=vµ − aµ ,
vµ and aµ are external vector and axial vector fields.
Renormalization using CMS.
Power counting rules:
Pion propagator ∼ O(q −2) if it does not carry large external momenta
and ∼ O(q 0) if it does.
Vector-meson propagator ∼ O(q 0) without large external momenta and
∼ O(q −1) - otherwise.
Pion mass ∼ O(q 1), ρ-meson mass ∼ O(q 0), and the width ∼ O(q 1).
(n)
Vertices generated by Lπ
count ∼ O(q n).
Derivatives acting on heavy vector mesons ∼ O(q 0).
Investigate all possible flows of external momenta through loop diagrams
and determine the chiral order for each flow.
Smallest order resulting from various assignments = chiral order of the
diagram.
a)
b)
Diagrams contributing to the pion form factor.
Diagrams of group a).
Diagrams of group b).
50
40
40
30
30
È FΠ È ^2
È FΠ È ^2
50
20
20
10
10
0
0.0
0.2
0.4
0.6
0.8
0
0.0
1.0
0.2
40
40
30
30
20
10
0
0.0
0.6
0.8
1.0
q^2 @GeV^2D
È FΠ È ^2
È FΠ È ^2
q^2 @GeV^2D
0.4
20
10
0.2
0.4
0.6
q^2 @GeV^2D
0.8
1.0
0
0.0
0.2
0.4
0.6
0.8
1.0
q^2 @GeV^2D
Pion form factor: Tree order - left; Tree + loop - right; Data - red dots;
Data:
Kloe Collab., Phys. Lett. B 670 (2009);
S. Schael et al. [ALEPH Collaboration], Phys. Rept. 421, 191 (2005).
Some aspects of complex-mass scheme
CMS leads to complex-valued renormalized parameters.
Issue of perturbative unitarity of the S-matrix in the CMS is still open.
Lagrangian does not change → Unitarity is not violated in exact theory.
Perturbation theory is based on order-by-order approximation to the
exact results → Not obvious that the approximate expressions to the
S-matrix satisfy unitarity.
Branching points are determined by poles of full propagators.
Branching points corresponding to unstable particles are complex.
Cuts can be placed arbitrarily.
Sometimes branching points, corresponding to unstable states, are placed
on real axis.
If poles are not very far from the real axis this is a reasonable approximation.
CMS places the poles and branching points at exact (complex) positions
already at the leading order.
Imaginary parts of loop integrals with complex masses
Standard causal propagator of a scalar particle
³
´
1
p2 − m2
2
2
S(p) = 2
=³
− iπδ p − m .
´2
2
p − m + i²
p2 − m2 + ²2
² → 0 limit is assumed.
Corresponding advanced and retarded propagators:
"
#
1
1
1
SA(p) =
−
2 Ep p0 − Ep − i ² p0 + Ep − i ²
´
³
p2 − m2
2
2
+ i π δ p − m σp,
= ³
´2
p2 − m2 + ²2
"
#
1
1
1
−
2 Ep p0 − Ep + i ² p0 + Ep + i ²
³
´
p2 − m2
2
2
= ³
− i π δ p − m σp,
´2
p2 − m2 + ²2
SR (p) =
q
where σp = sign(p0) and Ep =
p
~ 2 + m2.
Propagator of an unstable scalar particle in CMS
1
S 0(p) = 2
p − M 2 + i (M Γ + ²)
p2 − M 2
i (M Γ + ²)
=³
−
.
´2
³
´2
p2 − M 2 + (M Γ + ²)2
p2 − M 2 + (M Γ + ²)2
² can be neglected for finite Γ, but important for Γ → 0.
”Advanced” and ”retarded” propagators:
"
#
"
#
1
1
1
=
−
,
w(p) + w(p)∗ p0 − w(p)∗ p0 + w(p)
p2 − M 2 − M 2Γ2/(2 x(p)2)
iM Γ
p0
=
+
,
³
´2
³
´2
p2 − M 2 + M 2 Γ2
p2 − M 2 + M 2Γ2 x(p)
0 (p)
SA
1
1
1
−
w(p) + w(p)∗ p0 − w(p) p0 + w(p)∗
p2 − M 2 − M 2Γ2/(2 x(p)2)
iM Γ
p0
=
−
,
³
´2
³
´2
p2 − M 2 + M 2 Γ2
p2 − M 2 + M 2Γ2 x(p)
0 (p) =
SR
w(p) = x(p) − i y(p) ,
r
³
´1/2
1
4
2
2
x(p) = √
Ep + M Γ
+ Ep2 = Ep + O(Γ2) ,
2r
³
´1/2
1
4
2
2
y(p) = √
Ep + M Γ
− Ep2 = M Γ/(2 Ep) + O(Γ3) ,
2
q
Ep =
p
~ 2 + M2 .
Consider one-loop integral
Z
I1 = i
d4k d4q
4 δ 4 (k + q − p)S(k)S 0 (q).
(2π)
(2π)4 (2π)4
(5)
Imaginary part of this integral is given by
Z
d4k d4q
4δ 4(k + q − p)
Im[I1] =
(2π)
(2π)4 (2π)4
h
k2 − m2
q2 − M 2
× ³
´2
³
´2
2
2
2
2
2
k −m
+² q −M
+ M 2Γ2
³
´
i
MΓ
2
2
− πδ k −m ³
.
´2
q 2 − M 2 + M 2Γ2
(6)
To rewrite the first term consider the equality
Z
0=i
d4k d4q
4δ 4(k + q − p)S (k)S 0 (q)
(2π)
A
R
(2π)4 (2π)4
(7)
and take the imaginary parts of both sides
Z
d4k d4q
4 δ 4 (k + q − p)
0 =
(2π)
(2π)4 (2π)4
h
k 2 − m2
q2 − M 2
× ³
´2
³
´2
2
2
2
2
2
k −m
+² q −M
+ M 2 Γ2
³
´
i
q0
MΓ
2
2
2
+ π δ k − m σk
+ O(Γ ) .
³
´2
Eq q 2 − M 2 + M 2Γ2
(8)
Subtracting Eq. (8) from Eq. (6) obtain
⇒ −π
i
Z
³
´
(M Γ + ²) 1 + σp−k σk
d4k
2
2
2),
δ
k
−
m
+
O(Γ
h
i
2
(2π)4
(p − k)2 − M 2 + (M Γ + ²)2
Z
³
´
h
i
d4k
2
2
2
2
δ k − m δ((p − k) − M ) 1 + σp−k σk ,
4
(2π)
Im[I1] = −π
Γ→0
h
where the second line corresponds to the standard cutting formula for
loop integrals with real masses.
Generalization to any one-loop integral with complex masses leads to:
• For each cut of the line of an unstable particle one obtains one
overall factor of Γ.
• For each cut of the line of the stable particle one gets a deltafunction.
• For the integrals containing only propagators of stable particles the
usual cutting rules apply.
The Model
Consider a model of an unstable vector boson interacting with a fermion
M02
1
µν
µ
L = − F0µν F0 +
B0µB0
4
2
µ
/ − m0) ψ0 + g0 ψ̄0 γµψ0 B0 ,
+ ψ̄0 (i∂
0 = ∂ B 0 − ∂ B 0.
where Fµν
µ ν
ν µ
Vector boson decays into a fermion-antifermion pair.
Renormalization in two steps: first get rid off the divergences by applying
the dimensional regularization with the M S scheme.
Next express the renormalized masses of M S scheme in terms of poles
of the dressed propagators.
Substitutions into the Lagrangian
µ
B0 → B µ,
ψ0→ψ,
m0 → m + δm,
g0→g,
M02 → M 2 − i M Γ + δz ≡ z + δz ,
(9)
result in
L = Lmain + Lct ,
1
z
µν
Lmain = − Fµν F + BµB µ+ψ̄ (i∂
/ − m) ψ + g ψ̄ γµψ B µ,
4
2
δz
Lct =
BµB µ − δm ψ̄ ψ .
2
Counter term Lagrangian Lct is treated perturbatively and the propagators read
i
,
i SF (p) =
/
p − m + i²
gµν − pµpν /z
.
i Sµν (p) = −i
(10)
2
p −z
Perturbative unitarity of the S-matrix
Diagrams contributing in the f¯f → f¯f amplitude:
a)
k)
e)
d)
g)
f)
j)
c)
b)
h)
l)
i)
m)
n)
Imaginary part of the tree-order amplitude shown in Fig. a)
"
Im [Ta] = Im V µ
#
s − z∗
MΓ
µ
V
=−V
Vµ . (11)
µ
∗
∗
(s − z)(s − z )
(s − z)(s − z )
Imaginary parts of one-loop diagrams c) and i)
Im [Ti + Tc] =
Vµ
³ ´
Im [Π(s) + δz1]
3 .
V
+
O
Γ
µ
(s − z)(s − z ∗)
(12)
×
The ”square” of the tree-order amplitude above reads
TT†
=
2V µ
Im [Π(s)]
Vµ .
(s − z)(s − z ∗)
(13)
2 ), follows
From Eqs. (11), (12) and (13), ³using
Im[δz
]
=
M
Γ
+
O(h̄
1
´
3
that unitarity is satisfied up to O Γ .
Summary
• Magnetic moment of the Roper resonance in the framework of the
low-energy EFT of QCD.
• Vector form factor of the pion.
• Perturbative unitarity of the scattering amplitude within CMS for
one-loop diagrams.
• Result obtained under assumption that the expansion parameter remains real.
• Unstable particles do not appear as asymptotic states.
• Generalization of cutting rules to multi-loop diagrams is straightforward, analysis of perturbative unitarity - more involved.