Resonance matrix elements
Akaki Rusetsky, University of Bonn
In collaboration with A. Agadjanov, V. Bernard, D. Hoja and U.-G. Meißner
JHEP 1209 (2012) 023; NPB 886 (2014) 1199; NPB 910 (2016) 387
DESY Zeuthen, 16 January 2017
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.1
Plan
• Introduction: Matrix elements of the resonances
• Lattice QCD and the effective field theories in a finite volume
• Lüscher-Lellouch formula
• Continuation to the pole and fixing the photon virtuality
• Limit of an infinitely narrow resonance
• Conclusions, outlook
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.2
Resonances
• Resonances correspond to S -matrix poles on the unphysical
Riemann sheets after continuation into the complex plane
Threshold
Bound state
Resonance
• Resonances are characterized by their mass, their lifetime, . . .
These are the intrinsic properties of a resonance that should not
depend neither on a particular experiment nor a particular
theoretical model which is used to describe the data
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.3
Resonance matrix elements
γ
π
∆
N
N
In complex plane, in the vicinity of the resonance pole. . .
⇒
⇒
i
Z
i2
4
iP x
d xe
Z
¯
h0|T ∆(x)∆(0)|0i
→
ZR
+ ···
sR − P 2
d4 xd4 y eiP x−iQy h0|T ∆(x)J(0)N̄ (y)|0i
1/2
1/2
ZR
ZN
+ ···
h∆|J(0)|N i 2
→
sR − P 2
mN − Q2
֒→
model-independent definition of the quantity h∆|J(0)|N i
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.4
The problem
• How does one determine the position of a pole in a complex
plane, using lattice data on real energy spectrum?
• How are the matrix elements, measured on the lattice, continued
into the complex plane?
• What is the virtuality of the photon q 2 at the pole?
• What is the relation to the decay amplitudes, measured at the
real energies? How a comparison to the experiment is carried
out?
• What is the volume-dependence of the measured matrix
elements (Lüscher-Lellouch formula)?
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.5
Stable states in lattice field theory
The two-point function → spectrum
Z
Cp (t) = dU dψdψ̄ e−SQCD (U,ψ,ψ̄) Op (t)Op† (0) → Zp exp(−E(p)t)
Three-point function → form factors
Cp,q (t, t′ )
=
Z
dU dψdψ̄ e−SQCD (U,ψ,ψ̄) Op (t) jµ (0) Oq† (t′ )
→ Zp1/2 Zq1/2 exp(−E(p)|t| − E(q)|t′ |) hp|jµ (0)|qi
The method does not apply to:
Resonances, e.g., the mass and width of the ρ meson . . .
Decays, e.g., K → 2π , ∆ → N γ ∗ , B → K ∗ ℓ+ ℓ− . . .
→ Resonances do not correspond to an isolated energy level of the
lattice Hamiltonian!
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.6
Lüscher’s finite-volume approach to the scattering problems
M. Lüscher, lectures given at Les Houches (1988); NPB 364 (1991) 237, · · ·
The Lüscher equation for the two-particle elastic scattering:
• The box size L is much larger than the interaction range R
→ scattering phase shifts from the measured lattice levels
det (δll′ δmm′ − tan δl (p)Mlm,l′ m′ ) = 0
• Mlm,l′ m′ is a linear combination of Zlm [partial-wave mixing]
2
Zlm (1, q ) = lim
s→1
X
n∈Z3
Ylm (n)
,
2
2
s
(n − q )
q=
pL
2π
• The resonance pole position in the complex plane is obtained by
using certain parameterization of a scattering phase
(e.g., effective range expansion)
• Generalizations: coupled-channel scattering (e.g., K -matrix
parameterization)
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.7
Where are the resonance poles?
Assume, e.g., the effective range expansion (S-wave):
p cot δ(p) = A0 + A1 p2 + · · · ,
Analytic continuation to the resonance pole:
pR cot δ(pR ) = −ipR
⇒
A0 , A1 , · · · are measured on the lattice (real)
⇒
Resonance pole pR in the complex momentum plane
Can one perform effective-range expansion for the matrix
elements?
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.8
Why effective field theories?
Most general effective Lagrangian: all terms with assumed symmetry
֒→
Most general S -matrix consistent with analyticity, perturbative
unitarity, cluster decomposition and symmetries (Weinberg, 1979)
QCD
→
• Lattice QCD
ChPT (p ∼ Mπ )
→
→
non-rel. EFT (p ≪ Mπ )
EFT in a finite volume
• Short-distance behavior not changed: the same Lagrangian
֒→
A bridge between the measured spectrum and scattering sector:
•
•
•
•
•
Multichannel resonances
Twisted boundary conditions
Lüscher-Lellouch formula
Few-body systems
Resonance formfactors
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.9
Relativistic vs non-relativistic EFT
2−particle irreducible
2−particle reducible
s
s th > 2m
Exponentially suppressed for s < s th
• Well below inelastic threshold, finite-volume corrections to the
2-particle irreducible diagrams are exponentially suppressed
∼ exp(−∆E · L)
• 2-particle irreducible diagrams
֒→
→ L-independent local vertices
Particle number is conserved in intermediate states
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.10
Method: covariant NREFT in the infinite volume
G. Colangelo, J. Gasser, B. Kubis and A. Rusetsky, PLB 638 (2006) 187
J. Gasser, B. Kubis and A. Rusetsky, NPB 850 (2011) 96
The Lagrangian:
• Contains non-relativistic field operators
• Particle number is conserved
• Counting rules observed after applying “threshold expansion”
• Electromagnetic and weak interactions can be systematically
included
LI = C0 Φ†1 Φ†2 Φ1 Φ2 + derivatives
The propagator with the relativistic dispersion law:
1
1
,
D(p) =
2w(p) w(p) − p0 − i0
p
w(p) = M 2 + p2
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.11
Lippmann-Schwinger equation
• Threshold expansion: in Feynman integrals, expand all
integrands in powers of three-momenta, integrate in the
dimensional regularization and sum up again
֒→
T=
֒→
ip
loop = √ ,
8π s
+
λ1/2 (s, M12 , M22 )
√
p=
2 s
+
+ ...
Scattering amplitude is Lorentz-invariant:
√
8π s
,
Tl =
p cot δl (p) − ip
p
2l+1
1
1
cot δl (p) = − + rl p2 + · · ·
al
2
• Important in non-rest frames (formfactors, 3-body scattering)
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.12
Covariant NREFT in a finite volume
EFT in a finite volume with the same Lagrangian ⇒ lattice QCD
Feynman loops are modified in a finite volume
Z
d3 p
1 X
→ 3
(2π)3
L p
(finite box with periodic b.c.)
Lippmann-Schwinger equation in a finite volume
TL (E) = V + V GL (E)TL (E)
• The potential V is a low-energy polynomial, only exponentially
suppressed finite-volume corrections → phase shift
• The finite-volume Green function GL (E) contains a tower of the
real poles.
• Poles of TL (E) = spectrum in a finite volume
֒→
Lüscher equation
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.13
Relativistic vs non-relativistic
• Matching is performed on mass shell. Consequently, only
on-shell vertices appear in the perturbative expansion
• Threshold expansion automatically puts the amplitudes in the
loops on energy shell:
Z
3
2
d k f (k )
=
(2π)3 k2 − q02
Z
3
d k
(2π)3
f(k 2 )
=
f (q02 )
k2 − q02
regular
| {z }
+
=0, threshold expansion
f(q 20 )
+ ...
• All results are the same as in the relativistic framework +
skeleton expansion, come at a less effort
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.14
Example:
B → K ∗ ℓ+ ℓ− decays
• Rare B decays provide one of the best opportunities in search of
BSM physics (anomalies)
• Theoretical uncertainties arise in the calculated form factors
→ lattice QCD
• In general, the decay products contain resonances, e.g.
B → K ∗ ℓ+ ℓ− . A procedure for the extraction of the resonance
form factors on the lattice should be defined
• Analytic continuation to the resonance pole should be studied.
What is the photon virtuality q 2 at the pole?
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.15
The effective weak Hamiltonian for B
→ K ∗ ℓ+ ℓ− decay
Effective Hamiltonian:
10
Heff
X
4GF ∗
= − √ Vts Vtb
Ci W i
2
i=1
W7
=
W9
=
W10
=
mb e µν 1
(1 + γ5 )b Fµν
s̄σ
2
16π
2
e2
µ1
(1 − γ5 )b ℓ̄γµ ℓ
s̄γ
2
16π
2
e2
5
µ1
(1
−
γ
)b
ℓ̄γ
γ
ℓ
s̄γ
5
µ
2
16π
2
Seven form factors:
V (q 2 ),
A0 (q 2 ),
A1 (q 2 ),
A2 (q 2 ),
T1 (q 2 ),
T2 (q 2 ),
T3 (q 2 )
Photon virtuality: q 2 = (pB − pK ∗ )2
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.16
Effective non-relativistic Lagrangian for the B -decays
• Coupled-channel problem:
1) πK ,
2) ηK
†
†
Lstr = (OK
∗ X1 πK + OK ∗ X2 ηK + h.c.) + L4−meson
O
Xα
• Irreducible vertices describing the decay
†
†
† † †
† † †
Lweak = (OK
∗ J Γ0 B + h.c.) + (π K J F̄1 B + η K J F̄2 B + h.c.)
J†
J†
B
Γ0
B
Fα
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.17
Long-distance contributions?
There are no long-distance contributions in the matrix elements
of the flavor changing neutral corrents:
?
֒→
No vertices corresponding to B → Kπ or B → Kη without
lepton-antilepton pair
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.18
Lattice setting
• The K ∗ is unstable: need to scan the energy of the πK pair,
photon 3-momentum fixed
• πK pair in the CM frame: no mixing
• Asymmetric box: L × L × L′ , varying L, fixed L′
• Projection onto the irreps:
(±)
OE
1
= √ (O1 ∓ O2 ) ,
2
OA1 = O3
• Extraction of the form factors:
1
2iE V (q 2 )
hV (+)| √ s̄(γ1 + iγ2 )b|Bi = −
,
m
+
E
2
B
etc
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.19
Multi-channel amplitude in a finite volume
TL =
√
8π s
f (E)

1
[t1 τ1 (t2 + τ2 ) + s2ε τ1 τ2 t]
p
 1
− √p1 p cε sε τ1 τ2 t
1 2
• ti = tan δi (p),
• cε = cos ε,

− √p1 p cε sε τ1 τ2 t
1 2

1
[t τ (t + τ1 ) − s2ε τ1 τ2 t]
p2 2 2 1
t = t2 − t1 : scattering phases
sε = sin ε: mixing angle
• τi = tan φi : expressed through Lüscher zeta-functions
Lüscher equation in the two-channel case:
f (E) = (t1 + τ1 )(t2 + τ2 ) + s2ε t(τ2 − τ1 ) = 0
Factorization of the amplitude near the eigenvalue:
TLαβ (E)
fα fβ
→
+ ···
En − E
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.20
Lüscher-Lellouch formula, two-channel case
γ∗
K
K
O
X1
π
+
O
X1
···
F̄1M
···
F̄2M
B
+
O
X2
···
F̄1M
X2
···
F̄2M
+
η
+
O
V −1
−1
|F (En , |q|)| =
p1 τ1 f1 F̄1 + p2 τ2 f2 F̄2 8πE
−1
E=En
A1 = F̄1 (c2ε cos δ1 eiδ1 + s2ε cos δ2 eiδ2 ) +
r
p1
F̄2 cε sε (cos δ1 eiδ1 − cos δ2 eiδ2 )
p2
A2 = F̄2 (c2ε cos δ2 eiδ2 + s2ε cos δ1 eiδ1 ) +
r
p2
F̄1 cε sε (cos δ1 eiδ1 − cos δ2 eiδ2 )
p1
• F̄1 , F̄2 : Irreducible amplitudes, exponentially suppressed volume dependence
• Agrees with: M. Hansen and S. Sharpe, PRD 86 (2012) 016007
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.21
Extraction of the form factors at the pole
• T -matrix on the second Riemann sheet:
αβ
TII
(s) →
hα hβ
+ ···
sR − s
• Form factors at the pole: residues in the pertinent Green
functions
i
(p1 h1 F̄1 − p2 h2 F̄2 )
FR (ER , q) =
8πE
E→ER
• Universal: do not depend on the process!
• Uniquely defined!
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.22
Effective-range expansion for the matrix elements
• Extract irreducible amplitudes F̄i (p, |Q|) from the measured
amplitude F (p, |Q|), multiplying by Lüscher-Lellouch factors
(measurement at multiple energies/volumes, |Q| fixed)
• If there are no nearby resonances, F̄i (p, |Q|) are low-energy
polynomials
• In the presence of a resonance, define:
√
√
ũ1 = t−1
(
p
c
+
p2 sε F̄2 )
F̄
1 ε 1
1
√
√
(
p
c
−
p1 sε F̄1 )
ũ2 = t−1
F̄
2 ε 2
2
Effective-range expansion: ũi (p, |Q|) = Ai (|Q|) + p2 Bi (|Q|) + · · ·
Fit the real coefficients Ai (|Q|), Bi (|Q|), . . . to the data
• Find the form factor at the pole by substituting p → pR in the
effective range expansion
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.23
Photon virtuality
(q0, q)
(
r
m2B
q 2 = (ER −
+
q2 , q)
q
(ER , 0)
m2B + q2 )2 − q2 ,
complex!
2
• Keeping q 2 real and performing the limit s → ER
does not
correspond to the resonance pole!
• Experimental data should be analysed in a similar manner!
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.24
Infinitely small width
• 1-channel case: FR = F up to normalization in the limit Γ → 0
• 2-channel case: normalization different in different channels. . .
→ Define
√
√
(
p
c
+
p2 sε F̄2 )
ũ1 = t−1
F̄
1 ε 1
1
√
−1 √
ũ2 = t2 ( p2 cε F̄2 − p1 sε F̄1 )
• ũ1 , ũ2 are low energy polynomials: do not contain the small
energy scale Γ
• At the resonance, the decay matrix element factorizes
→ ũ1 = O(Γ−1/2 ) and ũ2 = O(1)
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.25
The form factor in the limit Γ
→0
1
FR (ER , q) = √ (r1 ũ1 + r2 ũ2 )
4π
E→ER
r12
=
t2
t21
+ i − 2is2ε
,
′
h (E)
r22
=
t1
t22
− i + 2is2ε
h′ (E)
h(E) = (t1 − i)(t2 + i) + 2is2ε (t2 − t1 )
Resonance emerges in one channel:
t1 diverges, t2 stays finite, t′1 = O(Γ−1 )
֒→
FR (ER , q)
Γ→0
=
√
2En
F (En , q) + O(Γ1/2 )
V
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.26
Conclusions, outlook
• Using non-relativistic EFT in a finite volume, a framework for the
extraction of the resonance matrix elements from the lattice data
is constructed. The case of the B → K ∗ ℓ+ ℓ− form factors is
considered in great detail
• The multi-channel Lüscher-Lellouch formula is reproduced.
• The extraction of the form factors at the resonance pole is carried
out by using effective-range expansion for thematrix elements
• It is shown that, at the resonance, the photon virtuality q 2 must
be complex
• The limit Γ → 0 is studied in detail in the multi-channel case
A. Rusetsky, DESY Zeuthen, 16 January 2017 – p.27