BLED WORKSHOPS
IN PHYSICS
VOL. 13, NO. 1
p. 74
Proceedings of the Mini-Workshop
Hadronic Resonances
Bled, Slovenia, July 1 - 8, 2012
Scattering phase shifts and resonances from lattice
QCD
S. Prelovšeka,b
a Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
b Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000
Ljubljana, Slovenia
Most of hadrons are hadronic resonances - they decay quickly via the strong
interactions. Among all the resonances, only the ρmeson has been properly simu-
lated as a resonance within lattice QCD up to know. This involved the simulation
of the ππ scattering in p-wave, extraction of the scattering phase shift and deter-
mination ofmR and Γ via the Breit-Wigner like fit of the phase shift.
In the past year, we performed first exploratory simulations of Dπ,D∗π and
Kπ scattering in the resonant scattering channels [1, 2]. Our simulations are done
in lattice QCDwith two-dynamical light quarks at a mass corresponding tomπ ≃
266MeV and the lattice spacing a = 0.124 fm.
D D* D
0
* D1 D1 D2* D2
-100
0
100
200
300
400
500
600
700
800
900
1000
E
ne
rg
y 
di
ff
er
en
ce
 [
M
eV
]
energy levels
resonances
PDG values
new BaBar states
J
 P
  :     0
 - 
       1
 -       
                   0
+ 
        1
 + 
         1
 +  
       2
 + 
          2
 - 
Fig. 1. Energy differences∆E = E− 1
4
(MD+3MD∗) forDmeson states in our simulation [1]
and in experiment; the reference spin-averaged mass is 1
4
(MD + 3MD∗) ≈ 1971 MeV
in experiment. Magenta diamonds give resonance masses for states treated properly as
resonances, while those extracted naively assumingmn = En are displayed as blue crosses
[1].
Scattering phase shifts and resonances from lattice QCD 75
The masses and widths of the broad scalar D∗0(2400) and the axial D1(2430)
charmed-light resonances are extracted by simulating the corresponding Dπ and
D∗π scattering on the lattice [1]. The resonance parameters are obtained using a
Breit-Wigner fit of the elastic phase shifts. The resulting D∗0(2400) mass is 351 ±
21 MeV above the spin-average 1
4
(mD + 3mD∗), in agreement with the experi-
mental value of 347 ± 29 MeV above. The resulting D∗0 → Dπ coupling glat =
2.55 ± 0.21 GeV is close to the experimental value gexp = 1.92± 0.14 GeV, where
g parametrizes the width Γ ≡ g2p⋆/s. The resonance parameters for the broad
D1(2430) are also found close to the experimental values; these are obtained by
appealing to the heavy quark limit, where the neighboring resonance D1(2420)
is narrow. The simulation of the scattering in these channels incorporates quark-
antiquark as well as D(∗)π interpolators, and we use distillation method for con-
tractions. The resulting D-meson spectrum is compared to the experimental one
in Fig. 1.
In addition, the ground and several excited charm-light and charmonium
states with various JP are calculated using standard quark-antiquark interpola-
tors. The lattice results for the charmonium are compared to the experimental
levels in Fig. 2.
η
c Ψ hc χ c0 χ c1 χ c2 η c2 Ψ 2 Ψ 3 hc3 χ c3
-100
0
100
200
300
400
500
600
700
800
900
1000
1100
E
ne
rg
y 
di
ff
er
en
ce
 [
M
eV
]
J
PC
 :  0
- +    
  1
- -    
 1
+ -  
 0
+ +  
    1
+ + 
    2
+ + 
   2
- + 
    2
 - - 
    3
- - 
    3
+ - 
    3
++
Fig. 2. Energy differences ∆E = E− 1
4
(Mηc + 3MJψ) for charmonium states in our simula-
tion [1] and in experiment; reference spin-averaged mass is 1
4
(Mηc + 3MJψ) ≈ 3068MeV
in experiment. The magenta lines on the right denote relevant lattice and continuum
D̄(∗)D(∗) thresholds.
We also simulated Kπ scattering in s-wave and p-wave for both isospins I =
1/2, 3/2 using quark-antiquark and meson-meson interpolating fields [2]. Fig. 3
shows the resulting energy levels of Kπ in a box. In all four channels we observe
the expectedK(n)π(−n) scattering states, which are shifted due to the interaction.
In both attractive I = 1/2 channels we observe additional states that are related
76 S. Prelovšek
0 5 10 15
t
0.4
0.6
0.8
1
1.2
E
 a
s-wave, I=1/2
0 5 10 15
t
s-wave, I=3/2
K(0)π (0)
K(1)π (-1)
2 exp fit
K(2)π (-2)
0 5 10 15
t
0.4
0.6
0.8
1
1.2
E
 a
p-wave, I=1/2
0 5 10 15
t
p-wave, I=3/2
K(2)π (-2)
K(1)π (-1)
Fig. 3. The energy levels E(t)a of the Kπ in the box for all four channels (multiply by
a−1 = 1.59 GeV to get the result in GeV). The horizontal broken lines show the ener-
gies E = EK + Eπ of the non-interacting scattering states K(n)π(−n) as measured on our
lattice; K(n)π(−n) corresponds to the scattering state with p∗ =
√
n 2π
L
. Note that there is
no K(0)π(0) scattering state for p-wave. Black and green circles correspond to the shifted
scattering states, while the red stars and pink crosses correspond to additional states re-
lated with resonances.
to resonances; we attribute them to K∗0(1430) in s-wave and K
∗(892), K∗(1410)
and K∗(1680) in p-wave. We extract the elastic phase shifts δ at several values of
the Kπ relativemomenta. The resulting phases exhibit qualitative agreementwith
the experimental phases in all four channels, as shown in Fig. 4. In addition to the
values of the phase shifts shown in Fig. 4, we also extract the values of the phase
shift close to the threshold, which are expressed in terms of the scattering lengths
in [2].
Scattering phase shifts and resonances from lattice QCD 77
0.6 0.8 1 1.2 1.4 1.6 1.8
sqrt(s) [GeV]
0
30
60
90
120
150
180
δ 
 [
de
gr
ee
s]
lat: present work
exp: Estabrooks (elastic)
exp: Aston (elastic)
exp: Aston (almost elastic)
s-wave,  I=1/2
0.6 0.8 1 1.2 1.4 1.6 1.8
sqrt(s) [GeV]
-40
-30
-20
-10
0
δ 
 [
de
gr
ee
s]
lat: present work
exp: Estabrooks (elastic)
s-wave,  I=3/2
0.6 0.8 1 1.2 1.4 1.6 1.8
sqrt(s) [GeV]
0
30
60
90
120
150
180
δ 
 [
de
gr
ee
s]
lat: present work
exp: Estabrooks (elastic)
exp: Aston (elastic)
exp: Aston (almost elastic)
p-wave,  I=1/2
0.6 0.8 1 1.2 1.4 1.6 1.8
sqrt(s) [GeV]
-30
-20
-10
0
10
20
30
δ 
 [
de
gr
ee
s]
lat: present work
exp: Estabrooks (elastic)
p-wave,  I=3/2
Fig. 4. The extracted Kπ scattering phase shifts δIℓ in all four channels l = 0, 1 and I =
1/2, 3/2. The phase shifts are shown as a function of the Kπ invariant mass
√
s = MKπ =
√
(pπ + pK)2. Our results (red circles) apply for mπ ≃ 266 MeV and mK ≃ 552 MeV in
our lattice simulation. In addition to the phases provided in four plots, we also extract the
values of δ
1/2, 3/2
0 near threshold
√
s =mπ+mK, but these are provided in the form of the
scattering length in the main text (as they are particularly sensitive to mπ,K). Our lattice
results are compared to the experimental elastic phase shifts (both are determined up to
multiples of 180 degrees).
We believe that these simulations of the Dπ, D∗π and Kπ scattering in the
resonant channels represent encouraging step to simulate resonances properly
from first principle QCD. There are many other exciting resonances waiting to be
simulated along the similar lines.
References
1. D. Mohler, S. Prelovsek and R. Woloshyn,Dπ scattering andDmeson resonances from
lattice QCD, arXiv:1208.4059.
2. C. B. Lang, Luka Leskovec, Daniel Mohler, Sasa Prelovsek, Kπ scattering for isospin
1/2 and 3/2 in lattice QCD, Phys. Rev. D.86. (2012) 054508, arXiv:1207.3204.