1. Lattice QCD review of charmonium and open-charm meson spectroscopy
Sasa Prelovsek
University of Ljubljana & Jozef Stefan Institute, Ljubljana, Slovenia
(1 – 8: until non-precision spec: 5 min) Let me thank the organizers for the invitation. I was asked to review lattice QCD results on spectroscopy, in particular charmonium-like states and open charm mesons.
Charm 2013, , Manchester, UK
S. Prelovsek, CHARM 13, Lattice spectrum

2. S. Prelovsek, CHARM 13, Lattice spectrum
Experimental status
charmonium (like)
D-mesons
That is the experimental status for D meson spectrum: recent BABAR states are in orange and brand new LHCb are in red.
In charmonium, new conventional member was discovered this year by Belle. A take home message concerning lattice treatment of this spectra is as follows:
States well below strong decay threshold have been treated properly and precisely for several year now. I will concentrate on states near threshold and resonances above threshold: until one year ago they were treated naively and I will detail later what does this mean, but it basically means that effect of threshold and strong decays of states are ignored. The first pioneering simulations with rigorous treatment were done during the past year.
cc 2--: Belle, , PRL
D : LHCb,
S. Prelovsek, CHARM 13, Lattice spectrum

3. Lattice status – general overview: D, Ds , cc
States well below strong decay threshold:
proper treatment & precision calculations already available for some time
States near threshold and resonances above threshold:
★ simulations until 2012: naive treatment: - effect of threshold not taken into account
- strong decays of states ignored
exception: [Bali, Ehmann, Collins, 2011]
★ 2012, 2013, ...: first exploratory simulations with rigorous treatment
Experiment: charged charmonium-like
particle
decay
year
coll
Z+(4430)
ψ(2S) π+
2008
Belle, BABAR
Z+(4050), Z+(4250)
χc1 π+
Belle, unconfirmed
Zc+(3900)
J/ψ π+
2013
BESIII, Belle, CLEOc
Zc+(4020)
hc(1P) π+
BESIII preliminary
Zc+(4025)
( D* D*)+
BES III preliminary
The most exciting are of course the charged charmonium like states with this exotic quark structure, and simulations addressed some of them.
[BESIII, 2013, arXiv: ]
Zc+(3900) g J/Ψ π+
cc du
S. Prelovsek, CHARM 13, Lattice spectrum

4. S. Prelovsek, CHARM 13, Lattice spectrum
Overview
Spectrum of cc(like), D, Ds states from lattice QCD:
States well bellow threshold
Excited states:
★ naive treatment
★ rigorous treatment:
(1) resonances (above threshold)
(2) states slightly below threshold
(3) search for exotic states
So I will first review states well below th, then turn to excited states.
First with naive approach. Then with rigorous approach, where I will discuss resonances, near-threshold states and searches for exotic states.
S. Prelovsek, CHARM 13, Lattice spectrum

5. "Precision" spectrum: States well below strong decay threshold
So first states well below strong decay th
S. Prelovsek, CHARM 13, Lattice spectrum

6. States well below threshold
Lattice QCD already determined masses of these states
very reliably and precisely O(10 MeV):
m=E (for P=0)
extrapolation : ag0, Lg∞
extrapolation or interpolation : mqgmqphy
particular care needed for amc discretization errors:
several complementary methods give compatible results
[HPQCD: , PRD]
exp
[Briceno, Lin, Bolton, , PRD]
In this case, mass of a state is just equal to the energy level measured on the lattice.
Simulations are performed at several lattice spacing a and extrapolate to zero lattice spacing, and tipically hit the experimental value within ~ few MeV as shown in this HPQCD result. They also perform extrapolations to infinite lattice size L and extrapolate or interpolator to physical quark masses. The extrapolated masses are shown for HPQCD coll here, while chiQCD col provides ratios with respect to experimental value, and all results are for these ground state masses basically agree with experiment to about 10 MeV.
[HPQCD: , PRD]
m [GeV]
S. Prelovsek, CHARM 13, Lattice spectrum

7. States well below threshold
[χQCD: 2013, private com.]
Other recent precision lattice results:
J/ψ, ηc : spectrum and radiative decays:
★ HPQCD: , PRD
★ D. Becirevic and F. Sanfilippo: , JHEP
D, Ds spectrum:
★ PACS-CS: , PRD
★ HPQCD: , PRD
★ M. Kalinowski, M. Wagner : ,
charmonium spectrum:
★ D. CHARM 2012: [review]
★ D. Mohler et al., FNAL/MILC:
OFor your reference, I am collecting other related recent works here.
S. Prelovsek, CHARM 13, Lattice spectrum

8. "Non-precision" spectrum: excited states
only one or two a, L, mu/d
limits ag0, Lg∞, mu/dgmu/dphy usually not performed
(until rigorous treatment: duration 4 min) I will focus on excited states, which are usually simulated only at one lattice spacing, volume and light quark mass.
So for the most of the following results extrapolations are not performed.
S. Prelovsek, CHARM 13, Lattice spectrum

9. Excited states: naive treatment
only interpolating fields
assumptions: all energy levels correspond to "one-particle" states
m=E (for P=0)
these are strong assumptions ...
Naive treatment of meson states means that only quark antiquark interpolators are used. It is assumed that all energy levels correspond to one-particle states. And that mass of the state equals to the energy level. These are very strong assumptions for the resonances, as I will detail later.
S. Prelovsek, CHARM 13, Lattice spectrum

10. D spectrum: naive treatment
m-mref usually compared between lat and exp in order to cancel leading amc discretization effects
1S-2S splitting:
~ 700 MeV
This is resulting spectrum for D mesons.
The symbols denote lattice masses for various JP, while lines and boxes are experiment. Recent BABAR and LHCb results are side by side.
Meson masses minus some reference mass is comonly compared between lat and exp in order to cancel leading discretization effects related to charm. You can notice overall agreement. The splitting between 1S and 2S is around 700 MeV for both lat and exp.
red diamonds: rigorous treatment: discussed later
D. Mohler, S.P. , R. Woloshyn: , PRD:
mπ≈266 MeV, L≈2 fm, Nf=2
crosses: naive lat, diamonds: rigorous lat, lines & boxes: exp
S. Prelovsek, CHARM 13, Lattice spectrum

11. D spectrum: naive treatment
1S-2S:
~ 700 MeV
This is impressive D-meson spectrum result obtained by Hadron spectrum col. Green boxes are lattice results, while black are experiment. They determine JP very reliably, and they are able to extract highly excited states to a good precision. The caveat is that the results apply for relatively high mpi=400 MeV and it is all still within naive treatment. They identify states in terms of quark-antiquark multiplets 1S, 2S, 3S, 1P, 2P, 1D, 1F. The remaining states show strong overlap with interpolators that include gluon field tensor and are identified as hybrids. These unfortunately dont lead to exotic quantum number as charge parity is not a good quantum number in this channel. As you can see, the experiment has a way to go to get there.
G. Moir et al, HSC (Hadron Spectrum Coll.): , JHEP:
mπ≈400 MeV, L≈2.9 fm, Nf=2+1
reliable JP determination; many excited states
identification with n 2S+1LJ multiplets using <O|n>
green: lat, black: exp
Hybrids:
large overlap with O= q Fij q
gluonic tensor Fij=[Di , Dj ]
S. Prelovsek, CHARM 13, Lattice spectrum

12. Ds spectrum: naive treatment
This is analogous result for the Ds spectrum with similar multiplets and hibrids.
Hybrids:
large overlap with O= q Fij q
gluonic tensor Fij=[Di , Dj ]
G. Moir et al., HSC : , JHEP:
mπ≈400 MeV, L≈2.9 fm, Nf=2+1
reliable JPC determination
identification with n 2S+1LJ multiplets using <O|n>
green: lat, black: exp
S. Prelovsek, CHARM 13, Lattice spectrum

13. cc spectrum: naive treatment
Their impressive charmonium result was discussed at CHARM conference already last year.
Red boxes indicate states with good overlap with such hybrid interpolators.
Here C is good quantum number and some of the hybrids have manifestly exotic JPC, while some of them have non-exotic JPC.
HSC , L. Liu et al: , JHEP:
mπ≈400 MeV, L≈2.9 fm, Nf=2+1
reliable JPC determination
identification with n 2S+1LJ multiplets using <O|n>
green: lat, black: exp
Hybrids:
some of them have exotic JPC
large overlap with O= q Fij q
S. Prelovsek, CHARM 13, Lattice spectrum

14. Excited states: naive treatment
Other recent lattice results:
★ G. Bali, S. Collins, P. Perez-Rubio:
★ G. Bali et al.:
★ D. Mohler and R. Woloshyn: , PRD
★ M. Kalinowski, M. Wagner : ,
D. Mohler, S.P. , R. Woloshyn: , PRD
Further references are collected here.
S. Prelovsek, CHARM 13, Lattice spectrum

15. Excited states: rigorous treatment (1) resonances
(for resonances: duration 4 min) Let's proceed with the rigorous treatment, first for resonances. In experiment they are identified with peak in cross section, which corresponds to scattering phase shift Pi/2.
S. Prelovsek, CHARM 13, Lattice spectrum

16. D0*(2400) resonance in Dπ scattering: JP=0+, I=1/2
"rigorous" treatment illustrated on this example
All states with JP=0+ appear in lat. spectrum:
Do*(2400)
D(p) π(-p) with p=n 2π/L : "two-particle" states
horizontal lines indicate their energies in absence of interaction
π(-p)
D(p)
Rigorous relation [M. Luscher , 1991]:
E g δ(E) phase shift for Dπ scattering in s-wave
Let me explain the main idea about the so-called rigorous treatment on an example of scalar D-meson resonance, where mass and strong decay width are extracted in the end. If you follow trough this one slide, you will get an idea where the major step forward came about during the last year. Lattice QCD extracts the energy levels. The main point is that all states with given quantum number in general lead to energy levels: in the case of scalar quantum number, this the scalar D-meson but also Dpi two-particle states. In experiment those are continous in energy, while on the lattice momenta of D and pi is multiple of 2*pi/L due to the periodic boundary conditions and horizontal lines will always indicate what are energies of two-particles states in non-interacting case . To see all these levels, in paractice one needs to implement conventional quark-antiquark interpolators as well as two-particle interpolating fields . We did that and this is the resulting spectrum. We D*pi state with both particles at rest, we see D*pi state with first non-zero back to back momentum and in the middle we see an additional state. That appears due to the scalar resonance. So, the extra level in addition to discrete two-particle levels indicates the presence of a state, in this case a resonance. One can go a step further. Each energy level provides renders the value of the scattering phase shift at this energy, according to rigorous Luscher's relation.. In this case fo s-wave scatterig of D-pi. So, for example this level gives phase shift close to 90 degrees indeed. This spectrum gave us three phase shift values at three energy points and we fitted these with Breit-Wigner. This renders the mass and the width of the resonance. Mass of scalar resonance agrees reasonably with exp.
m and Γ
for D0*(2400)
D. Mohler, S.P. , R. Woloshyn: , PRD

17. D-meson resonance masses and widths
g is compared to exp instead of Γ (Γ depends on phase sp. and mπ)
JP=1+ : D* π (analysis of spectrum in this case is based on an assumption given in paper below)
JP=0+ : D π
D0*(2400)
m - 1/4(mD+3 mD*) g
lat
351 ± 21 MeV
2.55 ± 0.21 GeV
exp
347 ± 29 MeV
1.92 ± 0.14 GeV
D1(2430)
m - 1/4(mD+3 mD*) g
lat
381 ± 20 MeV
2.01 ± 0.15 GeV
exp
456 ± 40 MeV
2.50 ± 0.40 GeV
So this is the result for the scalar D resonance mass, which is graphically presented here. It agrees with experiment nicely. The width depends on phase space and so on pion mass, so the coupling defined in this way is compared between lattice and experiment and they agree reasonably. Similar excarsize was done for the broad axial resonances, with reasonable agreement again.
first lattice result for strong decay width of a hadron containing charm quark
[D. Mohler, S.P. , R. Woloshyn: , PRD]
mπ≈266 MeV, L≈2 fm, Nf=2
S. Prelovsek, CHARM 13, Lattice spectrum

18. JPC=0++ charmonuim resonance(s): χc0' ?
PRELIMINARY
χc0' ??
χc2'
By simulating DD scattering in s-wave we find:
(1) narrow resonance in DD scattering [we call it χc0' ]
PDG12: χc0'=X(3915) ?! Why X(3915)gDD in exp ?!
perhaps there is a hit of it [D. Chen et al, , PRD]
(2) additional enhancement of σ(DD) near th. :
could it be related to broad structures ?
[see also F. Guo, U. Meissner, , PRD]
Belle
BABAR
two B fit [*]
DD scattering was also simulated in order to find first excited scalar charmonium.
Two features are observed: the first is a narrow resonance with mass at about 3930 width a DD partial decay width of about 30 MeV. Interestingly the corresponding exprimental state X(3915) has not been seen in DD decay experimentaly, but this theoretical reference indicates there might be a hint of it here. In addition to the relatively narrow resonance we have an indication for another structure near threshold and this could be possibly related to these structures in Belle and Babar data, as pointed out by Gou and Meissner. In any case, a lot remains to be understood here in spite of this PDG recent assignment.
S.P. , L. Leskovec and D. Mohler,
to appear in Lat 2013 proc:
mπ≈266 MeV, L≈2 fm, Nf=2
S. Prelovsek, CHARM 13, Lattice spectrum

19. Excited states: rigorous treatment (2) states slightly below threshold
note: most of interesting states are found near threshold:
Ds0*(2317), X(3872), Zc+(3900), Zb+
(for threshold states: 4 min) Let me proceed to state slightly below threshold. As you know, most of interesting states are found near threshold.
S. Prelovsek, CHARM 13, Lattice spectrum

20. Ds0*(2317) and DK scattering (JP=0+)
δ for DK scattering in s-wave
extracted using Luscher's relation
a0<0 indicates a state below th.
relation above gives pole position and
the mass of Ds0*(2317)
This is the first simulation of scalar Ds that incorporates quark-antiquak as well as DK interpolating fields. Note that the simulation is done at almost physical pion mass of 156 MeV. And we observe two energy levels due to DK discrete states and additional level below threshold due to scalar resonance 2317.
If we used just quark-antiquark interpolators, as always done up to now, note that there is just energy level and its value is missleading. The levels also lead to info related to DK scattering phase shift. They lead to negative scattering length, which is another indication for the presence the state below threshold in view of Levinson's theorem. These two parameters lead to the pole position below threshold and therefore the mass of a state in infinite volume, which is quite close to experiment, while it was always much higher in the past.
Ds0*(2317)
m - ¼ (mDs+3mDs*)
lat
± 16±4 MeV
exp
± MeV
Candidate for Ds0*(2317) is found in addition to the DK states for the first time.
D. Mohler, C. Lang, L. Leskovec, S.P. , R. Woloshyn:
: mπ≈156 MeV, L≈2.9 fm, Nf=2+1
S. Prelovsek, CHARM 13, Lattice spectrum

21. S. Prelovsek, CHARM 13, Lattice spectrum
X(3872) : JPC=1++ [LHCb 2013] , I=0
δ for DD* scattering in s-wave
extracted using Luscher's relation
large and a0<0 indicates a state
slightly below DD* threshold: X(3872)
pole position gives mass of X(3872)
Similar exercise was done to search for the well known X(3872). Previous simulation used only quark antiquark interpolators and you get chi_c1 and another level here. But you can not tell whether it is X or DD* scattering state, that is why X has never been established on the lattice so far. We performed simulation using cbar c, DD* and J/psi omega interpolators. We find All expected scattering states and these are these three states above. On top of this we find two extra states: chi_c1 and a candidate for X. From the extracted energies we extract phase shift for DD* staring and we find large and negative a0, which is another indication for a state slightly below threshold, namely X. Mass of X is determined using the pole condition, and we find X 11 MeV below threshold, as compared to exp which sits on threshold. We believe this is the first time X has been unambigously identified from lattice in addition to all the scattering states, but reliable determination of mass will require simulation on the larger volumes.
X(3872)
m - (mD0+mD0*)
lat
± MeV
exp
± MeV
Candidate for X(3872) is found in addition to the expected two-particle states for the first time.
S. P. and L. Leskovec :
mπ≈266 MeV, L≈2 fm, Nf=2
lat: simulations on larger L required
exp: Tomaradze et al.,
S. Prelovsek, CHARM 13, Lattice spectrum

22. Composition of established X(3872) with I=0
it has sizable coupling with cc as well as DD* interpolating fields
overlaps of X with interpolators : [see S. P. and L. Leskovec : ]
Search for X(3872) with JPC=1++ and I=1
Only expected two-particle states observed.
No candidate for X(3872) found.
In agreement with interpretation
isospin breaking: D0 D0* , D+D-* splitting
In simulation: mu=md
As for the composition of established X: it couples significantly to cbar c as well as DD* interpolators. Overlaps with all interpolators can be found in his paper.
In the I=1 channel, candidate for X is not found, which is in agreement with the popular interpretation that X is dominantly I=0, while I=1 is proportional to isospin breaking. Isospin is exact in the simulation, that is possible reason why X is not seen.
S. P. and L. Leskovec :
mπ≈266 MeV, L≈2 fm, Nf=2
S. Prelovsek, CHARM 13, Lattice spectrum

23. (3) Searches for exotic states: rigorous treatment
(duration: 4 min) Naive treatment of meson states means that only quark antiquark interpolators are used. And it is assumed that all energy levels correspond to one-particle states. And that mass of the state equals to the energy level. These are very strong assumptions for the resonances, as I will detail later.
S. Prelovsek, CHARM 13, Lattice spectrum

24. Search for Zc+(3900) in JPC=1+- , I=1 channel
[BesIII, Belle, CleoC, 2013]
Only expected two-particle states observed.
No candidate for Zc+(3900) with JPC=1+- is found.
Possible reasons:
perhaps JPC≠1+- (exp unknown)
perhaps our interpolators (all of scat. type) are not diverse enough : calls for further simulations ??
This is a serch for interesting Zc(3900) in 1+- channel, which is the most popular quantum number, although experimentally it is unknown. We find only expected DD* and J/psi pi scattering states, and no additional state. We conclude we do not find a candidate for Zc in this channel. One possible reason is the it has other JPC, the other is that our basis with such scattering interpolators is not diverse enough and further simulations are required.
S. P. and L. Leskovec :
mπ≈266 MeV, L≈2 fm, Nf=2
S. Prelovsek, CHARM 13, Lattice spectrum

25. Search for Y(4140) in J/Ψ Φ scattering
Experiment:
Y(4140) found in J/Ψ Φ , Γ ≈ 11 MeV [CDF 2009]
not seen in DsDs
not seen by Belle, LHCb
Lattice:
method to get δ at more E:
twisted BC for valence q.
instead of periodic BC (conventional)
conclusion: no resonant structure found at energies reported by CDF
Caveats:
★ s-quark annihilation ignored
★ twisting is partial: only on valence quarks
J/Ψ Φ scattering phase shift [radians]
Ozaki and Sasaki simulated J/psi phi scattering to search for controversial Y(4140), observed by CDF but not confirmed by Belle or LHCb.
This lattice simulation uses a clever trick with twisted boundary conditions, insetead of conventional periodic ones. This allows determination of the phase shift at a number of energy values. But the phase shift remains small and never reaching Pi/2 at the position where the state was seen by CDF, so they do not find this state on the lattice.
S. Ozaki and S. Sasaki, , PRD
mπ≈156 MeV, L≈2.9 fm, Nf=2+1
S. Prelovsek, CHARM 13, Lattice spectrum

26. Search for bound double charm tetraquark with JP=1+, I=0
(1) determine potential between D and D*
at distance r: HALQCD method: Ishii et al., PLB712, 437 (2012)
(2) Solve Schrodinger equation with given V(r)
and determine DD* scattering phase shift
D r D*
Conclusion:
potential is attractive
no bound tetraquark state at simulated mπ
in case of one bound state one would expect δ(E=0)=π due to Levinson's theorem
Open double charm tetraquark was searched by HALQCD collaboration. First they determine the potential of D and D* meson at the distance r, which turns out to be attractive. Then Schrodinger equation is solved using this potential and phase shift for DD* scattering is calculated, as given here. The conclusion is that the potential is attractive but it does not support a bound state, as in case of a bound state the phase shift would start at pi.
Y. Ikeda et al, HAL QCD coll. , 2013, private com.
mπ≈ MeV, L≈2.9 fm, Nf=2+1
S. Prelovsek, CHARM 13, Lattice spectrum

27. S. Prelovsek, CHARM 13, Lattice spectrum
Related topics
(until the end : duration 4 min) I will finish with two related topics.
S. Prelovsek, CHARM 13, Lattice spectrum

28. Interactions of charmed mesons with light pseudoscalars
(1) Five channels that do not include Wick contractions are simulated
(2) Scattering lengths for four mπ extracted
(3) simultaneous fit using SU(3) unitarized ChPT is performed and LEC's are determined
(4) using these LEC's indirect predictions for:
scattering length of two resonant-channels
with contractions
DK (S=1,I=0): pole in the first Riemann sheet found
Interactions of charmed mesons with light pseudoscalars were simulated in five non-resonant channels and scattering lenght was extracted for four values of mpi. Then global fit was made using unitarized ChPT and LOC were determined. Using info on those, scattering lenght was predicted for two resonant channels and a pole was found in DK channel. The pole is found at this mass, so it corresponds to a experimental state Ds0*(2317). This indirect result agrees with direct result for this channel, discussed earlier.
Ds0*(2317) m
Γ [Ds0*gDs π]
indirect lat
MeV
133±22 keV
exp
±0.6 MeV
< 3.8 MeV
L. Liu, Orginos, Guo, Hanhart, Meissner, , PRD, mπ≈ MeV, Nf=2+1
S. Prelovsek, CHARM 13, Lattice spectrum

29. Charmonium potential at finite temperature
Potential between c and c in charmonium with physical mc at finite T: HAL QCD method
Nf=2+1 [C. Allton, private com.]
Tc: deconfining temperature
Tc ~ 220 MeV (from Polyakov loop)
Conclusions:
T<Tc : linearly rising V(r) at large r
T>Tc : flattening of V(r) at large r
Another interesting study calculated potential between c and cbar in charmonium at finite temperature. This is the result for several temperatures. As you can see, for temperatures below deconfining temperature, potential is linearly rising. For temperatures larger than Tc, the potential is flat at large distances.
earlier Nf=2 simulation:
Evans, Allton, Skullerud,
S. Prelovsek, CHARM 13, Lattice spectrum

30. S. Prelovsek, CHARM 13, Lattice spectrum
Conclusions & outlook
Present status of lattice results for D, Ds, cc spectra :
states well below strong decay threshold determined reliably and with good precision
excited states: naive treatment
spectra with a number of full qq multiplets and hybrids calculated during 2012, 2013
excited states: rigorous treatment: first simulations during 2012, 2013
★ D0*(2400), D1(2430), Ds0*(2317) , X(3872) identified
★ Zc+(3900), Y(4140), ccud not (yet) found
Precision simulations of these channels will have to be performed in the future.
This brings me to the conclusions. First let me recap the present status of lattice results for open charm and charmonium spectra. States well below strong decay thresholds have been determined with good precision. For excited states with naive treatment, we have seen the impressive results with a number of full qq multiplets and hybrids. During the past year, the excited states were rigorously treated for the first time, in particular D-meson resonances, scalar Ds and X(3872) were identified, while these states were not yet found.
S. Prelovsek, CHARM 13, Lattice spectrum

31. S. Prelovsek, CHARM 13, Lattice spectrum
Conclusions & outlook
Outlook for lattice simulations of D, Ds, cc spectra :
Which excited states can one treat rigorously in the near future?
states not to far above strong decay threshold that have one (dominant) decay mode
example: Zc+(3900) is less challenging than Z+(4430)
states that are not accompanied by many lower states of the same quantum number
example: higher lying 1– charmonium states would be very challenging for rigorous treatment
Lots of exciting experimental results prompt for lots of exciting lattice simulations
in the near future, encouraged by the pioneering exploratory steps made during the last year!
S. Prelovsek, CHARM 13, Lattice spectrum