In Figure 4.2 the extracted spectrum of charm-strange excitations is shown in lattice units labelled by lattice irrep, ΛP, for both positive and negative parity.
As discussed earlier, mesons that are not eigenstates of the charge conjugation operator, such as the Ds meson (c¯s), are grouped by their JP quantum numbers
as opposed to JP C.
The energies are extracted from the principal correlators and spins are assigned according to the pattern of operators overlaps as detailed in the previous chapter. The various different colors in Figure4.2 correspond to different continuum spins. States with J = 0 are grey, J = 1 are red, J = 2 are green, J = 3 are blue and
J = 4 are orange.
The pattern of subduction into the different irreps for all J ≥ 2, as described in Table3.2 is evident. For example, for both positive and negative parity, in theA2, T2 andT1 irreps the appearance of two almost degenerate blue states can be seen, as expected for a spinJ = 3 state. As discussed before, it is checked that the states have similar operator overlap values, to ensure the matching across the different irreps is correct, shown in Figure 3.5. These degeneracies should become more exact as the continuum limit is approached. This overlap method is necessary as
Chapter 4. Ds and D Meson Spectrum 52
Figure 4.3: Ds meson spectrum labelled byJP. Green and red boxes are the
masses computed on theMπ ∼240 MeV ensemble while black boxes are
experimental masses of the neutral Ds mesons from the PDG summary
tables [65]. The vertical size of the boxes indicates the one-sigma statistical (or experimental) uncertainty on either side of the mean. Red boxes show states identified as constituting the lightest hybrid supermultiplet, as
described in the text. Dashed lines indicate theDK threshold using computed (coarse green dashing) and experimental (fine grey dashing) masses.
higher in the spectrum there is a large amount of overlapping states which would be impossible to untangle by eye alone.
Having identified the spin of the states, a joint fit to the principal correlators is performed across the various subduced irreps to extract a single energy for the spin identified spectrum. In Figure4.3 the energies of the different excitations are shown in MeV for each value of J ≤ 4. As was stated before, the energies are shown with half of the mass of theηcsubtracted. This is to reduce any systematic
uncertainty which may arise due to the tuning of the charm quark mass.
The energy threshold for strong decays into DK has also been included in Figure 4.3. Above this line the various excitations should be thought of as resonances, however as was stated before, there are no operators that interpolate two-meson states in the operator basis for the GEVP, so these states are treated as stable. The energies of the experimentally observed states have also been added in black. On inspection, among the states a clear pattern can be seen. They follow the simple quark model n2S+1L
J pattern. For instance the lowest two states with
Chapter 4. Ds and D Meson Spectrum 53 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 A−1 A − 2 E− T − 2 T − 1 A+1 A+2 E+ T2+ T1+
Figure 4.4: As Figure4.2 but for theD spectrum.
about 700 MeV higher. In the positive parity sector the lowest four states are a P wave singlet and triplet. Above these excited S and P as well as D and F
multiplets can be identified.
There are however some states which do not fit in with this pattern in the negative parity sector, labelled in red in Figure 4.3. These states have relatively strong overlap with operators that are proportional to the gluonic field strength tensor ie. states where the gluonic field has explicit excited degrees of freedom. It is proposed that these are members of the lightest hybrid supermultiplet, [(0−,1−,2−),1−].
This is consistent with a 1+− gluonic excitation coupled to an S wave meson.
In Figure 4.4 the spectrum computed on the same gauge configurations for the
D meson is shown labelled by lattice irrep. The D meson correlators used two different time sources as opposed to one for the Ds meson. The spectrum can be
seen to be qualitatively the same as the previous Ds meson spectrum shown in
4.2. The states of spin J ≥2 are subduced across the appropriate irreps and the overall pattern is the same. Some higher lying states have not been included in the plot due to noisy principal correlators.
The spin identified D meson spectrum is shown in 4.5. This time two relevant decay thresholds are included, that of Dπ and D∗π, using both experimental and
calculated masses. A similar spin structure can be seen and S, P, D and F wave multiplets can be identifed. A hybrid multiplet in the negative parity sector is
Chapter 4. Ds and D Meson Spectrum 54
Figure 4.5: As Figure 4.2but for theD meson spectrum. Dashed lines show theDπ and D∗π thresholds using computed (coarse green dashing) and experimental (fine grey dashing) masses.
also identified in the same position in the spectrum as before. The energies shown in both the Ds and D spin plots are tabulated in Appendix B.
It is worth noting that the data agrees quite closely with the few available ex- perimentally observed masses. The fact that the calculation was performed at an unphysically heavy pion mass may go some way towards explaining any discrep- ancies. There may also be discretisation effects due to the finite lattice spacing which lead to increased uncertainty.