arXiv:1311.5010v1 [hep-lat] 20 Nov 2013 d h g b a e c f i an,Germany Mainz, 2013 3, August LATT - - 29 Theory July Field Lattice on Symposium International 31st Collaborations QCDSF-UKQCD E-mail: iepo 6 B,UK 3BX, L69 Liverpool Schierholz
.Horsley R. states charmed and breaking symmetry flavour SU(3) c SM colo hmsr n hsc,Uiest fAdelai of University Physics, and Chemistry of School CSSM, einlsRceznrm nvriä abr,216Ham 20146 Hamburg, Universität Rechenzentrum, Regionales etce lkrnnSnhornDS,263Hmug Ge Hamburg, 22603 DESY, Elektronen-Synchrotron Deutsches hoeia hsc iiin eateto Mathematical of Department Division, Physics Theoretical Germany Jülich, 52425 Jülich, Forschungszentrum JSC, IE dacdIsiuefrCmuainlSine Kobe, Science, Computational for Institute Advanced RIKEN ntttfrTertsh hsk nvriä epi,04 Leipzig, Universität Physik, Theoretische für Institut ntttfrTertsh hsk nvriä Regensburg Universität Physik, Theoretische für Institut colo hsc n srnm,Uiest fEibrh E Edinburgh, of University Astronomy, and Physics of School oyih we yteato()udrtetrso h Creat the of terms the under author(s) the by owned Copyright ∗ siaino h eetyitoue isnflwscales. flow Wilson introduced recently the to of and estimation quantities, singlet using proced scale the the of of determination potential the a demonstrate singly and and encouraging states are meson pseudoscalar isospin charmed QCD open p some possible we for including masses masses quark hadron valence quark quenched charmed f partially expansion to breaking masses symmetry quark flavour SU(3) the extending By Speaker. [email protected] ∗ g a .Schiller A. , .Najjar J. , b .Nakamura Y. , d .Stüben H. , v omn trbto-oCmeca-hrAieLicen Attribution-NonCommercial-ShareAlike Commons ive h c n .M Zanotti M. J. and .Perlt H. , 0 epi,Germany Leipzig, 109 34 eesug Germany Regensburg, 93040 , d .Pleiter D. , cecs nvriyo Liverpool, of University Sciences, C 2013 ICE ibrhE93Z UK 3JZ, EH9 dinburgh yg 5-07 Japan 650-0047, Hyogo e dlieS 05 Australia 5005, SA Adelaide de, hsedw logv eeapreliminary a here give also we end this ug Germany burg, rmany r.Esnilfrtemto sthe is method the for Essential ure. i ddul hre aynstates baryon charmed doubly nd raigefcs nta results Initial effects. breaking o p onadsrnesea strange and down up, rom ooeamto odetermine to method a ropose eb .E .Rakow L. E. P. , ce. o(ATC 2013)249 PoS(LATTICE iepo T 993 LTH Liverpool 2013/30 Edinburgh ADP-13-23/T843 http://pos.sissa.it/ EY13-220 DESY f G. , SU(3) flavour symmetry breaking and ... R. Horsley
1. Motivation and Strategy
At present there is considerable interest in open charm masses. While much is known about C = 1 charmed meson masses and singly charmed baryon masses, the situation is far less clear for doubly charmed quark baryons (i.e. C = 2 or ccq with q = u, d, s). Here many masses are unknown, but as stable states in QCD they must exist (presently only some candidate states are seen by the SELEX Collaboration [1], but not by BaBar [2], or BELLE [3]). Also there are possible relations to tetraquark states. For example in the nc ∞, [4], mc ∞ limit we expect the relation → → M(ccud) MΞ++ + MΛ+ M 0 (M + + M 0 )/4, [5]. ≈ cc c − D − D∗ D The charm quark, c is considerably heavier than the up u, down d and strange s quarks, which has hampered its direct simulation using lattice QCD. However as the available lattice spacings become finer, this is becoming less of an obstacle. The sea quarks in present day n f = 2+1 flavour dynamical lattice simulations consist of two mass degenerate (i.e. mu = md) light flavours u, d and a heavier flavour s. Their masses are typically larger than the ‘physical’ masses necessary to reproduce the experimental spectrum. How can we usefully approach the ‘physical’ u, d, s quark masses? One possibility suggested in [6] is to consider an SU(3) flavour breaking expansion from a point m0 on the flavour symmetric line keeping the average quark mass m = (mu + md + ms)/3 constant (= m0). This not only significantly reduces the number of expansion coefficients allowed, but the expansion coefficients remain the same whether we consider mu = md or mu = md. Thus 6 we can also find the pure QCD contribution to isospin breaking effects with just one n f = 2 + 1 numerical simulation. The SU(3) flavour breaking expansion can also be extended to valence quark masses, i.e. the quarks making up the meson or baryon have not necessarily the same mass as the sea quarks. The valence quarks are called ‘Partially Quenched’ or PQ quarks in distinction to the sea or dynamical sea quarks. We call the ‘Unitary Limit’ when the masses of the valence quarks coincide with the sea quarks. PQ determinations have the advantage of being cheap compared to dynamical simulations and including them allows a better determination of the expansion coefficients over a wider range of quark masses. (This was the strategy pursued in [7].) In addition because the charm quark, c is much heavier than the u, d and s quarks, it contributes little to the dynamics of the sea and so we can regard the charm quark as a PQ quark.
2. Method
We presently only consider hadrons which lie on the outer ring of their associated multiplet and not the central hadrons. (So we need not consider either any mixing or numerical evaluation of quark–line disconnected correlation functions.) The SU(3) flavour symmetry breaking expansion for the pseudoscalar mesons with valence quarks a and b up to cubic or NNLO terms in the quarks masses is given by
2 2 α δ µ δ µ M (ab) = M0π + ( a + b) 1 2 2 2 2 2 2 +β0 (δm + δm + δm )+ β1(δ µ + δ µ )+ β2(δ µa δ µb) 6 u d s a b − γ δ δ δ γ δ µ δ µ δ 2 δ 2 δ 2 + 0 mu md ms + 1( a + b)( mu + md + ms ) 3 2 +γ2(δ µa + δ µb) + γ3(δ µa + δ µb)(δ µa δ µb) , (2.1) −
2 SU(3) flavour symmetry breaking and ... R. Horsley
with δ µq = µq m, q = a,b,... u,d,s,c being valence quarks of arbitrary mass, µq and δmq = − ∈ { } mq m, q u,d,s being sea quarks. (These have the automatic constraint δmu +δmd +δms = 0.) − ∈ { } Note that we have some mixed sea/valence mass terms. The unitary limit occurs when δ µq δmq. 2 α → The expansion coefficients are M0π (m), (m), ... so if m is held constant then we have constrained fits to the numerical data. In particular a n f = 2 + 1 flavour simulation, when δmu = δmd δml is ≡ enough to determine the expansion coefficients. We now use the PQ (and unitary) data to determine the expansion coefficients (i.e. αs, βs, γs). This in turn leads to a determination of the ‘physical’ exp exp exp δ δ δ δ µ η quark masses mu∗, md∗, ms∗ and c∗ by fitting to e.g. Mπ+ (ud), MK+ (us) and M c (cc). We can now describe pseudoscalar open charm states with the same wavefunction (and hence expansion) M γ 0 + + δ δ δ δ µ = q 5c (q = u, d, s) i.e. D (cu), D (cd) and Ds (cs). Using mu∗, md∗, ms∗ and c∗ gives estimates of their physical masses. Similarly for the baryon octet the same procedure can be applied. We have the SU(3) flavour symmetry breaking expansion
2 2 M (aab) = M + A1(2δ µa + δ µb)+ A2(δ µb δ µa) 0N − 1 2 2 2 2 2 2 2 2 +B0 (δm + δm + δm )+ B1(2δ µ + δ µ )+ B2(δ µ δ µ )+ B3(δ µb δ µa) 6 u d s a b b − a − 2 2 2 +C0δmuδmdδms + [C1(2δ µa + δ µb)+C2(δ µb δ µa)](δm + δm + δm ) − u d s 3 2 +C3(δ µa + δ µb) +C4(δ µa + δ µb) (δ µa δ µb) − 2 3 +C5(δ µa + δ µb)(δ µa δ µb) +C6(δ µa δ µb) , (2.2) − −
(so for example Mp = M(uud)). Again we use PQ (and unitary) data to first determine the expan- sion coefficients (i.e. the As, Bs, Cs). We can then describe charm states with the same nucleon like wavefunction and hence same expansion. For example for single open charm (C = 1) states Σ++ Σ0 Ω0 ε T γ we have c (uuc), c(ddc), c(ssc) which all have the wavefunction B = (q C 5c)q (q = u, Ξ++ Ξ+ Ω+ d, s) while for double open charm (C = 2) states we have cc (ccu), cc(ccd), cc(ccs) which all B ε T γ δ δ δ δ µ have the wavefunction = (c C 5q)c (q = u, d, s). In both cases using mu∗, md∗, ms∗ and c∗ gives estimates of these physical masses.
3. Lattice details
We use a tree level Symanzik gluon action and an O(a) improved clover fermion action, in- cluding mild stout smearing [8]. Thus the quark mass is given by
1 1 1 mq = , (3.1) 2 κq − κ0c where κq is the hopping parameter, κ0 is the hopping parameter along the symmetric line with κ0c being its chiral limit. We shall consider two lattice spacings (β = 5.50 on 323 64 lattices and × preliminary results for β = 5.80 on 483 96 lattices). × We shall now briefly mention here our progress in defining and determining the scale using singlet quantities, collectively denoted here by XS. There are many possibilities such as pure gluon quantities like the r0 Sommer scale: Xr0 = 1/r0, or the √t0 [9], w0 [10] scales based on the Wilson gauge action flow: Xt0 = 1/√t0, Xw0 = 1/w0, or quantities constructed using fermions. One simple possibility in this case is to take the ‘centre of mass’ of the hadron octet. In all these cases it can
3 SU(3) flavour symmetry breaking and ... R. Horsley
easily be shown that linear terms in δmq are absent, [6]. We then have from eqs. (2.1), (2.2) in the δ 2 δ 2 δ 2 δ 2 unitary limit (with m = ( mu + md + ms )/3) 2 1 2 2 2 2 2 2 2 1 2 Xπ = (M + + M 0 + Mπ+ + Mπ + M 0 + M )= M π + β0 + 2β1 + 3β2 δm + ..., (3.2) 6 K K − K K− 0 2 and 2 1 2 2 2 2 2 2 2 1 δ 2 X = (M + M + MΣ+ + MΣ + M 0 + MΞ )= M + (B0 + B1 + B3) m + .... (3.3) N 6 p n − Ξ − 0N 2 2 In the left panel of Fig. 1 we plot various singlet quantities (aSXS) for S = t0, w0, π, ρ, N. It is
S = t0 β=5.50, κ =0.12090 S = N 0.007 β=5.50 0.3 0 S = w0 S = ρ S = π ] 2 0.006
0.2 [fm 2 ) exp2 S S X /X S 2 ) (a S X
S 0.005
S = π = (a
0.1 2
S S = N
a S = ρ
S = w0 S = t 0.004 0
0 −0.010 −0.005 0.000 8.264 8.266 8.268 8.270 8.272 κ δ 1/ 0 ml
2 Figure 1: Left panel: (aSXS) for S = t0, N, w0, ρ and π along the unitary line, from the symmetric point δ δ δ δ ml = 0 down to the physical point ml∗ = ( mu∗ + md∗)/2 (vertical dashed line) together with constant fits β κ 2 π ρ κ (for = 5.50, 0 = 0.12090). Right panel: Values of aS for S = , N, , w0 and t0 using eq. (3.4) for 0 values on the symmetric line from κ0 = 0.12090 to κ0 = 0.12099, for β = 5.50, together with quadratic fits. The crossing of the horizontal and vertical dashed lines and circle gives an estimate for the common scale.
2 apparent that the constancy (aSXS) holds over the complete range from the symmetric point down exp to the physical point. Using this enables us to use XS to determine the lattice spacing by 2 2 (aSXS) aS = exp 2 . (3.4) XS We shall define our lattice spacing here using S = N: aN . Of course depending on how well we have chosen our initial κ0 point, using a different singlet quantity (i.e. S = N) will give a slightly different 6 lattice spacing. More ambitiously we can vary κ0 to try to find a point where we have a common 2 scale. We have initiated a programme to investigate this. In the right panel of Fig. 1 we plot aS π ρ κ 2 π ρ again for S = , N, , w0, t0 against various 0. The crossing of the aSs for S = , N and give exp exp an estimation of the common scale as a 0.074(2)fm. We can now adjust Xt0 , Xw0 to also cross ≈ exp at this point to find a preliminary estimate for these ‘intermediate scales’ of √t0 0.153(7)fm, ≈ wexp 0.179(6)fm. 0 ≈ Practically it is numerically advantageous to form dimensionless ratios (within a multiplet): 2 2 2 2 2 M M /X and re-write eqs. (2.1), (2.2) in terms of α α/M π , ... and Ai Ai/M , ... in the ≡ S ≡ 0 ≡ 0N expansions. About O(80) PQ and unitary masses are used to determine these expansion coeffi- e e ∼ δ δ e δ δ µ cients and hence the ‘physical’ quark masses mu∗, md∗, ms∗ and c∗ as described in section 2.
4 SU(3) flavour symmetry breaking and ... R. Horsley
4. Results and Conclusions
We now discuss our results. In Fig. 2 left panel, we show the diagonal pseudoscalar mesons
90 1.8
pure QCD 80 β=5.50 β=5.50 N(lll’) Σ(lls) 1.6 Ξ(ssl) N (sss’) 70 s
60 1.4 2 π 50
[Octet] 1.2 2 a’)/X N
(a 40 2 /X 2 M O N
30 M 1.0
20 0.8 10
0 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 −0.010 −0.005 0.000 δµ δ a ml
2 2 2 Figure 2: Left panel: M (aa′)= M (aa′)/Xπ versus δµa for β = 5.50, together with the fit from eq. (2.1). The vertical dashed line represents the symmetric point, while the horizontal dashed line is the physical value e 2 δ of Mηc . Right panel: ‘fan’ plot for the baryon octet, from the symmetric point ml = 0 to the physical point δm = (δm + δm )/2 (vertical dashed line and stars) for β = 5.50. The filled triangles are from 323 64 l∗e u∗ d∗ × sized lattices, while the open triangles are from 243 48 sized lattices (not used in the fits). The fits are × again given from eq. (2.1).
2 M (aa′) (to avoid a three dimensional plot) versus δ µa together with the fit from eq. (2.1) (using the prime notation a to mean a distinct quark from a but degenerate in mass). The horizontal dashed e ′ 2 line represents the physical value of Mηc , the intersection with the fit curve gives a determination 2 2 2 2 of δ µ . In the right panel we show a ‘fan’ plot of M , MΣ, MΞ and M against δml, together c∗ e N Ns with the fit using eq. (2.1). Note that the scales involved are rather different, for the unitary masses e e e e δml 0.01 and the LO terms in eq. (2.1) or (2.2) dominate, [6], while for the PQ masses (reaching | |∼ up to the charm masses) we have δ µa 0.4 but still with rather moderate curvature. 0 + ∼ + 2 In Fig. 3 we show D (cd), D (cd) and Ds (cs) against our aN lattice spacings (left panel) and their mass differences (right panel). These mass differences in particular are sensitive to unknown QED effects (the present computation is for pure QCD only). As we currently have only two lattice spacings (and are also increasing their statistics) the results are to be regarded as preliminary and we do not presently attempt a continuum extrapolation. However there do not seem to be strong scaling violations present. In Figs. 4 and 5 we show the C = 1 and C = 2 charmed baryons (from the spin 1/2 20-plet). Again while we do not see significant lattice effects in either case, we gain an impression that those present are a little larger for the doubly charmed baryons than for the singly charmed mesons. In conclusion we note that we have developed a method to determine some open charm states using a precise SU(3) flavour symmetry breaking expansion enabling u, d, s quarks to approach the physical point while the c quark is treated as PQ. The expansion appears to be highly convergent. The method can be extended to other states. In a 2 + 1 world there is no Σ0 - Λ0 mixing, but the determined coefficients can be used to compute Σ0(uds) - Λ0(uds) mixing, [12]. Therefore
5 SU(3) flavour symmetry breaking and ... R. Horsley
2000 125 + D +(cs)−D+(cd) Ds (cs) s 100
1750 75 2000 125 + + 0 D (cd) D s(cs)−D (cu) 100
1750 75 2000 15 0 D (cu) D+(cd)−D0(cu) 10
5
1750 0 0.0000 0.0025 0.0050 0.0075 0.0000 0.0025 0.0050 0.0075 a 2 [fm2] 2 2 N aN [fm ]
Figure 3: Left panel: D0(cu), D+(cd) and D+(cs). Right panel: D+(cd) D0(cu), D+(cs) D0(cu) and s − s − D+(cs) D+(cd) mass splittings. (All values in MeV.) The experimental values are given as red stars. To s − guide the eye, we extend these values as horizontal dashed lines.
3000 300 Ω 0 Ω 0(ssc)−Σ 0(ddc) c (ssc) c c 2750 250
2500 200 2750 300 Σ 0 Ω 0 Σ ++ c (ddc) c (ssc)− c (uuc) 2500 250
2250 200 2750 15
++ Σ (uuc) Σ 0(ddc)−Σ ++(uuc) c 10 c c 2500 5
2250 0 0.0000 0.0025 0.0050 0.0075 0.0000 0.0025 0.0050 0.0075 a 2 [fm2] 2 2 N aN [fm ]
Figure 4: Left panel: Σ++(uuc), Σ0(ddc), Ω0(ssc). Right panel: Σ0(ddc) Σ++(uuc), Ω0(ssc) Σ++(uuc), c c c c − c c − c Ω0(ssc) Σ0(ddc) mass splittings. c − c
Σ+ Λ+ Ξ0 Ξ 0 computing e.g. c - c , c - c′ mixing is possible. Furthermore the method can be applied to the baryon decuplet and QED effects can be introduced, [13].
Acknowledgements
The numerical configuration generation (using the BQCD lattice QCD program [11]) and data analysis (using the Chroma software library) was carried out on the IBM BlueGene/Q us- ing DIRAC 2 resources (EPCC, Edinburgh, UK), the BlueGene/P and Q at NIC (Jülich, Germany), the SGI ICE 8200 at HLRN (Berlin–Hannover, Germany) and on the NCI National Facility in Canberra, Australia (supported by the Australian Commonwealth Government). The BlueGene/P codes were optimised using Bagel. This investigation has been supported partly by the DFG under contract SFB/TR 55 (Hadron Physics from Lattice QCD) and by the EU grants 227431 (Hadron Physics2), 283826 (Hadron Physics3) and 238353 (ITN STRONGnet). JN was partially supported
6 SU(3) flavour symmetry breaking and ... R. Horsley
4000 150 + + Ω +(ccs) Ω (ccs)−Ξ (ccd) cc 125 cc cc 3500 100
3000 75 4000 150 + ++ Ξ +(ccd) Ω (ccs)−Ξ (ccu) cc 125 cc cc 3500 [] 100
3000 75 4000 15 ++ Ξ (ccu) Ξ +(ccd)−Ξ ++(ccu) cc 10 cc cc 3500 5
3000 0 0.0000 0.0025 0.0050 0.0075 0.0000 0.0025 0.0050 0.0075 a 2 [fm2] 2 2 N aN [fm ]
Figure 5: Left panel: Ξ+ (ccd) Ξ++(ccu). The SELEX Collaboration result, [1], for Ξ+ is shown as a red cc − cc cc star. Right panel: Ω+ (ccs) Ξ++(ccu), Ω+ (ccs) Ξ+ (ccd) mass splittings. cc − cc cc − cc by EU grant 228398 (HPC-EUROPA2). JMZ is supported by the Australian Research Council grant FT100100005. We thank all funding agencies.
References
[1] M. Mattson et al. [SELEX Collaboration], Phys. Rev. Lett. 89 (2002) 112001, [arXiv:hep-ex/0208014]; A. Ocherashvili et al. [SELEX Collaboration], Phys. Lett. B628 (2005) 18, [arXiv:hep-ex/0406033]. [2] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D74 (2006) 011103, [hep-ex/0605075]. [3] R. Chistov et al. [BELLE Collaboration], Phys. Rev. Lett. 97 (2006) 162001, [hep-ex/0606051]. [4] S. Weinberg, Phys. Rev. Lett. 110 (2013) 261601, [arXiv:1303.0342[hep-ph]]. [5] M. Karliner et al. JHEP 1307 (2013) 153, [arXiv:1304.0345[hep-ph]]. [6] W. Bietenholz et al. [QCDSF–UKQCD Collaboration], Phys. Rev. D 84 (2011) 054509, [arXiv:1102.5300[hep-lat]]. [7] R. Horsley et al. [QCDSF–UKQCD Collaboration], Phys. Rev. D 86 (2012) 114511, [arXiv:1206.3156[hep-lat]]. [8] N. Cundy et al. [QCDSF–UKQCD Collaboration], Phys. Rev. D 79 (2009) 094507, [arXiv:0901.3302[hep-lat]]. [9] M. Lüscher, JHEP 1008 (2010) 071, [arXiv:1006.4518[hep-lat]]. [10] S. Borsanyi et al. [BMW Collaboration], JHEP 1209 (2012) 010, [arXiv:1203.4469[hep-lat]]. [11] Y. Nakamura et al., PoS(Lattice 2010) 040 (2010), arXiv:1011.0199[hep-lat]. [12] QCDSF Collaboration, in preparation. [13] R. Horsley et al. [QCDSF–UKQCD Collaboration], PoS(LATTICE 2013) 499 (2013), arXiv:1311.4554[hep-lat].
7