Lattice QCD for Hyperon Spectroscopy David Richards Jefferson Lab KLF Collaboration Meeting, 12th Feb 2020 Outline • Lattice QCD - the basics….. • Baryon spectroscopy – What’s been done…. – Why the hyperons? • What are the challenges…. • What are we doing to overcome them… Lattice QCD • Continuum Euclidean space time replaced by four-dimensional lattice, or grid, of “spacing” a • Gauge fields are represented at SU(3) matrices on the links of the lattice - work with the elements rather than algebra iaT aAa (n) Uµ(n)=e µ Wilson, 74 Quarks ψ, ψ are Grassmann Variables, associated with the sites of the lattice Gattringer and Lang, Lattice Methods for Work in a finite 4D space-time Quantum Chromodynamics, Springer volume – Volume V sufficiently big to DeGrand and DeTar, Quantum contain, e.g. proton Chromodynamics on the Lattice, WSPC – Spacing a sufficiently fine to resolve its structure Lattice QCD - Summary Lattice QCD is QCD formulated on a Euclidean 4D spacetime lattice. It is systematically improvable. For precision calculations:: – Extrapolation in lattice spacing (cut-off) a → 0: a ≤ 0.1 fm – Extrapolation in the Spatial Volume V →∞: mπ L ≥ 4 – Sufficiently large temporal size T: mπ T ≥ 10 – Quark masses at physical value mπ → 140 MeV: mπ ≥ 140 MeV – Isolate ground-state hadrons Ground-state masses Hadron form factors, structure functions, GPDs Nucleon and precision matrix elements Low-lying Spectrum ip x ip x C(t)= 0 Φ(~x, t)Φ†(0) 0 C(t)= 0 e · Φ(0)e− · n n Φ†(0) 0 h | | i h | | ih | | i <latexit sha1_base64="equo589S0nhIsB+xVFPhW0b3Pck=">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</latexit> X~x X~x,n 2 E t = 0 Φ(0) n e− n |h | | i| <latexit sha1_base64="sJGqnNNHAx+9mldEOoTnaCkf0gg=">AAAC3XicdVJbi9NAFJ7E2xpvVR99OVgsXdGSVhd3HxYWFsHHCnZ3sWnLZHKaDjuZxJnJsiUN+OKDIr76v3zzf/gDnKRdjaIHBg7fd75znTATXBvf/+64ly5fuXpt67p34+at23dad+8d6TRXDEcsFak6CalGwSWODDcCTzKFNAkFHoenhxV/fIZK81S+McsMJwmNJZ9zRo2FZq0fQYgxlwW+k1Qpunxceoddsw2dfYBOoPNkVgRnyIrz8oksIRBUxgLBhyDhEeC04JBBwKLUwLmlhwve9bcr/Clv4lWwhECt1RdZJHg1U8mmRRDROEZVVglq2P8tCLwO7EMH6oYkrBqNrH5VXTVKrKaDuouXNtqUXoAyaow4a7X9nl8bNJy9ned7O8+gv0HaZGPDWetbEKUsT1AaJqjW476fmUlBleFMoM2fa8woO6Uxjq0raYJ6UtTXKeGRRSKYp8o+aaBGm4qCJlovk9BGJtQs9N9cBf6LG+dmvjspuMxyg5KtC81zASaF6tQQcYXMiKV1KFPc9gpsQRVlxn4Izy7hYlL4v3M06PX9Xv/1oH2wu1nHFnlAHpIu6ZMX5IC8IkMyIsx567x3Pjqf3Jn7wf3sflmHus5Gc5/8Ye7XnxXv3sg=</latexit> n X Durr et al., BMW Science 2008 Collaboration Now with electro-magnetic splittings included Need physical “ratios” to fit: mu/d, ms Science 2015 Variational Method Subleading terms ➙ Excited states Construct matrix of correlators with judicious choice of operators N N Z ⇤Z 1 i j EN t Cij(t, 0) = i(~x, t) j†(~y, 0) = e− V3 hO O i 2EN X~x,~y XN Delineate contributions using variational method: solve (N) (N) C(t)v (t, t0)=λN (t, t0)C(t0)v (t, t0). E (t t ) ∆E(t t ) λ (t, t ) e− N − 0 (1 + (e− − 0 )) N 0 ! O Can pull out excited-state energies - but pion and nucleon only states stable under strong interactions! <latexit sha1_base64="BDzGSyysIj74Il4VIe/fgoZZLQo=">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</latexit> Baryon Operators Aim: interpolating operators of definite (continuum) JM: OJM Starting point JM J 0 O J 0,M0 = Z δ δ h | | i J,J 0 M,M0 Flavor Spin Orbital Edwards et al., Phys.Rev. D84 (2011) i D! m= 1 = D! x i D! y 074508 − p2 − Introduce circular basis: ⇣ ⌘ D! m=0 = i D! z Chromomagnetic D! = i D! + i D! . m=+1 − p2 x y ⇣ ⌘ [D ,D ] F Straighforward to project to definite spin: J = 1/2, 3/2, 5/2 i j ⌘ ij Positive-parity Baryon Spectrum [2] 3.0 Dl=1,M 2.5 Hybrid: gluons structural 2.0 1.5 1.0 Dudek, Edwards, PRD85, 054016arXiv:1201.2349 Putting it Together 2000 1500 1000 500 0 Subtract ρ Subtract N Common mechanism in meson and baryon hybrids: chromomagnetic field with Eg ∼ 1.2 - 1.3 GeV Spectrum is at least as rich as quark model - plus hybrid states across flavor channels in P=+ R. Edwards et al., Phys. Rev. D87 (2013) 054506 Evidence for many charmed Baryons Bazavov et al, PLB 737, 210 (2014) All charmed mesons/baryons Charged charmed mesons/baryons Strange charmed mesons/baryons HRG with richer spectrum of states than PDG to describe lattice calculations Thermal Conditions at Freeze-out Bazavov et al, PRL 113, 072001 (2014) 0.30 µS/µB T=161 MeV T=170 MeV T=166 MeV Including additional 0.25 T=158 MeV T=155 MeV T=162 MeV strange states → T=155 MeV T=149 MeV lower freeze out LQCD: T=155(5) MeV T=152.5 MeV LQCD: T=145(2) MeV T=147 MeV temperature 0.20 T=150 MeV PDG-HRG T=145 MeV QM-HRG 39 GeV (STAR prlim.) 17.3 GeV (NA57) µB/T 0.15 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Can we learn about nature of the states? 4 Hints at structure of Λ(1405)? component within the ⇤(1405). The first term provides the full QCD contribution while the second term subtracts half of the weight of the disconnected sea-quark loop associated with photon couplings to the disconnected loop. Similarly, for the d-quark contribution, we obtain a value of 1 (2d 2 d +d ), 2 n − 3 n p and under charge symmetry, the two light quark contributions become equal, conn 1 2 ⇤⇤ µˆ ⇤⇤ = 2u u + u . (12) h | ` | i 2 p − 3 p n ✓ ◆ To test the KN model prediction of Eq. (12), we draw on the same set of configurations explored in Ref. [10]. where conn the left-hand side of the equation, ⇤⇤ µˆ ⇤⇤ , was calcu- h | ` | i lated. These calculations are based on the 323 64 full-QCD HallFIG. et 3. al,The Phys. light (u orRev.d) and D strange 95, 054510 quark contributions to the magnetic ⇥ form factor of the ⇤(1405) at Q2 0.16 GeV2/c2 from Ref. [10] are ensembles created by the PACS-CS collaboration [1], made ' presented as a function of the light u- and d-quark masses, indicated by the available through the International Lattice Data Grid (ILDG) squared pion mass, m2 . Sector contributions are for single quarks of unit 3 ⇡ [71]. The ensembles provide a lattice volume of (2.9 fm) charge. The lattice calculations are compared to the predictions of the con- with five different masses for the light u and d quarks, and nected KN model developed herein and summarized in Eq. (12). The vertical dashed line indicates the physical pion mass. The strange form factor results constant strange-quark simulation parameters. We simulate 2 are offset a small amount from the light sector in the m⇡-axis for clarity. the valence strange quark with a hopping parameter of s = 0.13665, reproducing the correct kaon mass in the physical limit [72]. We use the squared pion mass as a renormaliza- loop is significant. In the case where such couplings are in- tion group invariant measure of the quark mass. The scale cluded, the prediction of the KN model is significantly larger is set via the Sommer parameter [73] with r =0.492 fm 0 at ⇤⇤ µˆ ⇤⇤ =(2u + u )/2=1.03(2) µ . Thus, it is h | ` | i p n N [1]. The nucleon magnetic form factors are determined on important for the lattice community to continue to work to- these lattices using the methods introduced in Ref. [68] and wards a determination of these disconnected-loop contribu- refined in Ref. [70], providing values of up =1.216(17) µN tions, particularly for resonances where coupled channel dy- and u = 0.366(19) µ at the lightest pion mass. Results n − N namics play an important role. are reported for the lowest nontrivial momentum transfer of The light-quark sector contributions to the magnetic form Q2 0.16 GeV2/c2. ' factor of the ⇤(1405) calculated in lattice QCD [10] have been Lattice QCD results from Ref. [10] for the light- and examined in the context of a molecular KN model in which strange-quark magnetic form factors of the ⇤(1405) are plot- the quark-flow connected contributions to the magnetic form ted as a function of pion mass in Fig. 3. As mentioned earlier, factor have been identified. This enables a quantitative anal- the flavor symmetry present at heavy quark masses is broken ysis of the extent to which the light-quark contributions are as the u and d masses approach the physical point, where consistent with a molecular bound-state description. the strange magnetic form factor drops to nearly zero. The Identification and removal of the quark-flow disconnected light quark sector contribution differs significantly from the contributions to the KN model have been made possible via molecular KN model prediction until the lightest quark mass a recently developed graded-symmetry approach [44]. It is is reached. At this point, the direct matrix element calculation, conn interesting to note that the relative contribution of connected ⇤⇤ µˆ ⇤⇤ of Ref. [10], “⇤(1405) light sector” in Fig. 3, h | ` | i to disconnected contributions is in the ratio 1:2for both agrees with the prediction of the “connected