Meson Radiative Transitions on the Lattice hybrids and charmonium Jo Dudek, Robert Edwards, David Richards & Nilmani Mathur Jefferson Lab 1
JLab, GlueX and photocouplings GlueX plans to photoproduce mesons especially exotic J PC mesons γ resonance, X exchange meson, M p p γm X 2
photoproduction of exotics? exotic quantum numbers be explicable as hybrid meson states 1 +, 0 +, 2 + may photoproduction an untested method relies upon reasonably large couplings γ e.g. π ρa2 π 1 large in some model estimations 3
Lattice QCD estimation? relatively straightforward in principle; evaluate three-point function with a vector current in practice, not so easy truly light quarks unfeasible transitions involve unstable states experimental data is limited and imprecise (even for conventional meson transitions) 4
pragmatic approach try out an untested method in a region where approximations are controllable and where there is good experimental data to compare with charmonium 5
Charmonium - expt. multiple states below DD threshold have narrow widths radiative transitions are big branching fractions precision measurements 6
Charmonium - lattice states are small - small volumes OK quenched theory not sick* - just not expt. disconnected diagrams perturbatively suppressed need a fine lattice spacing a 3 GeV? η c J/ψ m c 1.5 GeV α s (O(m c )) * one heavy flavour QCD ; will only notice non-unitary up near 6 GeV 7
anisotropy charm quark mass scale requires a fine lattice but only in the temporal direction? spatial scale p 500 MeV so space direction can be more coarse introduce anisotropy param into fermion action and tune to get meson disp n rel ns right 8
our initial simulation anisotropic Wilson glue with at ξ = 3 β = 6.1 12 3 x48 gives a 1.2 fm box Domain-Wall fermions (L5 = 16) Ginsparg-Wilson ensures O(a) improvement vector current only multiplicatively renorm d spectroscopic splittings came out reasonably - usual quenched problem of hyperfine too small scale setting by Sommer parameter, but 1P-1S very similar 9
three-point functions Γ(t f, t ; p f, q) = x, y e i p f x e i q y ϕ f ( x, t f )j µ ( y, t)ϕ i ( 0, 0) we use gaussian smeared fermion bilinears as interpolating fields F ( z) ψ x+ z,t Γψ x z,t connected diagram constructed from forward propagator and sequential sink propagator with the simple point-like vector current 10 z new inversion for each change of the sink, but all possible momenta inserted at the current
three-point functions and matrix elements Γ(t f, t ; p f, q) = x, y e i p f x e i q y ϕ f ( x, t f )j µ ( y, t)ϕ i ( 0, 0) inserting two complete sets of states Γ(t f, t ; p f, q) = Z iz f 4E i E f e E f (t f t) e E it f( p f ) j µ (0) i( p i ) obtained from fits to two-point functions to be extracted 11
η c form-factor strictly speaking this does not exist due to charge conjugation invariance 0 + / 0 + 1 + 2 3 + = 0 2 3 0 0 1 + 0 12
η c form-factor η c ( p f ) j µ (0) η c ( p i ) = f(q 2 )(p i + p f ) µ Γ(t f = 24, t) Z i Z f 4E i E f e E f (t f t) E i t (p f + p i ) µ (111) (210) (100) (110) plateaux observed p f = (000) 13
η c form-factor 14
η c form-factor ( 1 + Q 2 ) 1 m 2 ψ r2 = 0.25 fm exp [ Q2 16β 2 (1 + αq2 ) ] 15
η c form-factor - not VMD? Q 2 ρ (1450) ρ Q 2 π ππ Ps. Ps. ψ (3770) ψ(3s) ψ J/ψ... D D Q 2 η c 16
J/ψ form-factors vector particle has three form-factors (c.f. deuteron) charge magnetic quadrupole V ( p f, r f ) j µ (0) V ( p i, r i ) = (p f + p i ) µ[ G 1 (Q 2 )ɛ ( p f, r f ) ɛ( p i, r i ) ] + G 3(Q 2 ) 2m 2 ɛ ( p f, r i ).p i ɛ( p i, r i ).p f [ɛ V ] + G 2 (Q 2 ) µ ( p i, r i )ɛ ( p f, r f ).p i + ɛ µ ( p f, r f )ɛ( p i, r i ).p f 17
J/ψ form-factors charge magnetic r2 0.25 fm κ c 0 quadrupole 18 a D (J/ψ) 10 3
χ c0 form-factors r2 0.3 fm larger radius due to centripetal barrier in P-wave meson 19
so much for unobservables, how about observables? 20
J/ψ η c γ transition η c ( p ) j µ (0) J/ψ( p, r) = 2V (Q2 ) m ηc + m ψ ɛ µαβγ p αp β ɛ γ ( p, r) very sensitive to hyperfine m J/ψ m ηc expt. = 117 MeV m J/ψ m ηc our lat. 80 MeV 21
Γ(J/ψ η c γ) = α J/ψ η c γ q 3 64 ˆV (0) 2. (m ηc + m ψ ) 2 27 transition phase space physical lattice beware - nobody gets this number right Crystal Ball expt. V (Q 2 ) = V (0)e Q 2 16β 2 new CLEO-c number soon! 22
P-wave to S-wave transitions many good measurements 23
χ c0 J/ψγ transition our poster boy covariant multipole decomposition of matrix element S( p S ) j µ (0) V ( p V, r) = ( [ Ω 1 (Q 2 ) E 1 (Q 2 ) Ω(Q 2 )ɛ µ ( ( p V, r) ɛ( p V, r).p S p µ V p V.p S m 2 V p µ ) ] S + C 1(Q 2 ) q 2 m V ɛ( p V, r).p S [p V.p S (p V + p S ) µ m 2 Sp µ V m2 V p µ S] ). E1 - electric dipole, expt ally measured at Q 2 =0 C1 - longitudinal, only non-zero at non-zero Q 2 24
χ c0 J/ψγ E1 transition our poster boy not used in the fit PDG CLEO lat. E 1 (Q 2 ) = E 1 (0) (1 + Q2 ρ 2 ) e Q2 16β 2 25
χ c0 J/ψγ C1 transition 26
χ c1 J/ψγ transition covariant multipole decomposition of matrix element A( p A, r A ) j µ (0) V ( p V, r V ) = [ i 4 2Ω(Q 2 ) ɛµνρσ (p A p V ) σ E 1 (Q 2 )(p A + p V ) ρ (2m A [ɛ ( p A, r A ).p V ]ɛ ν ( p V, r V ) + 2m V [ɛ( p V, r V ).p A ]ɛ ν( p A, r A ) ) + M 2 (Q 2 )(p A + p V ) ρ (2m A [ɛ ( p A, r A ).p V ]ɛ ν ( p V, r V ) 2m V [ɛ( p V, r V ).p A ]ɛ ν( p A, r A ) + C1(Q2 ) ( 4Ω(Q 2 )ɛ q 2 ν( p A, r A )ɛ ρ ( p V, r V ) ) + (p A + p V ) ρ [(m 2 A m 2 V + q 2 )[ɛ ( p A, r A ).p V ] ɛ ν ( p V, r V ) + (m 2 A m 2 V q 2 )[ɛ( p V, r V ).p A ] ɛ ν( p A, r A )]) ]. E1 - electric dipole, expt ally measured at Q 2 =0 M2 - magnetic quadrupole, expt ally measured at Q 2 =0 C1 - longitudinal, only at non-zero Q 2 27
χ c1 J/ψγ transition E1 M2 28
h c η c γ transition 29
quark potential model? our fitting form inspired by NR potential model with rel. corrections: E 1 (Q 2 ) = E 1 (0) (1 + Q2 ρ 2 ) e Q2 16β 2 χ c0 J/ψγ E1 β = 542(35) MeV ρ = 1.08(13) GeV χ c1 J/ψγ E1 β = 555(113) MeV ρ = 1.65(59) GeV h c η c γ E1 β = 689(133) MeV ρ 30
what about χ c2 J/ψγ? can t get at spin 2 with point-like fermion bilinears we have to extend our interpolating field set 31
extended interpolators Operator O h rep. lowest J P C name remark 1 A 1 0 ++ a 0 3 P 0 (χ c0 ) γ 5 A 1 0 + π 1 S 0 (η c ) γ i T 1 1 ρ 3 S 1 (J/ψ) γ 5 γ i T 1 1 ++ a 1 3 P 1 (χ c1 ) γ i γ j T 1 1 + b 1 1 P 1 (h c ) γ 5 i T 1 1 + π i T 1 1 a 0 γ 4 i T 1 1 + a 0 γ i i A 1 0 ++ ρ A 1 3 P 0 (χ c0 ) ɛ ijk γ j k E 1 ++ ρ T 1 3 P 1 (χ c1 ) s ijk γ j k T 2 2 ++ ρ T 2 3 P 2 (χ c2 ) γ 5 γ i i A 1 0 a 1 A 1 exotic γ 5 s ijk γ j k T 2 2 a 1 T 2 γ 5 S αjk γ j k T 2 2 a 1 E γ 4 γ 5 ɛ ijk γ j k T 1 1 + b 1 T 1 exotic γ 4 s ijk j k T 2 2 + a 0 D exotic γ 5 γ i D i A 2 3 ++ a 1 D A 2 γ 5 S αjk γ j D k E 2 ++ a 1 D E γ 5 s ijk γ j D k T 1 1 ++ a 1 D T 1 γ 5 ɛ ijk γ j D k T 2 2 ++ a 1 D T 2 γ 4 γ 5 s ijk γ i j k A 2 3 + b 1 D A 2 γ 4 γ 5 S αjk γ j D k E 2 + b 1 D E γ 4 γ 5 s ijk γ j D k T 1 1 + b 1 D T 1 γ 4 γ 5 ɛ ijk γ j D k T 2 3 + b 1 D T 2 γ i D i A 2 3 ρ D A 2 s ijk γ j D k T 1 1 ρ D T 1 ɛ ijk γ j D k T 2 2 ρ D T 2 γ 4 γ 5 s ijk j k T 2 2 + π D T 2 γ 5 B i T 1 1 π B T 1 ɛ ijk γ j B k T 1 1 + ρ B T 1 exotic s ijk γ j B k T 2 2 + ρ B T 2 γ 5 γ i B i A 1 0 + a 1 B A 1 exotic γ 5 ɛ ijk γ j B k T 1 1 + a 1 B T 1 γ 5 s ijk γ j B k T 2 2 + a 1 B T 2 exotic higher spins and the J PC exotics Table 1: Meson operators, names and quantum numbers. s ijk = ɛ ijk and S αjk = 0(j k), S 111 = 1, S 122 = 1, S 222 = 1, S 233 = 1.D i = s ijk j k, B i = ɛ ikj j k 32
extended interpolators exotics non-exotics Effective mass (2 + + ) Effective mass (0 + - ) 0.7 0.68 0.66 0.64 0.62 0.6 0.58 0.56 0.54 0.52 0.5 1.2 1.1 1 0.9 0.8 0.7 0.6! "_T 2 (SS) :! "_T 2 (SP) : 5 10 15 20 25 0.95 2 ++ 0.9! D_T 2 (SS) : 0.74 2 +! D_T 2 (SP) : 0.72 3 0.85 0.7 Effective mass (2 -+ ) t t 1.2 1.2 1.1 1.1 0 + 1 + 2 + Effective mass (1 - + ) 0.8 0.75 0.7 0.65 0.6 0.55 1 0.9 0.8 0.7 0.6 5 10 15 20 25 Effective mass (3 - - ) Effective mass (2 + - ) 0.68 0.66 0.64 0.62 0.6 0.58 0.56 1 0.9 0.8 0.7 0.6! D_A 2 (SS) :! D_A 2 (SP) : 5 10 15 20 25 t 0.5 0.5 0.5 0.4 0.3 a 1 B_ A 1 (SS) : a 1 B_ A 1 (SP) : 5 10 15 20 25 t 0.4 0.3 b 1!_ T 1 (SS) : b 1!_ T 1 (SP) : 5 10 15 20 25 t 0.4 0.3 a 1 B_T 2 (SS) : a 1 B_T 2 (SP) : 5 10 15 20 25 t 33
next up? radiative transitions with this extended set think we can do two-photon decays charmonium for now dynamical lattices for precision & maybe multi-particle (DD) states start turning down the quark mass if it all works to get at JLab physics 34
extra slides for the inquisitive 35
χ c0 J/ψγ E1 transition a t ^E1 (Q 2 ) -0.1-0.15 spat. p f = (000) J/ψ snk. spat. p f = (100) J/ψ snk. spat. p f = (000) χ c0 snk. spat. p f = (100) χ c0 snk. PDG phys. mass PDG lat. mass CLEO phys. mass CLEO lat. mass -0.2 0 0.5 1 36 Q 2 (GeV 2 )
some two-point functions J/ψ χ c0 37
multiple form-factors? pick out the three-point functions with the same Q 2 - various momentum and Lorentz index combinations Γ(a; t) P (a; t)k 1 (a) P (a; t)k 2 (a)... Γ(b; t) Γ(c; t) = P (b; t)k 1 (b) P (b; t)k 2 (b) P (c; t)k 1 (c) P (c; t)k 2 (c)..... invert this system with SVD f 1 (Q 2 )[t] f 2 (Q 2 )[t],. P K are known quantities 38
ZV set using meson form-factors at zero Q 2 1.22 1.34 1.2 1.18 1.32 1.3 1.28 1.26 spat. curr. ZV(clover) Z V 1.16 1.14 Z V 1.24 1.22 1.2 1.18 temp. p f =(000) temp. p f =(100) spat. p f =(100) 1.12 1.1 ZV(dwf) temp. p f =(000) temp. p f =(100) spat. p f =(100) 1.16 1.14 1.12 1.1 temp. curr. temp. curr. 1.08 1.08 η c J/ψ χ c0 χ c1 h c η c J/ψ χ c0 χ c1 h c 39
quenched? scale setting ambiguity - running coupling non-unitarity a negligible issue above threshold states rendered stable - they were narrow anyway 40
anisotropy - disp n rel n 1.15 ~6% deviation, could easily be 1.1 1.05 reduced c 2 1 0.95 0.9 η c J/ψ χ c0 0.85 c 2 ( p 2 ) E2 ( p 2 ) m 2 1 2 3 4 5 6 p 2 p 2 41
spectrum PDG lat η c ψ h c χ c0 χ c1
wrap-around pollution ti=0 t 0 ϕ f (t f = 24) f f j µ (t) i i ϕ i (t i = 0) 0 Z i Z f f j µ (0) i e E f (24 t) E i t tf=24 0 t 0 j µ (t) V V ϕ i (0) f f ϕ f (t i = 24) 0 Z V Z f V ϕ i (0) f e E V t E f 24-24 43
wrap-around pollution wrap Z V V ϕ i (0) f Z i f j µ (0) i e (E V δe if )t E V m J/ψ 3 GeV good δe if m χ m ψ 600 MeV so wrap around should fall off relatively sharply. if amplitude is large this will be a nasty pollution (prevents excited state extraction) 44
wrap-around pollution -0.8-0.9-1 -1.1-1.2 Q 2 = 0.99 GeV 2 rapid fall-off near t=0 indicative of the wrap-around pollution ^V(Q 2 )[t] -1.3-1.4-1.5-1.6-1.7-1.8 0 2 4 6 8 10 12 14 16 18 20 22 t we resorted to fitting the pollution with a single exponential f n (Q 2 )[t] = f n (Q 2 ) + f i e m it + f f e m f (24 t) ˆV (Q 2 ) = 1.55(1), f i = 1.45(3), 45f f = 0.42(14), m i = 0.41(1), m f = 0.27(7)
finite-size effects? previous charmonium spectrum studies saw no significant finite volume effects with QCD-TARO collabn. L s 1.1 fm we extracted from the form-factors that radius of charmonium states is 0.2 0.3 fm finite-size should be no problem for us @ Ls = 1.2 fm 46