Exotic & Radiative Charmonium

Radiative & Exotic Charmonium Physics

in Lattice QCD

Jo Dudek

Jefferson Lab / Old Dominion University

Hirschegg

Exotic & Radiative Charmonium

Charmonium @ JLab ?

eventual aim is the exotic, radiative light-quark meson physics of GlueX

studying charmonium as a test-bed has a number of advantages

lattice constraints are less critical

experimental data is superiorsuccessful models to compare tointeresting in its own right(especially given PANDA)

!

p N

!!M |X"

Exotic & Radiative Charmonium

lattice method

gauge-field configurations generated in advance by Monte Carlo - in this case they are quenched123x48, as = 0.1 fm, at = 0.033 fm - anisotropic

radiative transitions extracted from three-point functions

interpolating fields are some combination of quark and gluon fieldsWick contraction expresses this in terms of propagators

propagator = inversion of Dirac matrix for a given gauge-field configurationrequires a choice of discretised Dirac matrix - for this study we used the Domain Wall Fermion action [on 300 configurations]

(a) (b) (c)

!(tf , t; !p, !q) =!

!x,!y

e!i!p·!x ei!q·!y !"f (!x, tf )jµ(!y, t)"(!0, 0)"

OZI (perturbative?) charges / no vector mix

NB no continuum (a→0) extrapolation yet done, but DWF should have no O(a)

Exotic & Radiative Charmonium

lattice method

three-point function related to the radiative transition matrix element

for tf >> t and t >> 0 the ground states (n = m = 0) dominate

can relate transition matrix element to a Lorentz invariant form-factor and then to the width, e.g.

!(tf , t; !p, !q)!

!

n,m

e!Efn (tf!t)"0|!f (0)|fn("p)#

!"im(!p + !q)|"i(0)|0#e!Eim t

!"fn(!p)| jµ(0) |im(!p + !q)#

!(! ! "c#) = $em|%q|3

(m!c + m")26427

|V (0)|2

!!c("p !)|jµ(0)|#("p, r)" =2V (Q2)

m!c + m"$µ#$%p!

#p$$%("p, r)

Exotic & Radiative Charmonium

J/ψ→ηc γ transition

statistically most precise channel, but very sensitive to the hyperfine splitting which is not correct on this quenched lattice (δmlat. ≈ 80 MeV, δmexpt. ≈ 117 MeV)

the Crystal Ball experimental value needs confirmation

!(! ! "c#) = $em|%q|3

(m!c + m")26427

|V (0)|2

physical

lattice

phase space

V (Q2) = V (0)e!Q2

16!2

! = 540(10) MeV

Exotic & Radiative Charmonium

χc0→J/ψ γ transition

derived the covariant multipole decomposition

E1(Q2) - electric dipole - experimentally measured at Q2 = 0C1(Q2) - longitudinal - goes to zero at Q2 = 0

this lattice δm(χc0 - J/ψ) close to experiment, so small phase-space ambiguity

!S(!pS)|jµ(0)|V (!pV , r)" =

!!1(Q2)

!

E1(Q2)

"

!(Q2)"µ(!pV , r) # "(!pV , r).pS

#

pµV pV .pS # m2

V pµS

$

%

+C1(Q2)&

q2mV "(!pV , r).pS

"

pV .pS(pV + pS)µ # m2

SpµV # m2

V pµS

%

'

.

Exotic & Radiative Charmonium

χc0→J/ψ γ E1 transition

E1(Q2) = E1(0)

!

1 + Q2

!2

"

e!

Q2

16"2

not used in the fit

PDG

CLEOlat.

Exotic & Radiative Charmonium

χc1→J/ψ γ transition

derived the covariant multipole decomposition

E1(Q2) - electric dipole - experimentally measured at Q2 = 0M2(Q2) - magnetic quadrupole - experimentally measured (via photon angular dependence) at Q2 = 0

C1(Q2) - longitudinal - goes to zero at Q2 = 0

this lattice δm(χc1 - J/ψ) close to experiment, so small phase-space ambiguity

!A(!pA, rA)|jµ(0)|V (!pV , rV )" = i4!

2!(Q2)"µ!"#(pA # pV )#$

$

!

E1(Q2)(pA + pV )"

"

2mA[""( !pA, rA).pV ]"!( !pV , rV ) + 2mV ["(!pV , rV ).pA]""!(!pA, rA)#

+ M2(Q2)(pA + pV )"

"

2mA[""( !pA, rA).pV ]"!( !pV , rV ) # 2mV ["(!pV , rV ).pA]""!(!pA, rA)#

+C1(Q2)

$

q2

"

# 4!(Q2)""!(!pA, rA)""(!pV , rV )

+ (pA + pV )"

%

(m2A # m2

V + q2)[""(!pA, rA).pV ] "!(!pV , rV ) + (m2A # m2

V # q2)["(!pV , rV ).pA] ""!( !pA, rA)&#

'

.

Exotic & Radiative Charmonium

χc1→J/ψ γ transition

no Q2 < 0 points owing to kinematical structure of matrix element

0 1 2 3 4 5 6

Q2 (GeV

2)

-0.25

-0.2

-0.15

-0.1

-0.05

0

a tM

2(Q2 )

spat. pf = (000) χc1 snk.

spat. pf = (100) χc1 snk.

0 1 2 3 4 5 6

Q2 (GeV

2)

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

a tE

1(Q2 )

spat. pf = (000) χc1 snk.

spat. pf = (100) χc1 snk.

PDG phys. massPDG lat. massCLEO phys. massCLEO lat. mass

E1 M2

Exotic & Radiative Charmonium

hc→ηc γ transition

prediction: Γ(hc→ηc γ) = 630(100)(30)(...) keV

Location: Name of talk

charmonium results - radiative transitions

non-rel. quark potential model should work reasonably here - do our results match?

E1(Q2) = E1(0)

!

1 + Q2

!2

"

e!

Q2

16"2

!c0 → J/"#E1

$ = 542(35) MeV

% = 1.08(13) GeV

!c1 → J/"#E1

$ = 555(113) MeV

% = 1.65(59) GeV

hc ! !c"E1

# = 689(133) MeV

$ ! "

simplest quark model has all β equal and ρ(χc0) = 2 β, ρ(χc1) = √2 ∙ ρ(χc0), ρ(hc) →∞

Exotic & Radiative Charmonium

higher spins & exotics

using local fermion bilinears , one is limited to JPC =

to get higher spins one needs non-local operatorswe implemented a version of the Manke-Liao operators, correcting them to make charge-conjugation eigenstates at

discretised versions of

tested this set on the quenched anisotropic lattice with Clover quarks (cheaper, but possibly larger scaling errors [in a]) [on 2000 configurations]

used a rather complete set of operators, more than one per ‘JPC’ where possible

!(x)!!(x)0!+, 0++, 1!!, 1++, 1+!

!p != (000)

! !!"Dk ! ! Γ

←→Dj←→Dk !

Exotic & Radiative Charmonium

S-wave

P-wave

D-wave

F-wave

exotics

higher spins, exotics

crudely displayed results - one effective mass for each ‘JPC’ - gross structure looks ok

t

meff ameff

3.0

GeV

4.0

5.0

!c

J/"#c0

#c1#c2

hc

"2

"3 !c2

#c3hc3

!c1

hc0

hc0

Exotic & Radiative Charmonium

higher spins, exoticsdetailed fitting of the large data set now ongoing - trying a variety of fitting schemes

some interesting results, e.g. Manke & Liao’s 1-+ mass estimate appears to be too high

3500

4000

4500

5000

5500

0 2 4 6 8 10 12 14

meff / MeV

t

Manke & LiaoNew

Manke fit

lattices are essentially equivalent

Exotic & Radiative Charmonium

excited states ?

using a variational method with a large basis of interpolating fields, seem to be able to see excited states corresponding to ψ(3686), ψ(3770) ?

0.4

0.5

0.6

0.7

0.8

0.9

5 10 15 20 25 30

3000

3600

4200

meff

t

ground state

2nd excited?1st excited?

garbage?

Exotic & Radiative Charmonium

two-photon decays

not obvious how to do this in Euclidean lattice QCDphotons are not eigenstates of QCDbut photons can be expressed as a sum of QCD vector eigenstates

expressed field-theoretically in the LSZ reduction

take advantage of pert. thy. in QED to (approximately) integrate out the photon - left with an integral over the ‘source timeslice’

!!(q1,"1)!(q2,"2)|M(p)" = # limq!1!q1

q!2!q2

#"µ(q1,"1)#"!(q2,"2)

$ q#21 q#22

!d4xd4y eiq!

1.y+iq!2.x!0|T

"Aµ(y)A!(x)

#|M(p)",

limtf!t"#

e2 !µ(q1,"1)!!(q2,"2)ZM (p)2EM (p)e

!EM (p)(tf!t)

!dtie

!"1(ti!t)

!"0|T! "

d3!x e!i!p.!x"M (!x, tf )"

d3!y ei !q2.!yj"(!y, t)jµ(!0, ti)#

|0#

Exotic & Radiative Charmonium

two-photon decays

for this more computationally expensive project, used isotropic Clover243x48, a = 0.066 fmconserved lattice vector current

plus improvement term to reduce O(a) errorsDirichlet boundary conditions in the temporal direction

some concern about possible O(mca) errorsnew technique should still be trialled - worry about systematic errors later

Exotic & Radiative Charmonium

-4 -2 0 2 4

Q2 / GeV

2

0

0.1

0.2

0.3

0.4

0.5

F(0

, Q2 )

one pole fitpf = (100) cons. curr

pf = (110) cons. curr

pf = (100) cons. imp. curr

pf = (110) cons. imp. curr

PDG

-6 -4 -2 0 2 4

Q2 / GeV

2

0

0.2

0.4

G(0

, Q2 )

one pole fitpf = (100) cons. curr

pf = (110) cons. curr

pf = (100) cons. imp. curr

pf = (110) cons. imp. curr

PDG

0 10 20 30 40ti

0

0.01

0.02

0.03

0.04

0.05

t = 4 t = 16 t = 32 tf = 37

0 10 20 30 40t

0.1

0.15

0.2

0.25

F(0

, Q

2)

tf = 37

Q2 = -2.10 GeV

2

Q2 = 0.60 GeV

2

Q2 = 2.68 GeV

2

Q2 = 4.43 GeV

2

two-photon decays

integrand - can capture the integral within the available timeslicesintegral - plateaus observed

!c0 ! ""!

!c ! ""!

Exotic & Radiative Charmonium

coming soon...

excited spectrum from variational method - multiple excited states?radiative transitions involving higher spins and exotics

excited states in radiative transitions

need to accurately extract sub-leading exponentials - tough, but possible

important physics of unquenching - effect of open/closed decay channelsuncontrolled (?) mass shifts in quark models from closed channels in self- energy

need to address the lattice systematicsextrapolate a→0 limit

reduce quark masses to address the light-quark physics of GlueX

DD