overview of heavy quark spectroscopy from a lattice...

Overview of Heavy Quark Spectroscopy froma Lattice Perspective

Carleton DeTar(MILC collaboration)

University of Utah

January 28, 2015

C. DeTar (University of Utah) Royal Society January 28, 2015 1 / 24

Purpose of this talk

I Provide an overview of T = 0 heavy-quark spectroscopy

I Review current lattice methods: possibilities and limitations

I Not a review of results

I See talks by Prelovsek, Thomas

I First, the challenge


our formerly comfortable

world (cf. 1932 e,p,n)

Charmonium spectroscopy before the B-factories

Mariana Nielsen (CHARM 2010)


JP C

Charmonium Spectroscopy after the B-factories – p.22/34

Charmonium spectroscopy after the B-factories

Mariana Nielsen (CHARM 2010)


Theoretical challenges

I Test of our ability/methodology for solving QCD.

I Precision spectroscopy of the low levels.

I Characterize and predict states.I Spin exotics: (JPC inaccessible through QQ).I Hybrids: QQ + glueI Tetraquarks: QQqq as in the Z+

c (?)I Molecules: deuteron-like D − D


Lattice QCD

I Our only ab initio method

I LimitationsI Including multihadronic states is difficult.I Real-time behavior is only indirectly accessible (see next talk)

I To extend its reach: supplement with models, effective field theory.


Lattice Spectroscopy Methods

I Hadronic correlator. Hermitian interpolating operators Oi .

Cij(t) = 〈0|Oi (t)O†j (0)|0〉

I Spectral decomposition

Cij(t) =∑n

〈0|Oi (0)|n〉e−Ent〈n|O†j (0)|0〉


Low lying charmonium spectrum

0−+ 1−− 1+− 0++ 1++

JPC

3.0

3.2

3.4

3.6

3.8

Mas

s/

GeV χc0

χc1

ηc

η′c

hc

J/Ψ

Ψ′

expt (PDG)

m`/ms = 1/5

m`/ms = 1/10

m`/ms = phys

I HPQCD Galloway et al. [1411.1318]

I Two lattice spacings, three sea-quark mass sets, two smearings

I Bayesian multiexponential fit (up to 9 exponentials)


Variational MethodI Spectral decomposition

Cij(t) =∑n

〈0|Oi (0)|n〉e−Ent〈n|O†j (0)|0〉

I Matrix formC (t) = ZT tZ † ,

I Effective transfer matrix and overlap matrix

T = diag exp(−En) Zi,n = 〈0|Oi (0)|n〉I Choose a large basis set of interpolating operators.I Solve the generalized eigenvalue problem (reference time t0)

C (t)un = λn(t, t0)C (t0)un

I If we have N linearly independent interpolating operators, and exactly Nenergy levels, we get, exactly,

λn(t, t0) = exp[−En(t − t0)].

I Otherwise, there are corrections that die exponentially ast0 > t − t0 →∞.

I Low-lying levels are more reliable, higher levels are difficult.C. DeTar (University of Utah) Royal Society January 28, 2015 9 / 24

Low-lying charmonium: FNAL/MILC result

I 5 lattice spacings, 2 sea quark masses. Can take physical limit.

I Spin-averaged 1P and 1S masses:

M1P = (Mχc0 + 3Mχc1 + 5Mχc2)/9

M1S = (Mηc + 3MJ/ψ)/4

I Splittings

1S hyperfine = MJ/ψ −Mηc

1P− 1S splitting = M1P −M1S

1P spin− orbit = (5Mχc2 − 3Mχc1 − 2Mχc0)/9

1P tensor = (3Mχc1 −Mχc2 − 2Mχc0)/9

1P hyperfine = 1P−M(hc)



0 0.05 0.1 0.15 0.2a [fm]

0.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

∆MH

F r

1

lattice data 0.2 mh

lattice data 0.1 mh

fit results at lattice parameters

leading shape

χ2/d.o.f. = 4.46/6

PRELIMINARY

0 0.05 0.1 0.15 0.2a [fm]

0.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

∆MH

F r

1

lattice data 0.2 mh

lattice data 0.1 mh

fit results at lattice parameters

leading + subleading shapes

χ2

aug/d.o.f.

aug = 2.84/7

PRELIMINARY

I Mohler, Lattice 2014 [1412.1057]

I Fit to shapes derived from NRQCD power counting.

I Experiment ∆MHF r1 = 0.178.

I Note width of ηc is 32 MeV!



Mass difference This analysis [MeV] Experiment [MeV]

1P-1S splitting 457.3± 3.6 457.5± 0.31S hyperfine 118.1± 2.1−1.5−4.0 113.2± 0.71P spin-orbit 49.5± 2.5 46.6± 0.1

1P tensor 17.3± 2.9 16.25± 0.071P hyperfine −6.2± 4.1 −0.10± 0.22


Higher-lying lattice cc spectroscopy

DDDD

DsDsDsDs

0-+0-+ 1--1-- 2-+2-+ 2--2-- 3--3-- 4-+4-+ 4--4-- 0++0++ 1+-1+- 1++1++ 2++2++ 3+-3+- 3++3++ 4++4++ 1-+1-+ 0+-0+- 2+-2+-0

500

1000

1500

M-

MΗc

HMeV

L

Hadron Spectrum Collaboration 1204.5425 (2012)


Problems

I Not safe to omit open charm (multihadronic states).

I Only one lattice spacing (as = 0.12 fm, at ≈ 0.032 fm)

I Heavy up/down quarks: 400 MeV pion

I Got charmonium HFS = 80 MeV vs 113.2(7) MeV (expt).


Optimized interpolating operators

I Variationally optimized state vector

Zi,n = 〈0|Oi (0)|n〉

|Ψn〉 =∑i

Z−1n,i Oi 〈0|

I |Ψn〉 is an approximate eigenstate with eigenvalue En.

I Could be helpful.


Optimized interpolating operators

I Customized smearing.

I von Hippel et al [1306.1440]: “Free form” interpolating operator

I Gauge invariant form giving complete control over relative wave function

I Replace Gaussian smearing with a more realistic wave function.

I e.g. Mark Wurtz et al [1409.7103] for first excited D-wave bottomonium.

ψij(x) = sin(2πxi/L) sin(2πxj/L)(r − b) exp(r/a)


Crafting interpolating operators

0 1 1 0 1 2 2 1 2 3 3 3 4 49.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8m

ass

(GeV

)

this workexperiment

BB threshold (experiment)

S P D F G

_ + _ _ _ _ _ _ _ _ _ _ _ _ _ _+ ++ ++ ++ + + ++ +


Including two-hadron statesI Discrete scattering states at finite volume, center of mass.

En = 2√m2 + p2 p = (2π/L)n (Noninteracting, equal mass, two-body)

I Luscher method. With interaction, En is shifted.

I As long as the box is larger than the size of the interaction regionL/2 > R, and scattering is only elastic, the shift in energy carriesinformation about the elastic scattering phase shift.

I Replace p = (2π/L)q from (noninteracting q = n).

E = 2√m2 + (2π/L)2q2

I Then we get the phase shift at momentum p = (2π/L)q from

p cot δ(p) =2Z00(1; q2)√

πL

I This is for S-wave. Generalizations apply to higher angular momentumand moving frames.


Near-threshold bound states

I Partial wave scattering amplitude.

T = 1/[p cot δ(p)− ip]

I Effective range formula

p cot δ(p) = 1/a0 +1

2r20 p

2

I So use lattice result at discrete p to determine a0 and r20 . Then doanalytic continuation.

I Poles in T represent bound states (a < 0) or resonances.

I For example, Lang et al [1301.7670] use this method to conclude thatthe Ds1 is a D∗K bound state and the D∗s0 is a DK bound state.

I Also applied to charmonium excitations (Prelovsek talk.)

I For a recent example, Lang et al. [1501.01646] [1403.8103]


Perturbative mixing model

I Treat light quark annihilation/production perturbatively

H = H0 + V

I Unperturbed states H0: DD∗ and cc

I Mixing term V connects these two sectors.

I Use Euclidean lattice correlators to compute matrix elements of V .

I Compute level shifts in conventional real-time perturbation theory.


Application to X(3872)

I Model: The X(3872) is the result of mixing the χc1(2P) with DD∗

scattering states.

I Second order energy shift (continuum) of the unperturbed χc1(2P):

E = E (0)(χc1(2P)) +

∫d3p

x(p)2

E − E (p)

where x(p) is the transition hamiltonian matrix element taking theχc1(2P) to the DD∗ scattering state with momentum p, and E (p) is theenergy of the DD∗ scattering state.

I Measure x(p) and E (0)(χc1(2P)) on the lattice. (Work in progress)


Conclusions

I Lattice QCD is needed to characterize, predict states

I State of the artI Multiple interpolating operators, including multihadronic states where

appropriateI Multiple lattice spacings permitting a continuum extrapolationI Ability to reach physical quark massesI Sufficiently large volumeI Careful tuning of the heavy quark masses.

I Variational method

I Elastic scattering phase shift determination

I Effective range approximation for states near threshold

I Lattice-inspired phenomenology


Thank you

University of Glasgow above the River Kelvin, photo by Laurel Casjens


overview of heavy quark spectroscopy from a lattice...

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