Properties of ground and excited state hadrons from ...Excited states only in the 1−− and 0−+ channels. Quenching effects are visible in the a 0 and ρchannels Daniel Mohler

Properties of ground andexcited state hadrons

from lattice QCD

Daniel Mohler

TRIUMF, Theory GroupVancouver, B.C., Canada

Newport News,March 22 2010

Co-Workers: Christof Gattringer, Christian Lang, Georg Engel,Markus Limmer, Leonid Glozman, Sasa Prelovsek, Richard Woloshyn

Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March 22 2010 1 / 30

Outline

1 Excited state spectroscopyExcited states and the latticeThe variational methodSuitable sources and sinks

2 Spectroscopy with Chirally Improved quarksLight-quark mesonsSpotlight: Scalar mesonsLight tetraquark states?

3 Baryon axial chargesResults for Nucleon and Hyperon axial charges


Motivation: ground state spectrum

Recent (impressive) results: Postdiction of the ground state spectrum

BMW-collaboration 2008

What about excited states?Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March 22 2010 3 / 30

Euclidean space correlators

Euclidean correlator of two Hilbert-space operators O1 and O2.⟨

O2(t)O1(0)⟩

T=

1ZT

tr(

e−T HetHO2e−tHO1

)

T→∞−→

∑

n

e−tEn

⟨

0|O2|n⟩ ⟨

n|O1|0⟩

Can also be expressed as a Euclidean path integral⟨

O2(t)O1(0)⟩

T=

1ZT

∫

D[ψ, ψ,U]e−SE O2[ψ, ψ,U]O1[ψ, ψ,U],

ZT =

∫

D[ψ, ψ,U]e−SE .

No field operators appear on the right.”Simple” integral over the classical Euclidean actionCan be evaluated with an (importance sampling) Markov chainMonte Carlo




O2(t)O1(0)⟩

T=

1ZT

tr(


)

T→∞−→

∑

n

e−tEn

⟨

0|O2|n⟩ ⟨

n|O1|0⟩


O2(t)O1(0)⟩

T=

1ZT

∫


ZT =

∫

D[ψ, ψ,U]e−SE .





O2(t)O1(0)⟩

T=

1ZT

tr(


)

T→∞−→

∑

n

e−tEn

⟨

0|O2|n⟩ ⟨

n|O1|0⟩


O2(t)O1(0)⟩

T=

1ZT

∫


ZT =

∫

D[ψ, ψ,U]e−SE .



The problem with excited states

From the analysis of Euclidean correlators we found:⟨

O2(t)O1(0)⟩

T∝

∑

n

e−tEn < 0|O2|n >< n|O1|0 >

The whole tower of statescontributes

Ground state is dominant at large t

Exited states appear as sub-leadingexponentials

Noisy background from limitedstatistics

. . .

E0

E1

E2

E3

Fit to several exponentials leads to poor results/ is often unstable→ Advanced methods needed for excited states!


Variational method for hadron masses

Variational method (Michael; Lüscher and Wolff; Blossier et al.)Matrix of correlators projected to fixed momentum (will assume 0)

C(t)ij =∑

n

e−tEn 〈0|Oi |n〉⟨

n|O†j |0

⟩

Solve the generalized eigenvalue problem:

C(t)~ψ(k) = λ(k)(t)C(t0)~ψ(k)

λ(k)(t) ∝ e−tEk

(

1 +O(

e−t∆Ek

))

At large time separation: only a single mass in each eigenvalue.Eigenvectors can serve as a fingerprint.


Example operators

Need: Interpolating field operator that creates states with correctquantum numbers.

Example I: Pseudoscalar Mesons with IJPC = 10−+

O(1)π = uγ5d

O(2)π = u

←→D γiγtγ5d

Example II: Nucleon

ON = ǫabc Γ1 ua(

uTb Γ2 dc − dT

b Γ2 uc)

In practice: Many (slightly different) constructions possible!


Example operators

Need: Interpolating field operator that creates states with correctquantum numbers.

Example I: Pseudoscalar Mesons with IJPC = 10−+

O(1)π = uγ5d

O(2)π = u

←→D γiγtγ5d

Example II: Nucleon

ON = ǫabc Γ1 ua(

uTb Γ2 dc − dT

b Γ2 uc)

In practice: Many (slightly different) constructions possible!


Angular momentum (mesons)

Reminder: No unique spin assignment on the lattice.Five irreducible representations:

Irrep of O J Spinors in irrep

A1 0,4,. . . 1,γt ,γ5,γtγ5

A2 3,6,. . .

E 2,4,5,. . .

T1 1,3,4,5,. . . γi ,γtγi , γ5γi ,γtγ5γi

T2 2,3,4,5,. . .

Classification of interpolator basis by representations

Identify spin by degeneracies/ through continuum extrapolation


A variational basis for meson spectroscopy

Jacobi smeared quark sources, e.g., us ≡ (S u)x

S = M S0 with M =

N∑

n=0

κnHn

H(~n, ~m ) =

3∑

j=1

(

Uj(

~n, 0)

δ(

~n + j, ~m)

+ Uj

(

~n − j , 0)†

δ(

~n − j , ~m)

)

.

Combination of different widths allows nodes in the interpolatingoperatorsDerivative quark sources Wdi

:

Di(~x , ~y) = Ui(~x , 0)δ(~x + i , ~y)− Ui(~x − i , 0)†δ(~x − i, ~y) ,

Wdi = Di Sw .

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Dynamical Chirally Improved fermions

Dmn =

16∑

α=1

Γα

∑

p∈Pαm,n

cαp

∏

l∈p

Ul δn,m+p

Insert above ansatz into Ginsparg-Wilson-equationTruncate the length of the contributions⇒ Set of algebraicequations

1s s4s2 s3+ + + ....

Wilson

v1 v3v2

a1γµγν t1+ ....+

+

++

−−

−−

γµγνγρ γ5

p1

+ + + ....+ +

+

+

−

−

− −µ

....

γ

+ + ....

(Gattringer, Hip, Lang, 2001)Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March 22 2010 10 / 30

CI fermions - simulation details

nf = 2 mass degenerate flavors of CI quarks

Lüscher- Weisz gauge actionHybrid Monte Carlo simulation

Mass preconditioning with 2 pseudofermionsChronological inverterMixed precision inverter (Dürr et al. PRD 79 014501)

Multiple ensembles for 163 × 32

set βLW m0 # config’s a[fm] mπ[MeV] mAWI [MeV]A 4.70 -0.050 50/100 0.151(2) 525(7) 42.8(4)B 4.65 -0.060 50/200 0.150(1) 470(4) 34.1(2)C 4.58 -0.077 200/200 0.144(1) 322(5) 15.3(4)

We are currently analyzing further ensembles and extendingstatistics


Example I: 0−+ and 1−− channels

For the lack of a systematic approach: linear fits to guide the eye

More data needed in all channels! (in progress)

0 0.4 0.8 1.2

Mπ2 [GeV

2]

0

0.5

1

1.5

2

2.5

mas

s [G

eV]

A from 3,8,11B from 3,8,11C from 3,8,11π(1300)

0 0.4 0.8

Mπ2 [GeV

2]

0

0.5

1

1.5

2

2.5

mas

s [G

eV]

ABC

ρ(770)

ρ(1450)

Figure: First excited states in the pion and vector meson channels


Example I: 0−+ and 1−− channels

For the lack of a systematic approach: linear fits to guide the eyeMore data needed in all channels! (in progress)

0 0.4 0.8 1.2

Mπ2 [GeV

2]

0

0.5

1

1.5

2

2.5

mas

s [G

eV]

A from 3,8,11B from 3,8,11C from 3,8,11π(1300)

0 0.4 0.8 1.2

Mπ2 [GeV

2]

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

mas

s [G

eV]

A from 1,4B from 1,4C from 1,4ρ(770)quenched

ρ meson

Figure: First excited states in the pion and vector meson channels


Example II: 2++ channel

0 0.4 0.8 1.2

Mπ2 [GeV

2]

0

0.5

1

1.5

2

2.5

mas

s [G

eV]

a2(1320)

A from 1,2,3,4B from 1,2,3,4C from 2,3

0 0.4 0.8 1.2

Mπ2 [GeV

2]

0

0.5

1

1.5

2

2.5

mas

s [G

eV]

a2(1320)

A from 1,2,5B from 1,2,5C from 1,2,5

Figure: Ground state of the a2 from the T2 (lhs.) and E (rhs.) irreduciblerepresentations


Light hadron masses

π a0 ρ a

1 b1

0

0.5

1

1.5

2

mas

s [G

eV]

CI lattice results

π2

a2 ρ

20

0.5

1

1.5

2

2.5

mas

s [G

eV]

T2 irrep

E irrep

Errors are purely statistical and systematical effects are notnegligible

Excited states only in the 1−− and 0−+ channels.

Quenching effects are visible in the a0 and ρ channels


Isovector scalar mesons: Some lattice history

Most quenched results indicated the ground state to be consistent withthe a0(1450)

Group ma0 [GeV]

Bardeen et al. 1.34(9)Mathur et al. 1.42(13)Burch et al. ≈1.45

0.0 0.4 0.8 1.2

Mπ2 [GeV

2]

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

MS [G

eV]

systematic errora0 (1450)a0 (980)8,10,11

Gattringer et al., PRD 78 (2008) 034501


Recent results I

0 0.4 0.8

Mπ2 [GeV

2]

0

0.5

1

1.5

2

mas

s [G

eV]

a0(980)

run A from 8run B from 8run C from 8

a0 meson

0 0.4 0.8

Mπ2 [GeV

2]

0

0.5

1

1.5

2

mas

s [G

eV]

a0(980)

run A from 8run B from 8run C from 8quenched from 5,7,8

a0 meson

Caveat: Broad range of values for differentinterpolators/ensembles

Role of scattering states?

Supporting evidence: mb1−ma0 ≈ 140− 200MeV


A different view: The scalar meson puzzle

Low lying scalars could be tetraquark states

−1 −1/2 0 1/2 1

I=1/2

I=0

I=0,1 a0(980)

?

3I

?

(below 1 GeV) mass Observed scalars

σ(600)

κ(800)

f0(980)

−1 −1/2 0 1/2 1

dssdussu

ussd

udds

udud

mass Tetraquark nonet

I=1/2

I=0

I=0,1

3I

3

I=0,1

I=0

I=1/2

uduudd

us

ss

mass qq nonet (vector meson case)

K*

ρ,ω

φ

−1 −1/2 0 1/2 1 I

quark models would place qq with L = 1 above 1GeV

mκ < ma0 hard to reconcile with us and ud

a0(980) couples well with K K


Light scalars as tetraquark states?

Question: Do the light scalars have a sizable qqqq component?

In the following, we do not distinguish between tetraquark statesand mesonic molecules

A succesful lattice identification will need to simultaneouslyidentify the lowest scattering states

EP1P2≈ EP1

(k) + EP2(−k) , EP(k) =

√

m2P + ~k2 ~k = 2π

L~n

Results are of a qualitative nature and we do not strive to measurethe width of light ressonances

Will focus on I = 0 and I = 12 channels (I = 1 channel contains

two towers of low-lying scattering states)


Scalar tetraquarks: Interpolating fields

Isospin 0 (flavor content 2duud − uudd + uuuu + dddd )

O1 = PP′; O2 =∑

i

Vi V ′i ; O3 =

∑

i

AiA′i

O4 = [q1Cγ5q2][q3Cγ5q4]; O5 = [q1γ5q2][q3γ5q4]

Isospin 2 (flavor content dudu)

O1 = PP′; O2 =∑

i

ViV ′i ; O3 =

∑

i

Ai A′i

When performing the contractions we neglect single and doubleannihilation diagrams.→ No qqqq ↔ qq ↔ vac ↔ glue mixing

(a) (b) (c)

��

��

��

��

��

��


Scalar tetraquarks: Results at a glance

0.2 0.3 0.4 0.5mπ [GeV]

0.40.60.8

11.21.41.61.8

2

E [G

eV]

n=1n=2n=3

0.2 0.3 0.4 0.5mπ [GeV]

dynamical simulationI=0 I=2

0.2 0.3 0.4 0.5mπ [GeV]

0.40.60.8

11.21.41.61.8

2

0.2 0.3 0.4 0.5mπ [GeV]

quenched simulation, L=16I=0 I=2

S.Prelovsek et al., in preparation

Possible interpretation as σ and κ or artifact from omissions?

We checked several things but have no final answer



0.2 0.3 0.4 0.5mπ [GeV]

0.6

0.8

1

1.2

1.4

1.6

1.8

E [G

eV]

n=1n=2n=3

0.2 0.3 0.4 0.5mπ [GeV]

dynamical simulationI=1/2 I=3/2

0.2 0.3 0.4 0.5mπ [GeV]

0.6

0.8

1

1.2

1.4

1.6

1.8

0.2 0.3 0.4 0.5mπ [GeV]

quenched simulation, L=16 I=1/2 I=3/2






0.2 0.3 0.4 0.5mπ [GeV]

0.6

0.8

1

1.2

1.4

1.6

1.8

E [G

eV]

n=1n=2n=3

0.2 0.3 0.4 0.5mπ [GeV]

dynamical simulationI=1/2 I=3/2

0.2 0.3 0.4 0.5mπ [GeV]

0.6

0.8

1

1.2

1.4

1.6

1.8

0.2 0.3 0.4 0.5mπ [GeV]

quenched simulation, L=16 I=1/2 I=3/2





Scalar tetraquarks: Discussion of results

Idea: Volume dependence to distinguish between one and twoparticle states

Z ni (16) ≃ ( 12

16 )3/2Z ni (12) when |n〉 is two-particle state P1P2

Z ni (16) ≃ Z n

i (12) in case when |n〉 is a one-particle state(resonance)

Condition to apply this problematic (See Niu et al PRD 80 114509):La should be much larger than the range of interaction between P1

and P2

resonance width Γ≪ ∆E with energy spacing ∆E

The time dependence of Eigenavlues on a periodic lattice mayhelp us to identify contributions from scattering states.

In the presence of a scattering state we should have the form

λn(t) = wn [e−En t+e−En(T−t)]+wn [e−mP1t e−mP2

(T−t)+e−mP2t e−mP1

(T−t)]

This fit form leads to stable effective mass plateaus




Z ni (16) ≃ ( 12


Z ni (16) ≃ Z n



and P2






(T−t)]





Z ni (16) ≃ ( 12


Z ni (16) ≃ Z n



and P2






(T−t)]



Scalar tetraquarks: Some further consistency checks

Extracted energies En and couplings Z ni = 〈0|On

i |n〉 in the I = 12

channel for several interpolator choices.

|Z ni | = |〈0|Oi |n〉| =

|∑

k Cik(t) unk (t)|

√∑

lm |un∗l (t) Clm(t) un

m(t)|eEnt/2

0 1 2 3 4 5 6 7 8choice

0.0

5.0×105

1.0×106

1.5×106

2.0×106

|Zin |

n=1, t=[7,10]

0 1 2 3 4 5 6 7 8choice

0.0

2.5×105

5.0×105

7.5×105

1.0×106

n=2, t=[7,10]

1 2 3 4 5 6 7 8choice

0.0

1.0×106

2.0×106

3.0×106

i=1i=2i=3i=4i=5

n=3, t=[6,9]

0 1 2 3 4 5 6 7 8 9choice

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

E a

I=1/2 , dyn. simulation, mπ=469 MeV, fit t=[7,10]


Outline

1 Excited state spectroscopyExcited states and the latticeThe variational methodSuitable sources and sinks

2 Spectroscopy with Chirally Improved quarksLight-quark mesonsSpotlight: Scalar mesonsLight tetraquark states?

3 Baryon axial chargesResults for Nucleon and Hyperon axial charges


Baryon axial charges

Axial charge: value of the axial form factor at zero momentumtransfer Ga,BB′(q2 = 0)

< B|Aµ(q)|B >

= uB(p′)

(

γµγ5Ga,BB′(q2) + γ5qµGp(q2)

2MB

)

uB(p)e−iq·x

No disconnected contributions in isovector combinations

Nucleon Ga is related to polarized quark distributions in the proton(assuming CVC): Ga = ∆u −∆dInteresting issues

χPT description↔ lattice dataEffects from excited states?Axial charges of negative parity nucleon excitations↔ speculations about chiral restauration


Variational method II - three point functions

We extract the axial charge from ratios of three-point functions

GA =ZA

ZV

∑

i∑

j ψ(k)i TA(t , t ′)ijψ

(k)j

∑

l∑

m ψ(k)l TV (t , t ′)lmψ

(k)m

TA and TV are the three-point functions with axial and vectorcurrent insertions. Burch et al.PRD79:014504,2009

The eigenvectors ψ are obtained from the variational analysis ofthe two-point functions

Taking one set of eigenvectors, one assumes translationinvariance and reflection symmetry


Nucleon effective masses

Example plot for effective masses (run B)

0 3 6 9 12 15t

0

0.2

0.4

0.6

0.8

1

1.2

1.4E

ffec

tive

mas

ses

- am

Nucleon groundstate

m=m-0.06


Results for Σ and Ξ masses

0 0.1 0.2 0.3 0.4 0.5 0.6

Mπ2 [GeV

2]

0

0.5

1

1.5

2

mas

s [G

eV]

GS run AGS run BGS run CExperimental value

Σ masses

0 0.1 0.2 0.3 0.4 0.5 0.6

Mπ2 [GeV

2]

0

0.5

1

1.5

2

mas

s [G

eV]

GS run AGS run BGS run CExperimental value

Ξ masses

Reminder: Strange quark mass set with the Omega Baryon


Preliminary results: Nucleon axial charge

0 0.1 0.2 0.3 0.4 0.5

Mπ2 [GeV

2]

0.6

0.8

1

1.2

GA

2 flavor CI 2.4 fm2+1 flavor domain wall 2.7fm2+1 flavor domain wall 1.8fmExperiment

Domain Wall results from Yamazaki et al., PRL 100, 171602 (2008)


Preliminary results: Σ and Ξ axial chrages

0 0.1 0.2 0.3 0.4 0.5

Mπ2 [GeV

2]

0

0.1

0.2

0.3

0.4

0.5

GA

2 flavor CI2 flavor mixed action

0 0.1 0.2 0.3 0.4 0.5

Mπ2 [GeV

2]

-0.4

-0.3

-0.2

-0.1

0

GA

2 flavor CI2 flavor mixed action

Mixed action results from Lin & Orginos, PR D79 034507 (2009)


Summary

We used the variational method to extract excited states and tosuppress contaminations from excited states

Good signals for most ground states; weak signals for excitedstates towards small pion masses

Basis with non standard interpolators significantly improves theresults in the scalar and pseudovector channels

Higher spin states can be obtained with derivative sources

We studied light scalar mesons using tetraquark interpolators

The variational method can be used for baryon three-pointfunctions


Summary

We used the variational method to extract excited states and tosuppress contaminations from excited states

Good signals for most ground states; weak signals for excitedstates towards small pion masses

Basis with non standard interpolators significantly improves theresults in the scalar and pseudovector channels

Higher spin states can be obtained with derivative sources

We studied light scalar mesons using tetraquark interpolators

The variational method can be used for baryon three-pointfunctions

Thank you!


Documents

Properties of ground and excited state hadrons from ...Excited states only in the 1−− and 0−+ channels. Quenching effects are visible in the a 0 and ρchannels Daniel Mohler