rho resonance parameters from lattice qcdrho resonance parameters from lattice qcd dehua guo, andrei...

Rho Resonance Parameters from Lattice QCD

Dehua Guo, Andrei Alexandru, Raquel Molina and Michael Doring

Jefferson Lab Seminar

Oct 03, 2016

Dehua Guo, Andrei Alexandru, Raquel Molina and Michael Doring (GWU)Rho Resonance Parameters from Lattice QCD Oct 03, 2016 1 / 26

Overview

1 Introduction to Lattice QCD

2 Method

3 Results

4 Conclusion


Introduction to Lattice QCD

Most visible matter in the universe are made up of particles called hadrons.

The interaction between hadrons is dominated by the strong force.

Quantum Chromodynamics (QCD) is a theory to describe the strong interactionbetween quarks and gluons which make up hadrons.

LQCD = −1

2Tr FµνF

µν −∑f

ψf γµ [∂µ − igAµ]ψf −

∑f

mf ψf ψf , (1)

Some techniques to work with QCD: Perturbation theory, Effective field theory,Lattice QCD and so on.



For light hadron study, non-perturbative approach is needed. Lattice QCD is anon-perturbative approach to QCD. It formulates QCD in a discrete way.

Inputs:

lattice geometry N

lattice spacing a set indirectly through thecoupling constant g

quark mass represented by pion mass mπ

For light hadron study, only light quarks u and d are important. s quark introduces onlysmall correction.The role of Lattice QCD in resonance study is to extract the energy spectrum for twohadron states.



Why we study resonances from Lattice QCD?

Lattice QCD offers us a way to study the resonances in terms of quark and gluondynamics. It serves as a test of QCD for well determined resonance parameters.

The techniques can be used to investigate systems where the experimental situationis less clear.

Validate effective models used to describe hadron scattering.

How?

We start from meson resonance because they have better signal-to-noise ratio.

ρ(770) resonance in I = 1, J = 1 π-π scattering channel.


Symmetries on the latticeOn the lattice, the energy eigenstates |n > of the system are computed in a given irrep ofthe lattice symmetry group.

ψn(R−1x) = ψn(R−1(x + nL));⟨O2(t)O†1 (0)

⟩=∑n

⟨0|O2|n

⟩⟨n|O1|0

⟩e−tEn (2)

Isospin, color and flavor symmetries are similar to the continuum.

Table: Irreducible representation in SO(3),O and D4

SO(3) cubic box(Oh) elongated box(D4h)irep label Ylm; l = 0, 1...∞ A1,A2,E ,F1,F2 A1,A2,E ,B1,B2

dim 1, 3, ..., 2l + 1, ...∞ 1, 1, 2, 3, 3 1, 1, 2, 2, 2

Table: Angular momentum mixing among the irreducible representations of the lattice group

Oh D4h

irreducible representation l irreducible representation lA1 0,4,6, ... A1 0,2,3, ...A2 3,6, ... A2 1,3,4, ...F1 1,3,4,5,6, ... B1 2,3,4, ...F2 2,3,4,5,6, ... B2 2,3,4, ...E 2,4,5,6, ... E 1,2,3,4, ...


Symmetries of the elongated box

ρ resonance is in I = 1,JP = 1− channel for pion-pion scattering. Elongated box methodtunes the momentum of the scattering particles on the lattice p ∝ ( 2π

ηL).

The SO(3) symmetry group reduce to discrete subgroup Oh or D4h

J Oh D4h

0 A+1 A+

1

1 F−1 A−2 ⊕ E−

2 E+ ⊕ F+2 A+

1 ⊕ B+1 ⊕ B+

2 ⊕ E+

3 A−2 ⊕ F−1 ⊕ F−2 A−2 ⊕ B−1 ⊕ B−2 ⊕ 2E−

4 A+1 ⊕ E+ ⊕ F+

1 ⊕ F+2 2A+

1 ⊕ A+2 ⊕ B+

1 ⊕ B+2 ⊕ 2E+

For the p-wave(l = 1) scattering channel, we only need to construct the interpolatingfields in F−1 in the Oh group, A−2 representations in D4h group because the energycontribution from angular momenta l ≥ 3 is negligible.


Luscher’s formula for elongated box [1]Phase shift for l = 1, rest frame (P = 0):

A−2 : cot δ1(k) = W00 +2√5W20 (3)

(4)

Wlm(1, q2, η) =Zlm(1, q2, η)

ηπ32 ql+1

; q =kL

2π; η =

Nel

N: elongation factor (5)

Zeta functionZlm(s; q2, η) =

∑n

Ylm(n)(n2 − q2)−s ; n ∈ m (6)

Total energy

E = 2√

m2 + k2; k =

√(E

2

)2

−m2 (7)

[1] X. Feng, X. Li, and C. Liu, Phys.Rev. D70 (2004) 014505


Luscher’s formula for boost frame

In order to obtain new kinematic region, we boost the resonance along the elongateddirection.

P→

A−2 : cot δ1(k) = W00 +2√5W20 (8)

(9)

Wlm(1, q2, η) =ZP

lm(1, q2, η)

γηπ32 ql+1

; η =Nel

N: elongation factor; γ : boost factor; (10)

Z Plm(s; q2, η) =

∑n

Ylm(n)(n2 − q2)−s ; n ∈ 1

γ(m +

P

2); (11)


Interpolating field construction for ρ resonanceFour qq operators and two scattering operators ππ in A−2 sector.

ρJ(tf ) = u(tf )Γtf Atf (p)d(tf ); ρJ†(ti ) = d(ti )Γ†tiA†ti

(p)u(ti ) (12)

N Γtf Atf Γ†ti A†ti1 γi e ip −γi e−ip

2 γ4γi e ip γ4γi e−ip

3 γi ∇jeip∇j γi ∇†j e

−ip∇†j4 1

2{e ip,∇i} − 1

2{e−ip,∇i}

(ππ)P,Λ,µ =∑

p1∗,p2∗

C(P,Λ, µ; p1; p2)π(p1)π(p2), (13)

x

z y

x

z y

ππ100(p1, p2, t) =1√2

[π+(p1)π−(p2)− π+(p2)π−(p1)]; p1 = (1, 0, 0) p2 = (−1, 0, 0)

ππ110 =1

2(ππ(110) + ππ(101) + ππ(1− 10) + ππ(10− 1))


6 × 6 correlation matrix

C =

CρJ←ρJ′ CρJ←ππ100CρJ←ππ110

Cππ100←ρJ′ Cππ100←ππ100 Cππ100←ππ110

Cππ110←ρJ′ Cππ110←ππ100 Cππ110←ππ110

. (14)

The correlation functions: u(ti ) u(tf )

Cρi←ρj= −

⟨ ΓJtf, (p, tf )

ΓJ′†ti

, (−p, ti )

⟩= −

⟨Tr[M−1(ti , tf )ΓJ

tfe ipM−1(tf , ti )ΓJ′†

tie−ip]

⟩. (15)

Cρi←ππ =

⟨−

⟩P=0

= 2

⟨ ⟩. (16)

Cππ←ππ = −⟨

+ − − + −⟩

(17)

P=0= −

⟨2 − 2 + −

⟩(18)


Laplacian Heaviside smearing [2]

To estimate all-to-all propagators:The 3-dimensional gauge-covariant Laplacian operator

∆ab(x , y ;U) =3∑

k=1

{Uabk (x)δ(y , x + k) + Uba

k (y)∗δ(y , x − k) − 2δ(x , y)δab}. (19)

SΛ(t) =Λ∑λ(t)

|λ(t)〉〈λ(t)| ; u(t) = S(t)u(t) =∑λt

|λt〉〈λt | u(t). (20)

NΛ=25

NΛ=50

NΛ=75

NΛ=100

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

r�a

ÈΨ2

�ÈΨ0

2

40 60 80 100 120

3.0

3.2

3.4

3.6

3.8

4.0

N

<r2>

[2] C. Morningstar, J. Bulava, J. Foley, K. J. Juge, D. Lenkner, et al., Phys.Rev. D83 (2011)

114505Dehua Guo, Andrei Alexandru, Raquel Molina and Michael Doring (GWU)Rho Resonance Parameters from Lattice QCD Oct 03, 2016 12 / 26

Energy spectrum

We implement the calculation in 3 ensembles (η = 1.0, 1.25, 2.0) at mπ ≈ 310 MeV and3 ensembles (η = 1.0, 1.17, 1.33) at mπ ≈ 227 MeV with nHYP-smeared clover fermionsand two mass-degenerated quark flavor.

2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1.0

t

ameff

Figure: The lowest three energy states with their error bars for η = 1.0,mπ = 310 MeV ensemble

We extract energy E by using double exponential f (t) = we−Et + (1− w)e−E ′t to do theχ2 fitting for each eigenvalues.


Energy spectrum

12: 123: 1234: 1:5 12:5 123:5 1234:5 :56 1:56 12:56 123:56

0.485

0.490

0.495

0.500

0.505 aE112: 123: 1234: 1:5 12:5 123:5 1234:5 :56 1:56 12:56 123:56

0.655

0.660

0.665

0.670

0.675 aE212: 123: 1234: 1:5 12:5 123:5 1234:5 :56 1:56 12:56 123:56

0.85

0.90

0.95

1.00

1.05aE3

no ΠΠ operator ΠH®¬LΠ ΠH®¬LΠ and ΠH��LΠ

Oi 1 2 3 4 5 6ρ1 ρ2 ρ3 ρ4 ππ100 ππ110

(21)


Expectation for energy states

1.0 1.5 2.0 2.5

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Η

s@G

eV

D

80, 0, 0<

81, 0, 0<

82, 0, 0<

81, 1, 0<

81, 1, 1<

82, 1, 0<

Figure: The lowest 3 energy states prediction fromUχPTmodel. When η = 2.0 the 3rd state isfrom operator ππ200 instead of ππ110


Phaseshift fitted with Breit Wigner Form

amΠ=0.1934H5L

amΡ=0.4875H4LgΡΠΠ=5.57H6L

Χ2�dof=26.8�16=1.7


Χ2�dof=5.35�5=1.07

2.0 2.5 3.0 3.5 4.0 4.50

50

100

150

Ecm�mΠ

∆1@°

D

amΠ=0.1390H5L


Χ2�dof=33.5�9=3.7


Χ2�dof=8.04�6=1.34

2.0 2.5 3.0 3.5 4.0 4.50

50

100

150

Ecm�mΠ

∆1@°

Dcot(δ1(E)) =

M2R − E 2

EΓr (E), Γr (E) ≡ g 2

R12

6π

p3

E 2. (22)

δ1(E) = arccot6π(M2

R − E 2)E

g 2p3(23)


Phaseshift fitted with UχPTmodel

Figure: mπ ≈ 315 MeV and mπ ≈ 227 MeV data fitted with UχPT model.

mπ mρ Γρ g χ2/dofBreit Wigner 315 794.6(6) 37.0(2) 5.57(11) 2.16

UχPT 795.2(3) 36.1(1) 1.26Breit Wigner 227 748.4(1.6) 71.0(8) 5.70(12) 1.46

UχPT 748.2(7) 77.0(5) 1.53


KK channel contribution to ρ resonance

We study the KK effect using UχPTmodel with input parameters mπ, fπ, fK and lowenergy constant l1,2.

Nf=2 extrapolation

Nf=2+1 estimate

400 600 800 1000 12000

50

100

150

Ecm@MeVD

∆1

@°D

��

��

��

��

100 150 200 250 300 350 400

700

750

800

850

900

mπ[MeV]mρ[MeV

]

Figure: (Left) Chiral extrapolation of the phase shift to the physical mass (red band), obtainedfrom the simultaneous fit to pion masses. The blue band: phaseshift with KK . Open circles:experiment data [3]


mρ and gρππ comparison

Lang et al. ETMC

PACS-CS JLAB

Bali et al. this work

150 200 250 300 350 400 450 500

600

700

800

900

1000

1100

mΠ@MeVD

mΡ

@MeV

D

Lang et al. ETMC

PACS-CS JLAB

Bali et al. this work

150 200 250 300 350 400 450 5002

3

4

5

6

7

8

mΠ@MeVD

gΡ

ΠΠ

Figure: Comparison of different lattice calculation for the ρ resonance mass (left) and widthparameter gρππ (right). The errors included here are only stochastic. The band in the left plotindicates a Nf = 2 + 1 expectation from UχPT model constrained by some older lattice QCDdata and some other physical input [4].

The results of ETMC are taken from [5]. PACS result is from [6].


Conclusions

We complete a precision study of ρ resonance with LapH smearing method andobtain the resonance parameters at mπ ≈ 310 MeV and mπ ≈ 227 MeV.

For precise energy spectrum results, both Breit Wigner form and UχPTmodel workwell in the resonance region. Modification to the BW is needed when applied to awider energy region.

The extrapolation of mρ to physical pion mass is smaller than mphyρ = 775 MeV in a

Nf = 2 situation, we believe that this comes from the absence of strange quark andthe KK channel which is supported by our UχPTstudy.


X. Feng, X. Li, and C. Liu, Two particle states in an asymmetric box and the elasticscattering phases, Phys.Rev. D70 (2004) 014505, [hep-lat/0404001].

C. Morningstar, J. Bulava, J. Foley, K. J. Juge, D. Lenkner, et al., Improvedstochastic estimation of quark propagation with Laplacian Heaviside smearing inlattice QCD, Phys.Rev. D83 (2011) 114505, [arXiv:1104.3870].

S. D. Protopopescu, M. Alston-Garnjost, A. Barbaro-Galtieri, S. M. Flatte, J. H.Friedman, T. A. Lasinski, G. R. Lynch, M. S. Rabin, and F. T. Solmitz, Pi pi PartialWave Analysis from Reactions pi+ p —¿ pi+ pi- Delta++ and pi+ p —¿ K+ K-Delta++ at 7.1-GeV/c, Phys. Rev. D7 (1973) 1279.

J. R. Pelaez and G. Rios, Chiral extrapolation of light resonances from one andtwo-loop unitarized Chiral Perturbation Theory versus lattice results, Phys. Rev.D82 (2010) 114002, [arXiv:1010.6008].

X. Feng, K. Jansen, and D. B. Renner, Resonance Parameters of the rho-Mesonfrom Lattice QCD, Phys.Rev. D83 (2011) 094505, [arXiv:1011.5288].

PACS Collaboration, S. Aoki et al., ρ Meson Decay in 2+1 Flavor Lattice QCD,Phys. Rev. D84 (2011) 094505, [arXiv:1106.5365].

P. Estabrooks and A. D. Martin, pi pi Phase Shift Analysis Below the K anti-KThreshold, Nucl. Phys. B79 (1974) 301.


http://arxiv.org/abs/hep-lat/0404001

http://arxiv.org/abs/1104.3870




Symmetries on the lattice

The SO(3) symmetry group reduce to discrete subgroup Oh or D4h

Table: Resolution of 2J + 1 spherical harmonics into the irreducible representations of Oh and D4h

J Oh D4h

0 A+1 A+

1

1 F−1 A−2 ⊕ E−

2 E+ ⊕ F+2 A+

1 ⊕ B+1 ⊕ B+

2 ⊕ E+

3 A−2 ⊕ F−1 ⊕ F−2 A−2 ⊕ B−1 ⊕ B−2 ⊕ 2E−

4 A+1 ⊕ E+ ⊕ F+

1 ⊕ F+2 2A+

1 ⊕ A+2 ⊕ B+

1 ⊕ B+2 ⊕ 2E+

Assume that the energy contribution from angular momenta l ≥ 3 is negligible. Forexample, if we study the p-wave(l = 1) scattering channel, we should construct theinterpolating field in F−1 in the Oh group, A−2 and E− representations in D4h group.


Variational method [?]

Variational method is used to extract energy of the excited states.Construct correlation matrix in the interpolator basis

C(t)ij =< Oi (t)O†j (0) >; i , j = 1, 2, ..., number of operators (24)

The eigenvalues of the correlation matrix are

λ(n)(t, t0) ∝ e−Ent(1 +O(e−∆Ent)), n = 1, 2, ..., number of operators (25)

where ∆En = ENumber of operators + 1 − En.

Larger energy gap makes the high lying energy decay faster and effective mass plateauappear in an earlier time slice.


Appendix-B: LapH smearingBenefit from LapH smearing:

Keep low frequency mode up to Λ cutoff to compute the all to all propagators,u(x)u(y). The number of propagators M−1(tf , ti ) need to be computed reduce

from 6.34× 1013 in position space to 3.7× 108 in momentum space for the 24348ensemble.

The effective mass reach a plateau in an earlier time slice.

0 2 4 6 8 10 12 14

0.2

0.3

0.4

0.5

0.6

t

meff

Figure: pion effective mass with (red) and without LapH smearing (blue)


Appendix-C: Fitting phase-shift

Figure: χ2 fitting for the phase shift data to Breit Wigner form

χ2 = ∆TCOV−1∆ (26)

where

∆i =√

scurvei −

√sdatai (27)


Appendix-D: Experiment data [7]

500 600 700 800 900 1000 1100 12000

50

100

150

EHMevL

∆1

HEL

Gauss Form

Centrifugal form

Breit Wigner

Figure: ππ phase shift below KK threshold in experiment

ΓBW (E) =g 2

6π

p3

E 2

ΓCF (E) =g 2

6π

p3

E 2

1 + (pRR)2

1 + (pR)2

ΓGA(E) =g 2

6π

p3

E 2

e−p2/6β2

e−p2R/6β2

[7] Estabrooks, P. and Martin, Alan D. Nucl.Phys. B79 (1974) 301


K-matrix method

T−1 = V−1 − G =−3(f 2 − 8l1m

2π + 4l2W

2)

2p2− ReG(W ) +

ip

8πW(28)

For K-matrix method the ReG(W ) = 0.

T =−8πW

p cot δp − ip(29)

l1 =1

8π2(−1

2

m2ρ

g 2ρππ

+ f 2) (30)

l2 = − 1

8g 2ρππ

(31)


rho resonance parameters from lattice qcdrho resonance parameters from lattice qcd dehua guo, andrei...

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