# Nuclear Forces from Lattice QCD - Institute for · PDF fileNuclear Forces from Lattice QCD...

date post

19-Sep-2018Category

## Documents

view

221download

0

Embed Size (px)

### Transcript of Nuclear Forces from Lattice QCD - Institute for · PDF fileNuclear Forces from Lattice QCD...

Nuclear forces and their impact on structure, reactions and astrophysics

TALENT 2013

Lecture on

Nuclear Forces from Lattice QCD

Zohreh Davoudi

University of Washington

Wednesday, July 10, 2013

2 1 1 2 3

20

10

10

p Cot 1

a

p Cot 1

a

1

2 32 L S p2 L2

4 2

S-wave Luescher formula - a demonstration

Wednesday, July 10, 2013

NN spectrum in coupled 3S1-3D1 channel

L [fm]

E NR[M

eV]

E [M

eV]

Saturday, June 8, 2013

Based on hep-lat/1305.4903.

Wednesday, July 10, 2013

Nucleon-nucleon scattering

hep-lat/1301.5790v1.

UNH-11-3NT@UW-11-12ICCUB-11-159UCB-NPAT-11-008NT-LBNL-11-012

The I = 2 S-wave Scattering Phase Shift from Lattice QCD

S.R. Beane,1, 2 E. Chang,3 W. Detmold,4, 5 H.W. Lin,6 T.C. Luu,7

K. Orginos,4, 5 A. Parreno,3 M.J. Savage,6 A. Torok,8 and A. Walker-Loud9

(NPLQCD Collaboration)1Albert Einstein Zentrum fur Fundamentale Physik,

Institut fur Theoretische Physik, Sidlerstrasse 5, CH-3012 Bern, Switzerland2Department of Physics, University of New Hampshire, Durham, NH 03824-3568, USA

3Departament dEstructura i Constituents de la

Materia. Institut de Ciencies del Cosmos (ICC),

Universitat de Barcelona, Mart i Franques 1, E08028-Spain4Department of Physics, College of William and Mary,

Williamsburg, VA 23187-8795, USA5Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA

6Department of Physics, University of Washington,

Box 351560, Seattle, WA 98195, USA7N Division, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA

8Department of Physics, Indiana University, Bloomington, IN 47405, USA

9Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

AbstractThe ++ s-wave scattering phase-shift is determined below the inelastic threshold using LatticeQCD. Calculations were performed at a pion mass of m 390 MeV with an anisotropic nf = 2+1clover fermion discretization in four lattice volumes, with spatial extent L 2.0, 2.5, 3.0 and3.9 fm, and with a lattice spacing of bs 0.123 fm in the spatial direction and bt bs/3.5 inthe time direction. The phase-shift is determined from the energy-eigenvalues of ++ systemswith both zero and non-zero total momentum in the lattice volume using Luschers method. Ourcalculations are precise enough to allow for a determination of the threshold scattering parameters,the scattering length a, the effective range r, and the shape-parameter P , in this channel and toexamine the prediction of two-flavor chiral perturbation theory: m2ar = 3 + O(m

2/

2). Chiral

perturbation theory is used, with the Lattice QCD results as input, to predict the scatteringphase-shift (and threshold parameters) at the physical pion mass. Our results are consistent withdeterminations from the Roy equations and with the existing experimental phase shift data.

1

arX

iv:1

107.

5023

v2 [

hep-

lat]

15

Sep

2011

0 0.1 0.2 0.3 0.4 0.5 0.6 0.750

0

50

100

150

200

km

3 S 1 deg

rees ExperimentalLevinson's TheoremL32 , P1L24 , P1L32 , P0L24 , P0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.750

0

50

100

150

200

km

3 S 1 deg

rees ExperimentalLevinson's TheoremL32 , P1L24 , P1

L32 , P0L24 , P0

FIG. 10: The phase shift in the 3S1 channel. The left panel is a two-parameter fit to the ERE, whilethe right panel is a three-parameter fit to the ERE, as described in the text. The inner (outer)shaded region corresponds to the statistical uncertainty (statistical and systematic uncertaintiescombined in quadrature) in two- and three-parameter ERE fit to the results of the Lattice QCDcalculation. The vertical (red) dashed line corresponds to the start of the t-channel cut and theupper limit of the range of validity of the ERE. The light (green) dashed line corresponds to thephase shift at the physical pion mass from the Nijmegen phase-shift analysis [38].

V. NUCLEON-NUCLEON EFFECTIVE RANGES

Unlike the scattering length, the size of the effective range and the higher-order contributionsto the ERE are set by the range of the interaction. The leading estimate of the effective rangefor light quarks is r 1/m, and higher order contributions are expected to be suppressedby further powers of the light-quark masses. It is natural to consider an expansion of theproduct mr in the light-quark masses. While the most general form of the expansioncontains terms that are non-analytic in the pion mass [4043], for instance of the formmq logmq, with determinations at only two pion masses (including the experimental value)a polynomial fit function is chosen,

mr = A + B m + ... . (7)

In fig. 11, the results of our LQCD calculations of mr are shown, along with the experi-mental value in each channel and a fit to the form given in eq. (7). While the uncertaintiesin the lattice determinations are somewhat large compared to those of the experimental de-termination, it appears that there is modest dependence upon the light-quark masses. Thefit values are

A(1S0) = 1.348+0.0800.080

+0.0790.083 , B

(1S0) = 4.23+0.550.56+0.590.57 GeV

1

A(3S1) = 0.726+0.0650.059

+0.0720.059 , B

(3S1) = 3.70+0.420.47+0.420.52 GeV

1 . (8)

The two-parameter fit is clearly over simplistic and more precise LQCD calculations arerequired at smaller light-quark masses to better constrain the light-quark mass dependenceof the effective ranges.

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7100

50

0

50

100

150

200

km

1 S 0 deg

rees ExperimentalLevinson's TheoremL24 , P1

L32 , P0L24 , P0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7100

50

0

50

100

150

200

km

1 S 0 deg

rees ExperimentalLevinson's TheoremL24 , P1

L32 , P0L24 , P0

FIG. 5: The phase shift in the 1S0 channel. The left panel is a two-parameter fit to the ERE, whilethe right panel is a three-parameter fit to the ERE, as described in the text. The inner (outer)shaded region corresponds to the statistical uncertainty (statistical and systematic uncertaintiescombined in quadrature) in two- and three-parameter ERE fits to the results of the Lattice QCDcalculation. The vertical (red) dashed line corresponds to the start of the t-channel cut and theupper limit of the range of validity of the ERE. The light (green) dashed line corresponds to thephase shift at the physical pion mass from the Nijmegen phase-shift analysis [38].

IV. NUCLEON-NUCLEON SCATTERING IN THE 3S1 3D1 COUPLED CHAN-NELS

At the SU(3)-symmetric point, the deuteron is bound [13] by B = 19.5(3.6)(3.1)(0.2) MeVwhich, as with the bound di-neutron in the 1S0 channel, provides a constraint on k cot . Thedeuteron has a d-wave component induced by the tensor interaction, however mixing betweenthe s-wave and the d-wave is higher order in the ERE and first appears at the same orderas the shape parameter [39]. Therefore, while the scattering length and effective range arepurely s-wave, the shape parameter is contaminated by the d-wave admixture. The EMPsassociated with the first excited states with |P| = 0 are shown in fig. 6, and with |P| = 1are shown in fig. 7. The correlation functions calculated on the L = 6.7 fm ensemble have

4 8 12 16 20

0.02

0

0.02

0.04

0.06

tb

bE

4 8 12 16 20

0.02

0

0.02

0.04

0.06

tbbE

4 8 12 16 20

0.02

0

0.02

0.04

0.06

tb

bE

FIG. 6: The EMPs of the first excited states with |P| = 0 in the 3S1 channel in the L = 3.4 fm, L =4.5 fm and L = 6.7 fm ensembles, respectively. Twice the nucleon mass has been subtracted fromthe energy. The dark (light) shaded regions correspond to the statistical uncertainty (statisticaland systematic uncertainties combined in quadrature) of the fit to the plateau over the indicatedtime interval.

energies that are too close to, or straddle, the singularities of Luschers eigenvalue equation

8

3S1 channel

m 800 MeV

Wednesday, July 10, 2013

Fine tuning and naturalness from LQCD?

hep-lat/1301.5790v1.

UNH-11-3NT@UW-11-12ICCUB-11-159UCB-NPAT-11-008NT-LBNL-11-012

The I = 2 S-wave Scattering Phase Shift from Lattice QCD

S.R. Beane,1, 2 E. Chang,3 W. Detmold,4, 5 H.W. Lin,6 T.C. Luu,7

K. Orginos,4, 5 A. Parreno,3 M.J. Savage,6 A. Torok,8 and A. Walker-Loud9

(NPLQCD Collaboration)1Albert Einstein Zentrum fur Fundamentale Physik,

Institut fur Theoretische Physik, Sidlerstrasse 5, CH-3012 Bern, Switzerland2Department of Physics, University of New Hampshire, Durham, NH 03824-3568, USA

3Departament dEstructura i Constituents de la

Materia. Institut de Ciencies del Cosmos (ICC),

Universitat de Barcelona, Mart i Franques 1, E08028-Spain4Department of Physics, College of William and Mary,

Williamsburg, VA 23187-8795, USA5Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA

6Department of Physics, University of Washington,

Box 351560, Seattle, WA 98195, USA7N Division, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA

8Department of Physics, Indiana University, Bloomington, IN 47405, USA

9Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

AbstractThe ++ s-wave scattering phase-shift is determined below the inelastic threshold using LatticeQCD. Calculations were performed at a pion mass of m 390 MeV with an anisotropic nf = 2+1clover fermion discretization in four lattice volumes, with spatial extent

*View more*