Xiu-Lei Ren (任修磊)

Institute of theoretical physics II, Ruhr-Universität Bochum

Relativistic chiral nuclear force up to NLO

8th International Conference on

Quarks and Nuclear PhysicsTsukuba, Japan (Nov.13-17, 2018)

Introduction

Theoretical framework

Results and discussion

Summary and Outlook

Outline

Nuclear force (residual quark-gluon strong interaction)

• Acts between two or more nucleons

• Binds protons and neutrons into atomic nuclei

• Plays an important role in all nuclear physics

• Ab-initio calculation

Nuclear force

Nuclear structure

Nuclear reaction

Precise understanding of the strong nuclear force is essential!

Realistic Nuclear force

Many-Body Hamiltonian

ONLY input:

Exactly Solve:

e.g. NN: AV18, CD-Bonn, Reid93, N3LO… 3N: Tucson-Melbourne, NNLO…

e.g. no-core shell model, Green’s function MC method, Nuclear Lattice EFT…

NF from phenomenological models (since 1935, Yukawa OPE )

• Meson “theory”: CD-Bonn R. Machleidt,PRC2001

• Operators parameterization: Reid93, V. Stoks, PRC(1994)

AV18, R. Wiringa,PRC(1994)

NF from lattice QCD simulations (since 2006)

• HAL QCD collaboration

• NPLQCD collaboration • CalLat coll. / T. Yamazaki et al.

Nuclear force studies (I)

Plenary talk: Tetsuo Hatsuda, “Baryon interactions from lattice QCD”

VNN = Vc(r) 1 + Vσ(r) σ1 ⋅ σ2 + VLS(r) L ⋅ S+ VT(r) σ1 ⋅ qσ2 ⋅ q +⋯

N. Ishii et al. PRL(2007)

NF from chiral effective field theory (EFT of low-energy QCD) • Hierarchy of chiral force in non-relativistic scheme (Weinberg p.c.)

Nuclear force studies (II)

P. F. Bedaque, U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52 (2002) 339 E. Epelbaum, H.-W. Hammer, Ulf-G. Meißner, Rev. Mod. Phys. 81 (2009) 1773 R. Machleidt, D. R. Entem, Phys. Rept. 503 (2011) 1

1990

2007 N3LO: 3N

2015 N4LO: 2N

LO

2003 N3LO: 2N

In future N4LO: 3N & 4NShort-range loop contrib. still missing

Not yet…

High precision chiral force

Phenomenological forces Non-Rel. Chiral nuclear force

Reid93 AV18 CD-Bonn LO NLO NNLO N3LO N4LO+

No. of para. 50 40 38 2+2 9+2 9+2 24+2 (3 redundant)

24+3+4 (3 redundant)

np 0-300 MeV 1.03 1.04 1.02

94 36.7 5.28 1.27 1.10

75 14 4.2 2.01 1.06

Non-Rel. Chiral NF has achieved a great success!

• NNLO/N3LO chiral forces have been extensively applied in the study of nuclear structure and reactions within the few/many-body theories.E. Epelbaum, et al., PRL 106(2011) 192501, PRL109(2012)252501, PRL112(2014)102501; S. Elhatisari, et al., Nature 528 (2015) 111; G. Hagen, et al., PRL109(2012)032502; H. Hergert, et al., PRL110(2013)24501; G.R. Jansen, et al., PRL113(2014)102501; S.K.Bogner, et al., PRL113(2014)142501; J.E. Lynn, et al., PRL113(2014)192501; V. Lapoux, et al.,PRL117(2016)052501

P. Reinert, et al., EPJA54(2018)86

χ2/datum

Description of neutron-proton scattering observables

include 4 ct. in F-waves

D. Entem, et al., PRC96(2017)024004

The principle of relativity: law of nature • Elastic NN scattering (0–-300 MeV)

• Non-relativistic is a good approximation

• Small relativistic effects could be detectable • via the onging/future high precision experiments

Investigate the relativistic effects

• Unsolvable problems: Ay puzzle in N-d scattering, …

• Improve the description of polarization observables

Relativistic chiral force could provide a new view

• Renormalization group invariance: still controversial …

• Convergence of chiral expansion: relatively faster convergence?

Key inputs for the relativistic many-body calculations • Relativistic Breuckner-Hartree-Fock theory

Why reformulate chiral NF in relativistic scheme?

m2N + p2 ≤ m2

N 1+0.16

R. Brockmann and R. Machleidt, PLB 149, 283 (1984)

J. Golak et al. EPJA50(2014)177

We employ covariant ChEFT (tentatively) to construct a relativistic chiral nuclear force • Interaction kernel obtained from covariant ChEFT with the (first

attempt) naive dimensional analysis (NDA)

• Observables obtained by solving the relativistic 3D-reduced Bethe-Salpeter equation • i.e. Kadyshevsky, Blankenbecler-Sugar equations

Project: relativistic chiral nuclear force

ESI 2018 May/June

Relativistic chiral nucleon-nucleon interaction up to leading order

Description the phase shifts of J=0, 1 partial waves

Relativistic chiral NF up to LO

• Improve description of 1S0 and 3P0 phase shifts

• Quantitatively similar to the nonrelativistic case up NLO

XLR, K.-W. Li, L.-S. Geng, B. Long, P. Ring, J. Meng, Chin.Phys.C 42 (2018) 014103

• (maybe) a more efficient formulation • should be verified beyond LO

Relativistic chiral NF

Description the phase shifts of J=0, 1 partial waves

Relativistic chiral NF up to LO

• Improve description of 1S0 and 3P0 phase shifts

• Quantitatively similar to the nonrelativistic case up NLO

XLR, K.-W. Li, L.-S. Geng, B. Long, P. Ring, J. Meng, Chin.Phys.C 42 (2018) 014103

• (maybe) a more efficient formulation • should be verified beyond LO

Relativistic chiral NF

We study the relativistic chiral force up to next-to-leading order! • Expect a good description of phase shifts.

Introduction

Theoretical framework

• Concept of potential and Definition of relativistic chiral force

• Relativistic chiral force up to next-to-leading order

Results and discussion

Summary and Outlook

Outline

Often-thought as the nonrelativistic quantity

• Appear in the Schroeinger equation

• (or) Appear in the Lippmann-Schwinger equation

Generalize the definition of potential

• An interaction quantity appearing in a three-dimensional scattering equation can be referred as a NN potential.

• If one employs a relativistic scattering equation, one can define a relativistic NN potential.

NN potential concept

M.H. Partovi, E.L. Lomon, PRD2 (1970) 1999 K. Erkelenz, Phys.Rept. 13C(1974) 191

Blankenbecler-Sugar equation (used in rel. Bonn potentials)

• Relativistic potential definition:

• Interaction kernel in covariant chiral EFT:

• Perturbatively expand according to naive dimensional analysis

• Up to next-to-leading order

Relativistic scheme for chiral force

T (p0,p) = V (p0,p) +

Zdk

(2⇡)3V (p0,k)

m2N

Ek

1

E2p � E2

k + i✏T (k,p).

Blankenbeckler, Sugar, Phys.Rev.(1966).

( QΛχ )

nχ

Covariant chiral Lagrangians

• LO contact Lagrangian

• NLO contact Lagrangian • Increase chiral order via partial derivative on nucleon field • Employ the equation of motion to obtain the minimum terms

NN interaction up to NLO

Le↵ = L(2)⇡⇡ + L(1)

⇡N +L(0)NN +L(2)

NN .

H. Polinder, J. Haidenbauer, U.-G. Meißner,NPA779, 244 (2006)

5 LECs

Y. Xiao, L. Geng, XLR, … in preparation

12 LECs

Preliminary

Two-pion exchange contribution in covariant ChEFT

• Evaluate 3-/4-point functions with complicated Lorentz structures

• Keep the four external legs off-shell

• Subtract the power-counting breaking terms using extended-on-mass-shell scheme

• Remove the iterated one-pion exchange contribution

Complicated evaluation of TPE

FeynCalc

LoopTools Numerical evaluation

Perturbatively calculate the phase shifts of F, G, H partial waves • No free-parameters

• OPE vs. TPE vs. Nijmegen93

• TPE contribution is smaller than OPE (except 3F2 wave)

• TPE can (slightly) improve the description of the phase shifts of F,G,H waves

Two-pion exchange contribution

PreliminarygA = 1.267, fπ = 92.4 MeV

Perform the partial wave projection, one can obtain the BbS equation in |LSJ> basis

Cutoff renormalization for BbS equation

• Potential regularized using an exponential regulator function

On-shell S matrix and phase shifts

Scattering equation and Phase shifts

TSJL0,L(p

0,p) = V SJL0,L(p

0,p)

+X

L00

Z +1

0

k2dk

(2⇡)3V SJL0,L(p

0,k)M2

Npk2 +M2

N

1

p2 � k2 + i✏TSJL00,L(k,p).

Blankenbeckler, Sugar, Phys.Rev.(1966).

E.Epelbaum et al., NPA(2000)

SSJL0L = �L0L � i

8⇡2

M2N |p|Ep

TSJL0L. S = exp(2i�)

Couple channel: Stapp parameterization

17 LECs are determined by fitting

• NPWA: p-n scattering phase shifts of Nijmegen 93• 12 partial waves: J=0, 1, 2

• 72 data points: 6 data points for each partial wave (Tlab ≤ 100 MeV)

• Fit-chi-square:

Best fitting results

•

• Other parameters:

Numerical results

V. Stoks et al., PRC48(1993)792

χ2/datum = 0.47

gA = 1.267, fπ = 92.4 MeVmπ = 138.03 MeV, mN = 938.0 MeV

Λ = 600 MeV

in unit of 104 GeV-2

Preliminary

Description of phase shifts: J=0,1,2

Preliminary

A better description of phase shifts!• e.g. 1P1, mixing angle and 3P2 p.w. • Potential 1D2 is still weak, similar as NR case

More cross-check is working on!

We have studied the relativistic chiral nuclear force up to next-to-leading order

• Constructed the NLO contact Lagranagians

• Evaluated the two-pion exchange contributions

• Presented a better description of phase shifts than LO

We are working on …

• Important cross-check: reproduce the non-relativistic results of TPE at the static limit

• Employ a more suitable regular function for long-range interaction

• Calculate the properties of Deuteron

Our final goal is to construct a high-precision relativistic chiral nuclear force

• Study the two-nucleon force up to NNLO/N3LO

• Study the three-nucleon force at NNLO/N3LO

Summary and outlook

P. Reinert, et al., EPJA54(2018)86

Collaborators

Beijing Lisheng Geng

Kai-Wen Li

Jie Meng

Chunxuan Wang

Yang Xiao

Munich Peter Ring

Chengdu Bingwei Long

Orsay Bira van Kolck

Bochum Evgeny Epelbaum

Jambul Gegelia

Collaborators

Beijing Lisheng Geng

Kai-Wen Li

Jie Meng

Chunxuan Wang

Yang Xiao

Munich Peter Ring

Chengdu Bingwei Long

Orsay Bira van Kolck

Bochum Evgeny Epelbaum

Jambul Gegelia

Thank you for your attention!

Relativistic nucleon-nucleon scattering

• Bethe-Salpeter equation with an operator form:

•T, Invariant Scattering amplitude

•A, Interaction kernel (sum all the irreducible diagrams)

• G, Two-nucleon’s Green function

Bethe-Salpeter equation

Introduce a three-dimensional Green function g

• Maintain the same elastic unitarity of G at physical region

• Choose the Blankenbecler-Sugar propagator

• Positive energy projection operator for two nucleons

• No retardation effects of pion-exchange contribution

Introduce the effective interaction kernel V• BS equation equivalently to two couple equations

Reduction of BS equation

g = 2⇡m2

N

Ek

⇤(1)+ (k)⇤(2)

+ (�k)14s� E2

k + i✏�(k0)

T = A+AGT .T = V + V g T .

V = A+A (G� g) V.

Blankenbeckler, Sugar, Phys.Rev.(1966).

BS equation reduces to the BbS equation

• Sandwiched by Dirac spinors:

• Relativistic potential definition:

Reduction of BS equation

T = V + V g T

= V +

Zdk

(2⇡)3

Zdk02⇡

V ⇥ 2⇡m2

N

Ek

⇤(1)+ (k)⇤(2)

+ (�k)14s� E2

k + i✏�(k0)⇥ T

= V +

Zdk

(2⇡)3V ⇥ m2

N

Ek

⇤(1)+ (k)⇤(2)

+ (�k)14s� E2

k + i✏⇥ T , with k0 = 0.

T (p0,p) = V (p0,p) +

Zdk

(2⇡)3V (p0,k)

m2N

Ek

1

E2p � E2

k + i✏T (k,p).

Blankenbeckler, Sugar, Phys.Rev.(1966).

V (p0,p) = u(p0, s1)u(�p0, s2)V[p00 = 0,p0; p0 = 0,p]u(p, s1)u(p0, s2).

To obtain the relativistic potential V

Need to solve the iterated equation perturbatively

Calculate the interaction kernel A, in covariant ChEFT

• First attempt, we employ the naive dimensional analysis to guide the power expansion

Calculate potential in ChEFT

A =X

i

C[gi(µ)]

✓Q

⇤�

◆n�

Relativistic potential definition:

• Using the relativistic kinematics

• Keep the small component of Dirac spinor

Relativistic potential

Spin orbit interaction is appeared at Lowest order!!

LO contact potential

Leading order partial waves with cutoff 1—9 GeV

Cutoff dependence

Take left-triangle diagram for examplep In the momentum space

pTransform to the Helicity basis• Apply four identities related to Dirac spinor to simplify the

tensor structures

• Perform the one loop integration in FeynCalc (D-dimension)

“(off-shell)Dirac equations”

M.J. Zuilhof et al., PRC26, 1277 (1982) Left triangle contributions

Compact form of TPE contributionsp Using the aforementioned “(off-shell) Dirac eqs.”,one can express TPE diagrams in terms of twentytensor structures

VTPE =X

n

on Fn(A0,B0,C0,D0, ...)

• Superposition of PaVe functions

• Remove the power-counting breaking (PCB)• Numerical evaluation using LoopTools

Term 10:

Term 11:

NLO Lagrangians