Chiral two- and three-nucleon forces Chiral two- and three-nucleon forces with explicit Delta degree of freedomwith explicit Delta degree of freedomChiral two- and three-nucleon forces Chiral two- and three-nucleon forces with explicit Delta degree of freedomwith explicit Delta degree of freedom

A. M. Gasparyan, Ruhr-Universität Bochum

in collaboration with

H. Krebs, E. Epelbaum,D. Siemens, V. Bernard, Ulf-G. Meißner

February 28, 2017, TRIUMF

989OutlineOutlineOutlineOutline

➔Introduction&Motivation

➔2-N forces with explicit

➔3-N forces with explicit

➔N scattering with explicit

➔Summary and Outlook

EFT with explicit EFT with explicit EFT with explicit EFT with explicit

➔Standard chiral expansion:

EFT with explicit EFT with explicit EFT with explicit EFT with explicit

➔Standard chiral expansion:

➔Small scale expansion: Hemmert, Holstein, Kambor '98

EFT with explicit EFT with explicit EFT with explicit EFT with explicit

➔Standard chiral expansion:

➔Small scale expansion: Hemmert, Holstein, Kambor '98

➔ gives a large contribution to LEC (c3,c

4) via resonance saturation

Bernard, Kaiser, Meißner '97

EFT with explicit EFT with explicit EFT with explicit EFT with explicit

➔Standard chiral expansion:

➔Small scale expansion: Hemmert, Holstein, Kambor '98

➔ gives a large contribution to LEC (c3,c

4) via resonance saturation

Bernard, Kaiser, Meißner '97

➔Explicit decoupling of makes comparison with less theory more transparent Bernard, Fearing, Hemmert, Meißner '98

finite parts of LECs can be always chosen such that Appelquist, Carrazone '74 (Decoupling theorem)

Small scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NF

Δ-less theory Δ-full theory: additional graphs

...

... Krebs, Epelbaum, Meißner '07

Keiser, Gerstendorfer, Weise '98

Small scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NF

Δ-less theory Δ-full theory: additional graphs

...

... Krebs, Epelbaum, Meißner '07

Keiser, Gerstendorfer, Weise '98

Small scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NF

Δ-less theory Δ-full theory: additional graphs

...

... Krebs, Epelbaum, Meißner '07

Keiser, Gerstendorfer, Weise '98

Small scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NFSmall scale expansion of 2NF

Δ-less theory Δ-full theory: additional graphs

...

...

Higher orders

Krebs, Epelbaum, Meißner '07

Keiser, Gerstendorfer, Weise '98

Preliminary results for NPreliminary results for N33LOLO2N forces with explicit 2N forces with explicit

Preliminary results for NPreliminary results for N33LOLO2N forces with explicit 2N forces with explicit

➔Only 2-pion-exchange contribution are considered (the long range part)

➔1/mN corrections are not yet included

➔Results for peripheral phases, no refitting of LEC's, no cut offs

➔No additional parameters, h

A and g

1 (N and )

are extracted from the fit to N scattering

F and G wavesF and G wavesF and G wavesF and G waves

LO

NLO

N2LO

N3LO

N4LO

-less -full -less -full

Data:Nijmegen PWA

F and G wavesF and G wavesF and G wavesF and G waves

LO

NLO

N2LO

N3LO

N4LO

-less -full -less -full F-waves might be sensitive to the short-range physics

Data:Nijmegen PWA

F and G wavesF and G wavesF and G wavesF and G waves

LO

NLO

N2LO

N3LO

N4LO

-less -full -less -full F-waves might be sensitive to the short-range physics

Significant improvement compared with -less case

Data:Nijmegen PWA

H and I wavesH and I wavesH and I wavesH and I waves

LO

NLO

N2LO

N3LO

N4LO

-less -full -less -full

Data:Nijmegen PWA

ixing angles ixing angles , , , , , , ixing angles ixing angles , , , , , ,

LO

NLO

N2LO

N3LO

N4LO

-less -full

Data:Nijmegen PWA

NLO

N2LO

Δ-less theory Δ-full theory: additional graphs

N3LO

N4LO

Small scale expansion of 3NFSmall scale expansion of 3NFSmall scale expansion of 3NFSmall scale expansion of 3NF

NLO

N2LO

Δ-less theory Δ-full theory: additional graphs

N3LO

N4LO

Small scale expansion of 3NFSmall scale expansion of 3NFSmall scale expansion of 3NFSmall scale expansion of 3NF

NLO

N2LO

Δ-less theory Δ-full theory: additional graphs

N3LO

N4LO

Small scale expansion of 3NFSmall scale expansion of 3NFSmall scale expansion of 3NFSmall scale expansion of 3NF

Large contributions to the ringand 2 topologies saturating some of the N5,6LO graphs in the less theory

Long-range 3NFLong-range 3NFLong-range 3NFLong-range 3NF

exchange

ringexchange

Long-range 3NFLong-range 3NFLong-range 3NFLong-range 3NF

exchange

ringexchange

Long-range 3NFLong-range 3NFLong-range 3NFLong-range 3NF

➔Only the long range part considered (coordinate space)

➔Scheme independent

➔No unknown parameters

Most general structure of a local 3NFMost general structure of a local 3NFMost general structure of a local 3NFMost general structure of a local 3NF

Krebs, Gasparyan, Epelbaum '13

Constraints:➔ Locality➔ Isospin symmetry➔ Parity and time-reversal invariance

Up to N4LO all considered contribution are local

Most general structure of a local 3NFMost general structure of a local 3NFMost general structure of a local 3NFMost general structure of a local 3NF

Krebs, Gasparyan, Epelbaum '13

Constraints:➔ Locality➔ Isospin symmetry➔ Parity and time-reversal invariance

Up to N4LO all considered contribution are local

Most general structure of a local 3NFMost general structure of a local 3NFMost general structure of a local 3NFMost general structure of a local 3NF

Krebs, Gasparyan, Epelbaum '13

Constraints:➔ Locality➔ Isospin symmetry➔ Parity and time-reversal invariance

Up to N4LO all considered contribution are local

dTwo-pion-exhcange 3NF in Two-pion-exhcange 3NF in full andfull and

less approach (preliminary)less approach (preliminary)Two-pion-exhcange 3NF in Two-pion-exhcange 3NF in full andfull and

less approach (preliminary)less approach (preliminary)

TPE “structure functions” Fi in MeV”

in equilateral-triangle configuration

Krebs, Gasparyan, Epelbaum, in preparation

dTwo-pion-exhcange 3NF in Two-pion-exhcange 3NF in full andfull and

less approach (preliminary)less approach (preliminary)Two-pion-exhcange 3NF in Two-pion-exhcange 3NF in full andfull and

less approach (preliminary)less approach (preliminary)

TPE “structure functions” Fi in MeV”

in equilateral-triangle configuration

Krebs, Gasparyan, Epelbaum, in preparation

➔similar results for large contributions

dTwo-pion-exhcange 3NF in Two-pion-exhcange 3NF in full andfull and

less approach (preliminary)less approach (preliminary)Two-pion-exhcange 3NF in Two-pion-exhcange 3NF in full andfull and

less approach (preliminary)less approach (preliminary)

TPE “structure functions” Fi in MeV”

in equilateral-triangle configuration

Krebs, Gasparyan, Epelbaum, in preparation

➔similar results for large contributions➔slightly different for small contributions

Two-pion-one-pion-exhcange 3NF in Two-pion-one-pion-exhcange 3NF in full full andand less approach (preliminary)less approach (preliminary)

Two-pion-one-pion-exhcange 3NF in Two-pion-one-pion-exhcange 3NF in full full andand less approach (preliminary)less approach (preliminary)

Bands indicate physicsnot described by explicitcontributions

Two-pion-one-pion-exhcange 3NF in Two-pion-one-pion-exhcange 3NF in full full andand less approach (preliminary)less approach (preliminary)

Two-pion-one-pion-exhcange 3NF in Two-pion-one-pion-exhcange 3NF in full full andand less approach (preliminary)less approach (preliminary)

➔Dominant effects come from N3LO-N4LO

Bands indicate physicsnot described by explicitcontributions

Two-pion-one-pion-exhcange 3NF in Two-pion-one-pion-exhcange 3NF in full full andand less approach (preliminary)less approach (preliminary)

Two-pion-one-pion-exhcange 3NF in Two-pion-one-pion-exhcange 3NF in full full andand less approach (preliminary)less approach (preliminary)

➔Dominant effects come from N3LO-N4LO➔The largest N4LO contribution is saturated by

Bands indicate physicsnot described by explicitcontributions

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

➔Narrow bands:higher order contributionsbeyond are small

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

➔Narrow bands:higher order contributionsbeyond are small

➔Strong central isoscalar 3NFdue to double- excitation

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

Ring-topology 3NF in Ring-topology 3NF in full andfull and less approach (preliminary)less approach (preliminary)

➔Narrow bands:higher order contributionsbeyond are small

➔Strong central isoscalar 3NFdue to double- excitation

➔Explicit-approach is more efficient !

exchange

ringexchange

N input for 3-Nucleon ForcesN input for 3-Nucleon ForcesN input for 3-Nucleon ForcesN input for 3-Nucleon Forces

➔Longest-range contributions

➔ Intermediate-range contributions

➔Short-range contributions

exchange

ringexchange

N input for 3-Nucleon ForcesN input for 3-Nucleon ForcesN input for 3-Nucleon ForcesN input for 3-Nucleon Forces

➔Longest-range contributions

➔ Intermediate-range contributions

➔Short-range contributions

N scattering up to N scattering up to N scattering up to N scattering up to

11

Siemens et al. In preparation

N scattering up to N scattering up to N scattering up to N scattering up to

11

redundant, can be absorbed by redefining other LEC's

Siemens et al. In preparation

N scattering up to N scattering up to N scattering up to N scattering up to

11

redundant, can be absorbed by redefining other LEC's

only two new parameters

Siemens et al. In preparation

N differential cross sectionN differential cross section N differential cross sectionN differential cross section

N differential cross sectionN differential cross section N differential cross sectionN differential cross section

➔Theoretical error-bands are narrower

Quality of the fit to Quality of the fit to N data N data in the in the less and less and full full PT(without theoretical errors)PT(without theoretical errors)

Quality of the fit to Quality of the fit to N data N data in the in the less and less and full full PT(without theoretical errors)PT(without theoretical errors)

HB-NN HB-N covariant

Quality of the fit to Quality of the fit to N data N data in the in the less and less and full full PT(without theoretical errors)PT(without theoretical errors)

Quality of the fit to Quality of the fit to N data N data in the in the less and less and full full PT(without theoretical errors)PT(without theoretical errors)

sizable reduction of the for a wider energy range

HB-NN HB-N covariant

eSummarySummarySummarySummary

➔Preliminary results for -full chiral 2-nucleon and 3-nucleon forces at N3LO are presented

➔ 2-nucleon forces (peripheral phases): significant improvement compared to the -less case

➔3-nucleon forces: indication of a better convergence; sizable -contributions missing in -less N4LO 3NF ~O(1/)

➔New results for N scattering at order 4: much better fit to data

➔Completing construction of -full chiral 2N and 3N forces at N3LO and moving forward to even more precise nuclear forces.

OutlookOutlook