1. Francesca Sammarruca University of Idaho
How well does the chiral expansion converge in nuclear and neutron matter?
Francesca Sammarruca
University of Idaho
8th International Workshop on Chiral Dynamics
June 29 - July 3, 2015
Pisa, Italy
Research supported in part by the
US Department of Energy.

2. Collaborators:
L. Coraggio (INFN, Naples)
J.W. Holt (University of Washington)
N. Itaco (INFN and University of Naples)
R. Machleidt (University of Idaho)
L.E. Marcucci (INFN, Pisa)

3. Nuclear/neutron matter as a testing ground for many-body theories.
OUTLINE:
Nuclear/neutron matter as a testing ground
for many-body theories.
Different approaches to predicting nuclear/neutron
matter properties: advantages/disadvantages.
Chiral EFT and error quantification.
Chiral forces: Overview.
Order-by-order nuclear/neutron matter calculations.
Conclusions, outlook.

4. After more than 8 decades of nuclear physics,
still a lot is unknown about the nuclear chart.
Particularly true for systems with large isospin asymmetry

5. Experimental programs at RIBF
(and the upcoming FRIB) will have widespread
impact, filling some of the gaps in our incomplete
knowledge of the nuclear chart.
Isospin-Asymmetric Nuclear Matter (IANM)
is closely related to neutron-rich nuclei
and is a convenient theoretical laboratory.
Studies of IANM (particularly the symmetry
energy contribution to the EoS),
are now particularly timely,
as they support rich on-going
and future experimental effort.
The EoS of highly neutron-rich matter
is crucial to understand wide-ranging
questions in nuclear structure.

6. The goal of microscopic nuclear physics:
To derive the properties of nuclear systems
from the basic nuclear interactions
(AB INITIO)

7. Meson-theoretic interactions have been used extensively
in nuclear matter calculations (and still are).
Problem:
The 3NFs employed in these calculations
have only a lose connection to the associated 2NFs.
How to select the contributions to be retained?

8. (virtual nucleon-antinucleon excitation)
Relativistic approaches have focused on the
Dirac-Brueckner-Hartree-Fock (DBHF) framework together with relativistic OBEPs, and are suitable to address a broad range of momenta/densities.
DBHF: An efficient way to include a particular class of 3NF.
(virtual nucleon-antinucleon
excitation)

9. Problem shared by all non-EFT-based approaches:
It is essentially impossible to estimate reliably the
actual uncertainty associated with a particular
prediction.
The philosophy of EFT and power counting:
When correctly implemented, EFT provides
a well-defined path to calculate observables whose
truncation error should decrease systematically as higher
orders are included.

10. EFT: A framework in which the properties governed by
low-energy physics are specified by the choice
of degrees of freedom and symmetries, and can be
computed systematically.
Short-range physics is included through the processes
of regularization and renormalization
Power counting: an organizational scheme to rank-order
the various diagrams.
Nuclear two- and few-body forces emerge on equal
footing in a controlled hierarchy.

12. Sources of uncertainty:
QUANTIFICATION OF THE UNCERTAINTY IN
NUCLEONIC MATTER
Sources of uncertainty:
1) Many-body method (not inherent to EFT)
2) ERRORS IN THE LECs AND THEIR PROPAGATION IN THE
HAMILTONIAN:
Short-range (NN) LECs
Long-range (pi-N) LECs
3) Regulator dependence
4) Truncation error

13. Nuclear matter has been approached with
a broad spectrum of many-body methods:
Coupled-cluster
Many-body perturbation
Monte Carlo:
Variational MC
Green’s function MC

14. Uncertainty from the many-body method:
We adopt infinite summation of particle-particle (pp)
ladder diagrams.
From recent coupled-cluster calculations
extended beyond pp and hh ladders
[Hagen et al., PRC89, (2014)],
we conclude:
Negligible effect in neutron matter
About +/-1 MeV in symmetric nuclear matter
We determine that the pp ladder approximation
provides sufficient accuracy for our purposes.

15. Findings from the Granada group (arXiv:1407.7784 (2014)):
Applied 205 MC samples of smooth local potentials
(all with chi-squared/datum of about 1) and found an
uncertainty of +/- 15 keV for the triton B.E.
Uncertainty from the NN LECs:
In a standard BHF calculation of nuclear matter
with local high-precision potentials from the Nijmegen
group, we find an uncertainty of +/- 0.6 MeV in the E/A
at normal density.
(Consistent with a similar study of the Granada group
using Skyrme forces.)
Thus:
For the NN LECs, we believe that the uncertainty arising from errors in the NN data has only
negligible impact in the many-body system.

16. Uncertainty in Pi-N LECs:
They impact mostly high partial waves (those where
no contacts are available).
At NLO: D waves and higher
At N3LO: F waves and higher
Therefore, we expect this to cause only small uncertainty
in nuclear matter.
However, a systematic investigation with consideration
of pi-N LECs uncertainty in both 2NF and (consistent) 3NF
has not yet been done.

17. 17

18. REGULARIZATION and CUTOFF DEPENDENCE:
Chiral NN potentials are multiplied by a regulator function:
which sets the “UV scale”.
Potentials with different cutoffs constitute a family
with the same long-range properties.
Cutoff dependence is expected to decrease with
increasing order.

19. NLO
Cutoff = MeV

20. NLO NLO NNLO Cutoff = 450-800 MeV R. Machleidt R. Machleidt
Chiral NFs NTSE2014, June 23-27, 2014
Chiral NFs NTSE2014, June 23-27, 2014 20 20

21. NLO NLO NLO NNLO NNLO N3LO Cutoff = 450-800 MeV Cutoff = 450-600 MeV
R. Machleidt
R. Machleidt
R. Machleidt
Chiral NFs NTSE2014, June 23-27, 2014
Chiral NFs NTSE2014, June 23-27, 2014
Chiral NFs NTSE2014, June 23-27, 2014 21 21 21

22. At the 2N level:
Excellent order-by-order improvement.
From NLO to N2LO:
No new contacts are generated.
At N2LO, subleading 2PE contributions allow
for a better description of the intermediate-range
attraction (they encode the important physics of
correlated 2PE).
Only at N3LO high-precision quality description of
the NN data becomes possible (additional contacts).

23. Moving on to nuclear matter:
We proceed with the particle-particle ladder
approximation (see above),
with NN forces as described above,
and 3NF…..

24. 3NFs make their first appearance at N2LO:
The 3NF LECs are fitted to the binding energies
of 3H and 3He and the Gamow-Teller matrix element
for triton beta-decay.
(Marcucci et al.).

25. Predictions at NLO (yellow), N2LO (red), N3LO (blue),
varying in each case the cutoff in the regulator
function applied to the chiral NN potential between 450
and 600 MeV.

26. Observations from the nuclear matter (SNM) calculations:
Large spread at NLO and N2LO;
Bands do not overlap. Thus, their width is not a suitable
representation of the uncertainty in the respective orders.
Cutoff dependence is reduced in our N3LO calculation.
But, still large uncertainty. Missing contributions (3NF at
N3LO) could be a source of uncertainty.
Better convergence pattern with the lower ( MeV)
cutoffs.

27. Same study, for neutron matter..
Yellow: NLO
Red: N2LO
Blue: N3LO

28. For neutron matter (NM):
Generally weaker cutoff dependence than in SNM.
Similar observations apply as in SNM with regard to the
bands not overlapping.
Results are not well converged.
Overall:
Although cutoff sensitivity decreases with increasing order,
cutoff variations generally underestimate the uncertainty.

29. …and, similar study for the symmetry energy
Yellow: NLO
Red: N2LO
Blue: N3LO

30. Summary and Conclusions:
Order-by-order convergence studies with chiral forces
are very important.
Our goal is to lay the foundation for such calculations
of nuclear many-body systems.
At this time, we conclude:
At low orders in the chiral expansion, no indication
of good convergence.
Our N3LO calculations indicate a pattern of slow
convergence for the lower values of the cutoff.

31. Program to make the analysis broader and more systematic:
OUTLOOK
Program to make the analysis broader and
more systematic:
Vary the pi-N LECs.
For each set of ci, construct NN potentials and use them
consistently in the 2NF and the 3NF. For each set,
refit the cD and cE LECs to the A=3 system.
Start series of calculations at N4LO!
N4LO is very promising. See Few-Body talks this
afternoon…