Perspectives of ab Initio Computations of Medium/Heavy Hypernuclei

Francesco Pederiva

Physics Department - University of Trento INFN - TIFPA, Trento Institute for Fundamental Physics and Applications LISC - Interdisciplinary Laboratory for Computational Science - Trento

1

Collaboration

Diego Lonardoni (NSCL/MSU-LANL) Alessandro Lovato (ANL) Stefano Gandolfi (LANL)

2

Outline:

• Quantum Monte Carlo: what can be computed by it? What is the state of the art? What is its relevance?

•A few words on our approach to the hyperon-nucleon potential, and some (exciting) perspectives for the future.

The non relativistic many-body problem

Many problems of interest in physics can be addressed by solving a non-relativistic quantum problem for N interacting particles:

where is the N particle state, and

The potential can be as simple as the Coulomb potential, or as complicated as, for instance, the Argonne AV18 + UIX or some EFT nucleon-nucleon force (local or non local).

3

A general solutionThere is an interesting, general way of solving the many-body Schroedinger problem, at least for the ground state. Let us consider the following operator, that we call “propagator”( =1):

where is the ground state eigenvalue. If we apply it to an arbitrary state we obtain:

where:

4

A general solutionIn the limit of large it is easily seen that:

All this is very general: no mention is made either of the details of H or of the representation of the states.

provided that the initial state is not orthogonal to the ground state.

Projection Monte Carlo algorithms are based on a stochastic implementation of this “imaginary time propagation”. Different flavours correspond to the choice of a specific representation of the propagator and/or of the specific Hilbert space used.

NB: This is an example within the more general class of “power methods”.

5

Projection Monte CarloThe stochastic implementation of the imaginary time propagator is made by sampling a sequence of states in some Hilbert space. Each state is sampled starting from the previous one with a probability given by the propagator.

For instance, if the potential depends on the coordinates of the particles only, the formulation is relatively simple. First, we approximate our state with an expansion on a finite set of points in space:

Particle coordinates

Wavefunction in coordinate space

6

AFDMCThe computational cost can be reduced in a Monte Carlo framework by introducing a way of sampling over the space of states, rather than summing explicitly over the full set. For simplicity let us consider only one of the terms in the interaction. We start by observing that:

Then, we can linearize the operatorial dependence in the propagator by means of an integral transform:

Linear combination of spin operators for different particles

Hubbard-Stratonovich transformationK. E. Schmidt and S. Fantoni, Phys. Lett. B 446, 99 (1999).S. Gandolfi, F. Pederiva, S. Fantoni, and K. E. Schmidt,Phys. Rev. Lett. 99, 022507 (2007)

auxiliary fields→Auxiliary Field Diffusion Monte Carlo

Stefano Fantoni & Kevin Schmidt, 1999

7

AFDMCThe crucial advantage of AFDMC is that the scaling of the required computer resources is no longer exponential: the cost scales as A3 (the scaling required by the computation of the determinants in the antisymmetric wave functions) LARGER SYSTEMS ACCESSIBLE!

Progress

• The HS transformation can be used ONLY FOR THE PROPAGATOR Accurate wave functions require an operatorial dependence! “Cluster expansion” introduced and working!(Gandolfi, Lovato, Schmidt)

• Some problems in treating nuclear spin-orbit have been addressed. • Three-body forces are now implemented in a quasi-perturbative way,

but results are very promising.

0 1024 2048 3072 4096 5120 6144 7168 8192# nodes

0

0.05

0.1

0.15

0.2

0.25

1/t

(se

c-1)

16 MPI ranks per node

AFDMC scaling @ Mira (ANL)32,768 configurations, 25 steps, 28 nucleons in a periodic box, ρ=0.16 fm

-3

Great for parallel computing!

8

9

AFDMCWhat can we do?This is a crucial question if we want to address the questions relevant for a possible hyper nuclear program at J-Lab.

• Currently we can efficiently do calculations up to A=90/91. Not all most recent improvements are implemented yet (e.g. CVMC-like variational functions). In principle the use of more realistic potentials (up to AV8’) in the nucleon sector is possible, at least for checking purposes.

• We could in principle push the calculations further. For instance 208Pb is computable, but with an expected use of computer time (to reach a sufficient statistics) of order 107 core hours. This means a substantial investment in computational resources.

• “Cheaper” models (maybe even more useful for astrophysical applications) might be based on a neutron rich matter, and compared e.g. with Pb results.

𝜦 Hyperon

nucleus (e.g. 90Zr) neutron sea (e.g. implemented by means of periodic boundary conditions)

Fock space calculations

The stochastic power method can also be used in Fock space. In this case the propagator acts on the occupation number of a basis set used to span the Hilbert space of the solution of a given Hamiltonian. In particular, given two basis states |m⟩ and |n⟩ the quantity:

hm|P�⌧ |ni = hm|1� (H � E0)�⌧ |niis interpreted as the probability of the system of switching the occupation of the state |n⟩ into the occupation of the state |m⟩. This propagation has in principle the same properties of the coordinates space version.

10

Fock space calculations

Unfortunately matrix elements for a many-Fermion systems are not positive definite. It is possible, however, to introduce an importance sampling using a variational ansatz of the wave function to circumvent this problem. First one redefines the Hamiltonian as:

hm|H� |ni =⇢ ��hm|H|ni s(m,n) > 0

hm|H|ni otherwise

for the off-diagonal terms and

hn|H� |ni = hn|H|ni+ (1 + �)X

m 6=n

s(m,n)>0

s(m,n) .

for the diagonal terms, with:

s(m,n) = �G(m)hm|H|ni/�G(n)

Variational function (explicit)

parameter: if > 0, no sign problem, but biased result. Extrapolation to -1 gives the exact result (or at least a rigorous upper bound)

11

Fock space calculations

We now define a new propagator:

hm|P� |ni = 1��⌧�G(m)hm|H� � ET |ni/�G(n) .

The propagator , by construction, is free from the sign problem for 𝛾≥0, and filters out the wave function , where is the ground state wave function of

P�

�G(n) �(n) �(n)H�

12

As previously mentioned, the choice of the representation of the Hilbert space is arbitrary!

FINITE SYSTEMS ➔ H.O. basis, Gaussians, HH.. INFINITE SYSTEMS ➔ Plane waves, BCS,…

Configuration Interaction Monte Carlo

0 0.5 1 1.5 2

kF [fm

-1]

-1

0

Bpoln

dn /

EF

pp resummed N3LO

GFMCCIMC NNLOPT2 NNLOHF NNLOSAMISkOSkO’SkPBSk20BSk21SLy4

SLy5

SLy6

SLy7

SkM*UNEDF0UNEDF1UNEDF1’

0 0.04 0.08 0.12 0.16 0.2 0.24

ρ (fm-3

)

0

4

8

12

16

20

24

E/N

(M

eV

)

200 400 600

kmax

(MeV)

9

9.5

10

10.5

ρ = 0.08 fm-3

One of the main advantages of using an algorithm in Fock space is the possibility of using non-local Hamiltonians, such as the chiral EFT based Hamiltonians. Here on the left, the computation of the equation of state of pure neutron matter with the 2-body N2LO(opt) interaction of Machleidt et al. and using as importance functions the result of a CC calculation at the SD level.

On the right the energy of a neutron polaron, always computed with the same Hamiltonian and CIMC.

Alessandro Roggero, Abhishek Mukherjee, Francesco PederivaPRL, in press (2014)

Alessandro Roggero, Abhishek Mukherjee, Francesco Pederiva

13

Open questions…

The fine tuning of the hyperon-nucleon interaction is essential to understand the behaviour of matter in extreme conditions.

Example: Neutron stars

> 2.0M�

⇠ 2÷ 3⇢0

31

soft

stiff

⇢th⇤

M

R12 km

E

⇢b⇢0

⇠ 2M�

obs:

< 2.0M�

R ⇠ 12 km

M ⇠ 1.4 M�

?⇤ ⌃ ⌅ ⇡c Kc qp

TOV

Neutron stars: the hyperon puzzle

µ⇤ = µn

npeµ

HNM

Far away from any possible perturbative treatment..

Internal composition still largely unknown

Equation of state

Neutron star structure

Hyperon PuzzleA few NS with a large mass were observed by using Shapiro delay measurements. The first (2010) was PSRJ1614-2230 pulsar with M=1.97(4)M⊙.(P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts and J. W. T. Hessels. A two-solar-mass neutron star measured using Shapiro delay measurements, Nature 467, 1081 (2010).

In a non relativistic framework (= pure baryonic stars)

hyperons are problematic

Before 2010: Maximum mass observed: 1.6M⊙ Maximum mass predicted without hyperons: 2.3⊙ (still ok in principle)Maximum mass predicted with hyperons: 1.4-1.6M⊙ (good!)After 2010: Observed mass: 2.0M⊙ Maximum mass predicted without hyperons: 2.3M⊙ (good!)Maximum mass predicted with hyperons: 1.4-1.6M⊙ (very bad…)

Many possible description of the YN interaction

NON RELATIVISTIC:write an Hamiltonian including some potential and try to solve a many-body Schroedinger equation.• The potential energy is not an observable: several different equivalent descriptions

are possible. • The interaction can be based on some more or less phenomenological scheme (fit

the existing experimental data, rely on some systematic meson exchange model), or can be inferred from EFT systematic expansions.

• Only accurate many-body calculations can help distinguishing among different realizations of the potential.

RELATIVISTIC:write a Lagrangian including relevant fields, and try to solve the field theoretical problem (usually RMF calculations are performed).

22 S. Aoki et al. (HAL QCD Collaboration),

-0.04

-0.02

0.00

0.02

0.04

0.0 0.5 1.0 1.5 2.0 2.5

V (

GeV

)

r (fm)

ΛN, 1S0, t-t0=8, 3pt

VC

-0.04

-0.02

0.00

0.02

0.04

0.0 0.5 1.0 1.5 2.0 2.5

V (

GeV

)

r (fm)

ΣN(I=3/2), 1S0, t-t0=8, 3pt

VC

Fig. 10. Left: The central potential in the 1S0 channel of the ΛN system in 2+ 1 flavor QCD as afunction of r. Right: The central potential in the 1S0 channel of the ΣN(I = 3/2) system as afunction of r.

-0.04

-0.02

0.00

0.02

0.04

0.0 0.5 1.0 1.5 2.0 2.5

V (

GeV

)

r (fm)

ΛN, 3S1-3D1, t-t0=8, 3pt

VCVT

-0.04

-0.02

0.00

0.02

0.04

0.0 0.5 1.0 1.5 2.0 2.5

V (

GeV

)

r (fm)

ΣN(I=3/2), 3S1-3D1, t-t0=8, 3pt

VCVT

Fig. 11. Left: The central potential (circle) and the tensor potential (triangle) in the 3S1 −3 D1

channel of the ΛN system as a function of r. Right: The central potential (circle) and the tensorpotential (triangle) in the 3S1 −

3 D1 channel of the ΣN(I = 3/2) system as a function of r.

ΛN central potential in the 1S0 channel, while the tensor potential itself (triangle)is weaker than the tensor potential in the NN system.44)

The right panel of Fig. 11 shows the central potential (circle) and the tensorpotential (triangle) of the ΣN(I = 3/2) system in the 3S1 −3 D1 channel. Dueto the isospin symmetry, this channel belongs solely to the flavor 10 representationwithout mixture of 10 or 8a As seen from the figure, there is no clear attractive well inthe central potential (circle). This repulsive nature of the ΣN(I = 3/2,3 S1 −3 D1)central potential is consistent with the prediction from the naive quark model.45)

The tensor force is a little stronger that that of the ΛN system but is still weaker inmagnitude than that of the NN system.

5.2. ΞN potential in quenched QCD

Experimentally, not much information is available on the NΞ interaction ex-cept for a few studies: a recent report gives the upper limit of elastic and inelasticcross sections46) while earlier publications suggest weak attractions of Ξ− nuclearinteractions.47)–49) The Ξ−nucleus interactions will be soon studied as one of the

Some hints from LQCD……

S. Aoki et al. (HAL-QCD collaboration)

Hard cores seem to be unavoidable in

a realisticdescription!

Model Hyperon-nucleon interactionIn order to gain some understanding, we need to set up some scheme.

9

✓ 2-body interaction: AV18 & Usmani

⇤N scattering

A = 4 CSB⇤N

NN

8

>

>

<

>

>

:

vij =X

p=1,18

vp(rij)O pij

O p=1,8ij =

n

1,�ij , Sij ,Lij · Sij

o

⌦n

1, ⌧ijo

8

>

>

<

>

>

:

v�i =X

p=1,4

vp(r�i)O p�i

O p=1,4�i =

n

1,��i

o

⌦n

1, ⌧zi

o

NNscattering

deuteron

< v

b

R.H. Dalitz et al., s-shell hypernuclei 123

~ t i i i i i - 7 r i \ \ × 0 25

CSB potential (2"9) j ~ -~'7 ~ ~ {

o [ I I I ] I I I I 0 2 4 6 8 I0 12 14 16 18

E (MeV)

Fig. 1. Total cross section for Ap scattering as a function of c.m. kinetic energy E(MeV). The solid line is obtained with CSB potential of the form (2.9), while the dashed line is obtained

with CSB potential of the form (3.16).

20

a large intrinsic range (b ~ 2.0 fm) for t he /AN = 0 in teract ion, this Ap potent ia l gives rise to significant p-wave scattering, suff icient to cause appreciable forward- backward asymmet ry through interference with the dominat ing s-wave ampli tudes, at relative low c.m. energies, far in excess o f the observed F/B ratios. As a result, it is necessary to allow for the possibility of a p-wave potent ia l weaker than that required for the s-wave. Thus, in this investigation, we shall assume the fol lowing form for the Ap potent ial ,

U P ( r ) = [ (1-x)+xprAp ] [½( l+P'aAp)Upt(r)+½(1-P~p)Us p(r)] , (3 .15)

where PEp is the space exchange operator , and U p and U p are the Ap potent ia ls in tr iplet and singlet s -s ta tes .The addit ional parameter x is a reduct ion factor ¢; the

* This is of course not the most general possibility, even for central potentials, since the param- eter x could take different values for the singlet and the triplet states. The absence of data depending on the A and proton spins leaves only one empirical number accessible at each energy, namely the F/B ratio, and there is no possibility of determining more than one theo- retical parameter from the data available.

R. H. Dalitz, R. C. Herndon, Y. C. Tang, Nucl. Phys. B47 (1972) 109-137

36 J. Haidenbauer et al. / Nuclear Physics A 915 (2013) 24–58

Fig. 2. “Total” cross section σ (as defined in Eq. (24)) as a function of plab. The experimental cross sections are takenfrom Refs. [54] (filled circles), [55] (open squares), [69] (open circles), and [70] (filled squares) (Λp → Λp), from [56](Σ−p → Λn, Σ−p → Σ0n) and from [57] (Σ−p → Σ−p, Σ+p → Σ+p). The red/dark band shows the chiral EFTresults to NLO for variations of the cutoff in the range Λ = 500, . . . ,650 MeV, while the green/light band are results toLO for Λ = 550, . . . ,700 MeV. The dashed curve is the result of the Jülich ’04 meson-exchange potential [37].

also for Λp the NLO results are now well in line with the data even up to the ΣN threshold.Furthermore, one can see that the dependence on the cutoff mass is strongly reduced in the NLOcase. We also note that in some cases the LO and the NLO bands do not overlap. This is partlydue to the fact that the description at LO is not as precise as at NLO (cf. the total χ2 values inTable 5). Also, the error bands are just given by the cutoff variation and thus can be consideredas lower limits.

A quantitative comparison with the experiments is provided in Table 5. There we list theobtained overall χ2 but also separate values for each data set that was included in the fittingprocedure. Obviously the best results are achieved in the range Λ = 500–650 MeV. Here, inaddition, the χ2 exhibits also a fairly weak cutoff dependence so that one can really speak ofa plateau region. For larger cutoff values the χ2 increases smoothly while it grows dramatically

J. Haidenbauer et al., Nucl. Phys. A 915

(2013) 24–58

1968

19681972

Hyperon-nucleon interaction

9

✓ 2-body interaction: AV18 & Usmani

⇤N scattering

A = 4 CSB⇤N

NN

8

>

>

<

>

>

:

vij =X

p=1,18

vp(rij)O pij

O p=1,8ij =

n

1,�ij , Sij ,Lij · Sij

o

⌦n

1, ⌧ijo

8

>

>

<

>

>

:

v�i =X

p=1,4

vp(r�i)O p�i

O p=1,4�i =

n

1,��i

o

⌦n

1, ⌧zi

o

NNscattering

deuteron

< v

b

R.H. Dalitz et al., s-shell hypernuclei 123

~ t i i i i i - 7 r i \ \ × 0 25

CSB potential (2"9) j ~ -~'7 ~ ~ {

o [ I I I ] I I I I 0 2 4 6 8 I0 12 14 16 18

E (MeV)

Fig. 1. Total cross section for Ap scattering as a function of c.m. kinetic energy E(MeV). The solid line is obtained with CSB potential of the form (2.9), while the dashed line is obtained

with CSB potential of the form (3.16).

20

a large intrinsic range (b ~ 2.0 fm) for t he /AN = 0 in teract ion, this Ap potent ia l gives rise to significant p-wave scattering, suff icient to cause appreciable forward- backward asymmet ry through interference with the dominat ing s-wave ampli tudes, at relative low c.m. energies, far in excess o f the observed F/B ratios. As a result, it is necessary to allow for the possibility of a p-wave potent ia l weaker than that required for the s-wave. Thus, in this investigation, we shall assume the fol lowing form for the Ap potent ial ,

U P ( r ) = [ (1-x)+xprAp ] [½( l+P'aAp)Upt(r)+½(1-P~p)Us p(r)] , (3 .15)

where PEp is the space exchange operator , and U p and U p are the Ap potent ia ls in tr iplet and singlet s -s ta tes .The addit ional parameter x is a reduct ion factor ¢; the

* This is of course not the most general possibility, even for central potentials, since the param- eter x could take different values for the singlet and the triplet states. The absence of data depending on the A and proton spins leaves only one empirical number accessible at each energy, namely the F/B ratio, and there is no possibility of determining more than one theo- retical parameter from the data available.

R. H. Dalitz, R. C. Herndon, Y. C. Tang, Nucl. Phys. B47 (1972) 109-137

36 J. Haidenbauer et al. / Nuclear Physics A 915 (2013) 24–58

Fig. 2. “Total” cross section σ (as defined in Eq. (24)) as a function of plab. The experimental cross sections are takenfrom Refs. [54] (filled circles), [55] (open squares), [69] (open circles), and [70] (filled squares) (Λp → Λp), from [56](Σ−p → Λn, Σ−p → Σ0n) and from [57] (Σ−p → Σ−p, Σ+p → Σ+p). The red/dark band shows the chiral EFTresults to NLO for variations of the cutoff in the range Λ = 500, . . . ,650 MeV, while the green/light band are results toLO for Λ = 550, . . . ,700 MeV. The dashed curve is the result of the Jülich ’04 meson-exchange potential [37].

also for Λp the NLO results are now well in line with the data even up to the ΣN threshold.Furthermore, one can see that the dependence on the cutoff mass is strongly reduced in the NLOcase. We also note that in some cases the LO and the NLO bands do not overlap. This is partlydue to the fact that the description at LO is not as precise as at NLO (cf. the total χ2 values inTable 5). Also, the error bands are just given by the cutoff variation and thus can be consideredas lower limits.

A quantitative comparison with the experiments is provided in Table 5. There we list theobtained overall χ2 but also separate values for each data set that was included in the fittingprocedure. Obviously the best results are achieved in the range Λ = 500–650 MeV. Here, inaddition, the χ2 exhibits also a fairly weak cutoff dependence so that one can really speak ofa plateau region. For larger cutoff values the χ2 increases smoothly while it grows dramatically

J. Haidenbauer et al., Nucl. Phys. A 915

(2013) 24–58

1968

19681972

Hyperon-nucleon interaction

OUR CHOICE

• NON RELATIVISTIC APPROACH (should be fine if the central density is not too large)

• YN INTERACTION CHOSEN TO FIT EXISTING SCATTERING DATA (with a hard-core)

• PHENOMENOLOGICAL YNN THREE-BODY FORCES with few parameters to be adjusted to reproduce light hypernuclei binding energies

• ALL THE OTHER RESULTS ARE PREDICTIONS WITH NO OTHER ADJUSTABLE PARAMETERS obtained from an accurate solution of the Schroedinger equation.

THIS IS ONE OF MANY POSSIBLE WAY TO ATTACK THE PROBLEM.EMPHASIS IS ON EXPERIMENTALLY AVAILABLE INFORMATION.

Model Hyperon-nucleon interaction

Model interaction (Bodmer, Usmani, Carlson):

V⇤i

(r) = v0(r) + v0(r)"(Px

� 1) +1

4v�

T 2⇡

(m⇡

r)�⇤ · �i

from Kaon exchange terms (not considered explicitly in our calculations)

⇤⌃⇧

⌃⌅

V 2��ij = CSW

2� O2�,SW�ij + CPW

2� O2�,PW�ij

V D�ij = WD T 2

� (m�r�i) T2� (m�r�j)

�1 +

1

6�� · (�i + �j)

⇥

V�ij = V 2��ij + V D

�ij

Two-body potential: accurately fitted on p-𝛬 scattering data

Parameters to be determined from calculations

A. Bodmer, Q. N. Usmani, and J. Carlson, Phys. Rev. C 29, 684 (1984).

Q. N. Usmani and A. R. Bodmer, Phys. Rev. C 60, 055215 (1999).

Non trivial isospin dependence in the three-body sector?

In hypernuclei it is possible that the 𝛬NN interaction is not well constrained, especially in the isospin triplet channel:

np

nn

𝝠 𝝠

On can try o do the exercise of re-projecting the interaction in the isospin singlet and triplet channels and try to explore the dependence of the hypernuclei binding energy on the relative strength.

⇤NN potential resolved in the NN isospin singlet and triplet

F. Pederiva

The ~⌧i · ~⌧j part of the three-body potential can be written as:

v2⇡,P = �CP6 {Xi�, X�j}~⌧i · ~⌧j

v2⇡,S = CSO2⇡,Sij� ~⌧i · ~⌧j

We want to rewrite these contributions in such a way that they are splitted intoan isospin triplet and an isospin singlet channels, adding then a parameter tocontrol the first with respect to the second.

As always, let us notice that:

~⌧i · ~⌧j = 1� 4PT=0ij = 4PT=1

ij � 3.

We can sum the two expressions multiplying the first by 3, and obtain thefollowing identity:

~⌧i · ~⌧j = �3PT=0ij + PT=1

ij

Now, defining:

vPij� ⌘ vPij�(CS , CP ) = �CP

6{Xi�, X�j}+ CSO

2⇡,Sij�

the isospin-dependent three body potential then becomes:

v⌧⌧ij� = �3vPij�PT=0ij + vPij�P

T=1ij .

We define a new potential by inserting a parameter A that controls the strengthof the potential projected on the isospin triplet channel:

v⌧⌧ij� = �3vPij�PT=0ij +AvPij�P

T=1ij .

A = 0 is the case in which the isospin triplet channel is suppressed. A = 1 is thepresent potential case. However, I think that in this context A could assumearbitrarily large values, and even change sign. Actually, it can be inferred thatif PT=0

ij is the most contributing channel in hypernuclei as expected, the expec-

tation of vPij� should be mostly negative in order to give the observed reductionof B(⇤). This means that under this hypothesis some repulsion might be gainedin neutron matter withouta↵ecting the results in hypernuclei by playing withnegative values of A.

This potential can be easily recast in the usual form useful for AFDMCcalculations in this way:

v⌧⌧ij� =3

4(A� 1)vPij� +

1

4(3 +A)vPij�~⌧i · ~⌧j .

Notice that there is a contribution that has to be added to the isospin indepen-dent part of the interaction as well.

Please check coe�cients, signs etc.

1

v⌧⌧ij� = �3vPij�PT=0ij + CT v

Pij�P

T=1ij

CT=1 gives the original potential, but we can choose an arbitrary value. CT < 1 ⇒ more repulsion

NN isospin singlet NN isospin triplet

Pauli repulsion

must be negative on averageto give repulsion

v⌧⌧ij� =3

4(CT � 1)vPij� +

1

4(3 + CT )v

Pij�~⌧i · ~⌧j

Input from experiment

We need to fit the three body interaction against some experimental data. There are available several measurements of the binding energy of 𝛬-hypernuclei, i.e. nuclei containing a hyperon. The idea is to compute such binding energies. We can then compute the hyperon separation energy:

⇤

B⇤ = Bhyp �Bnuc

where is the total binding energy of a hypernucleus with A nucleons and one , and is the total binding energy of the corresponding nucleus with A nucleons. This number can be used to gauge the coefficients in the nucleon- interaction.

Bhyp

Bnuc⇤

⇤

3Hypernuclei: experiments

��

�

940 MeV

1116 MeV

1200 MeV

1300 MeV

n

np

p

4He 4⇤He

np

p⇤

AZ�e, e0K+

�A⇤[Z � 1]

AZ�K�,⇡0

�A⇤[Z � 1]

AZ�⇡�,K0

�A⇤[Z � 1]

AZ�K�,⇡��A

⇤Z

AZ�⇡+,K+

�A⇤Z

AZ�⇡�,K+

�A+1

⇤[Z � 2]

AZ�K�,⇡+

�A+1

⇤[Z � 2]

✓ Charge conserving

✓ Single charge exchange

✓ Double charge exchange

⌧(⇤) = 2.6 · 10�10s

Hypernuclei data6

Proceedings of

The IX International Conference on Hypernuclear and Strange

Particle Physics

HYP 2006

October 10-14, 2006 Mainz, Germany

edited by

J. Pochodzalla and Th. Walcher

NZ

|S|

binding energies:

|S| = 0

|S| = 1

|S| = 2

scattering data:

NN : ⇠ 4300

⇤N : ⇠ 52

nuc : ⇠ 3340

⇤ hyp : ⇠ 41

⇤⇤ hyp : ⇠ 5

⌃ hyp : ⇠ (1)

Hypernuclei: experiments4

15 28ȁ P 29

ȁ P 30ȁ P 31

ȁ P 32ȁ P 33

ȁ P 34ȁ P 35

ȁ P 36ȁ P 37

ȁ P 38ȁ P

14 24ȁ Si 25

ȁ Si 26ȁ Si 27

ȁ Si 28ȁ Si 29

ȁ Si 30ȁ Si 31

ȁ Si 32ȁ Si 33

ȁ Si 34ȁ Si 35

ȁ Si 36ȁ Si 37

ȁ Si

13 24ȁ Al 25

ȁ Al 26ȁ Al 27

ȁ Al 28ȁ Al 29

ȁ Al 30ȁ Al 31

ȁ Al 32ȁ Al 33

ȁ Al 34ȁ Al 35

ȁ Al 36ȁ Al

12 20ȁ Mg21

ȁ Mg 22ȁ Mg 23

ȁ Mg 24ȁ Mg 25

ȁ Mg 26ȁ Mg 27

ȁ Mg 28ȁ Mg 29

ȁ Mg 30ȁ Mg 31

ȁ Mg 32ȁ Mg 33

ȁ Mg 34ȁ Mg 35

ȁ Mg

11 20ȁ Na 21

ȁ Na 22ȁ Na 23

ȁ Na 24ȁ Na 25

ȁ Na 26ȁ Na 27

ȁ Na 28ȁ Na 29

ȁ Na 30ȁ Na 31

ȁ Na 32ȁ Na 33

ȁ Na 34ȁ Na

10 17ȁ Ne 18

ȁ Ne 19ȁ Ne 20

ȁ Ne 21ȁ Ne 22

ȁ Ne 23ȁ Ne 24

ȁ Ne 25ȁ Ne 26

ȁ Ne 27ȁ Ne 28

ȁ Ne 29ȁ Ne 30

ȁ Ne 31ȁ Ne 32

ȁ Ne 33ȁ Ne

9 16ȁ F 17

ȁ F 18ȁ F 19

ȁ F 20ȁ F 21

ȁ F 22ȁ F 23

ȁ F 24ȁ F 25

ȁ F 26ȁ F 27

ȁ F 28ȁ F 29

ȁ F 30ȁ F 31

ȁ F 32ȁ F

8 13ȁ O 14

ȁ O 15ȁ O 16

ȁ O 17ȁ O 18

ȁ O 19ȁ O 20

ȁ O 21ȁ O 22

ȁ O 23ȁ O 24

ȁ O 25ȁ O 26

ȁ O 27ȁ O

7 12ȁ N 13

ȁ N 14ȁ N 15

ȁ N 16ȁ N 17

ȁ N 18ȁ N 19

ȁ N 20ȁ N 21

ȁ N 22ȁ N 23

ȁ N 24ȁ N

6 10ȁ C 11

ȁ C 12ȁ C 13

ȁ C 14ȁ C 15

ȁ C 16ȁ C 17

ȁ C 18ȁ C 19

ȁ C 20ȁ C 21

ȁ C

5 9ȁB 10

ȁ B 11ȁ B 12

ȁ B 13ȁ B 14

ȁ B 15ȁ B 16

ȁ B 17ȁ B 18

ȁ B

4 7ȁBe 8

ȁBe 9ȁBe 10

ȁ Be 11ȁ Be 12

ȁ Be 13ȁ Be 14

ȁ Be 15ȁ Be : ( , );( , );( , )stopn K K KS S S� � � � � �o /

3 6ȁLi 7

ȁLi 8ȁLi 9

ȁLi 10ȁ Li 11

ȁ Li 12ȁ Li 0: ( , ); ( , )stopp e e K K S� �co /

2 4ȁHe 5

ȁHe 6ȁHe 7

ȁHe 8ȁHe

9ȁHe : ( , )pp n KS � �o /

1 3ȁH 4

ȁH 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Number of Neutrons

Num

ber o

f Pro

trons

J. Pochodzalla, Acta Phys. Polon. B 42, 833– 842 (2011)

Hypernuclei: experiments

5

S. N. Nakamura, Hypernuclear workshop, JLab, May 2014 updated from: O. Hashimoto, H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006)

Present Status of Λ Hypernuclear Spectroscopy

Updated from: O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. 57 (2006) 564.

52ΛV

Hypernuclei: experiments

• The available data are very limited. • There are several planned and ongoing

systematic measurements. • At present no proposals for gathering more 𝛬-nucleon scattering data • Essentially no information on 𝛬𝛬

interaction • (Almost) nothing on 𝛴 or 𝛯 hypernulcei

Gravitational wavesThe EoS of dense matter is one of the ingredients needed in the solution of Einstein’s equation when studying the dynamics of neutron star mergers.

How sensitive is the spectrum of GW on the details of the EoS?

There is a region within a few ms away from the actually merging where the spectrum seems to become rather sensitive on the stiffness, at the post that the GW spectrum might be used in this case to determine the NS radius with an accuracy of about 1Km.

If such events will be experimentally observed, a completely new kind of constraints will be provided. Will we be ready for that?

Warning: temperature, neutrinos… These were two BHs. Too bad…

24

Gravitational waves

Different EoS fitted with polytropic functions…

Conclusions• AFDMC calculations are evolving. Better accuracy, better

performance. This reflects on the work on hypernuclei (see Diego Lonardoni’s talk).

• Accessible systems: definitely A=90. For heavier systems one can possibly use alternative approaches.

• Our philosophy in attacking the problem of the hyperon-nucleon interaction: we do not want to add more information than the one that the experiments can give us. Having too many parameters will result in a substantially arbitrary prediction of the EoS, and consequently adjustable predictions on the Neutron Star structures.

25

26

SUPPLEMENTAL MATERIAL

27

Projection Monte CarloWe should also write the propagator in coordinates space, so that:

In the limit of “short” 𝜏 (let us call it “𝜟𝜏”), the propagator can be broken up as follows (Trotter-Suzuki formula):

Kinetic term Potential term (“weight”)

Sample a new point from the Gaussian kernel

Create a number of copies proportional to the weight

If the weight is small, the points are canceled.

28

Projection Monte CarloOnce the convergence to the ground state is reached it is possible to use the sampled configurations to evaluate expectations of observables of interest in a Monte Carlo way. For example, if we want to compute the energy, we can use some test function and evaluate the following ratio:

This is the Monte Carlo estimate of:

29

Known issues• The naive algorithm does not work for any realistic

potential. In general the random walk needs to be guided by an “importance function”. In a correct formulation there is no bias on the results.

• The algorithm works (strictly speaking) only for the “mathematical” ground state of the Hamiltonian, which is always a symmetric (bosonic) wavefunction. Fermions live on an “mathematical excited state” of H! ➡ SIGN PROBLEM. Workarounds exist, but the results are biased. However, in some cases it is possible to estimate the bias.

30

Many-nucleon systems

EXAMPLE: One of the most celebrated model nucleon-nucleon (NN) interaction is the so-called Argonne AVX potential, defined by:

EX: AV8Spin-orbit

Spin representation in term of Pauli matrices

Isospin

Here Sij is the tensor operator

that characterises the “one-pion exchange” part of the interaction.

R B Wiringa, V G J Stoks, and R Schiavilla PRC 51, 38 (1995)

Nuclear physics experiments teach us that the nucleon-nucleon interaction depends on the relative spin and isospin state of nucleons. This fact can be formally related to the fundamental symmetry properties of QCD, and it is necessary in any realistic interactions that can be used in a many-body calculation

31

Projection MC many-nucleon systems

We can apply our (very general) propagator to a state that is now given by the particle positions (the “R”), and the spin/isospin state of each nucleon (the “S”).

ProblemIn the stochastic evolution, spins are subject to factors like:

The action of the propagators depends on the relative spin state of each pair of nucleons

But:

32

Projection MC many-nucleon systems

Multicomponent wave functions are needed! How large is the system space? For a system of A nucleons, Z protons, the number of states is

• Very accurate results, possibility of using accurate wave functions for the evaluation of general estimators (e.g. response functions

• Due to the high computational cost, application limited so far to A≤12: COMPUTATIONAL CHALLENGE!

Number of states in many nucleon wave functions for a few selected nuclei

33

AFDMCThe operator dependence in the exponent has become linear.

In the Monte Carlo spirit, the integral can be performed by sampling values of x from the Gaussian . For a given x the action of the propagator will become:

In a space of spinors, each factor corresponds to a rotation induced by the action of the Pauli matrices

The sum over the states has been replaced by sampling rotations!

34

35

Configuration Interaction Monte Carlo

A first test of this algorithm was the evaluation of the equation of state of the three-dimensional homogeneous electron gas, for which very accurate results are already available. In this case the Hamiltonian is very simple, and includes the contribution of a uniform cancelling background of positive charge.

As importance function we used the overlaps computed by COUPLED CLUSTERS at the doubles level (CCD) method.

Basis size-1

-0.58

-0.57

-0.56

-0.55

-0.51

-0.50

-0.49

-0.48

Corr

elat

ion e

ner

gy (

a.u.)

162-1

342-1

2378-1

-0.41

-0.40

-0.39

2378-1

342-1

162-1

-0.33

-0.32

-0.31

-0.30

rs = 0.5

rs = 2.0

rs= 1.0

rs = 3.0

Alessandro Roggero, Abhishek Mukherjee, and Francesco PederivaPhys. Rev. B 88, 115138 (2013)

36

Response FunctionsFrom QMC calculations

Alessandro Roggero, Francesco Pederiva, and Giuseppina Orlandini, Phys. Rev. B 88, 094302 (2013)

0 0.5 1 1.5 2 2.5

q [A-1

]

0

20

40

60

ωp

eak

[K]

0 0.5 1 1.5 2 2.5

q [A-1

]

0 1 2 3

q [A-1

]

0.1

1

∆ω

/ ω

(a) (b)

0 5 10 15 20 25ω [K]

0

0.2

0.4

0.6

0.8

1

S(q

,ω)/

S(q

) [K

-1]

0 25 50 75σ [Κ]

0

0.2

0.4

0.6

0.8

1Φ

(q,σ

)/S

(q)

x 10 Laplace kernel

Kernel as in Eq.(5) �(q,�) =

ZK(�,!)SO(q,!) d!.

The natural choice within QMC is a Laplace kernel, very inefficient and amplifying the ill-posedness of the inversion problem.

SO(q,!) =X

⌫

|h ⌫ |O(q)| 0i|2�(E⌫ � !)

= h 0|O†(q)�(H � !)O(q)| 0i

Better kernel: SUMUDU

KP (�,!) = N

e�µ!

�

�� e�⌫ !

�

�

�P

•Still “natural” in the language of QMC •“Bell shaped”, and therefore more efficient and less prone to inversion ambiguities.

0 15 30 45 60 75σ [K]

0

0.05

0.1

0.15

0.2

0.25

σ K

P(σ

,ω)

2

3

5

10

2

3

5

11

10

37

Sign ProblemOne of the major issues in Quantum Monte Carlo calculations comes from the fact that Fermions live in an excited state (in mathematical sense) of the Hamiltonian. This means that if we want to preserve the normalisation of the Fermionic ground state (using for instance instead of the propagation:

leads to

therefore quantities that are symmetric (like the variance of any operator…) will grow exponentially in imaginary time compared to the expectation of any antisymmetric function. This is the essence of the so called the “sign problem”.

Symmetric (bosonic) ground state

Antisymmetric (fermionic) ground state

Always > 0 (it’s a theorem!)

38

Sign ProblemIn order to cope with the sign problem it is useful to introduce some approximations. In particular, the general idea is to solve a modified Schroedinger equation with additional boundary conditions.

• For real-valued wave functions, the nodes (zeros) of the solutions must correspond to the nodes of some trial wavefunctions

• (FIXED NODE APPROXIMATION)• For complex valued wave functions, we have two options:

A. Constrain the phase of the solution to be equal to the phase of some trial wave function (FIXED PHASE APPROXIMATION)

B. Constrain the sign of the real part of the wave function (or some suitable combination) to preserve the sign (CONSTRAINED PATH APPROXIMATION)

39

Thanks!

• Stefano Fantoni

• Malvin H. Kalos (LLNL)

• Kevin E. Schmidt (ASU)

• Alberto Ambrosetti (MPI-Potsdam)

• Paolo Armani (Trento)

• Omar Benhar (Roma I)

• Francesco Catalano (Trento)

• Lorenzo Contessi (Trento)

• Stefano Gandolfi (LANL)

• Alexey Yu. Illarionov (ETH)

• Diego Lonardoni (ANL)

• Alessandro Lovato (ANL)

• Abhishek Mukherjee (ECT*)

• Giuseppina Orlandini (Trento)

• Alessandro Roggero (Trento)

• Computer time: NERSC, LLNL, CINECA, ECT*

40

Effective InteractionsIn many cases it is possible to derive effective interactions obtained from the matrix elements of realistic Hamiltonians, computed using advanced many-body approaches.

vr

≠

vS ,T r

vS ,T r

vr

≠

vS ,T r

vS ,T rv

r

≠

vt,T r

vt,T r

41

Neutrino mean free path in neutron matter

The mean free path of non degenerate neutrinos at zero temperature is obtained from:

1

�=

G2F

4

⇢

Zd3q

(2⇡)3⇥(1 + cos ✓)S(q,!) + C2

A(3� cos ✓)S(q,!)⇤

where and are the density (Fermi) and spin (Gamow Teller) response, respectively

S S

⁄⁄

FG

E‹

Both long and short range correlations are important.

A Lovato, O. Benhar, S. Gandolfi & C. Losa, PRC 89, 025804 (2014)

42

Perspectives• Inclusion of explicit 𝜋 and 𝛥 degrees of freedom in many-nucleon

AFDMC calculations

• Use of AFDMC calculations in the interpretation of current large m𝜋 LQCD calculations via 𝜋-less EFT. (“Effective Field Theory for Lattice Nuclei”, N.Barnea, L.Contessi, D. Gazit, F. Pederiva, U. van Kolck, arXiv:1311.4966)

• Development of general formulations of DMC in Fock-space (e.g. in momentum space), to be used with strongly non-local Hamiltonians (e.g. 𝜒-EFT-based potentials), and wave functions derived from Coupled Cluster theory (useful in quantum chemistry and materials science). (Quantum Monte Carlo with coupled-cluster wave functions”Alessandro Roggero, Abhishek Mukherjee, and Francesco Pederiva Phys. Rev. B 88, 115138 (2013))

• Search for improved algorithms based on the propagation of multiplets of points in configuration space in order to eliminate the systematic bias due to the fixed-node/fixed-phase approximations.

43

Thanks!• Stefano Fantoni (ANVUR)

• Malvin H. Kalos (LLNL)

• Enrico Lipparini (U. Trento)

• Kevin E. Schmidt (ASU)

• Alberto Ambrosetti (MPI-Potsdam)

• Paolo Armani (Trento)

• Lorenzo Contessi (U. Trento)

• Stefano Gandolfi (LANL)

• Alexey Yu. Illarionov (ETH)

• Diego Lonardoni (ANL)

• Alessandro Lovato (ANL)

• Abhishek Mukherjee (ECT*)

• Alessandro Roggero (U. Trento)

• Computer time: NERSC, LLNL, CINECA, ECT*

44

Sign ProblemOne of the major issues in Quantum Monte Carlo calculations comes from the fact that Fermions live in an excited state (in mathematical sense) of the Hamiltonian. This means that if we want to preserve the normalisation of the Fermionic ground state (using for instance instead of the propagation:

leads to

therefore quantities that are symmetric (like the variance of any operator…) will grow exponentially in imaginary time compared to the expectation of any antisymmetric function. This is the essence of the so called the “sign problem”.

Symmetric (bosonic) ground state

Antisymmetric (fermionic) ground state

Always > 0 (it’s a theorem!)

45

Sign ProblemIn order to cope with the sign problem it is useful to introduce some approximations. In particular, the general idea is to solve a modified Schroedinger equation with additional boundary conditions.

• For real-valued wave functions, the nodes (zeros) of the solutions must correspond to the nodes of some trial wavefunctions

• (FIXED NODE APPROXIMATION)• For complex valued wave functions, we have two options:

A. Constrain the phase of the solution to be equal to the phase of some trial wave function (FIXED PHASE APPROXIMATION)

B. Constrain the sign of the real part of the wave function (or some suitable combination) to preserve the sign (CONSTRAINED PATH APPROXIMATION)

46

An alternative: AFDMC

• The crucial advantage of AFDMC is that the scaling of the required computer resources is no longer exponential, but scales as A3 (the scaling required by the computation of the determinants in the antisymmetric wave functions) LARGER SYSTEMS ACCESSIBLE!

• Non trivial technical issues make the method still non optimal with respect to the standard approach for small systems.

• ACCURATE COMPUTATIONS FOR NUCLEAR/NEUTRON MATTER FEASIBLE!

47

An alternative: AFDMCThe crucial advantage of AFDMC is that the scaling of the required computer resources is no longer exponential, but goes as A3 (the scaling required by the computation of the determinants in the antisymmetric wave functions) LARGER SYSTEMS ACCESSIBLE!

Problems• The HS transformation can be used ONLY FOR THE

PROPAGATOR No possibility of using accurate wave functions that require an operatorial dependence! Constraints used to cope with the sign problem less accurate.

• Extra variables larger fluctuations and autocorrelations. • Some problems in treating nuclear spin-orbit. • Three-body forces (extremely important in nuclear physics) can be

reduced by a HS transformation only for pure neutron systems.

Progress!

Progress!

Progress!

48

Neutron stars

P. Haensel, A.Y. Potekhin, D.G. Yakovlev, Neutron Stars 1, Springer 2007

NS

The structure of a neutron star can be determined by solving a set of equations describing the equilibrium between the competing effects of the gravitational force (tending to make the star collapse) and the neutron-neutron (or more generally baryon-baryon) interaction that at high density provides mutual repulsion among the particles. (Tolman-Oppenheimer-Volkov equations).

neutron star

outer core:

(� 9 km)

n p e µ

outer crust:

(0.3÷ 0.5 km)

e Z

inner crust:

(1÷ 2 km)

e Z n

inner core:

(0÷ 3 km)R ⇠ 10 kmM ⇠ 1.4M�

electronsions

n,p,e,𝜇 and maybe hyperons…

Necessary ingredient for NS theory: energy and pressure vs. density for dense matter!

49

Hyperon puzzleThe appearance of hyperons (particles including a strange quark) has an immediate consequence on the equation of state: it makes it softer, i.e. the pressure coming from the baryon-baryon interaction is reduced. This is due to the larger mass and to the fact that nucleons transforming into hyperons become distinguishable in the Fermi sea

H. Ðapo, B.-J. Schaefer, and J. Wambach. Appearance of hyperons in neutron stars. Phys. Rev. C, 81(3): 035803 (2010)

based on NN (“soft” and “stiff”) EoS from M.Heiselberg, M.Hjort-Jensen, Phys. Rep. 328, 237 (2000)

Until 2010 observed masses of NS were

distributed around the Chandrashekar mass

MS=1.4 M⊙

Use of the equation of state of a p,n,e,𝜇 leads to a maximum mass > 2 M⊙

Soft EoS allowed: hyperons ok!

➧Softer EoS ➠ lower star massMany hyperon-nucleon

model interactions, giving differen EoS and different predictions.

50

Hyperon PuzzleRecently a few NS with a large mass were observed. The first (2010) was PSR~J1614-2230 pulsar with M=1.97(4)M⊙. (P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts and J. W. T. Hessels. A two-solar-mass neutron star measured using Shapiro delay measurements)

Are there no hyperons in a NS???

8 9 10 11 12 13 14 15 16R (km)

0

0.5

1

1.5

2

2.5

3

M (

MO• )

Causality: R>2.9 (G

M/c2 )

ρ central=2ρ 0

ρ central=3ρ 0

error associated with Esym

35.1

33.7

32

Esym

= 30.5 MeV (NN)

1.4 MO•

1.97(4) MO•

Before 2010: Maximum mass observed: 1.6M⊙ Maximum mass predicted without hyperons: 2.3⊙ (still ok in principle) Maximum mass predicted with hyperons: 1.4-1.6M⊙ (good!)

After 2010: Observed mass: 2.0M⊙ Maximum mass predicted without hyperons: 2.3M⊙ (good!) Maximum mass predicted with hyperons: 1.4-1.6M⊙ (very bad…)

S. Gandolfi, J. Carlson, and Sanjay Reddy Phys. Rev. A 83, 041601 (2011)

Key problem: understand the hyperon-nucleon interaction!

51

NS structure

Hyperon puzzle possibly solvable!Key to success: the possibility of performing accurate,

realistic calculations for large nuclear systems.

PRELIMINARY RESULT WITH THE NEW PARAMETRIZATIONResults might become compatible with the more recent astronomical observations: predicted maximum mass exceeds 2M⊙

M [M

sun]

R [km]

PNM RNM : VRN RNM : VRN+VRNN1

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

10 11 12 13 14 15 16

causality2.47 Msun

0.69 Msun

1.33 Msun

M [M

sun]

R [km]

PNM RNM : VRN RNM : VRN+VRNN1

RNM : VRN+VRNN2

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

10 11 12 13 14 15 16

causality2.47 Msun

0.69 Msun

1.33 Msun

D. Lonardoni, A. Lovato, S. Gandolfi, F. Pederiva

52