Hpernuclei and Hypermatter: Quantum

Monte Carlo StudiesFrancesco Pederiva

Physics Department - University of Trento INFN - TIFPA, Trento Institute for Fundamental Physics and Applications

Collaborators• Diego Lonardoni (FRIB/MSU-LANL) • Alessandro Lovato (INFN-TIFPA,Trento & ANL) • Stefano Gandolfi (LANL)

Open questions…The fine tuning of the hyperon-nucleon interaction is essential to understand the behavior of matter in extreme conditions (see Fiorella’s talk)

Our favorite 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

Fortin, M., Zdunik, J. L., Haensel, P., & Bejger, M. (2015).Astronomy & Astrophysics, 576, A68.

So many different models!

Fortin, M., Zdunik, J. L., Haensel, P., & Bejger, M. (2015).Astronomy & Astrophysics, 576, A68.

(Nuclear)

(Hyperons)

So many different models!

Fortin, M., Zdunik, J. L., Haensel, P., & Bejger, M. Astronomy & Astrophysics, 576, A68 (2015)

(Nuclear)

(Hyperons)

All masses compatible with the 2M⊙ constraint

So many different models!

Fortin, M., Zdunik, J. L., Haensel, P., & Bejger, M. (2015).Astronomy & Astrophysics, 576, A68.

(Nuclear)

(Hyperons)

Radii roughly divided in two groups.

So many different models!

Fortin, M., Zdunik, J. L., Haensel, P., & Bejger, M. (2015).Astronomy & Astrophysics, 576, A68.

(Nuclear)

CONSTRAINING EoS

CONSTRAINING Nuclear Interactions

? ~ 18 orders of magnitude in between…

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 realisations 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)

NB: The potentialis NOT an observable!Features like the hard core depend e.g. on the method used to reconstruct the kinetic energy.

Tensor

Central

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 the fewest possible parameters to be adjusted to reproduce light hypernuclei binding energies

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

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

Many-Body theory: projection Monte CarloWe compute ground state energies of nuclei by means of projection Monte Carlo methods. The ground state of a many-body system is computed by applying an “imaginary time propagator” to an arbitrary state that has to be non-orthogonal to the ground state (power method):

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.

15

Many-nucleon systems

Very accurate results have been obtained in the years for the ground state and some excitation properties of nuclei with A≤12 by the Argonne based group (GFMC calculations by Pieper, Wiringa, Carlson, Schiavilla…). These calculations include two- and three-nucleon interactions. -100

-90

-80

-70

-60

-50

-40

-30

-20

Ener

gy (M

eV)

AV18AV18+IL7 Expt.

0+

4He0+2+

6He 1+3+2+1+

6Li3/2−1/2−7/2−5/2−5/2−7/2−

7Li

0+2+

8He 2+2+

2+1+

0+

3+1+

4+

8Li

1+

0+2+

4+2+1+3+4+

0+

8Be

3/2−1/2−5/2−

9Li

3/2−1/2+5/2−1/2−5/2+3/2+

7/2−3/2−

7/2−5/2+7/2+

9Be

1+

0+2+2+0+3,2+

10Be 3+1+

2+

4+

1+

3+2+

3+

10B

3+

1+

2+

4+

1+

3+2+

0+

2+0+

12C

Argonne v18with Illinois-7

GFMC Calculations10 January 2014

• IL7: 4 parameters fit to 23 states• 600 keV rms error, 51 states• ~60 isobaric analogs also computed

Courtesy of R. Wiringa, ANL

PROBLEMfor realistic many-nucleon Hamiltonians, propagators must be evaluated on wave functions that have a number of components exponentially growing with A (spin/isospin singlet/triplet state for each pair of nucleons)

Auxiliary Field Diffusion Monte Carlo (AFDMC)

The computational cost of GFMC can be reduced 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

17

The 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!

18

Auxiliary Field Diffusion Monte Carlo (AFDMC)

Input from experiment

B⇤ = Bhyp �Bnuc

3.5 4.0 4.5 5.0 5.5 6.0WD [ 10 -2 MeV ]0.5

1.01.5

2.02.5

CP [ MeV ]

3.0

4.5

6.0

7.5

B R

[ MeV

]

WD

[10

-2 M

eV]

CP [MeV]

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.5 1.0 1.5 2.0 2.5

Assumption: use of a simplified NN interaction cancel in the difference and therefore the estimate of BΛ is accurate

(verified!)

Only two parameters are relevant (one of them is essentially ineffective)

H = T +AV 40 + UIXcH = T +AV 40 + UIXc + V⇤N + V⇤NN

Hypernuclei

Diego Lonardoni, Alessandro Lovato, FP, Stefano Gandolfi

B Λ [M

eV]

A-2/3

s

emulsion

(K-,π-)

(π+,K+)

(e,e’K+)

ΛN

ΛN + ΛNN (I)

ΛN + ΛNN (II)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0.0 0.1 0.2 0.3 0.4 0.5

Hypernuclei

Diego Lonardoni, F.P. S. Gandolfi, Phys. Rev. C 89, 014314 (2014) F.P., F. Catalano, Diego Lonardoni, A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

Hypernuclei

Diego Lonardoni, F.P. S. Gandolfi, Phys. Rev. C 89, 014314 (2014) F.P., F. Catalano, Diego Lonardoni, A. Lovato, S. Gandolfi, arXiv:1506.04042 (2015)

Hypernuclei

Hypernuclei

Hypernuclei

Hypernuclei

Can we really constrain 𝛬NN interaction from hyper nuclear data?

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

np

nn

𝝠 𝝠

We are doing 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

Can we really constrain the interaction from hyper nuclear data?

Francesco Catalano, Diego Lonardoni, FP

small asymmetry (40Ca)

larger asymmetry (48Ca)

B Λ [M

eV]

A-2/3

208

89

40

28

1612

9

76

54

3

s

experiments

CT = 1.0

CT = 1.5

CT = 0.0

CT = -1.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0 0.1 0.2 0.3 0.4 0.5

New experiments needed

Strangeness exchange reaction

Associated production reaction

(⇡+,K+)reaction

K+⇡+

⇤

u u

uudd

d

ds

s

n⇢⇢

n n

⇢⇢

n n

(K�,⇡�)reaction

K� ⇡�

⇤u u

u u

dd d

ds

sn

⇢

n8><

>:(e, e0K+)reaction

K+

⇤

e e0

p

u

u

u

ud d

ss�⇤ AZ(e, e�K+)A� [Z � 1]

AZ(K�,��)A�Z

AZ(�+,K+)A�Z

Proposal presented and approved at JLAB: “A study of the 𝜦-N interaction through the high precision spectroscopy of 𝜦-hypernuclei with electron beam” (spokepersons: S. Nakamura, F. Garibaldi, P.E.C. Markowitz, J. Reinhold, L. Tang, G.M. Urciuoli)

Including mainly measurements of 48𝜦K and 40𝜦K, but hopefully also light hypernuclei and hyper-Pb (EoS…)

𝛬-neutron matter 33Results: hyper-neutron matter

n

n

n

nn

n

⇤

n

n

nnn

n

n

n

n

⇤

n

n

⇤

n n

p

p

p

p

hyper-nuclear matter

n

n

n

nn

n

⇤

n

n

nnn

n

n

n

n

⇤

n

n

⇤

n n

nn

nn

hyper-neutron matter

x⇤ ⌘ x⇤(⇢b)equilibrium condition µ⇤(⇢b, x⇤) = µn(⇢b, x⇤)

chemical potentials:

PNM hyperon fraction

EHNM ⌘ EHNM(⇢b, x⇤)energy per

particle

energy density EHNM ⌘ EHNM(⇢b, x⇤)

Neutron star structureM

[M0]

lc [fm-3]

PNM

RN

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.2 0.4 0.6 0.8 1.0

PSR J1614-2230

PSR J0348+0432

M [M

0]

lc [fm-3]

PNM

RN

RN + RNN (I)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.2 0.4 0.6 0.8 1.0

PSR J1614-2230

PSR J0348+0432

M [M

0]

lc [fm-3]

PNM

RN

RN + RNN (I)

RN + RNN (II)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

0.0 0.2 0.4 0.6 0.8 1.0

PSR J1614-2230

PSR J0348+0432Within this model the repulsion needed to correctly describe hypernuclear binding energy is so strong that hyperons wouldnot be present in 2M⊙ stars!

40

2.45(1)M�

0.66(2)M�

Results: hyper-neutron matter

M [M

0]

R [km]

PNM

RN

RN + RNN (I)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

10 11 12 13 14 15 16

PSR J1614-2230

PSR J0348+0432

1.36(5)M�

D. L., A. Lovato, S. Gandolfi, F. Pederiva, arXiv:1407.4448 [nucl-th], submitted to Phys. Rev. Lett.Diego Lonardoni, A. Lovato, S. Gandolfi, FP, arXiv:1407.4448 [nucl-th], submitted to Phys. Rev. Lett.

Neutron star structure

YN interaction only (from 𝛬N scattering data)

YN + YNN interaction (old VMC based parametrization)

YN +YNN interaction (more accurate description of hypernuclear BE)

Conclusions• Our philosophy in attacking the problem of the hyperon-nucleon

interaction: we do not want to add more information than the one that 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.

• AFDMC calculations are evolving. Better accuracy, better performance. This reflects on the work on hypernuclei Accessible systems: definitely A=90. For heavier systems one can possibly use alternative approaches.

• At this point there is real need of accurate experiments on hypernuclei in order to be able to gain more insight on NS interior at densities > 2𝜌0.

33