Strangeness in Baryonic Matter - T39 Physics Group

Hypernuclei

ECT* Workshop Strangeness in the Universe 21 October 2013

Strangeness in Baryonic Matterand

New Constraints from Neutron Stars

Trento and Technische Universität München

Strangeness in Low-Energy QCD: Chiral SU(3) Effective Field Theory

Hyperon-Nucleon Interaction from Chiral SU(3) Dynamics

Two-Solar-Mass Neutron Stars and their implications

Wolfram Weise

Conventional versus exotic degrees of freedom

ECT*

Role of strangeness (hyperons and related)

Author's personal copy

Available online at www.sciencedirect.com

Nuclear Physics A 915 (2013) 24–58www.elsevier.com/locate/nuclphysa

Hyperon–nucleon interaction at next-to-leading orderin chiral effective field theory

J. Haidenbauer a, S. Petschauer b, N. Kaiser b, U.-G. Meißner c,a,∗,A. Nogga a, W. Weise b,d

a Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics,Forschungszentrum Jülich, D-52425 Jülich, Germany

b Physik Department, Technische Universität München, D-85747 Garching, Germanyc Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn,

D-53115 Bonn, Germanyd ECT*, Villa Tambosi, I-38123 Villazzano (Trento), Italy

Received 22 April 2013; received in revised form 24 May 2013; accepted 11 June 2013

Available online 21 June 2013

Abstract

Results for the ΛN and ΣN interactions obtained at next-to-leading order in chiral effective field the-ory are reported. At the order considered there are contributions from one- and two-pseudoscalar-mesonexchange diagrams and from four-baryon contact terms without and with two derivatives. SU(3) flavorsymmetry is imposed for constructing the hyperon–nucleon interaction, however, the explicit SU(3) sym-metry breaking by the physical masses of the pseudoscalar mesons (π , K , η) and of the involved baryonsis taken into account. An excellent description of the hyperon–nucleon system can be achieved at next-to-leading order. It is on the same level of quality as the one obtained by the most advanced phenomenologicalhyperon–nucleon interaction models. 2013 Elsevier B.V. All rights reserved.

Keywords: Hyperon–nucleon interaction; Effective field theory

* Corresponding author at: Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universität Bonn, D-53115 Bonn, Germany.

E-mail address: [email protected] (U.-G. Meißner).

0375-9474/$ – see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysa.2013.06.008


Available online at www.sciencedirect.com

Nuclear Physics A 915 (2013) 24–58www.elsevier.com/locate/nuclphysa

Hyperon–nucleon interaction at next-to-leading orderin chiral effective field theory

J. Haidenbauer a, S. Petschauer b, N. Kaiser b, U.-G. Meißner c,a,∗,A. Nogga a, W. Weise b,d

a Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics,Forschungszentrum Jülich, D-52425 Jülich, Germany

b Physik Department, Technische Universität München, D-85747 Garching, Germanyc Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn,

D-53115 Bonn, Germanyd ECT*, Villa Tambosi, I-38123 Villazzano (Trento), Italy

Received 22 April 2013; received in revised form 24 May 2013; accepted 11 June 2013

Available online 21 June 2013

Abstract

Results for the ΛN and ΣN interactions obtained at next-to-leading order in chiral effective field the-ory are reported. At the order considered there are contributions from one- and two-pseudoscalar-mesonexchange diagrams and from four-baryon contact terms without and with two derivatives. SU(3) flavorsymmetry is imposed for constructing the hyperon–nucleon interaction, however, the explicit SU(3) sym-metry breaking by the physical masses of the pseudoscalar mesons (π , K , η) and of the involved baryonsis taken into account. An excellent description of the hyperon–nucleon system can be achieved at next-to-leading order. It is on the same level of quality as the one obtained by the most advanced phenomenologicalhyperon–nucleon interaction models. 2013 Elsevier B.V. All rights reserved.

Keywords: Hyperon–nucleon interaction; Effective field theory

* Corresponding author at: Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universität Bonn, D-53115 Bonn, Germany.

E-mail address: [email protected] (U.-G. Meißner).

0375-9474/$ – see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nuclphysa.2013.06.008

Chiral SU(3) Effective Field Theory


J. Haidenbauer et al. / Nuclear Physics A 915 (2013) 24–58 27

Fig. 1. Relevant Feynman diagrams up-to-and-including next-to-leading order. Solid and dashed lines denote octetbaryons and pseudoscalar mesons, respectively. The square symbolizes a contact vertex with two derivatives. From leftto right: LO contact term, one-meson exchange, NLO contact term, planar box, crossed box, left triangle, right triangle,football diagram.

2. Chiral potential at next-to-leading order

The derivation of chiral baryon–baryon potentials for the strangeness sector at LO using theWeinberg power counting has been outlined in Refs. [21,44–46]. The NLO contributions for theNN case are described in detail in Ref. [33], while the extension to baryon–baryon systems withany combination of octet baryons has been worked out in Ref. [47]. The LO potential consistsof four-baryon contact terms without derivatives and of one-pseudoscalar-meson exchanges. AtNLO contact terms with two derivatives arise, together with loop contributions from (irreducible)two-pseudoscalar-meson exchanges. The corresponding Feynman diagrams are shown in Fig. 1.

2.1. Contact terms

The spin dependence of the potentials due to leading order contact terms is given by [33]

V(0)BB→BB = CS + CT σ 1 · σ 2, (1)

where the parameters CS and CT are low-energy constants (LECs), depending on the consideredbaryon–baryon channel, which need to be determined in a fit to data. At next-to-leading orderthe spin and momentum dependence of the contact terms reads

V(2)BB→BB = C1q2 + C2k2 +

(C3q2 + C4k2)σ 1 · σ 2 + i

2C5(σ 1 + σ 2) · (q × k)

+ C6(q · σ 1)(q · σ 2) + C7(k · σ 1)(k · σ 2) + i2C8(σ 1 − σ 2) · (q × k), (2)

where Ci (i = 1, . . . ,8) are additional LECs. The transferred and average momenta, q and k,are defined in terms of the final and initial center-of-mass momenta of the baryons, p′ and p, asq = p′−p and k = (p′+p)/2. When performing a partial-wave projection, these terms contributeto the two S-wave (1S0, 3S1) potentials, the four P -wave (1P1, 3P0, 3P1, 3P2) potentials, andthe 3S1–3D1 and 1P1–3P1 transition potentials in the following way [29]:

V(1S0

)= 4π(CS − 3CT ) + π(4C1 + C2 − 12C3 − 3C4 − 4C6 − C7)

(p2 + p′ 2)

= C1S0+ C1S0

(p2 + p′ 2), (3)

V(3S1

)= 4π(CS + CT ) + π

3(12C1 + 3C2 + 12C3 + 3C4 + 4C6 + C7)

(p2 + p′ 2)

= C3S1+ C3S1

(p2 + p′ 2), (4)

V(1P1

)= 2π

3(−4C1 + C2 + 12C3 − 3C4 + 4C6 − C7)pp′ = C1P1

pp′, (5)

V(3P1

)= 2π

3(−4C1 + C2 − 4C3 + C4 + 2C5 − 8C6 + 2C7)pp′ = C3P1

pp′, (6)



Table 1SU(3) relations for the various contact potentials in the isospin basis. C27

ξ etc. refers to the corresponding irreducibleSU(3) representation for a particular partial wave ξ . The actual potential still needs to be multiplied by pertinent powersof the momenta p and p′.

Channel I V (ξ)

ξ = 1S0, 3P0, 3P1, 3P2 ξ = 3S1, 3S1–3D1, 1P1 ξ = 1P1–3P1

S = 0 NN → NN 0 – C10∗ξ –

NN → NN 1 C27ξ – –

S = −1 ΛN → ΛN 12

110 (9C27

ξ + C8sξ ) 1

2 (C8aξ + C10∗

ξ ) −1√20

C8s8aξ

ΛN → ΣN 12

310 (−C27

ξ + C8sξ ) 1

2 (−C8aξ + C10∗

ξ ) −3√20

C8s8aξ

ΣN → ΛN 1√20

C8s8aξ

ΣN → ΣN 12

110 (C27

ξ + 9C8sξ ) 1

2 (C8aξ + C10∗

ξ ) 3√20

C8s8aξ

ΣN → ΣN 32 C27

ξ C10ξ –

singlet representation (C1) is present in the strangeness S = −2 channels with isospin I = 0 [45]and there are four more LECs that contribute to the S = −2 sector at NLO [50].

2.2. Goldstone boson exchange

The one- and two-pseudoscalar-meson exchange potentials follow from the SU(3)-invariantmeson–baryon interaction Lagrangian

LMB = tr(B

(iγ µDµ − M0

)B

)− D

2tr(Bγ µγ5{uµ,B}

)− F

2tr(Bγ µγ5[uµ,B]

), (14)

with DµB = ∂µB + [Γµ,B], Γµ = 12 (u†∂µu + u∂µu†) and uµ = i(u†∂µu − u∂µu†), and where

the trace is taken in flavor space. The constant M0 denotes the baryon mass in the three-flavorchiral limit. The coupling constants F and D satisfy the relation F + D = gA & 1.26, where gA

is the axial-vector strength measured in neutron β-decay. The pseudoscalar mesons and octetbaryons are collected in traceless 3 × 3 matrices [51]

P =

π0√2

+ η√6

π+ K+

π− − π0√2

+ η√6

K0

K− K0 − 2η√6

, B =

Σ0√2

+ Λ√6

Σ+ p

Σ− −Σ0√2

+ Λ√6

n

−Ξ− Ξ0 − 2Λ√6

.

(15)

For the pseudoscalar mesons we use the usual non-linear realization of chiral symmetry withU(x) = u2(x) = exp(i

√2P(x)/f0), where f0 is the Goldstone boson decay constant in the chiral

limit. These fields transform under the chiral group SU(3)L × SU(3)R as U → RUL† and B →KBK† with L ∈ SU(3)L, R ∈ SU(3)R and the SU(3) valued compensator field K = K(L,R,U),cf. Ref. [52]. After an expansion of the interaction Lagrangian in powers of P one obtains fromthe terms proportional to D and F the pseudovector coupling term

L1 = −√

22f0

tr(DBγ µγ5{∂µP,B} + FBγ µγ5[∂µP,B]

), (16)





Channel I V (ξ)

ξ = 1S0, 3P0, 3P1, 3P2 ξ = 3S1, 3S1–3D1, 1P1 ξ = 1P1–3P1

S = 0 NN → NN 0 – C10∗ξ –

NN → NN 1 C27ξ – –

S = −1 ΛN → ΛN 12

110 (9C27

ξ + C8sξ ) 1

2 (C8aξ + C10∗

ξ ) −1√20

C8s8aξ

ΛN → ΣN 12

310 (−C27

ξ + C8sξ ) 1

2 (−C8aξ + C10∗

ξ ) −3√20

C8s8aξ

ΣN → ΛN 1√20

C8s8aξ

ΣN → ΣN 12

110 (C27

ξ + 9C8sξ ) 1

2 (C8aξ + C10∗

ξ ) 3√20

C8s8aξ

ΣN → ΣN 32 C27

ξ C10ξ –




LMB = tr(B

(iγ µDµ − M0

)B

)− D


)− F


), (14)




P =

π0√2

+ η√6

π+ K+

π− − π0√2

+ η√6

K0

K− K0 − 2η√6

, B =

Σ0√2

+ Λ√6

Σ+ p

Σ− −Σ0√2

+ Λ√6

n

−Ξ− Ξ0 − 2Λ√6

.

(15)




L1 = −√

22f0


), (16)





Channel I V (ξ)

ξ = 1S0, 3P0, 3P1, 3P2 ξ = 3S1, 3S1–3D1, 1P1 ξ = 1P1–3P1

S = 0 NN → NN 0 – C10∗ξ –

NN → NN 1 C27ξ – –

S = −1 ΛN → ΛN 12

110 (9C27

ξ + C8sξ ) 1

2 (C8aξ + C10∗

ξ ) −1√20

C8s8aξ

ΛN → ΣN 12

310 (−C27

ξ + C8sξ ) 1

2 (−C8aξ + C10∗

ξ ) −3√20

C8s8aξ

ΣN → ΛN 1√20

C8s8aξ

ΣN → ΣN 12

110 (C27

ξ + 9C8sξ ) 1

2 (C8aξ + C10∗

ξ ) 3√20

C8s8aξ

ΣN → ΣN 32 C27

ξ C10ξ –




LMB = tr(B

(iγ µDµ − M0

)B

)− D


)− F


), (14)




P =

π0√2

+ η√6

π+ K+

π− − π0√2

+ η√6

K0

K− K0 − 2η√6

, B =

Σ0√2

+ Λ√6

Σ+ p

Σ− −Σ0√2

+ Λ√6

n

−Ξ− Ξ0 − 2Λ√6

.

(15)




L1 = −√

22f0


), (16)


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

which leads to a vertex between two baryons and one meson. In the same way, the term involvingthe chiral connection Γµ gives

L2 = 1

4f 20

tr(iBγ µ

[[P, ∂µP ],B

]), (17)

which describes a (Weinberg–Tomozawa) vertex between two baryons and two mesons.Writing the pseudovector interaction Lagrangian L1 explicitly in the isospin basis, one gets

L1 = −fNNπ Nγ µγ5τN · ∂µπ + ifΣΣπ Σγ µγ5 × Σ · ∂µπ

− fΛΣπ

[Λγ µγ5Σ + Σγ µγ5Λ

]· ∂µπ − fΞΞπΞγ µγ5τΞ · ∂µπ

− fΛNK

[Nγ µγ5Λ∂µK + h.c.

]− fΞΛK

[Ξγ µγ5Λ∂µK + h.c.

]

− fΣNK

[Nγ µγ5τ∂µK · Σ + h.c.

]− fΣΞK

[Ξγ µγ5τ∂µK · Σ + h.c.

]

− fNNη8Nγ µγ5N∂µη − fΛΛη8Λγ µγ5Λ∂µη

− fΣΣη8Σ · γ µγ5Σ∂µη − fΞΞη8Ξγ µγ5Ξ∂µη. (18)

Here, we have introduced the isospin doublets

N =(

p

n

), Ξ =

(Ξ0

Ξ−

), K =

(K+

K0

), K =

(K0

−K−

). (19)

The signs have been chosen according to the conventions of Ref. [48], such that the inner productof the isovector Σ (or π ) defined in spherical components reads

Σ · Σ =∑

m

(−1)mΣmΣ−m = Σ+Σ− + Σ0Σ0 + Σ−Σ+. (20)

Since the original interaction Lagrangian in Eq. (16) is SU(3)-invariant, the various couplingconstants are related to each other by [48]

fNNπ = f, fNNη8 = 1√3(4α − 1)f, fΛNK = − 1√

3(1 + 2α)f,

fΞΞπ = −(1 − 2α)f, fΞΞη8 = − 1√3(1 + 2α)f, fΞΛK = 1√

3(4α − 1)f,

fΛΣπ = 2√3(1 − α)f, fΣΣη8 = 2√

3(1 − α)f, fΣNK = (1 − 2α)f,

fΣΣπ = 2αf, fΛΛη8 = − 2√3(1 − α)f, fΞΣK = −f. (21)

Evidently, all coupling constants are given in terms of f ≡ gA/2f0 and the ratio α = F/(F +D).The expression for the one-pseudoscalar-meson exchange potential is similar to the standard

one-pion-exchange potential [33]

V OBEB1B2→B3B4

= −fB1B3P fB2B4P(σ 1 · q)(σ 2 · q)

q2 + m2P

IB1B2→B3B4 . (22)

Here, mP is the mass of the exchanged pseudoscalar meson. In the present calculation we usethe physical masses mπ , mK , mη in Eq. (22). Thus, the explicit SU(3) breaking reflected in themass splitting between the pseudoscalar mesons is taken into account. The η meson is identifiedwith the octet-state η8. The isospin factors IB1B2→B3B4 are given in Table 2.

. . . generate Nambu-Goldstone boson exchange processes

interaction terms involving baryon and pseudoscalar meson octets . . .

LO NLO





Channel I V (ξ)

ξ = 1S0, 3P0, 3P1, 3P2 ξ = 3S1, 3S1–3D1, 1P1 ξ = 1P1–3P1

S = 0 NN → NN 0 – C10∗ξ –

NN → NN 1 C27ξ – –

S = −1 ΛN → ΛN 12

110 (9C27

ξ + C8sξ ) 1

2 (C8aξ + C10∗

ξ ) −1√20

C8s8aξ

ΛN → ΣN 12

310 (−C27

ξ + C8sξ ) 1

2 (−C8aξ + C10∗

ξ ) −3√20

C8s8aξ

ΣN → ΛN 1√20

C8s8aξ

ΣN → ΣN 12

110 (C27

ξ + 9C8sξ ) 1

2 (C8aξ + C10∗

ξ ) 3√20

C8s8aξ

ΣN → ΣN 32 C27

ξ C10ξ –




LMB = tr(B

(iγ µDµ − M0

)B

)− D


)− F


), (14)




P =

π0√2

+ η√6

π+ K+

π− − π0√2

+ η√6

K0

K− K0 − 2η√6

, B =

Σ0√2

+ Λ√6

Σ+ p

Σ− −Σ0√2

+ Λ√6

n

−Ξ− Ξ0 − 2Λ√6

.

(15)




L1 = −√

22f0


), (16)






2.1. Contact terms


V(0)BB→BB = CS + CT σ 1 · σ 2, (1)


V(2)BB→BB = C1q2 + C2k2 +

(C3q2 + C4k2)σ 1 · σ 2 + i

2C5(σ 1 + σ 2) · (q × k)



V(1S0

)= 4π(CS − 3CT ) + π(4C1 + C2 − 12C3 − 3C4 − 4C6 − C7)

(p2 + p′ 2)

= C1S0+ C1S0

(p2 + p′ 2), (3)

V(3S1

)= 4π(CS + CT ) + π

3(12C1 + 3C2 + 12C3 + 3C4 + 4C6 + C7)

(p2 + p′ 2)

= C3S1+ C3S1

(p2 + p′ 2), (4)

V(1P1

)= 2π

3(−4C1 + C2 + 12C3 − 3C4 + 4C6 − C7)pp′ = C1P1

pp′, (5)

V(3P1

)= 2π

3(−4C1 + C2 − 4C3 + C4 + 2C5 − 8C6 + 2C7)pp′ = C3P1

pp′, (6)






2.1. Contact terms


V(0)BB→BB = CS + CT σ 1 · σ 2, (1)


V(2)BB→BB = C1q2 + C2k2 +

(C3q2 + C4k2)σ 1 · σ 2 + i

2C5(σ 1 + σ 2) · (q × k)



V(1S0

)= 4π(CS − 3CT ) + π(4C1 + C2 − 12C3 − 3C4 − 4C6 − C7)

(p2 + p′ 2)

= C1S0+ C1S0

(p2 + p′ 2), (3)

V(3S1

)= 4π(CS + CT ) + π

3(12C1 + 3C2 + 12C3 + 3C4 + 4C6 + C7)

(p2 + p′ 2)

= C3S1+ C3S1

(p2 + p′ 2), (4)

V(1P1

)= 2π

3(−4C1 + C2 + 12C3 − 3C4 + 4C6 − C7)pp′ = C1P1

pp′, (5)

V(3P1

)= 2π

3(−4C1 + C2 − 4C3 + C4 + 2C5 − 8C6 + 2C7)pp′ = C3P1

pp′, (6)

Hyperon - Nucleon Interaction(contact terms)






2.1. Contact terms


V(0)BB→BB = CS + CT σ 1 · σ 2, (1)


V(2)BB→BB = C1q2 + C2k2 +

(C3q2 + C4k2)σ 1 · σ 2 + i

2C5(σ 1 + σ 2) · (q × k)



V(1S0

)= 4π(CS − 3CT ) + π(4C1 + C2 − 12C3 − 3C4 − 4C6 − C7)

(p2 + p′ 2)

= C1S0+ C1S0

(p2 + p′ 2), (3)

V(3S1

)= 4π(CS + CT ) + π

3(12C1 + 3C2 + 12C3 + 3C4 + 4C6 + C7)

(p2 + p′ 2)

= C3S1+ C3S1

(p2 + p′ 2), (4)

V(1P1

)= 2π

3(−4C1 + C2 + 12C3 − 3C4 + 4C6 − C7)pp′ = C1P1

pp′, (5)

V(3P1

)= 2π

3(−4C1 + C2 − 4C3 + C4 + 2C5 − 8C6 + 2C7)pp′ = C3P1

pp′, (6)






2.1. Contact terms


V(0)BB→BB = CS + CT σ 1 · σ 2, (1)


V(2)BB→BB = C1q2 + C2k2 +

(C3q2 + C4k2)σ 1 · σ 2 + i

2C5(σ 1 + σ 2) · (q × k)



V(1S0

)= 4π(CS − 3CT ) + π(4C1 + C2 − 12C3 − 3C4 − 4C6 − C7)

(p2 + p′ 2)

= C1S0+ C1S0

(p2 + p′ 2), (3)

V(3S1

)= 4π(CS + CT ) + π

3(12C1 + 3C2 + 12C3 + 3C4 + 4C6 + C7)

(p2 + p′ 2)

= C3S1+ C3S1

(p2 + p′ 2), (4)

V(1P1

)= 2π

3(−4C1 + C2 + 12C3 − 3C4 + 4C6 − C7)pp′ = C1P1

pp′, (5)

V(3P1

)= 2π

3(−4C1 + C2 − 4C3 + C4 + 2C5 − 8C6 + 2C7)pp′ = C3P1

pp′, (6)

see also:S. Petschauer, N. Kaiser

Nucl. Phys. A 916 (2013) 1-29

note: moderate attraction at low momentastrong repulsion at higher momenta


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



Fig. 6. The Λp 1S0 and 1P1 phase shifts δ as a function of plab. The red/dark band shows the chiral EFT results toNLO for variations of the cutoff in the range Λ = 500, . . . ,650 MeV, while the green/light band shows results to LO forΛ = 550, . . . ,700 MeV. The dashed curve is the result of the Jülich ’04 meson-exchange potential [37].

Fig. 7. The Λp phase shifts for the coupled 3S1–3D1 partial wave as a function of plab. Same description of curves asin Fig. 6.

state in the ΣN system. It should be said, however, that the majority of the meson-exchangepotentials [36,38,39] produce an unstable bound state, similar to our NLO interaction. The onlycharacteristic difference of the chiral EFT interactions to the meson-exchange potentials mightbe the mixing parameter ε1 which is fairly large in the former case and close to 45◦ at the ΣN

threshold, see Fig. 7. It is a manifestation of the fact that the pertinent Λp T -matrices (for the3S1 → 3S1, 3D1 → 3D1, and 3S1 ↔ 3D1 transitions) are all of the same magnitude.

The strong variation of the 3S1–3D1 amplitudes around the ΣN threshold is reflected inan impressive increase in the Λp cross section at the corresponding energy, as seen in Fig. 2.

LO

NLO

LO

NLO

repulsion

phase shift

Hyperon - Nucleon Interaction(contd.)

J. Haidenbauer, S. Petschauer, N. Kaiser,U.-G. Meißner, A. Nogga, W. W.

Nucl. Phys. A 915 (2013) 24

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VC

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#s"

0

1000

2000

3000

4000

5000

6000

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VC

-2500

-2000

-1500

-1000

-500

0

500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V(r)

[MeV

]

r [fm]

V!"# VC

-150

-100

-50

0

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$*"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#a"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

L ! 4 fm, mπ ! 470 MeV

same as NN 8s: strong repulsive core. repulsion only. 1: attractive instead of repulsive core ! attraction only .

same as NN 10: strong repulsive core. weak attraction. 8a: weak repulsive core. strong attraction.

Flavor dependences of BB interactions become manifest in SU(3) limit !

12年4月18日水曜日

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VC

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#s"

0

1000

2000

3000

4000

5000

6000

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VC

-2500

-2000

-1500

-1000

-500

0

500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V(r)

[MeV

]

r [fm]

V!"# VC

-150

-100

-50

0

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$*"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#a"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

L ! 4 fm, mπ ! 470 MeV





0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VC

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#s"

0

1000

2000

3000

4000

5000

6000

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VC

-2500

-2000

-1500

-1000

-500

0

500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V(r)

[MeV

]

r [fm]

V!"# VC

-150

-100

-50

0

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$*"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#$"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

0

500

1000

1500

2000

2500

3000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

V!r"

[MeV

]

r [fm]

V!#a"

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

u,d,s=0.13840

VCVT

L ! 4 fm, mπ ! 470 MeV





ΛN(1S0) =9

10[27] +

1

10[8s]

ΛN(3S1) =1

2[10

∗] +1

2[8a]

Hyperon - Nucleon Interactions from Lattice QCD

T. Inoue et al.(HAL QCD)

PTP 124 (2010) 591Nucl. Phys.

A881 (2012) 28

mps = 0.47GeV

π

π

Λ(!p − !q/2)

Λ(!p + !q/2)

Σ

Λ

Λ

Λ

Λ

Σ

π

π

medium

insertion

K

medium

insertions

Λ

Σ

Λ

π

π

Λ(!p − !q/2)

Λ(!p + !q/2)

Σ

Λ

Λ

Λ

Λ

Σ

π

π

medium

insertion

K

medium

insertions

Λ

Σ

Λ

Hypernuclei and Chiral SU(3) Effective Field Theory

P. Finelli, N. Kaiser, D. Vretenar, W.W. Nucl. Phys. A831 (2009) 163

Input:

Λ

Λ

short-distance (contact) termsΛ

Λ

Λ

Λ

Σ

π

π

medium

insertion

K

small:

K exchange Fock term

large:two-pion exchange mechanisms

N. Kaiser, W. W. Phys. Rev.

C71 (2005) 015203

= ζ

(

ζ !

1

2

)

3/40

Mass Radius Relationship

Lattimer & Prakash , Science 304, 536 (2004).39/40

NEUTRON STARS and the EQUATION OF STATE of

DENSE BARYONIC MATTER

Mass-Radius Relation

J. Lattimer, M. Prakash: Astrophys. J. 550 (2001) 426

Neutron Star Scenarios

STRANGEQUARK MATTER

NUCLEARMATTER

Tolman-Oppenheimer-Volkovequations

Phys. Reports 442 (2007) 109

dM

dr= 4πr

2E

c2

dP

dr= −

G

c2

(M + 4πPr3)(E + P)

r(r − GM/c2)

direct measurement of

neutron star mass from

increase in travel time

near companion

J1614-2230

most edge-on binary

pulsar known (89.17°)

+ massive white dwarf

companion (0.5 Msun)

heaviest neutron star

with 1.97±0.04 Msun

Nature, Oct. 28, 2010

New constraints from NEUTRON STARS

Many physically motivated extensions to general relativity (GR) predict sig-nificant deviations in the properties of spacetime surrounding massive neu-tron stars. We report the measurement of a 2.01±0.04 solar mass (M⊙) pul-sar in a 2.46-hr orbit with a 0.172±0.003 M⊙ white dwarf. The high pulsarmass and the compact orbit make this system a sensitive laboratory of a pre-viously untested strong-field gravity regime. Thus far, the observed orbitaldecay agrees with GR, supporting its validity even for the extreme conditionspresent in the system. The resulting constraints on deviations support the useof GR-based templates for ground-based gravitational wave detectors. Addi-tionally, the system strengthens recent constraints on the properties of densematter and provides insight to binary stellar astrophysics and pulsar recycling.

Neutron stars (NSs) with masses above 1.8 M⊙ manifested as radio pulsars are valuableprobes of fundamental physics in extreme conditions unique in the observable Universe andinaccessible to terrestrial experiments. Their high masses are directly linked to the equation-of-state (EOS) of matter at supra-nuclear densities (1, 2) and constrain the lower mass limitfor production of astrophysical black holes (BHs). Furthermore, they possess extreme internalgravitational fields which result in gravitational binding energies substantially higher than thosefound in more common, 1.4 M⊙ NSs. Modifications to GR, often motivated by the desire fora unified model of the four fundamental forces, can generally imprint measurable signatures ingravitational waves (GWs) radiated by systems containing such objects, even if deviations fromGR vanish in the Solar System and in less massive NSs (3–5).

However, the most massive NSs known today reside in long-period binaries or other systemsunsuitable for GW radiation tests. Identifying a massive NS in a compact, relativistic binaryis thus of key importance for understanding gravity-matter coupling under extreme conditions.Furthermore, the existence of a massive NS in a relativistic orbit can also be used to test currentknowledge of close binary evolution.

ResultsPSR J0348+0432 & optical observations of its companion PSR J0348+0432, a pulsar spin-ning at 39 ms in a 2.46-hr orbit with a low-mass companion, was detected by a recent sur-vey (6, 7) conducted with the Robert C. Byrd Green Bank Telescope (GBT). Initial timing ob-servations of the binary yielded an accurate astrometric position, which allowed us to identifyits optical counterpart in the Sloan Digital Sky Survey (SDSS) archive (8). The colors and fluxof the counterpart are consistent with a low-mass white dwarf (WD) with a helium core at a dis-tance of d ∼ 2.1 kpc. Its relatively high apparent brightness (g� = 20.71 ± 0.03 mag) allowed usto resolve its spectrum using the Apache Point Optical Telescope. These observations revealeddeep Hydrogen lines, typical of low-mass WDs, confirming our preliminary identification. Theradial velocities of the WD mirrored that of PSR J0348+0432, also verifying that the two starsare gravitationally bound.

2

Science 340 (2013) 6131

A Massive Pulsar in a Compact Relativistic Binary∗

John Antoniadis,1 †Paulo C. C. Freire,1 Norbert Wex,1 Thomas M. Tauris,2,1

Ryan S. Lynch,3 Marten H. van Kerkwijk,4 Michael Kramer,1,5 Cees Bassa,5

Vik S. Dhillon,6 Thomas Driebe,7 Jason W. T. Hessels,8,9 Victoria M. Kaspi,3

Vladislav I. Kondratiev,8,10 Norbert Langer,2 Thomas R. Marsh,11

Maura A. McLaughlin,12 Timothy T. Pennucci,13 Scott M. Ransom,14

Ingrid H. Stairs,15 Joeri van Leeuwen,8,9

Joris P. W. Verbiest,1 David G. Whelan,13

1Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany2Argelander Institut fur Astronomie, Auf dem Hugel 71, 53121 Bonn, Germany

3Department of Physics, McGill University, 3600 University Street, Montreal, QC H3A 2T8, Canada4Department of Astronomy and Astrophysics, University of Toronto,

50 St. George Street, Toronto, ON M5S 3H4, Canada5Jodrell Bank Centre for Astrophysics, The University of Manchester,

Alan Turing Building, Manchester, M13 9PL, UK6Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, UK

7Deutsches Zentrum fur Luft- und Raumfahrt e.V. (DLR),Raumfahrtmanagement, Konigswinterer Str. 522–524, 53227 Bonn, Germany

8ASTRON, the Netherlands Institute for Radio Astronomy,Postbus 2, 7990 AA Dwingeloo, The Netherlands

9Astronomical Institute “Anton Pannekoek”, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands

10Astro Space Center of the Lebedev Physical Institute,Profsoyuznaya str. 84/32, Moscow 117997, Russia

11Department of Physics, University of Warwick, Coventry, CV4 7AL, UK12Department of Physics, West Virginia University, 111 White Hall, Morgantown, WV 26506, USA

13Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904, USA14National Radio Astronomy Observatory, 520 Edgemont Rd., Charlottesville VA 22903, USA

15Department of Physics and Astronomy, University of British Columbia6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada

∗This is the authors’ version of the work. The definite version is published in Science Online, 26 April 2013,Vol: 340, Issue: 6131 doi: 10.1126/science.1233232

†Member of the International Max Planck Research School for Astronomy and Astrophysics at the Universitiesof Bonn and Cologne; e-mail: [email protected]

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A Massive Pulsar in a Compact Relativistic Binary∗

John Antoniadis,1 †Paulo C. C. Freire,1 Norbert Wex,1 Thomas M. Tauris,2,1

Ryan S. Lynch,3 Marten H. van Kerkwijk,4 Michael Kramer,1,5 Cees Bassa,5

Vik S. Dhillon,6 Thomas Driebe,7 Jason W. T. Hessels,8,9 Victoria M. Kaspi,3

Vladislav I. Kondratiev,8,10 Norbert Langer,2 Thomas R. Marsh,11

Maura A. McLaughlin,12 Timothy T. Pennucci,13 Scott M. Ransom,14

Ingrid H. Stairs,15 Joeri van Leeuwen,8,9

Joris P. W. Verbiest,1 David G. Whelan,13

1Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany2Argelander Institut fur Astronomie, Auf dem Hugel 71, 53121 Bonn, Germany

3Department of Physics, McGill University, 3600 University Street, Montreal, QC H3A 2T8, Canada4Department of Astronomy and Astrophysics, University of Toronto,

50 St. George Street, Toronto, ON M5S 3H4, Canada5Jodrell Bank Centre for Astrophysics, The University of Manchester,

Alan Turing Building, Manchester, M13 9PL, UK6Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, UK

7Deutsches Zentrum fur Luft- und Raumfahrt e.V. (DLR),Raumfahrtmanagement, Konigswinterer Str. 522–524, 53227 Bonn, Germany

8ASTRON, the Netherlands Institute for Radio Astronomy,Postbus 2, 7990 AA Dwingeloo, The Netherlands

9Astronomical Institute “Anton Pannekoek”, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands

10Astro Space Center of the Lebedev Physical Institute,Profsoyuznaya str. 84/32, Moscow 117997, Russia

11Department of Physics, University of Warwick, Coventry, CV4 7AL, UK12Department of Physics, West Virginia University, 111 White Hall, Morgantown, WV 26506, USA

13Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904, USA14National Radio Astronomy Observatory, 520 Edgemont Rd., Charlottesville VA 22903, USA

15Department of Physics and Astronomy, University of British Columbia6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada

∗This is the authors’ version of the work. The definite version is published in Science Online, 26 April 2013,Vol: 340, Issue: 6131 doi: 10.1126/science.1233232

†Member of the International Max Planck Research School for Astronomy and Astrophysics at the Universitiesof Bonn and Cologne; e-mail: [email protected]

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M = 2.01 ± 0.04

Many physically motivated extensions to general relativity (GR) predict sig-nificant deviations in the properties of spacetime surrounding massive neu-tron stars. We report the measurement of a 2.01±0.04 solar mass (M⊙) pul-sar in a 2.46-hr orbit with a 0.172±0.003 M⊙ white dwarf. The high pulsarmass and the compact orbit make this system a sensitive laboratory of a pre-viously untested strong-field gravity regime. Thus far, the observed orbitaldecay agrees with GR, supporting its validity even for the extreme conditionspresent in the system. The resulting constraints on deviations support the useof GR-based templates for ground-based gravitational wave detectors. Addi-tionally, the system strengthens recent constraints on the properties of densematter and provides insight to binary stellar astrophysics and pulsar recycling.

Neutron stars (NSs) with masses above 1.8 M⊙ manifested as radio pulsars are valuableprobes of fundamental physics in extreme conditions unique in the observable Universe andinaccessible to terrestrial experiments. Their high masses are directly linked to the equation-of-state (EOS) of matter at supra-nuclear densities (1, 2) and constrain the lower mass limitfor production of astrophysical black holes (BHs). Furthermore, they possess extreme internalgravitational fields which result in gravitational binding energies substantially higher than thosefound in more common, 1.4 M⊙ NSs. Modifications to GR, often motivated by the desire fora unified model of the four fundamental forces, can generally imprint measurable signatures ingravitational waves (GWs) radiated by systems containing such objects, even if deviations fromGR vanish in the Solar System and in less massive NSs (3–5).

However, the most massive NSs known today reside in long-period binaries or other systemsunsuitable for GW radiation tests. Identifying a massive NS in a compact, relativistic binaryis thus of key importance for understanding gravity-matter coupling under extreme conditions.Furthermore, the existence of a massive NS in a relativistic orbit can also be used to test currentknowledge of close binary evolution.

ResultsPSR J0348+0432 & optical observations of its companion PSR J0348+0432, a pulsar spin-ning at 39 ms in a 2.46-hr orbit with a low-mass companion, was detected by a recent sur-vey (6, 7) conducted with the Robert C. Byrd Green Bank Telescope (GBT). Initial timing ob-servations of the binary yielded an accurate astrometric position, which allowed us to identifyits optical counterpart in the Sloan Digital Sky Survey (SDSS) archive (8). The colors and fluxof the counterpart are consistent with a low-mass white dwarf (WD) with a helium core at a dis-tance of d ∼ 2.1 kpc. Its relatively high apparent brightness (g� = 20.71 ± 0.03 mag) allowed usto resolve its spectrum using the Apache Point Optical Telescope. These observations revealeddeep Hydrogen lines, typical of low-mass WDs, confirming our preliminary identification. Theradial velocities of the WD mirrored that of PSR J0348+0432, also verifying that the two starsare gravitationally bound.

2

arXiv:1304.6875 [astro-ph.HE]

PSR J0348+0432

– 37 –

6 8 10 12 14 16 18

0.5

1

1.5

2

2.5

0

20

40

60

80

100

310!

R (km)

)M

(M

=Rphr

6 8 10 12 14 16 18

0.5

1

1.5

2

2.5

0

20

40

60

80

100

120

140

160

180

310!

R (km)

)M

(M

>>Rphr

9 10 11 12 13 14 150.8

1

1.2

1.4

1.6

1.8

2

2.2

R (km)

)M

(M =Rphr

9 10 11 12 13 14 150.8

1

1.2

1.4

1.6

1.8

2

2.2

R (km)

)M

(M >>Rphr

Fig. 9.— The upper panels give the probability distributions for the mass versus radius curves implied bythe data, and the solid (dotted) contour lines show the 2-σ (1-σ) contours implied by the data. The lowerpanes summarize the 2-σ probability distributions for the 7 objects considered in the analysis. The leftpanels show results under the assumption rph = R, and the right panes show results assuming rph ! R. Thedashed line in the upper left is the limit from causality. The dotted curve in the lower right of each panelrepresents the mass-shedding limit for neutron stars rotating at 716 Hz.

News from NEUTRON STARS

K. Hebeler, J. Lattimer, C. Pethick, A. Schwenk:

Phys. Rev. Lett. 105 (2010) 161102

purely “nuclear” EoS

kaon condensate

quark matter

Constraints from neutron star observables

“Exotic” equations of state ruled out ?

A. Akmal, V.R. Pandharipande, D.G. RavenhallPhys. Rev. C 58 (1998) 1804

F. Özil, D. Psaltis: Phys. Rev. D80 (2009) 103003F. Özil, G. Baym, T. Güver: Phys. Rev. D82 (2010)101301

A.W. Steiner, J. Lattimer, E.F. Brown

Astroph. J. 722 (2010) 33

Mass-Radius Relation

26 J. W. HOLT, N. KAISER, AND W. WEISE

0 0.05 0.1 0.15 0.2 ! [fm-3]

0

10

20

30

40

E/A

[MeV

]

V2N+V3N (2.1 fm-1)

V2N+V3N (2.5 fm-1)APRQMC

FIG. 4: (Color online) Energy per particle of neutron matter from chiral and low-momentum two-

and three-body interactions. The cutoff scale associated with the bare chiral nuclear potential is

Λ = 2.5 fm−1, while that of the low-momentum interaction is Λ = 2.1 fm−1. The curve labeled

“APR” is taken from the variational calculations of Akmal et al. [34], and the curve labeled “QMC”

is taken from the quantum Monte Carlo calculations in Refs. [39, 40].

two- and three-nucleon forces. We begin by considering the leading-order (free-space) con-

tribution from both the bare chiral N3LO NN interaction [19] as well as the low-momentum

NN potential Vlow-k [22, 23] obtained by integrating out momenta beyond the cutoff scale of

2.1 fm−1. We use the general formula in Eq. (9) for computing the first-order perturbative

contribution to the quasiparticle interaction as well as the projection formulas in Eq. (7)

for extracting the scalar functions f, g, and h. In Table IV we show the results for neutron T4

matter with a Fermi momentum of kf = 1.7 fm−1 corresponding to a density of ρ0 = 0.166

fm−3. In both cases the Fermi liquid parameters decrease rapidly in magnitude with L for

all channels. For larger values of L the difference between bare and evolved two-neutron

NEUTRON MATTER

In-medium chiral effective field theory

perfect agreement with sophisticated many-body calculations(e.g. Quantum Monte Carlo computations (P. Armani et al., arXiv:1110.0993) )

Neutron matterbehaves almost (but not quite) likea unitary Fermi gas

Bertsch parameter

ξ =

E

EFermi gas! 0.5

Chiral NN + 3N

V(low-k) + 3N

Akmal et al. 1998

J.W. Holt, N. Kaiser, W. W. Phys. Rev. C 87 (2013) 014338

Prog. Part. Nucl. Phys. 73 (2013) 35

S. Gandolfi et al.PRC80 (2009) 045802

NEUTRON STAR MATTEREquation of State

Neutron star constraints plus Chiral Effective Field Theory at lower density

B. Röttgers, W. W. (2011)

W. W.Prog. Part. Nucl. Phys.

67 (2012) 299

6 RESULTS, CONCLUSION, AND OUTLOOK 30

Figure 6.3: PSR J1614-2230 and the observational constraints to the radii due to Steineret al. delimits the range for the physical EoS into the green area (for further explanationsee text). The horizontal dashed lines are the limits previously given for P2 according to[19]. The APR EoS [20] is shown for comparison.

Low-density (crust) + ChEFT (FKW)Constrained extrapolation (polytropes)Akmal, Pandharipande, Ravenhall (1998)

nuclear physicsconstraints

astrophysicsconstraints

M(R)

P = const · ρΓ

S. Fiorilla, N. Kaiser, W. W.

Nucl. Phys. A880 (2012) 65

pressure

energy density

NEUTRON STARS: MASS and RADIUS constraints

J. E. TrümperProg. Part. Nucl. Phys.

66 (2011) 674

!"#$%&'()%*#+%%,))#-./-!"#$%&'()%*#+%%,))#-./-

very stiff Equation of State required !

Ε0 Ε1 Ε2 Ε3

100 1000500200 300150 7000.1

1

10

100

Ε �MeV fm�3�

P�MeV

fm�3 �

NEUTRON STAR MATTEREquation of State (contd.)

low density (crust)+

nuclear ChEFT

P = const · ρΓi

Trümper constraint

Steiner et al.Hebeler et al. constraints

T. Hell, B. Röttgers,W. W.

arXiv:1307.4582

NEUTRON STAR MATTEREquation of State

In-medium Chiral Effective Field Theory up to 3 loops(reproducing thermodynamics of normal nuclear matter)

coexistence region:Gibbs conditions

n ↔ p + e, µ

3-flavor PNJL model at high densities (incl. strange quarks)

beta equilibrium

charge conservation

quark-nuclearcoexistence occurs(if at all)at baryon densities

ρ > 5 ρ0

ChEFT

PNJL, Gv � 0.5G

PNJL, Gv � 0

50 100 200 300 500 1000 20001

51020

100200

Ε �MeV fm�3�

P�MeV

fm�3 �

realistic“conventional” EoS

quark - nuclearcoexistence

T. Hell, W.W. (2013)

see also: K. Masuda, T. Hatsuda, T. Takatsuka

PTEP (2013) 7, 073D01

M � 1.95M�R � 11.39 km

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

r �km�

Ρ�fm�3

�

ρcentral

Mass

Mmax

Mmaxstiff

soft

EoS

EoS

ChEFT

ChEFT � PNJL �Gv � 0�

0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

r �km�

�r�R�2Ρ�Ρ 0

NEUTRON STAR MATTERDensity Profiles

neutron star central density: five times ρ0

M(R) =4π

c2

∫ R

0

dr r2E(r)

stiff EoS larger maximum mass lower central density

relevant quantity: r2ρ(r)

T. Hell, W.W. (2013)

M = 2MO

R = 11km

.

mostly neutrons

5

0.0001

0.001

0.01

0.1

1

1 2 3 4 5 6

x i

!B[!0]

"EFT600

p

n

#

e

µ

K0=200 MeVat=32 MeV

FIG. 4: Density ratios for different particles for a "soft" nu-cleonic EoS as a function of the baryon density using theχEFT600 model.

with the corresponding lepton density ratio xl = ρl/ρL.The total lepton density ρL is the sum over all three lep-tons. For nonrelativistic interacting baryons, the chemi-cal potential for species b reads

µb = Mb +k2Fb

2Mb+ Ub(kFb

) . (20)

For a given total baryon density ρB the equations (15)-(18) govern the composition of matter, i.e. the baryonicand leptonic concentrations. The corresponding solutionis referred to as β-stable matter.

For the sake of consistency we now have to treat thenucleonic part of the chemical potential µN in the sameway as the corresponding energy per particle. Since thechemical potential can be obtained as a derivative of theenergy density ε and is related to the energy per particlevia ε = ρBE/A, we use the definition

µb =∂ε

∂ρb, (21)

to have the appropriate replacement in the nucleonicchemical potential. Finally, we arrive at the expression

µN =∂εNN

∂ρN+ UY

N (kFY), (22)

where we have effectively replaced MN+k2FN

2MN+UN

N (kFN)

of Eq. (20) with the derivative ∂εNN/∂ρN . In this waythe parameterization Eq. (9) enters in the nucleonic partof the chemical potential.

Since we are only parameterizing the nucleonic sec-tor, no such replacement is necessary for the hyperons.However, since we have neglected the Y Y interactionUYY (kFY

) is zero and Eq. (20) reduces to

µY = MY +k2FY

2MY+ UN

Y (kFY) . (23)

0.0001

0.001

0.01

0.1

1

1 2 3 4 5 6

x i

!B[!0]

NSC97f

p

n

#

$-

e

µ K0=300 MeVat=32 MeV

$0

FIG. 5: Same as Fig. 4 but for "stiff" nucleonic EoS using theNSC97f model.

For the determination of the particle concentration thesingle-particle potential in equilibrium is used. For hy-perons below the threshold density it is given by UY (p =0), similar to the symmetric matter case. Above thethreshold density it depends on composition and den-sity. In Fig. 6 the density dependence of UΣ−(kF

Σ−) in

β-equilibrium for two different incompressibilities K0 isshown. In the figure a kink in the curves appears at thepoint where the hyperons appear.

Another observation is the relative ordering and themagnitudes which resemble those of the single-particlepotentials at zero momentum in symmetric matter asshown in Fig. 1. Essentially, the NSC97a, NCS97c,NSC97f and J04 interactions are still slightly attractivewhile the NSC89 and χEFT600 remain repulsive. Simi-lar observations hold for the Λ system. A new structurein form of a second inflection point emerges due to theappearance of the Σ− hyperon.

A better indicator at which densities hyperons start toappear is given by the concentrations of all particles andis displayed in Figs. 4 and 5. In Fig. 4 a “soft” nucle-onic EoS is used in combination with an attractive ΛNand a very repulsive ΣN interaction implemented by theχEFT600 model. In contrast in Fig. 5 a “stiff” EoS isused represented by the NSC97f model which has a simi-lar ΛN interaction compared to the χEFT600 model butalso an attractive ΣN interaction. This difference alreadyleads to very different density profiles. While in Fig. 4the Λ hyperon is the first one which appears and no Σ−

hyperons are present, the Σ− hyperon appears first inFig. 5.

One should note that with the appearance of the Σ−

hyperon the density of the negatively charged leptonsstarts to drop immediately. This is because their role inthe charge neutrality condition, Eq. (15), is now beingtaken over by the Σ−. Similarly, the appearance of theΛ hyperon will accelerate the disappearance of neutrons

NEUTRON STAR MATTERwith HYPERONS

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10 12 14

M [M

0]

R [km]

K0=200 MeVat=32 MeV

NNNSC97aNSC97cNSC97fNSC89

J04!EFT600

(a) K0 = 200 MeV

0

0.5

1

1.5

2

2.5

3

10 12 14 16

M [M

0]

R [km]

K0=300 MeVat=32 MeV


J04!EFT600

(b) K0 = 300 MeV

FIG. 10: Mass-radius relation of a neutron star for symmetry energy at = 32 MeV and different Y N interactions. Forcomparison the mass-radius curve obtained for the pure NN interaction is also shown. Left panel: soft EoS, right panel: stiffEoS.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

10 12 14

M [M

0]

R [km]

K0=200 MeVat=32 MeV


J04!EFT600

(a) K0 = 200 MeV,γ = 1 for ρ ≥ ρ0

0

0.5

1

1.5

2

2.5

3

10 12 14 16

M [M

0]

R [km]

K0=300 MeVat=32 MeV


J04!EFT600

(b) K0 = 300 MeV,γ = 1 for ρ ≥ ρ0

FIG. 11: Mass-radius relation of a neutron star for symmetry energy at = 32 MeV, γ = 1 for ρ ≥ ρ0 and different Y Ninteractions. For comparison the mass-radius curve obtained for the pure NN interaction is also shown. Left panel: soft EoS,right panel: stiff EoS.

cant softening of the EoS which in turn results in smallermaximum masses of a neutron star compared to a purelynucleonic EoS. Notably, the predicted maximum massesare well below the observed value of 1.4 M!, an outcomealso known from other works, e.g. [8, 33–35]. This posesa serious problem.

The softening of the EoS due to hyperons cannot becircumvented by stiffening the nucleonic EoS, i.e., by in-creasing K0, since this will cause hyperons to appear ear-

lier. Changing the high-density behavior of the symmetryenergy dependence or including the S = −2 sector doesnot alter this conclusion either. For more details aboutthe S = −2 sector, in particular Ξ hyperons in densebaryonic matter see e.g. [39–42]. This can only mean,that correlations beyond the one-loop level could be im-portant to stiffen the hyperon contributions to the EoS.This, however, is not sufficient as Brueckner-Hartree-Fock calculations indicate [9, 26]. As has been known

d quarks

u quarkss quarksprotons

neutrons

�

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

Ρ�Ρ0Ρ i�Ρ

p

n

Λ

du

s

ρ/ρ0

T. Hell, W.W. (2013)

H. Djapo, B.-J. Schaefer,J. Wambach

Phys. Rev. C81 (2010) 035803

with inclusion of hyperons:EoS far too soft to support 2 solar mass starunless strong repulsion in YN interaction



Fig. 6. The Λp 1S0 and 1P1 phase shifts δ as a function of plab. The red/dark band shows the chiral EFT results toNLO for variations of the cutoff in the range Λ = 500, . . . ,650 MeV, while the green/light band shows results to LO forΛ = 550, . . . ,700 MeV. The dashed curve is the result of the Jülich ’04 meson-exchange potential [37].

Fig. 7. The Λp phase shifts for the coupled 3S1–3D1 partial wave as a function of plab. Same description of curves asin Fig. 6.

state in the ΣN system. It should be said, however, that the majority of the meson-exchangepotentials [36,38,39] produce an unstable bound state, similar to our NLO interaction. The onlycharacteristic difference of the chiral EFT interactions to the meson-exchange potentials mightbe the mixing parameter ε1 which is fairly large in the former case and close to 45◦ at the ΣN

threshold, see Fig. 7. It is a manifestation of the fact that the pertinent Λp T -matrices (for the3S1 → 3S1, 3D1 → 3D1, and 3S1 ↔ 3D1 transitions) are all of the same magnitude.

The strong variation of the 3S1–3D1 amplitudes around the ΣN threshold is reflected inan impressive increase in the Λp cross section at the corresponding energy, as seen in Fig. 2.

NLO

LO

repulsion

phase shift

attraction

phenomenologicalpotential

hypernuclei

neutron star matter

E [M

eV]

b [fm-3]

AV8’

AV8’ + UIX

AV8’ + UIX + N(N) S nlin

AV8’ + UIX + N(N) S nquad

0

20

40

60

80

100

120

140

0.0 0.1 0.2 0.3 0.4 0.5 0.6

th ( S nlin ) = 1.97 0

th ( S nquad ) = 1.91 0

D. Lonardoni, F. Pederiva, S. Gandolfi (2013)

preliminary

B [M

eV]

A-2/3

exp

N

N + NN (I)

N + NN (II)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.0 0.1 0.2 0.3 0.4 0.5

D. Lonardoni, F. Pederiva, S. Gandolfisubmitted to Phys. Rev. C (2013)

Hypernuclei and Neutron Star EoS with inclusion of Hyperons

repulsive interaction important

ΛNN

two-body interactions notsufficient

state-of-the-art Quantum Monte Carlo calculationof hypernuclei and neutron star matter

hypernuclei Neutron Star EoS

incl. hyperons

nucleons

D. Lonardoni, F. Pederiva, S. Gandolfi (2013)

0.5 fm

1.2 fm

baryoniccore

pionicfield

ρB = 0.6 fm−3

normal nuclear matter: dilute

0.5 fm

2 fm

baryoniccore

pionicfield

ρB = 0.15 fm−3

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Ir!J]raL 0r 80 90 t0 z0 0

Z

[' u]l

\uoluos awtt$ls sD ''uoapnN I 'lD D /Duss!aI,! .c_.n

01rJ,4 UeC 3,4A

1

2

0.2 0.4 0.6 0.8 1.0 1.2 r [fm]

[fm−1]

4πr2

ρB(r)

4πr2

ρS(r)charge density

baryon density

isoscalar

0

Densities and Scales in Compressed Baryonic Matter

neutron star core matter: compressed but not superdense

recall:chiral (soliton) modelof the nucleon

compactbaryonic core

mesonic cloud

. . . treated properly in chiral EFT

N. Kaiser, U.-G. Meißner, W. W.Nucl. Phys. A 466 (1987) 685

〈r2〉1/2B

# 0.5 fm

〈r2〉1/2E,isoscalar # 0.8 fm

SUMMARY

Well established existence of two-solar mass neutron stars:

no ultrahigh central densities in neutron stars

“Conventional” (non-exotic) EoS works remarkably well (nuclear effective field theory + advanced many-body methods)

Small admixtures of strangeness possible with additional repulsive hyperon-nucleon and hyperon-hyperon interactions

very stiff equation-of-state of dense baryonic matter

Change of paradigm ? “The constraints strongly suggest that the objects . . . are really neutron stars and not strange quark stars.” ( J. Trümper)

next major step: consistent implementation of chiral SU(3) EFT basedhyperon-nucleon interaction in advanced many-body calculations

Strangeness in Baryonic Matter - T39 Physics Group

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