STRANGENESS in LOW ENERGY QCD and DENSE ......the low-energy pion-, kaon and antikaon scattering data was achieved earlier within the χ-BS(3) scheme upon incorporating chiral correction

LNF Seminar Frascati 11 March 2013

STRANGENESS in LOW-ENERGY QCDand

DENSE BARYONIC MATTER

Wolfram Weise

Role of strange quarksChiral symmetry breaking patterns in QCD

Strangeness in dense baryonic matterNew constraints from neutron stars

Baryonic interactions with strangenessShort-range repulsions from Lattice QCD

ECT* Trento and Technische Universität München

Antikaon interactions with nucleons and nucleiAntikaon-nucleon threshold observables

Nature and structure of

Testing ground: high-precision antikaon-nucleon threshold physics

Strange quarks are intermediate between “light” and “heavy”:

Role of strangeness in dense baryonic matter

Quest for quasi-bound antikaon-nuclear systems ?

interaction

BASIC ISSUES

Interplay between spontaneous and explicit chiral symmetry breaking in low-energy QCD

!(1405)

Three-quark valence structure vs. “molecular” meson-baryon system ?

(B = 1, S = !1, JP = 1/2!)

Strongly attractive low-energy KN

Kaon condensation ? Strange quark matter in neutron stars ?

LOW-ENERGY QCD

with STRANGE QUARKS:

Chiral SU(3) Dynamics

Part I.

Hierarchy of QUARK MASSES in QCD

0 !

1 GeVu,d s c

“light” quarks “heavy” quarks

mc ! 1.25 GeV

(µ ! 2 GeV)

mb ! 4.2 GeV

mt ! 174 GeV

LOW-ENERGY QCD:CHIRAL EFFECTIVEFIELD THEORY

expansion in and in powers of low momentum

Non-Relativistic QCD: HEAVY QUARKEFFECTIVE THEORY

expansion inpowers of

mq

1/mQ

mass

separation of scales

mu ! 2 " 3 MeV

mu/md ! 0.4 " 0.6

ms = 95 ± 5 MeV

PDG Phys. Rev. D 86 (2012)

Spectroscopic patterns: LIGHT versus HEAVY

Q Qq q[ ]1S0[ ]3S1

Mmeson ! mQ ! mq

[GeV]

0

0.2

0.4

0.6

0.8

1.0

1.2

B!

B D!

D

K!

K

!

!

bd cd sd ud

hyperfinesplitting

massgap

spontaneouschiral

symmetrybreaking

MASS SPLITTINGS in SINGLET and TRIPLETQuark-Antiquark States

perturbativeQCD

NAMBU - GOLDSTONE BOSONS:

Spontaneously Broken CHIRAL SYMMETRYSU(3)L ! SU(3)R

Pseudoscalar SU(3) meson octet {!a} = {!, K, K, "8}

DECAY CONSTANTSORDER PARAMETERS:

µ

!

axial current

!

K

f! = 92.4 ± 0.3 MeV

chiral limit: f = 86.2 MeV( )!0|Aµa(0)|!b(p)" = i"ab pµ

fb

fK = 110.0 ± 0.9 MeV

m2!f2!

= !mu + md

2"uu + dd#Gell-Mann,

Oakes,Rennerrelations m

2K f

2K = !

mu + ms

2"uu + ss#

+higher order corrections

SYMMETRY BREAKING PATTERN

PSEUDOSCALAR MESON SPECTRUM

U. Vogl, W. W. Prog. Part. Nucl. Phys. 27 (1991) 195

Calculation: Nambu & Jona-Lasinio model with N = 3 quark flavorsf

mq = 0

S. Klimt, M. Lutz, U. Vogl, W. W. Nucl. Phys. A 516 (1990) 429

!0

0

0.2

0.4

0.6

0.8

1.0

!, K, "0, "8 !, K, "8

!0!

!!

!

!

K

U(3) x U(3)

m = m = m = 0u d s

SU(3) x SU(3)

U(1)A

breaking

L R

L R

m = m = 5 MeVu d

m = 130 MeVs

MASS[GeV]

mass

[GeV]

0

0.2

0.4

0.6

0.8

1.0

SU(3)L ! SU(3)R

U(1)Abreaking

mu,d ! 5 MeV

ms ! 130 MeV

!

Kms ! 100 MeV

mu,d < 5 MeV

CHIRAL SU(3) EFFECTIVE FIELD THEORY

Interacting systems of NAMBU-GOLDSTONE BOSONS (pions, kaons) coupled to BARYONS

+ + ...

Low-Energy Expansion: CHIRAL PERTURBATION THEORY

“small parameter”: energy / momentum

+

p

4! f!

Leff = Lmesons(!) + LB(!, "B)

Leading DERIVATIVE couplings (involving )!µ!

determined by spontaneously broken CHIRAL SYMMETRY

!! ! !B B

works well for low-energy pion-pion and pion-nucleon interactions

... but NOT for systems with strangeness S = !1 (KN, !!, ...)

1 GeV

higher orders with additional

low-energy constants

!

(!)(B)

2 M.F.M. Lutz et al.

Thus, it is useful to review also in detail e!ective coupled-channel field theoriesbased on the chiral Lagrangian.

The task to construct a systematic e!ective field theory for the meson-baryonscattering processes in the resonance region is closely linked to the fundamental ques-tion as to what is the ’nature’ of baryon resonances. The radical conjecture10), 5), 11), 12)

that meson and baryon resonances not belonging to the large-Nc ground states aregenerated by coupled-channel dynamics lead to a series of works13), 14), 15), 17), 16), 18)

demonstrating the crucial importance of coupled-channel dynamics for resonancephysics in QCD. This conjecture was challenged by a phenomenological model,11)

which generated successfully non-strange s- and d-wave resonances by coupled-channeldynamics describing a large body of pion and photon scattering data. Of course,the idea to explain resonances in terms of coupled-channel dynamics is an old onegoing back to the 60’s.19), 20), 21), 22), 23), 24) For a comprehensive discussion of thisissue we refer to.12) In recent works,13), 14) which will be reviewed here, it was shownthat chiral dynamics as implemented by the !!BS(3) approach25), 10), 5), 12) providesa parameter-free leading-order prediction for the existence of a wealth of strange andnon-strange s- and d-wave wave baryon resonances. A quantitative description ofthe low-energy pion-, kaon and antikaon scattering data was achieved earlier withinthe !-BS(3) scheme upon incorporating chiral correction terms.5)

§2. E!ective field theory of chiral coupled-channel dynamics

Consider for instance the rich world of antikaon-nucleon scattering illustrated inFig. 1. The figure clearly illustrates the complexity of the problem. The KN statecouples to various inelastic channel like "# and "$, but also to baryon resonancesbelow and above its threshold. The goal is to bring order into this world seeking adescription of it based on the symmetries of QCD. For instance, as will be detailedbelow, the $(1405) and $(1520) resonances will be generated by coupled-channeldynamics, whereas the #(1385) should be considered as a ’fundamental’ degree offreedom. Like the nucleon and hyperon ground states the #(1385) enters as anexplicit field in the e!ective Lagrangian set up to describe the KN system.

The starting point to describe the meson-baryon scattering process is the chiralSU(3) Lagrangian (see e.g.26), 5)). A systematic approximation scheme arises due to asuccessful scale separation justifying the chiral power counting rules.27) The e!ectivefield theory of the meson-baryon scattering processes is based on the assumption

s1/2KN

[MeV]

poles

thresholdsΣηΛη

Λ**(1520)

Λ*(1405)Σ

*(1385)

Σ(1195)Λ(1116)

KNΣπΛπ

15001000

Fig. 1. The world of antikaon-nucleon scattering

KN_

!

s [MeV]

Low-Energy K N Interactions_

Chiral Perturbation Theory NOT applicable:!(1405) resonance 27 MeV below K p threshold-

Non-perturbative Coupled Channels

approach based on Chiral SU(3) Dynamics

N. Kaiser, P. Siegel, W. W. (1995)E. Oset, A. Ramos (1998)

Leading s-wave I = 0 meson-baryon interactions (Tomozawa-Weinberg)

0 !

1 GeVu,d s c


0 !

1 GeVu,d s c


K N

!

!

0 !

1 GeVu,d s c


K N

! !K N !

!

KN!!

channel coupling

CHIRAL SU(3) COUPLED CHANNELS DYNAMICS

Tij = Kij +

!

n

Kin Gn Tnj

input fromchiral SU(3)

meson-baryoneffective Lagrangian

loop functions(dim. regularization)

with subtraction constants encoding short distance dynamics

coupled channels:

K!p, K0n, !0!0

, !+!!

, !!!+

, !0", "", "!0

, K+#!

, K!#0

0 !

1 GeVu,d s c


0 !

1 GeVu,d s c


K N

!

!

0 !

1 GeVu,d s c


K N

! !

K N !

!


Tij = Kij +

!

n

Kin Gn Tnj

12 ! 21 :

|1! = |KN, I = 0! |2! = |!!, I = 0!

!!bound stateKN resonance

driving interactions individually strong enough to produce

strong channel coupling

K11 =3

2 f2K

(!

s " MN) K22 =2

f2!

(!

s " M!)

K12 =!1

2 f! fK

!

3

2

"

"

s !M! + MN

2

#

Note: ENERGY DEPENDENCE characteristic of Nambu-Goldstone BosonsLeading s-wave I = 0 meson-baryon interactions (Tomozawa-Weinberg)

f! = 92.4 ± 0.3 MeV

fK = 110.0 ± 0.9 MeV

(a) (b) (c) (d)

Figure 1: Feynman diagrams for the meson-baryon interactions in chiral perturbation theory.(a) Weinberg-Tomozawa interaction, (b) s-channel Born term, (c) u-channel Born term, (d)NLO interaction. The dots represent the O(p) vertices while the square denotes the O(p2)vertex.

where qi, Mi and Ei are the momentum, the mass and the energy of the baryon in channel i, and !!i isthe two-component Pauli spinor for the baryon in channel i. Applying the s-wave projection (11), weobtain the WT interaction

V WTij (W ) = !Cij

4f2(2W ! Mi ! Mj)

!Mi + Ei

2Mi

"Mj + Ej

2Mj. (15)

The Cij coe!cients express the sign and the strength of the interaction for this channel. With theSU(3) isoscalar factors [101, 102], it is given by [103, 104]

Cij =#

"

[6 ! C2(")]

$8 8 "

Ii, Yi Ii, Yi I, Y

%$8 8 "

Ij, Yj Ij, Yj I, Y

%, (16)

Y = Yi + Yi = Yj + Yj, I = Ii + Ii = Ij + Ij,

where " is the SU(3) representation of the meson-baryon system with C2(") being its quadratic Casimir,Ii and Yi are the isospin and hypercharge of the particle in channel i (i stands for the baryon and i forthe meson). Explicit values of Cij for the S = !1 meson-baryon scattering can be found in Ref. [8]. Itis remarkable that the sign and the strength of the interaction (15) are fully determined by the grouptheoretical factor Cij. This is because the low energy constant is absent in the Lagrangian (13), as itis derived from the covariant derivative. In the language of current algebra, this is the consequenceof the vector current conservation (Weinberg-Tomozawa theorem) [74, 75]. Indeed, at threshold ofthe #N " #N amplitude, Eq. (15) gives the scattering length (the relation of the T-matrix with thenonrelativistic scattering amplitude is summarized in Appendix)

a#N!#N =

&'''(

''')

MN

4#(MN + m#)

m#

f 2for I = 1/2

! MN

8#(MN + m#)

m#

f2for I = 3/2

,

in accordance with the low energy theorem.It is also remarkable that the phenomenological vector meson exchange potential [6] leads to the

same channel couplings with Cij when the flavor SU(3) symmetric coupling constants are used. In fact,with the KSRF relation g2

V = m2V /2f 2 [105, 106], the vector meson exchange potential reduces to the

contact interaction V # Cij/f2 in the limit mV " $.Another important feature of Eq. (15) is the dependence on the total energy W . This is a consequence

of the derivative coupling nature of the NG boson in the nonlinear realization. The energy dependenceis an important aspect for the discussion of the s-wave resonance state.

14

(a) (b) (c) (d)



V WTij (W ) = !Cij

4f2(2W ! Mi ! Mj)

!Mi + Ei

2Mi

"Mj + Ej

2Mj. (15)


Cij =#

"

[6 ! C2(")]

$8 8 "

Ii, Yi Ii, Yi I, Y

%$8 8 "

Ij, Yj Ij, Yj I, Y

%, (16)



a#N!#N =

&'''(

''')

MN

4#(MN + m#)

m#

f 2for I = 1/2

! MN

8#(MN + m#)

m#

f2for I = 3/2

,






14

(a) (b) (c) (d)



V WTij (W ) = !Cij

4f2(2W ! Mi ! Mj)

!Mi + Ei

2Mi

"Mj + Ej

2Mj. (15)


Cij =#

"

[6 ! C2(")]

$8 8 "

Ii, Yi Ii, Yi I, Y

%$8 8 "

Ij, Yj Ij, Yj I, Y

%, (16)



a#N!#N =

&'''(

''')

MN

4#(MN + m#)

m#

f 2for I = 1/2

! MN

8#(MN + m#)

m#

f2for I = 3/2

,






14

CHIRAL SU(3) COUPLED CHANNELS DYNAMICS:

- NLO hierarchy of driving terms -

leading order (Weinberg-Tomozawa) termsinput: physical pion and kaon decay constants

direct and crossed Born termsinput: axial vector constantsD and F from hyperon beta decays

Now we turn to the baryons which are introduced as matter fields in the nonlinear realization [93, 94].The octet baryon fields are collected as

B =

!

"#

1!2!0 + 1!

6" !+ p

!" ! 1!2!0 + 1!

6" n

#" #0 ! 2!6"

$

%& ,

which transforms under g " SU(3)R # SU(3)L as

Bg$ hBh†, B

g$ hBh†,

with h(g, u) " SU(3)V . For baryons, the mass term M0Tr(BB) is chiral invariant even if the quarkmasses are absent. The mass term brings the additional scale M0 in the theory, which causes problemsin the counting rule of Lagrangian and eventually in the systematic renormalization program. Anelegant method to avoid this di$culty is the heavy baryon chiral perturbation theory [95], where thebaryon fields are treated as heavy static fermions and the limit M0 $ % is taken. Here we followRefs. [96, 97, 98] to construct the relativistic chiral Lagrangian with keeping the common mass of theoctet baryons M0 finite.4 We define the following quantities

!+ = u!†u + u†!u†, !" = u!†u ! u†!u†,

uµ = i{u†("µ ! irµ)u ! u("µ ! ilµ)u†},

%µ =1

2{u†("µ ! irµ)u + u("µ ! ilµ)u†}.

The latter two quantities are related to the vector (Vµ) and axial vector (Aµ) currents as Aµ = !uµ/2

and Vµ = !i%µ. These quantities are transformed as Og$ hOh†, except for the chiral connection %µ,

which transforms as%µ

g$ h%µh† + h"µh

†.

Then the covariant derivatives for the octet baryon fields can be defined as

DµB = "µB + [%µ, B].

The power counting rule for baryon fields is given by

B, B : O(1), uµ, %µ, (i /D ! M0)B : O(p), !± : O(p2).

With these counting rules, we can construct the most general e&ective Lagrangian for meson-baryonsystem as

Le!(B, U) =#'

n=1

[LM2n(U) + LMB

n (B,U)],

where LMBn (B, U) consists of bilinears of B field with the chiral order O(pn). In the lowest order O(p),

we have

LMB1 = Tr

(B(i /D ! M0)B +

D

2(B#µ#5{uµ, B}) +

F

2(B#µ#5[uµ, B])

), (9)

4In this paper we utilize chiral perturbation theory for the meson-baryon scattering amplitude up to O(p2) where noloop diagram appears. At O(p3), an appropriate renormalization procedure in the relativistic scheme [99, 100] must beintroduced.

12

Now we turn to the baryons which are introduced as matter fields in the nonlinear realization [93, 94].The octet baryon fields are collected as

B =

!

"#

1!2!0 + 1!

6" !+ p

!" ! 1!2!0 + 1!

6" n

#" #0 ! 2!6"

$

%& ,

which transforms under g " SU(3)R # SU(3)L as

Bg$ hBh†, B

g$ hBh†,

with h(g, u) " SU(3)V . For baryons, the mass term M0Tr(BB) is chiral invariant even if the quarkmasses are absent. The mass term brings the additional scale M0 in the theory, which causes problemsin the counting rule of Lagrangian and eventually in the systematic renormalization program. Anelegant method to avoid this di$culty is the heavy baryon chiral perturbation theory [95], where thebaryon fields are treated as heavy static fermions and the limit M0 $ % is taken. Here we followRefs. [96, 97, 98] to construct the relativistic chiral Lagrangian with keeping the common mass of theoctet baryons M0 finite.4 We define the following quantities

!+ = u!†u + u†!u†, !" = u!†u ! u†!u†,

uµ = i{u†("µ ! irµ)u ! u("µ ! ilµ)u†},

%µ =1

2{u†("µ ! irµ)u + u("µ ! ilµ)u†}.

The latter two quantities are related to the vector (Vµ) and axial vector (Aµ) currents as Aµ = !uµ/2

and Vµ = !i%µ. These quantities are transformed as Og$ hOh†, except for the chiral connection %µ,

which transforms as%µ

g$ h%µh† + h"µh

†.

Then the covariant derivatives for the octet baryon fields can be defined as

DµB = "µB + [%µ, B].

The power counting rule for baryon fields is given by

B, B : O(1), uµ, %µ, (i /D ! M0)B : O(p), !± : O(p2).

With these counting rules, we can construct the most general e&ective Lagrangian for meson-baryonsystem as

Le!(B, U) =#'

n=1

[LM2n(U) + LMB

n (B,U)],

where LMBn (B, U) consists of bilinears of B field with the chiral order O(pn). In the lowest order O(p),

we have

LMB1 = Tr

(B(i /D ! M0)B +

D

2(B#µ#5{uµ, B}) +

F

2(B#µ#5[uµ, B])

), (9)

4In this paper we utilize chiral perturbation theory for the meson-baryon scattering amplitude up to O(p2) where noloop diagram appears. At O(p3), an appropriate renormalization procedure in the relativistic scheme [99, 100] must beintroduced.

12

next-to-leading order (NLO)input: 7 low-energy constants

gA = D + F = 1.26

O(p2)where D and F are low energy constants related to the axial charge of the nucleon gA = D + F !1.26, and M0 denotes the common mass of the octet baryons. Among many next-to-leading orderLagrangians [96, 97, 98], the relevant terms to the meson-baryon scattering are

LMB2 =bDTr

!B{!+, B}

"+ bF Tr

!B[!+, B]

"+ b0Tr(BB)Tr(!+)

+ d1Tr!B{uµ, [uµ, B]}

"+ d2Tr

!B[uµ, [uµ, B]]

"

+ d3Tr(Buµ)Tr(uµB) + d4Tr(BB)Tr(uµuµ), (10)

where bi and di are the low energy constants. The first three terms are proportional to the ! field andhence to the quark mass term. Thus, they are responsible for the mass splitting of baryons. Indeed,Gell-Mann–Okubo mass formula follows from the tree level calculation with isospin symmetric massesmu = md = m "= ms.

3.3 Low energy meson-baryon interaction

Here we derive the s-wave low energy meson-baryon interaction up to the order O(p2) in momen-tum space. In three flavor sector, several meson-baryon channels participate in the scattering, whichare labeled by the channel index i. The scattering amplitude from channel i to j can be written asVij(W, !,"i,"j) where W is the total energy of the meson-baryon system in the center-of-mass sys-tem, ! is the solid angle of the scattering, and "i is the spin of the baryon in channel i. Since weare dealing with the scattering of the spinless NG boson o" the spin 1/2 baryon target, the angulardependence vanishes and the spin-flip amplitude does not contribute after the s-wave projection andthe spin summation. Thus, the s-wave interaction depends only on the total energy W as

Vij(W ) =1

8#

#

!

$d! Vij(W, !, ","). (11)

In chiral perturbation theory up to O(p2), there are four kinds of diagrams as shown in Fig. 1. For thes-wave amplitude, the most important piece in the leading order terms is the Weinberg-Tomozawa (WT)contact interaction (a). The covariant derivative term in Eq. (9) generates this term which can also bederived from chiral low energy theorem. At order O(p), in addition to the WT term, there are s-channelBorn term (b) and u-channel term (c) which stem from the axial coupling terms in Eq. (9). Althoughthey are in the same chiral order with the WT term (a), the Born terms mainly contribute to the p-waveinteraction and the s-wave component is in the higher order of the nonrelativistic expansion [72]. Withthe terms in the next-to-leading order Lagrangian (10), the diagram (d) gives the O(p2) interaction. Insummary, the tree-level meson-baryon amplitude is given by

Vij(W, !,"i,"j) = V WTij (W, !,"i,"j) + V s

ij(W, !,"i,"j) + V uij (W, !,"i,"j) + V NLO

ij (W, !,"i,"j), (12)

where V WTij , V s

ij, V uij and V NLO

ij terms correspond to the diagrams (a), (b), (c) and (d) in Fig. 1,respectively. In the following we derive the amplitude Vij(W ) by calculating these diagrams.

Let us first consider the WT interaction (a). By expanding the covariant derivative term in Eq. (9)in powers of meson field #, we obtain the meson-baryon four-point vertex

LWT =1

4f 2Tr

!Bi$µ[#%µ# # (%µ#)#, B]

". (13)

The tree-level amplitude by this term is given by

V WTij (W, !,"i,"j) = # Cij

4f 2

%Mi + Ei

2Mi

&Mj + Ej

2Mj

$ (!!i)T

'2W # Mi # Mj + (2W + Mi + Mj)

qi · qj + i(qi $ qj) · !(Mi + Ei)(Mj + Ej)

(!!j , (14)

13

0

50

100

150

200

250

50 100 150 200 250

K- p ->

K- p(

mb)

Plab(MeV)

0

20

40

60

80

100

120

140

50 100 150 200 250

K- p ->

0

0 (m

b)

Plab(MeV)

!(K!p ! K!p) [mb]

!(K!p ! "0!0) [mb]

0

50

100

150

200

250

300

50 100 150 200 250

K- p ->

+

- (mb)

Plab(MeV)

!(K!p ! "+!!) [mb]

0 10 20 30 40 50 60 70 80 90

100

50 100 150 200 250

K- p ->

-

+ (m

b)

Plab(MeV)

UPDATED ANALYSIS of K!p LOW-ENERGY CROSS SECTIONS

!(K!p ! "!!+) [mb]

0

0.5

1

1.5

2

2.5

1340 1360 1380 1400 1420 1440

Im[T

K- p-K- p]

(fm

)

Ec.m. (MeV)

[fm] Im f(K!p ! K!p)

-1

-0.5

0

0.5

1

1.5

1340 1360 1380 1400 1420 1440

Re[

T K- p

- K- p]

(fm

)

Ec.m. (MeV)

Re f(K!p ! K!p)

[fm]

!

s [MeV]!

s [MeV]

. .Re a(K!p) Im a(K!p)

K!p SCATTERING AMPLITUDE

threshold region and subthreshold extrapolation:

complex scattering length (including Coulomb corrections)

f(K!p) =1

2

!

fKN(I = 0) + fKN(I = 1)"

Ima(K!p) = 0.81 ± 0.15 fmRe a(K!p) = !0.65 ± 0.10 fm

!(1405)

!(1405)

: KN (I = 0) quasibound state embedded in the !! continuum

Y. Ikeda, T. Hyodo, W. W. PLB 706 (2011) 63 NPA881 (2012) 98

TW

SIDDHARTA

CONSTRAINTS from SIDDHARTA

Kaonic hydrogen precision data

M. Bazzi et al. (SIDDHARTA) Phys. Lett. B 704 (2011) 113

strong interaction shift and width:

!E = 283 ± 36 (stat)±6 (syst) eV

! = 541 ± 89 (stat)±22 (syst) eV

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

/0%(1+23"#$%&'(! ! 31&3'$

Background estimation

/456 /7689-

:12/

:12/

/968

/958/468

/98;

/<=>5

EM value!"#$!

)1.-=,0('%-)?1,

@'-,'$1-.)*'+,$-.

B. Borasoy, R. Nißler, W. W. Eur. Phys. J. A25 (2005) 79

Theory: leading order

B. Borasoy, U.-G. Meißner, R. Nißler PRC74 (2006) 055201

(Tomozawa - Weinberg)

!"##!"$#!""#!"%#&$#&"#&%#&'##($

#("

#(%

#('#($

#("

#(%

#('

!"#$%&

!"#$%&#'"(%

)*#$%&#'"(%

+,+&#-.'"(%

9

Pole structure in the complex energy planeResonance state ~ pole of the scattering amplitude

!

Tij(!

s) " gigj!s # MR + i!R/2

D. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Nucl. Phys. A 723, 205 (2003)

Λ(1405) in meson-baryon scattering

T. Hyodo, D. Jido, arXiv:1104.4474

KN!!

The TWO POLES scenario

D. Jido et al. Nucl. Phys. A723 (2003) 205

dominantly!!

dominantlyKN

T. Hyodo, W. W. Phys. Rev. C 77 (2008) 03524

T. Hyodo, D. Jido Prog. Part. Nucl. Phys. 67 (2012) 55

Pole positions from chiral SU(3) coupled-channels calculation with SIDDHARTA threshold constraints:

E1 = 1424 ± 15 MeV

!1 = 52 ± 10 MeV !2 = 162 ± 15 MeV

E2 = 1381 ± 15 MeV

Y. Ikeda, T. Hyodo, W. W. : Nucl. Phys. A 881 (2012) 98

4

4

2

0

-2

FK

N [

fm]

1440140013601320

s1/2

[MeV]

Re F

KN(I=0)

1.5

1.0

0.5

0.0

-0.5

-1.0

F!" [

fm]

1440140013601320

s1/2

[MeV]

!"(I=0)

Im F

Fig. 2 Forward scattering amplitudes FKN (left) and F!" (right). Real parts are shown assolid lines and imaginary parts as dashed lines. The amplitudes shown are related to the Tij

in Eq. (2) by Fi = !MiTii/(4!"

s).

valance of the two attractive forces. As we emphasized in the previous section, themeson-baryon interaction is governed by the chiral low energy theorem. Hence, weconsider that the two-pole structure of the !(1405) is a natural consequence of chiralsymmetry.

4 E!ective single-channel interaction

Keeping the structure of the !(1405) in mind, we construct an e!ective single-channelKN interaction which incorporates the dynamics of the other channels 2-4 ("#, $N ,and K%). We would like to obtain the solution T11 of Eq. (2) by solving a single-channelequation with kernel interaction V e!, namely,

T e! = V e! + V e! G1 T e! = T11.

Consistency with Eq. (2) requires that V e! be the sum of the bare interaction inchannel 1 and the contribution V11 from other channels:

V e! = V11 + V11, V11 =4X

m=2

V1m Gm Vm1 +4X

m,l=2

V1m Gm T(3)ml Gl Vl1, (3)

T(3)ml = Vml +

4X

k=2

Vmk Gk T(3)kl , m, l = 2, 3, 4.

where T(3)ml is the 3 ! 3 matrix with indices 2-4, and expresses the resummation of

interactions other than channel 1. Note that V11 includes iterations of one-loop termsin channels 2-4 to all orders, stemming from the coupled-channel dynamics. This is anexact transformation, as far as the KN scattering amplitude is concerned.

The e!ective KN interaction V e! is calculated within a chiral coupled-channelmodel [14]. It turns out that the "# and other coupled channels enhance the strengthof the interaction at low energy, although not by a large amount. The primary e!ectof the coupled channels is found in the energy dependence of the interaction kernel. Inthe left panel of Fig. 2, we show the result of KN scattering amplitude T e!, which isobtained by solving the single-channel scattering equation with V e!. The full amplitudein the "# channel is plotted in the right panel for comparison. It is remarkable that theresonance structure in the KN channel is observed at around 1420 MeV, higher thanthe nominal position of the !(1405). What is experimentally observed is the spectrum

14051420

T. Hyodo, W. W. : Phys. Rev. C77 (2008) 03524

The TWO POLES scenario (contd.)

Note difference in spectral maxima of and !!KN

D. Jido et al. , NP A725 (2003) 263

KN !! amplitudesand

Johannes Siebenson 5 24.10.2012

Our !(1405) Signals

!(1405)+p+K+

"(1385)0+p+K+

!(1520)+p+K+

"+/-+#-/++p+K+

p+p !

"+#- + "-#+ channel

Johannes Siebenson 5 24.10.2012

Our !(1405) Signals

!(1405)+p+K+

"(1385)0+p+K+

!(1520)+p+K+

"+/-+#-/++p+K+

p+p !

"+#- + "-#+ channel

Interference Analysis of !(1405)

Johannes Siebenson

Technische Universität München and Excellence Cluster Universe

PRODUCTION of p p ! K+p !±!!

!(1405) in

HADES collaborationarXiv:1208.0205Nucl. Phys. A 881 (2012) 178

thanks to:J. Siebenson & L. Fabbietti

PHOTOPRODUCTION of !(1405)

16

)2 Invariant Mass (GeV/c! "1.3 1.4 1.5

b/G

eV)

µ/d

m (

#d

0

1

2

3 1.95<W<2.05 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(a)


b/G

eV)

µ/d

m (

#d0

1

2

3

4 2.05<W<2.15 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(b)


b/G

eV)

µ/d

m (

#d

0

0.5

1

1.5

2

2.52.15<W<2.25 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(c)


b/G

eV)

µ/d

m (

#d

0

0.5

1

1.52.25<W<2.35 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(d)


b/G

eV)

µ/d

m (

#d

0

0.5

1

1.52.35<W<2.45 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(e)


b/G

eV)

µ/d

m (

#d

0

0.5

12.45<W<2.55 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(f)


b/G

eV)

µ/d

m (

#d

0

0.2

0.4

0.6

0.8 2.55<W<2.65 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(g)


b/G

eV)

µ/d

m (

#d

0

0.2

0.4

0.62.65<W<2.75 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(h)


b/G

eV)

µ/d

m (

#d

0

0.1

0.2

0.3

0.4

0.52.75<W<2.85 (GeV)

thresholds! "

-! +"0! 0"+! -"

PDG Breit-Wigner

(i)

FIG. 17. (Color online) Mass distribution results for all three Σπ channels. The weighted average of the two Σ+π− channels is shown in the red circles, the Σ0π0

channel is shown as the blue squares, and the Σ−π+ channel is shown as the green triangles. The Σ0π0 and Σ−π+ channels have inner error bars representing thestatistical errors, and outer error bars that have the estimated residual discrepancy of the data and fit results added in quadrature. The dashed line represents arelativistic Breit-Wigner function with a mass-dependent width, with the mass and width taken from the PDG and with arbitrary normalization. The vertical dashedlines show the opening of each Σπ threshold.

at JLAB

CLAS collaboration

arXiv:1301.5000

Note difference in spectral maxima

for

photoproduction vs.

proton-induced production

! p ! K+"!

plus potentially important information about K-NN absorption


Predicted antikaon-neutron amplitudes at and below thresholdY. Ikeda, T. Hyodo, W. Weise : Phys. Lett. B 706 (2011) 63 , Nucl. Phys. A 881 (2012) 98

Needed: accurate constraints from antikaon-deuteron threshold measurements

complete information for both isospin I = 0 and channelsI = 1 KN

subtraction constants as in the Q = 0 sector, the calculated K−n scattering

lengths are:

a(K−n) = 0.29 + i 0.76 fm (TW) , (29)

a(K−n) = 0.27 + i 0.74 fm (TWB) , (30)

a(K−n) = 0.57 + i 0.73 fm (NLO) . (31)

The relatively large jump in Re a(K−n) when passing from “TW” and “TWB”

to the best-fit “NLO” scheme is strongly correlated to the corresponding

change in Re a(K−p). Thus, to determine the I = 1 component of the KNscattering length, it is highly desirable to extract the K−n scattering length,

e.g. from a precise measurement of kaonic deuterium [23, 16].

Next, consider the subthreshold extrapolation of the complex elastic K−namplitude. Fig. 5 shows the real and imaginary parts of this amplitude. Note

that the I = 1 KN interaction is also attractive but weaker than the I = 0

interaction so that f(K−n → K−n) is non-resonant. In the absence of

empirical threshold constraints for the K−n scattering length one still faces

relatively large uncertainties. Variation of the subtraction constants within

the range of Eq. (23) applied to the NLO scheme leads to the following

estimated uncertainties:

a(K−n) = 0.57+0.04−0.21 + i 0.72

+0.26−0.41 fm . (32)

√s [MeV]

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1360 1380 1400 1420 1440

TWTWBNLO

Re

f(K

−n

→K

−n)

[fm

]

√s [MeV]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1340 1360 1380 1400 1420 1440

TWTWBNLO

Imf(K

−n

→K

−n)

[fm

]

Figure 5: Real part (left) and imaginary part (right) of the K−n → K−n forward scat-tering amplitude extrapolated to the subthreshold region.

16

K!n scattering length and amplitude

Tetsuo Hyodo

October 24, 2011

The K!n scattering length is calculated as

aK!n = 0.29 + i0.76 fm (WT)

aK!n = 0.27 + i0.74 fm (WTB)

aK!n = 0.57 + i0.72 fm (NLO)

The scattering amplitude is shown in Fig. .The jump of the real part of the scattering length in the step WTB " NLO is correlated with

the jump of the K!p scattering length:

aK!p = !0.93 + i0.82 fm (WT)

aK!p = !0.94 + i0.85 fm (WTB)

aK!p = !0.70 + i0.89 fm (NLO)

Note that the results with the WT and WTB models are a bit o! the SIDDHARTA result.

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

re T

[fm

]

14401420140013801360

sqrt s [MeV]

real part

WT WTB NLO

0.8

0.6

0.4

0.2

0.0

im T

[fm

]

14401420140013801360

sqrt s [MeV]

imaginary part

WT WTB NLO

Figure 1: Scattering amplitude in K!n channel.

1

K!n scattering length and amplitude

Tetsuo Hyodo

October 24, 2011

The K!n scattering length is calculated as

aK!n = 0.29 + i0.76 fm (WT)

aK!n = 0.27 + i0.74 fm (WTB)

aK!n = 0.57 + i0.72 fm (NLO)

The scattering amplitude is shown in Fig. .The jump of the real part of the scattering length in the step WTB " NLO is correlated with

the jump of the K!p scattering length:

aK!p = !0.93 + i0.82 fm (WT)

aK!p = !0.94 + i0.85 fm (WTB)

aK!p = !0.70 + i0.89 fm (NLO)

Note that the results with the WT and WTB models are a bit o! the SIDDHARTA result.

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

re T

[fm

]

14401420140013801360

sqrt s [MeV]

real part

WT WTB NLO

0.8

0.6

0.4

0.2

0.0

im T

[fm

]14401420140013801360

sqrt s [MeV]

imaginary part

WT WTB NLO

Figure 1: Scattering amplitude in K!n channel.

1

Re f(K!

n ! K!

n) Im f(K!

n ! K!

n)[fm]

[fm]

!

s [MeV]!

s [MeV]

TW

TW

TW + Born

TW + Born

NLO

NLO

ANTIKAON - DEUTERON THRESHOLD PHYSICS

. . . looking forward to SIDDHARTA 2

Strategies: Multiple scattering (MS) theory vs. three-body (Faddeev) calculations with Chiral SU(3) Coupled Channels input

MS approach (fixed scatterer approximation): K!

d scattering length

K!

N N

ap =

!

1 +mK

MN

"

a(K!p ! K!p)

an =

!

1 +mK

MN

"

a(K!

n ! K!

n)

Table 4: Parameters of the deuteron wave functions [6].

j Cj (fm!1/2) Dj (fm!1/2)1 0.88472985 0.22623762 ! 10!1

2 "0.26408759 "0.504710563 "0.44114404 ! 10!1 0.562788974 "0.14397512 ! 102 "0.16079764 ! 102

5 0.85591256 ! 102 0.11126803 ! 103

6 "0.31876761 ! 103 "0.44667490 ! 103

7 0.70336701 ! 103 0.10985907 ! 104

8 "0.90049586 ! 103 "0.16114995 ! 104

9 0.66145441 ! 103 (11)10 "0.25958894 ! 103 (12)11 (10) (13)

0.8

0.6

0.4

0.2

0.0

wav

e fu

nctio

n [f

m^-

1/2]

43210r [fm]

u(r) w(r)

Figure 4: Wave functions of CD-Bonn [6].

Table 5: Estimate of K!d scattering length with Refs. [2, 3] and realistic deuteron wave functionsof CD-Bonn [6].

Input [2, 3] CD-BonnFull model AKd [fm] "1.54 + i1.64No charge exchange AKd [fm] "1.04 + i1.34Impulse approximation AKd [fm] "0.13 + i1.81

6

r [fm]

u(r)

w(r)

wave

functio

n[fm

!1/2]

a(K!d) =

!

1 +mK

Md

"

!1 #

"

0

dr$

u2(r) + w

2(r)%

A(r)

A(r) =ap + an + (2 ap an ! b2

x)/r ! 2b2x an/r2

1 ! ap an/r2 + b2x an/r3

K!p ! K0 n K0 n ! K0 nbx incorporates and

Using IHW input scattering lengths constrained by SIDDHARTA kaonic hydrogen:

K!

N N

Table 4: Parameters of the deuteron wave functions [6].

j Cj (fm!1/2) Dj (fm!1/2)1 0.88472985 0.22623762 ! 10!1

2 "0.26408759 "0.504710563 "0.44114404 ! 10!1 0.562788974 "0.14397512 ! 102 "0.16079764 ! 102

5 0.85591256 ! 102 0.11126803 ! 103

6 "0.31876761 ! 103 "0.44667490 ! 103

7 0.70336701 ! 103 0.10985907 ! 104

8 "0.90049586 ! 103 "0.16114995 ! 104

9 0.66145441 ! 103 (11)10 "0.25958894 ! 103 (12)11 (10) (13)

0.8

0.6

0.4

0.2

0.0

wav

e fu

nctio

n [f

m^-

1/2]

43210r [fm]

u(r) w(r)

Figure 4: Wave functions of CD-Bonn [6].

Table 5: Estimate of K!d scattering length with Refs. [2, 3] and realistic deuteron wave functionsof CD-Bonn [6].

Input [2, 3] CD-BonnFull model AKd [fm] "1.54 + i1.64No charge exchange AKd [fm] "1.04 + i1.34Impulse approximation AKd [fm] "0.13 + i1.81

6

full MSno charge exchangeimpulse approximation

a(K!

d) [fm]T. Hyodo, Y. Ikeda, W. W.(2012) preliminary

S.S. Kamalov, E. Oset, A. Ramos: Nucl. Phys. A 690 (2001) 494

ANTIKAON - DEUTERON SCATTERING LENGTH

Fig. 4. Uncertainty of the boundary of allowed values for AKd. The central value(solid line) corresponds to the average of ap from scattering data and the SID-DHARTA value; uncertainty (shaded area) from the combined errors of these twosources.

4. In summary, we have reanalysed the predictions for the kaon-deuteronscattering length in view of the new kaonic hydrogen experiment from SID-DHARTA. Based on consistent solutions for input values of the K!p scatter-ing length, we have explored the allowed ranges for the isoscalar and isovectorkaon-nucleon scattering lengths and explored the range of the complex-valuedkaon-deuteron scattering length that is consistent with these values. In partic-ular, the new SIDDHARTA measurement is shown to resolve inconsistenciesfor a0, a1, and AKd as they arose from the DEAR data. A precise measure-ment of the K!d scattering length from kaonic deuterium would thereforeserve as a stringent test of our understanding of the chiral QCD dynamicsand is urgently called for.

Acknowledgments

We thank Akaki Rusetsky for stimulating discussions. Partial financial supportfrom the EU Integrated Infrastructure Initiative HadronPhysics2 (contractnumber 227431) and DFG (SFB/TR 16, “Subnuclear Structure of Matter”)is gratefully acknowledged.

References

[1] M. Bazzi, G. Beer, L. Bombelli, A. M. Bragadireanu, M. Cargnelli, G. Corradi,C. Curceanu, A. d’U!zi et al., [arXiv:1105.3090 [nucl-ex]].

7

Recent calculations using SIDDHARTA - constrained input

Multiple scattering using IHW scattering lengthsT. Hyodo, Y. Ikeda, W. W.(2012-13) preliminary

Faddeev calculation separable “chiral” amplitudes

N.V. Shevchenko NPA 890-891 (2012) 50

M. Döring, U.-G. MeißnerPhys. Lett. B 704 (2011) 663

Non-relativisticeffective field theory

Predicted energy shift and width of kaonic deuterium (Faddeev calculation):

!E1s = !794 eV !1s(K!

d) = 1012 eV

Primary theoretical uncertainties from K!

n amplitude

Not included: K!

d ! YN absorption

UPDATE on QUASIBOUND K pp-

Variational calculations 3-Body (Faddeev) calculations

. . . now consistently using amplitudes from Chiral SU(3) coupled-channels dynamics including energy dependence in subthreshold extrapolations

K!pp calculated binding energies & widths (in MeV)

chiral, energy dependent non-chiral, static calculations

var. [1] var. [2] Fad. [3] var. [4] Fad. [5] Fad. [6] var. [7]

B 16 17–23 9–16 48 50–70 60–95 40–80

! 41 40–70 34–46 61 90–110 45–80 40–85

1. N. Barnea, A. Gal, E.Z. Liverts, PLB 712 (2012)

2. A. Dote, T. Hyodo, W. Weise, NPA 804 (2008) 197, PRC 79 (2009) 014003

3. Y. Ikeda, H. Kamano, T. Sato, PTP 124 (2010) 533

4. T. Yamazaki, Y. Akaishi, PLB 535 (2002) 70

5. N.V. Shevchenko, A. Gal, J. Mares, PRL 98 (2007) 082301

6. Y. Ikeda, T. Sato, PRC 76 (2007) 035203, PRC 79 (2009) 035201

7. S. Wycech, A.M. Green, PRC 79 (2009) 014001 (including p waves)

Robust binding & large widths; chiral models give weak binding.

18

Calculated binding energy and width (in MeV) of the K!pp system[1] [2] [3]

[3]

[1]

[2]

Variational (hyperspherical harmonics):

modest binding

large width

Variational (Gaussian trial wave functions):

Faddeev:

N. Barnea, A. Gal, E.Z. Livets ; Phys. Lett. B 712 (2012) 132

A. Doté, T. Hyodo, W. W. ; Phys. Rev. C 79 (2009) 014003

Y. Ikeda, H. Kamano, T. Sato ; Prog. Theor. Phys. 124 (2010) 533

remarkable degree of consistency

STRANGENESS

in BARYONIC MATTER

andNEWS from NEUTRON STARS

Part II.

Kaon spectrum in matter determined by:

Figure 3

Figure 4

19

!2! "q2

! m2

K ! !K(!,"q; #) = 0

!K! = 2!UK! = !4"!

fK!p #p + fK!n #n

"

+ ...

Pauli blocking, Fermi motion,

2N correlations

In-medium Chiral SU(3) Dynamics with Coupled Channels

K mass

+

-

K width / 100 MeV-

Note: In-medium K width drops when mass falls below _

threshold!!

T. Waas, N. Kaiser, W. W. :Phys. Lett. B 379 (1996) 34

T. Waas, W. W. :Nucl. Phys. A 625 (1997) 287

Kaons and Antikaons in Nuclear MatterBrief History, Part I

M. Lutz, C.L. Korpa, M. Möller Nucl. Phys. A 808 (2008) 124

V. KochPhys. Lett. B 337 (1994) 7

M. LutzPhys. Lett. B 426 (1998) 12

A. Ramos, E. OsetNucl. Phys. A 671 (2000) 481

-

m!

K(!)

mK

!/!0

K mass+

Figure 5

20

first suggested by D. Kaplan, A. Nelson (1985) on the basis of attractive K N Tomozawa - Weinberg term

T. Waas, M. Rho, W. W. : Nucl. Phys. A 617 (1997) 449

at high density, energetically favourable to condense K-

in-mediumchiral SU(3) dynamics

including NN correlations

electron chemical potential

. . . but: conversion of kaons to hyperons via K!

NN ! YN

-

Kaon Condensation in Neutron Star Matter ?

?

!

Brief History, Part II

m!

K(!)

mK

!/!0

µe/mK

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= 4!r

2E

c2

dP

dr= !

G

c2

(M + 4!Pr3)(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

– 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-! (1-!) contours implied by the data. The lowerpanes summarize the 2-! 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 PRL 105 (2010) 161102

realistic “nuclear” EoS

A.W. Steiner, J. Lattimer, E.F. Brown Astroph. J. 722 (2010) 33

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

NEUTRON STAR MATTEREquation of State

Including new 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 · !!

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

Nucl. Phys. A880 (2012) 65

pressure

energy density

HARD Equation of State required !

NEUTRON STAR Equation of State

T. Hell, B. Röttgers,S. Schultess, N. Kaiser

W. W. (2012-13)

ChEFT, resummed �n , p , e , Μ�PNJL, Gv � 0.7 G , T � 0 MeV�d , u , e ��d , u , e �PNJL, Gv � 0 , T � 0 MeV

SLy , FKW , polytropes

10 20 50 100 200 500 10000.1

1

10

100

Ε �MeV fm�3�

P�MeV

fm�3 �

pressure

energy density

(n, p, e, µ)

Gv/G = 0.7 (d, u, e)

Gv = 0 (d, u, e)

“green belt”permitted by M(R) constraints

realistic“conventional”

nuclear EoS

PNJL withvector coupling PNJL without

vector coupling

quark-nuclearmatching region

(2 versions of chiral quark model)

see also:

K. Masuda,T. Hatsuda,T.TakatsukaAstroph. J.

764 (2013) 12

M � 2.00 M�R � 11.94 km

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

r �km �

Ρ�fm�3

�densityprofile

NEUTRON STAR MATTERMass - Radius relation

Conventional hadronic (baryonic + mesonic) degrees of freedom

In-medium Chiral Effective Field Theory up to 3 loops S. Fiorilla, N. Kaiser, W. W. Nucl. Phys. A 880 (2012) 65

T. Hell, B. Röttgers, S. Schultess,

N. Kaiser, W. W. (2012-13)

(reproducing thermodynamics of normal nuclear matter) including beta equilibrium n ! p + e, µ

M = 2MO.

neutron star core density is less than four times !0

R = 11.9 km!0 = 0.16 fm

!3

(density of normal nuclear matter)

NEUTRON STARS: MASS and RADIUS constraints

from:J. Trümper

Irsee Symposium

2012

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

HARD Equation of State required !!

ChEFT�PNJL, Gv � 0.5G, �d, u, e�ChEFT�PNJL, Gv � 0, �d, u, e�SLy, FKW, polytropes

50 100 200 300 500 1000 20001

51020

100200

Ε �MeV fm�3�

P�MeV

fm�3 � “green” and “blue”belts

permitted byM(R) constraints

Chiral EFT EoS

quark-nuclear coexistence (3-flavor PNJL)

Gv = 0.5 G

Gv = 0

T. Hell, W. W.(2013) preliminary

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

! > 5 !0

PSR J1614�2230PSR J1614�2230

PNJL �Gv � 0 G � � ChEFT �n, p, e, Μ�, GibbsPNJL �Gv � 0.5 G � � ChEFT �n, p, e, Μ�, Gibbs

8 10 12 14 160.0

0.5

1.0

1.5

2.0

2.5

R �km�

M�M �

Chiral EFT + PNJL

T. Hell, W. W.(2013) prel.

Gv = 0

Gv = 0.5 G

Trümper

Steiner et al.

d quarks

u quarkss quarks

protons

neutrons

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

Ρ�Ρ0Ρ i�Ρ

neutrons

protons

quarks

du

s

NEUTRON STAR MATTERMass - Radius Relation and Composition

In-medium Chiral Effective Field Theory + 3-flavor PNJL model

Strangeness: small admixtures of hyperons possible at ! ! > 3 !0

but: strong hyperon-nucleon short-range repulsion and/or 3-body forces required !NN

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

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(27)

-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(1–0–)

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

!u,d,s=0.13840

VCVT

Fig. 13. The BB potentials in 27 (Left) and 10 (Right) representations extracted from the latticeQCD simulation at Mps = 469 MeV. Taken from Ref. 56).

is related to the !NN coupling as f!"" = !f!NN (1! 2") with the parameter " =F/(F +D), and g!NN = f!NN

m!2mN

. The empirical vales, " " 0.36 and g!NN/(4!) "14.0, are used for the plot. Unlike the NN potential, the OPEP in the presentcase has opposite sign between the spin-singlet channel and spin-triplet channel.In addition, the absolute magnitude is smaller due to the factor 1 ! 2". No clearsignature of the OPEP at long distance (r # 1.2 fm) is yet observed in Fig. 12 withinstatistical errors.

§6. Baryon interaction in the flavor SU(3) limit

6.1. Potentials in the flavor SU(3) limit

In order to reveal the nature of the hyperon interactions in various channels,it is more convenient to consider an idealized flavor SU(3) symmetric world, whereu, d and s quarks are all degenerate with a common finite mass. In this limit, onecan capture essential features of the interaction, in particular, the short range forcewithout contamination from the quark mass di!erence.

In the flavor SU(3) limit, the ground state baryon belongs to the flavor-octetwith spin 1/2, and two-baryon states with a given angular momentum can be labeledby the irreducible representation of SU(3) as

8$ 8 = 27% 8% 1! "# $

symmetric

% 10% 10% 8! "# $

anti!symmetric

, (6.1)

where ”symmetric” and ”anti-symmetric” stand for the symmetry under the ex-change of the flavor for two baryons. For the system with orbital S-wave, the Pauliprinciple for baryons imposes 27, 8 and 1 to be spin-singlet (1S0), while 10, 10 and8 to be spin-triplet (3S1 !3 D1). Calculations in the SU(3) limit allow us to extractpotentials for these six flavor irreducible multiplets as follows.

A two-baryon operator BB(X) which belongs to one definite flavor representationX, can be given in terms of the baryon base operator with the corresponding Clebsch-Gordan (CG) coe"cients CX

ij as BB(X) =%

ij CXij BiBj. By using this operator at

source and/or sink, the NBS wave function for two-baryon system in the flavor

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

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(27)

-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(1–0–)

-100

-50

0

50

100

0.0 0.5 1.0 1.5 2.0 2.5

!u,d,s=0.13840

VCVT

Fig. 13. The BB potentials in 27 (Left) and 10 (Right) representations extracted from the latticeQCD simulation at Mps = 469 MeV. Taken from Ref. 56).

is related to the !NN coupling as f!"" = !f!NN (1! 2") with the parameter " =F/(F +D), and g!NN = f!NN

m!2mN

. The empirical vales, " " 0.36 and g!NN/(4!) "14.0, are used for the plot. Unlike the NN potential, the OPEP in the presentcase has opposite sign between the spin-singlet channel and spin-triplet channel.In addition, the absolute magnitude is smaller due to the factor 1 ! 2". No clearsignature of the OPEP at long distance (r # 1.2 fm) is yet observed in Fig. 12 withinstatistical errors.

§6. Baryon interaction in the flavor SU(3) limit

6.1. Potentials in the flavor SU(3) limit

In order to reveal the nature of the hyperon interactions in various channels,it is more convenient to consider an idealized flavor SU(3) symmetric world, whereu, d and s quarks are all degenerate with a common finite mass. In this limit, onecan capture essential features of the interaction, in particular, the short range forcewithout contamination from the quark mass di!erence.

In the flavor SU(3) limit, the ground state baryon belongs to the flavor-octetwith spin 1/2, and two-baryon states with a given angular momentum can be labeledby the irreducible representation of SU(3) as

8$ 8 = 27% 8% 1! "# $

symmetric

% 10% 10% 8! "# $

anti!symmetric

, (6.1)

where ”symmetric” and ”anti-symmetric” stand for the symmetry under the ex-change of the flavor for two baryons. For the system with orbital S-wave, the Pauliprinciple for baryons imposes 27, 8 and 1 to be spin-singlet (1S0), while 10, 10 and8 to be spin-triplet (3S1 !3 D1). Calculations in the SU(3) limit allow us to extractpotentials for these six flavor irreducible multiplets as follows.

A two-baryon operator BB(X) which belongs to one definite flavor representationX, can be given in terms of the baryon base operator with the corresponding Clebsch-Gordan (CG) coe"cients CX

ij as BB(X) =%

ij CXij BiBj. By using this operator at

source and/or sink, the NBS wave function for two-baryon system in the flavor

Lattice QCD approach to Nuclear Physics 27

-500

0

500

1000

1500

2000

2500

3000

3500

0.0 0.5 1.0 1.5 2.0 2.5

V(r)

[MeV

]

r [fm]

-50

0

50

100

150

0.0 0.5 1.0 1.5 2.0 2.5

1S0

S=!2, I=0

"",""##,##N$,N$

-3000

-2000

-1000

0

1000

2000

0.0 0.5 1.0 1.5 2.0 2.5V(r)

[MeV

]

r [fm]

1S0

S=!2, I=0

"",##"",N$##,N$

Fig. 15. BB potentials in particle basis for the S=!2, I=0, 1S0 sector. Three diagonal(o!-diagonal)potentials are shown in left(right) panel.

0

1000

2000

3000

4000

5000

0.0 0.5 1.0 1.5 2.0 2.5

(!)V(r) [

MeV

]

r [fm]

VC

-50

0

50

100

150

0.0 0.5 1.0 1.5 2.0 2.5

V(r)

[MeV

]

1S0

S=!1, I=1/2

N",N"N#,N#N",N#

0

200

400

600

800

1000

1200

1400

0.0 0.5 1.0 1.5 2.0 2.5

(!)V(r) [

MeV

]

r [fm]

VC

-50

0

50

100

150

0.0 0.5 1.0 1.5 2.0 2.5

V(r)

[MeV

]

JP=1+

S=!1, I=1/2

N",N"N#,N#N",N#

-100

-80

-60

-40

-20

0

20

40

0.0 0.5 1.0 1.5 2.0

V(r)

[MeV

]

r [fm]

VTJP=1+

S=!1, I=1/2

N",N"N#,N#N",N#

Fig. 16. BB potentials in particle basis for the S=!1, I=1/2 sector, 1S0 (Left), spin-triplet VC

(Center) and spin-triplet VT (Right), extracted from the lattice QCD simulation at Mps = 469MeV.

attraction,

V (r) = b1e!b2 r2 + b3(1! e!b4 r2)2

!e!b5 r

r

"2

, (6.3)

with five parameters b1,2,3,4,5. The left panel of Fig. 15 shows the diagonal potentials.One observes that all three diagonal potentials have a repulsive core. The repulsionis most strong in the !!(I=0) channel, reflecting its largest CG coe!cient of the 8sstate among three channels, while the attraction in the 1 state is reflected most inthe N"(I=0) potential due to its largest CG coe!cient. The right panel of Fig. 15shows the o"-diagonal potentials, which are comparable in magnitude to the diagonalones, except for the ##-N" transition potential. Since the o"-diagonal parts arenot negligible in the particle basis, a fully coupled channel analysis is necessary tostudy observables in this system. This is important when we study the real worldwith the flavor SU(3) breaking, where only the particle base is meaningful.

As another example, we show BB potentials for S=!1, I=1/2 sector at MPS =469 MeV in Fig. 16. The left panel of Fig. 16 shows the potential in the 1S0 channel,while the center (right) panel of Fig. 16 shows the central (tensor) potential forJP = T+

1 spin-triplet channel. We observe that the o"-diagonal N#-N! potentialsare significantly large, especially in the spin-triplet channel, so that the full N#-N!coupled channel analysis is also necessary to study observables. In addition, therepulsive cores in the spin-single channel are much stronger than that in the spin-

Hyperon - Nucleon Potentials from Lattice QCD

PTEP 2012 (2012) 01A105 ; arXiv: 1206.5088 [hep-lat]

Strong repulsionat short distances

works against superdense strange matter

0.5 fm

1.2 fm

baryoniccore

pionicfield

!B = 0.6 fm!3

normal nuclear matter: dilute

0.5 fm

2 fm

baryoniccore

pionicfield

!B = 0.15 fm!3

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1

2

0.2 0.4 0.6 0.8 1.0 1.2 r [fm]

[fm!1]

4!r2

"B(r)

4!r2

"S(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 properlyin 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

Two-solar-mass neutron star sets strong constraints:

“Stiff”

equation-of-state required

“Conventional” EoS works best (nuclear physics + effective field theory + advanced many-body methods)

Small admixtures still possible, but only with strongly repulsive short-range hyperon-nucleon and hyperon-hyperon interactions ( see Lattice QCD)

Strangeness and hyperons in dense baryonic matter ?

Change of paradigm ? “Exotic” forms of matter (kaon condensates, quark matter with first-order phase transition, . . . ) unlikely to exist in n-star cores

Central baryon densities in neutron stars are not extreme(do not exceed 4 - 5 times the density of normal nuclear matter)

neutronmatter

quark matter ?

. . . looking forward to kaonic deuterium (SIDDHARTA 2)

SUMMARY

Chiral SU(3) Coupled-Channels approach based on symmetry breaking pattern of Low-Energy QCD

. . . is now a well established framework:

“Non-exotic” stiff equation-of-state works best !

New dense & cold matter constraints from neutron stars:

Mass - radius relation; observation of 2-solar-mass n-star

antiK-nucleon & antiK-deuteron scattering lengths

Remarkable consistency of variational and Faddeev K-pp calculations:weak binding, large width

K-deuteron threshold physics: predictions now available

Kaon condensation ruled out

Hyperon admixtures still possible but with requirement of strongly repulsive short-range YN interactions ( Lattice QCD)

!anks to :

Tetsuo Hyodo Yoichi Ikeda Salvatore Fiorilla Thomas Hell Jeremy Holt Norbert Kaiser Bernhard Röttgers Sebastian Schultess

"e End

Documents

STRANGENESS in LOW ENERGY QCD and DENSE ......the low-energy pion-, kaon and antikaon scattering data was achieved earlier within the χ-BS(3) scheme upon incorporating chiral correction