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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
“light” quarks “heavy” quarks
0 !
1 GeVu,d s c
“light” quarks “heavy” quarks
K N
!
!
0 !
1 GeVu,d s c
“light” quarks “heavy” quarks
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
“light” quarks “heavy” quarks
0 !
1 GeVu,d s c
“light” quarks “heavy” quarks
K N
!
!
0 !
1 GeVu,d s c
“light” quarks “heavy” quarks
K N
! !
K N !
!
CHIRAL SU(3) COUPLED CHANNELS DYNAMICS
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)
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)
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
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)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
b/G
eV)
µ/d
m (
#d0
1
2
3
4 2.05<W<2.15 (GeV)
thresholds! "
-! +"0! 0"+! -"
PDG Breit-Wigner
(b)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
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)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
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)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
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)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
b/G
eV)
µ/d
m (
#d
0
0.5
12.45<W<2.55 (GeV)
thresholds! "
-! +"0! 0"+! -"
PDG Breit-Wigner
(f)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
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)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
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)
)2 Invariant Mass (GeV/c! "1.3 1.4 1.5
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
CHIRAL SU(3) COUPLED CHANNELS DYNAMICS
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 , �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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01rJ,4 UeC 3,4A
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