Theory of ANTIKAON NUCLEON INTERACTIONS …...Theory of ANTIKAON - NUCLEON INTERACTIONS with new constraints from kaonic hydrogen Wolfram Weise Low-energy QCD with strange quarks:

50th Winter Meeting on Nuclear Physics

Theory of ANTIKAON - NUCLEON INTERACTIONS

with new constraints from kaonic hydrogen

Wolfram Weise

Low-energy QCD with strange quarks:Chiral SU(3) Effective Field Theory + Coupled Channels

KN threshold physics and kaonic hydrogen

25 January 2012

New next-to-leading-order (NLO) analysis with constraints from SIDDHARTA measurements

New determination of K - N scattering lengths -

Bormio

Implications and outlooks

LOW-ENERGY QCD

with STRANGE QUARKS

and SPONTANEOUSLY BROKEN

CHIRAL SYMMETRY

1.

. . . realized as an EFFECTIVE FIELD THEORY with SU(3) octet of pseudoscalar Nambu-Goldstone bosons coupled to the baryon octet

. . . explicit chiral symmetry breaking by non-zero quark masses (at a renormalization scale ):

mq =mu + md

2= 3 ! 5 MeV

µ ! 2 GeV

ms ! 25 mq

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

new constraints from neutron stars

m2!f2!

= !mq "!!# + O(m2

q)

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

f! = 120.1 ± 4.6 MeV

spontaneous symmetry breaking

explicit symmetry breaking

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

!

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


K N

!

!

0 !

1 GeVu,d s c


K N

! !K N !

!

KN!!

channel coupling

0 !

1 GeVu,d s c


0 !

1 GeVu,d s c


K N

!

!

0 !

1 GeVu,d s c


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)


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

(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


Tij = Kij +

!

n

Kin Gn Tnj

(contd.)

loop integrals (with meson-baryon Green functions) using dimensional regularization:

finite parts including subtraction constants a(µ) :

channels: K!p, K0n, !0!0

, !+!!

, !!!+

, !0", "", "!0

, K+#!

, K!#0

KN!!

The TWO POLES scenario

D. Jido et al. , Nucl. Phys. A725 (2003) 181

dominantly!!

dominantlyKN

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

THRESHOLD

and LOW-ENERGY OBSERVABLES

2.KN

Kaonic hydrogen

Threshold branching ratios

Low-energy scattering data

arX

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3090

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nucl

-ex]

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A New Measurement of Kaonic Hydrogen X rays

M. Bazzia, G. Beerb, L. Bombellic, A.M. Bragadireanua,d, M. Cargnellie,!,G. Corradia, C. Curceanu (Petrascu)a, A. d’U!zia, C. Fiorinic, T. Frizzic,

F. Ghiof, B. Girolamif, C. Guaraldoa, R.S. Hayanog, M. Iliescua,d,T. Ishiwatarie, M. Iwasakih, P. Kienlee,i, P. Levi Sandria, A. Longonic,

V. Lucherinia, J. Martone, S. Okadaa,!, D. Pietreanua, T. Pontad, A. Rizzoa,A. Romero Vidala, A. Scordoa, H. Shig, D.L. Sirghia,d, F. Sirghia,d,H. Tatsunog, A. Tudorached, V. Tudorached, O. Vazquez Docea,

E. Widmanne, J. Zmeskale

aINFN, Laboratori Nazionali di Frascati, Frascati (Roma), ItalybDep. of Phys. and Astro., Univ. of Victoria, Victoria B.C., Canada

cPolitecnico di Milano, Sez. di Elettronica, Milano, ItalydIFIN-HH, Magurele, Bucharest, Romania

eStefan-Meyer-Institut fur subatomare Physik, Vienna, AustriafINFN Sez. di Roma I and Inst. Superiore di Sanita, Roma, Italy

gUniv. of Tokyo, Tokyo, JapanhRIKEN, The Inst. of Phys. and Chem. Research, Saitama, Japan

iTech. Univ. Munchen, Physik Dep., Garching, Germany

Abstract

The KN system at threshold is a sensitive testing ground for low energyQCD, especially for the explicit chiral symmetry breaking. Therefore, wehave measured the K-series x rays of kaonic hydrogen atoms at the DA"NEelectron-positron collider of Laboratori Nazionali di Frascati, and have de-termined the most precise values of the strong-interaction energy-level shiftand width of the 1s atomic state. As x-ray detectors, we used large-areasilicon drift detectors having excellent energy and timing resolution, whichwere developed especially for the SIDDHARTA experiment. The shift andwidth were determined to be !1s = !283 ± 36(stat)± 6(syst) eV and #1s =541± 89(stat)± 22(syst) eV, respectively. The new values will provide vitalconstraints on the theoretical description of the low-energy KN interaction.

!Corresponding authors.Email addresses: [email protected] (M. Cargnelli),

[email protected] (S. Okada)

Preprint submitted to Physics Letters B August 10, 2011

arX

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105.

3090

v2 [

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-ex]

9 A

ug 2

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A New Measurement of Kaonic Hydrogen X rays

M. Bazzia, G. Beerb, L. Bombellic, A.M. Bragadireanua,d, M. Cargnellie,!,G. Corradia, C. Curceanu (Petrascu)a, A. d’U!zia, C. Fiorinic, T. Frizzic,

F. Ghiof, B. Girolamif, C. Guaraldoa, R.S. Hayanog, M. Iliescua,d,T. Ishiwatarie, M. Iwasakih, P. Kienlee,i, P. Levi Sandria, A. Longonic,

V. Lucherinia, J. Martone, S. Okadaa,!, D. Pietreanua, T. Pontad, A. Rizzoa,A. Romero Vidala, A. Scordoa, H. Shig, D.L. Sirghia,d, F. Sirghia,d,H. Tatsunog, A. Tudorached, V. Tudorached, O. Vazquez Docea,

E. Widmanne, J. Zmeskale

aINFN, Laboratori Nazionali di Frascati, Frascati (Roma), ItalybDep. of Phys. and Astro., Univ. of Victoria, Victoria B.C., Canada

cPolitecnico di Milano, Sez. di Elettronica, Milano, ItalydIFIN-HH, Magurele, Bucharest, Romania

eStefan-Meyer-Institut fur subatomare Physik, Vienna, AustriafINFN Sez. di Roma I and Inst. Superiore di Sanita, Roma, Italy

gUniv. of Tokyo, Tokyo, JapanhRIKEN, The Inst. of Phys. and Chem. Research, Saitama, Japan

iTech. Univ. Munchen, Physik Dep., Garching, Germany

Abstract

The KN system at threshold is a sensitive testing ground for low energyQCD, especially for the explicit chiral symmetry breaking. Therefore, wehave measured the K-series x rays of kaonic hydrogen atoms at the DA"NEelectron-positron collider of Laboratori Nazionali di Frascati, and have de-termined the most precise values of the strong-interaction energy-level shiftand width of the 1s atomic state. As x-ray detectors, we used large-areasilicon drift detectors having excellent energy and timing resolution, whichwere developed especially for the SIDDHARTA experiment. The shift andwidth were determined to be !1s = !283 ± 36(stat)± 6(syst) eV and #1s =541± 89(stat)± 22(syst) eV, respectively. The new values will provide vitalconstraints on the theoretical description of the low-energy KN interaction.

!Corresponding authors.Email addresses: [email protected] (M. Cargnelli),

[email protected] (S. Okada)

Preprint submitted to Physics Letters B August 10, 2011

Physics Letters B 704 (2011) 113 arXiv:1105.3090

“WT” approach

100 150 200 250 300 350 400 450 500

200

400

600

800

DEAR KEK

!E (eV)

"(e

V)

!2/d.o.f. < 1.4

1.4 < !2/d.o.f. < 1.6

1.6 < !2/d.o.f. < 2.33

2.33 < !2/d.o.f. < 4.0

4.0 < !2/d.o.f. < 6.0

6.0 < !2/d.o.f. < 9.0

“WTB” approach

100 150 200 250 300 350 400 450 500

200

400

600

800

DEAR KEK

!E (eV)

"(e

V)

!2/d.o.f. < 0.95

0.95 < !2/d.o.f. < 1.15

1.15 < !2/d.o.f. < 1.93

1.93 < !2/d.o.f. < 3.0

3.0 < !2/d.o.f. < 5.0

5.0 < !2/d.o.f. < 8.0

“full” approach

100 150 200 250 300 350 400 450 500

200

400

600

800

DEAR KEK

!E (eV)

"(e

V)

!2/d.o.f. < 0.8

0.8 < !2/d.o.f. < 1.0

1.0 < !2/d.o.f. < 1.76

1.76 < !2/d.o.f. < 3.0

3.0 < !2/d.o.f. < 5.0

5.0 < !2/d.o.f. < 8.0

Figure 4: Strong energy shift !E and width " of kaonic hydrogen for the three approaches.The shaded areas represent di#erent upper limits of the overall !2/d.o.f. The 1" confidenceregion is bordered by the dashed line. See text for further details.

10

TW

SIDDHARTA

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

R. Nißler PhD thesis (2008)

NEWS from SIDDHARTA

Kaonic hydrogen precision data Theory: leading order

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

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

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

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

Background estimation

/456 /7689-

:12/

:12/

/968

/958/468

/98;

/<=>5

EM value!"#$!

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

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

strong interaction shift and width:

!E = 283 ± 36 (stat)±6 (syst) eV

! = 541 ± 89 (stat)±22 (syst) eV

(Tomozawa - Weinberg)

![eV ]

[eV ]

Improved constraints on chiral SU(3) dynamicsfrom kaonic hydrogen

Yoichi Ikedaa,b,∗

, Tetsuo Hyodoa

and Wolfram Weisec

aDepartment of Physics, Tokyo Institute of Technology, Meguro 152-8551, Japan

bRIKEN Nishina Center, 2-1, Hirosawa, Wako, Saitama 351-0198, Japan

cPhysik-Department, Technische Universitat Munchen, D-85747 Garching, Germany

Abstract

A new improved study of K−-proton interactions near threshold is performed using coupled-channels dynamics based on the next-to-leading order chiral SU(3) meson-baryon effective La-grangian. Accurate constraints are now provided by new high-precision kaonic hydrogen mea-surements. Together with threshold branching ratios and scattering data, these constraints per-mit an updated analysis of the complex KN and πΣ coupled-channels amplitudes and an im-proved determination of the K−p scattering length, including uncertainty estimates.

Introduction. Within the hierarchy of quark masses in QCD, the strange quark plays

an intermediate role between “light” and “heavy”. Hadronic systems with strange quarks

and, in particular, antikaon-nucleon interactions close to threshold are therefore suitable

testing grounds for investigating the interplay between spontaneous and explicit chiral

symmetry breaking in low-energy QCD.

Methods of effective field theory with coupled-channels, based on the chiral SU(3)R×SU(3)L

meson-baryon effective Lagrangian, have become a well established framework for dealing

with low-energy KN interactions [1,2] (see also Ref. [3] for a recent review). However,

previous applications of such approaches, combining information from earlier kaonic hy-

drogen measurements [4,5] and older K−p scattering data, were still subject to consid-

erable uncertainties. The theoretical studies [6–9] gave strong indications for a possible

inconsistency between the DEAR K−hydrogen data [5] and the low-energy K−p elastic

scattering cross section. With the recent appearance of results from the SIDDHARTA

kaonic hydrogen measurements [10], a new level of accuracy has now been reached that

permits an improved analysis with updated constraints. The present work describes such

∗ Corresponding author at: Department of Physics, Tokyo Institute of Technology, Meguro 152-8551,Japan

Email address: [email protected] (Yoichi Ikeda).

Preprint submitted to Elsevier 24 October 2011

Physics Letters B 706 (2011) 63arXiv: 1109.3005

UPDATED ANALYSIS of K!p THRESHOLD PHYSICS

Chiral SU(3) coupled-channels dynamics Tomozawa-Weinberg + Born terms + NLO

!(K!p ! !+"!)

!(K!p ! !!"+)

!(K!p ! !+"!

, !!"+)

!(K!p ! all inelastic channels)

!(K!p ! !0")

!(K!p ! neutral states)

threshold branching ratios

2.36 ± 0.04

0.66 ± 0.01

0.19 ± 0.02

kaonic hydrogen shift & width theory (NLO) exp.

!E (eV)

! (eV)

best fit achieved with

scattering length

541 ± 89 ± 22

283 ± 36 ± 6306

0.19

Re a(K!p) = !0.65 ± 0.10(fm)

591

0.66

Y. Ikeda, T. Hyodo, W.W. Physics Letters B 706 (2011) 63

Ima(K!p) = 0.81 ± 0.15

2.37

!2/d.o.f. ! 0.9

(SIDDHARTA)

UPDATED ANALYSIS of K!p THRESHOLD PHYSICS

Non-trivial result: best NLO fit prefers physical values of decay constants:

Table 9: Summary of the subtraction constants and low-energy constants in BNWconvention[4] and meson decay constants in Model NLO3.

Channels Fock spaces ai(µ = 1GeV) ! 10!3

1, 2 KN "2.37813 !! "16.569

4, 5, 6 !" 4.34987 "! "0.00558668 "" 1.9014

9, 10 K# 15.829fK (MeV) 110.00f! (MeV) 118.82

b0 (GeV!1) "0.047876bD (GeV!1) 0.0047648bF (GeV!1) 0.040119d1 (GeV!1) 0.086461d2 (GeV!1) "0.10623d3 (GeV!1) 0.092194d4 (GeV!1) 0.063991

Table 10: Results of fitting in NLO3. The experimental values of branching ratios aretaken from [9, 10].

Observables Theory Experiment$E (eV) 306 283 ± 42% (eV) 591 541 ± 111

# 2.36 2.36 ± 0.04Rc 0.659 0.664 ± 0.011Rn 0.192 0.189 ± 0.015

aK!p (fm) "0.81 + i0.87Poles of the !(1405) (MeV) 1424.2 " i26.3, 1380.7 " i81.3

16

NLO parameters are non-negligible but small

Tomozawa-Weinberg terms dominant

Born terms significant

(f! = 92.4 MeV )

(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

(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

with SIDDHARTA constraints

Y. Ikeda, T. Hyodo, W.W. Physics Letters B 706 (2011) 63

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

10

20

30

40

50

60

50 100 150 200 250

K- p ->

0

(mb)

Plab(MeV)

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

0

10

20

30

40

50

60

50 100 150 200 250

K- p ->

bar

K0 n(m

b)

Plab(MeV)

!(K!p ! K0n) [mb]

0

50

100

150

200

1340 1360 1380 1400 1420 1440

arbi

trary

uni

t

Ec.m. (MeV)

Mass spectrum I=0

UPDATED ANALYSIS of K!p LOW-ENERGY CROSS SECTIONS

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

SCATTERING AMPLITUDE

threshold region and subthreshold extrapolation

K!

n

f(K!

n) = fKN(I = 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

0.0im

T [f

m]

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]

complex scattering length

Im a(K!

n) = 0.72+0.3

!0.4fmRe a(K!

n) = 0.57+0.1

!0.2fm

Y. Ikeda, T. Hyodo, W.W. : Nucl. Phys. A (2012), in print

TW

TW

TW + Born

TW + Born

NLO

NLO

Implications & Comments

K!p scattering length more accurately determined K

!

n

Uncertainties in KN (I = 1) interaction primarily from large uncertainties in the

(SIDDHARTA constraints)than

K!p ! !0! channel

Kaonic deuterium measurements important for providing further constraints on K!

n interaction

absorption into non-mesonic hyperon-nucleon final statesB = 2 systems KNN ! YN - key issue:

e.g.:

K!

p

n

!(1405)

n

!

!

N

Monday, October 31, 2011

K!

p

n

!(1405)

n

!

!

N

Monday, October 31, 2011

Nuclear force in SU(3) limit

T. Inoue (HAL QCD Coll.) PRL106 (2011).

mπ ~ 1 [GeV]

Corresponding to 1S0 NN potential

V (�r) =−H0ψ(t, �r)

ψ(t, �r)−

∂

∂tlogψ(t, �r)

−∂

∂tψ(t, �r) =

�H0 + V (�r)

�ψ(t, �r)

Λ* N interaction in SU(3) limit

mπ ~ 1 [GeV]

Core of potential appears!Size of core is almost same as

for pp interaction

PRELIMINARY

Repulsive short-distance interaction ?!!(uds) N

Y. Ikeda et al. (HAL QCD collaboration)

Lattice QCD

interaction between !(1405) and residual nucleon(s)

Outlooks:

New Constraintsfrom

NEUTRON STARS

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

Neutron Star Structure

Tolman-Oppenheimer-Volkov equations

dp

dr= !

G

c2

(m + 4!pr3)(" + p)

r(r ! 2Gm/c2)dm

dr= 4!

"

c2r2

!

!

!

! maximum mass

J.M. Lattimer, NFQCD 2010, 18 January 2010 – p. 4/36

Neutron Star Structure

Tolman-Oppenheimer-Volkov equations

dp

dr= !

G

c2

(m + 4!pr3)(" + p)

r(r ! 2Gm/c2)dm

dr= 4!

"

c2r2

!

!

!

! maximum mass

J.M. Lattimer, NFQCD 2010, 18 January 2010 – p. 4/36

Phys. Reports 442 (2007) 109

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

– 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.

2

13.0 13.5 14.0log 10 [g / cm3]

31

32

33

34

35

36

37

log 1

0P

[dyn

e/cm

2 ]

1

2

with ci uncertainties

crust EOS neutron star matter

121 0.2 0.4 0.6 0.8 1.0 [ 0]

0.5

1.0

1.5

2.0

2.5

P [1

033dy

ne/c

m2 ]

00

NN only, EM NN only, EGM

FIG. 1: Pressure of neutron star matter based on chiral low-momentum interactions for densities ! < !1 (correspondingto a neutron density !1,n = 1.1!0). The band estimates the theoretical uncertainties from many-body forces and from anincomplete many-body calculation. At low densities, the results are compared to a standard crust EOS [16], where the rightpanel demonstrates the importance of 3N forces. The extension to higher densities using piecewise polytropes (as explained inthe text) is illustrated schematically in the left panel.

< 5% for densities !0/8 < ! < !1 = 3.0 ! 1014 g cm!3

(!1 corresponds to a neutron density !1,n = 1.1!0). Weobtain the following symmetry energy parameters andproton fractions:

c1 [GeV!1] c3 [GeV!1] S2 [MeV] " x(!0)

"0.7 "2.2 30.1 0.5 4.8%"1.4 "4.8 34.4 0.6 7.2%

NN-only EM 26.5 0.4 3.3%NN-only EGM 25.6 0.4 2.9%

The resulting pressure of neutron star matter is shownin Fig. 1 for densities ! < !1. The comparison of theseparameter-free calculations to a standard crust EOS [16]shows good agreement to low densities ! ! !0/10 withinthe theoretical uncertainties. The band in Fig. 1 is domi-nated by the uncertainty in c3, which may seem large, butcan be expected at leading 3N order [17]. In addition, theright panel of Fig. 1 demonstrates the importance of 3Nforces. The pressure obtained from low-momentum NNinteractions only, based on the RG-evolved chiral N3LOpotentials of Entem and Machleidt (EM) [11] or of Epel-baum et al. (EGM) [12], di!er significantly from thecrust EOS at !0/2.Neutron stars.– The structure of neutron stars (non-

rotating and without magnetic fields) is determined bysolving the Tolman-Oppenheimer-Volkov (TOV) equa-tions. Because the central densities reach values higherthan !1, we need to extend the uncertainty band forthe pressure of neutron star matter beyond !1. To thisend, we introduce a transition density !12 that separatestwo higher-density regions, and describe the pressure bypiecewise polytropes, P (!) = #1!!1 for !1 < ! < !12, andP (!) = #2!!2 for ! > !12, where #1,2 are determined bycontinuity of the pressure. Ref. [18] has shown that sucha piecewise polytropic EOS can match a large set of neu-

tron star matter EOS taking 1.5 < "1,2 < 4.0 and transi-tion densities !12 # (2.0 . . . 3.5)!0. We therefore extendthe pressure of neutron star matter based on chiral EFTusing two general piecewise polytropes, as illustrated inFig. 1, with 1.5 < "1,2 < 4.5 and 1.5 < !12/!0 < 4.5.

We solve the TOV equations for the limits of the pres-sure band below nuclear densities continued by the piece-wise polytropes to higher densities. The range of "1,2

and !12 can be constrained further, first, by causalitythat limits the speed of sound to lightspeed, and second,by requiring the EOS to support a neutron star with atleast M = 1.65M" [19]. The resulting allowed rangeof polytropes is shown by the light blue band at higherdensity in Fig. 2. The comparison with a representa-tive set of EOS used in the literature [15] demonstratesthat the pressure based on chiral EFT interactions (thedarker blue band) sets the scale for the allowed higher-density extensions and is therefore extremely important.It also significantly reduces the spread of the pressure atnuclear densities from a factor 6 at !1 in current neutronstar modeling to a factor 1.5.

Results.– In Fig. 3 we show the neutron star M -Rcurves obtained from the allowed EOS range. The blueregion corresponds to the blue band for the pressure inFigs. 1 and 2. At the limits of this region, the pressureof neutron star matter continues in form of the piece-wise polytropes, and all curves end when causality isviolated. If this is reached before a maximum mass atdM/dR = 0, one could continue the M -R curves by en-forcing causality. This would lead to a somewhat largermaximum mass, but would not a!ect the masses andradii of neutron stars with lower central densities. Weobserve from the transition density points !12 in Fig. 3that the range of "1 dominates the uncertainty of thegeneral extension to high densities. Smaller values of "1

neutron star matter

(chiral EFT)

!0 !n

News from NEUTRON STARSK. Hebeler, J. Lattimer, C. Pethick, A. Schwenk

PRL 105 (2010) 161102

realistic “nuclear” EoS(Illinois 1998)

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

kaon condensate

quark matter

New constraintsfrom EFT and neutron star observables

“Exotic” equations of state ruled out ?

NEUTRON STAR MATTEREquation of State

Including new neutron star constraints plus Chiral Effective Field Theory at lower density

B. Röttgers, W. W. (2011)

Prog. Part. Nucl. Phys.(2012) in print

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)

S. Fiorilla, N. Kaiser, W. W.arXiv:1111.3688

NPA (2012) in print

P = const · !!

SUMMARY

New consistent analysis of KN threshold physics and scattering data

Substantial improvements through constraints provided by SIDDHARTA kaonic hydrogen measurements

based on chiral SU(3) effective Lagrangian at next-to-leading order

New more precise evaluation of

a(K!p) = !0.65 + 0.81 i [fm]

Future need kaonic deuterium at SIDDHARTA 2 to complete KN

and set constraints for KNN

(~ 15 % accuracy)

a(K!

n) ! 0.6 + 0.7 i [fm] (less accurate)deduced:

K!p scattering length:

Documents

Theory of ANTIKAON NUCLEON INTERACTIONS …...Theory of ANTIKAON - NUCLEON INTERACTIONS with new constraints from kaonic hydrogen Wolfram Weise Low-energy QCD with strange quarks: