Meson-baryon scattering lengths in HB$χ$PT

0.22 MB·English

by  Yan-Rui Liu

Meson-baryon scattering lengths in HBχPT
Yan-Rui Liu and Shi-Lin Zhu
∗ †
Department of Physics, Peking University, Beijing 100871, China
(Dated: February 2, 2008)
We calculate the chiral corrections to the s-wave pseudoscalar meson octet-baryon scattering
lengths a to O(p3) in the SU(3) heavy baryon chiral perturbation theory (HBχPT). Hopefully
MB
the obtained analytical expressions will be helpful in the chiral extrapolation of these scattering
lengths in thefuture lattice simulation.
PACSnumbers: 13.75.Gx, 13.75.Jz
Keywords: Scatteringlength,meson-baryoninteraction,heavybaryonchiralperturbationtheory
7
0
0 I. INTRODUCTION
2

n Heavy baryon chiral perturbation theory (HBχPT) [1, 2] is widely used to study low energy processes involving
a chiral fields and ground state baryons. Its Lagrangian is expanded simultaneously with p/Λ and p/M , where p
χ 0
J
represents the meson momentum or its mass or the small residue momentum of baryon in the non-relativistic limit,

8 Λ 1GeV is the scale of chiral symmetry breaking, and M is the baryon mass in the chiral limit.
χ 0
1 ∼
Pion-nucleon interactions are widely investigated in the two-flavor HBχPT. For processes involving kaons or hy-

perons, one has to employ the SU(3) HBχPT, where more unknown low-energy constants (LEC) appear at the same
3
chiralorderthan inthe SU(2) case. Determining these LECsneeds moreexperimentaldata whichareunavailableat
v
0 present. BrokenSU(3) symmetry results in large mK and mη. In certaincases,the chiralexpansionconvergesslowly
0 due to m /Λ m /Λ 1/2 [3, 4, 5, 6].
K χ η χ
∼ ∼
1 The scattering length is an important observable which contains important information about low-energy meson-
7 baryon strong interactions. As an effective theory of QCD, HBχPT provides a model-independent approach to
0
calculatemeson-baryonscatteringlengths. Inthispaper,wewillcalculatealls-wavemeson-baryonscatteringlengths
6
to the third chiral order in the SU(3) HBχPT.
0
/ Experimentalmeasurements ofπN scatteringlengths arerelativelyeasierthanthose for other processes. Recently,
h the results of precision X-ray experiments on pionic hydrogen [7] and deuterium [8] were published. These two
p
experiments together constrain the isoscalar and isovector πN scattering lengths [9].
-
p Since the prediction of πN scattering lengths with current algebra, chiral corrections have been calculated to high
e orders in the two-flavor HBχPT prior [10, 11]. There were many theoretical calculations of πN scattering lengths
h
[12]. As we will see below, πN scattering lengths play an important role in determining LECs in the SU(3) HBχPT.
:
v Experimental data is very scarce for kaon-nucleon scattering lengths. The recent DEAR measurements on kaonic
Xi hydrogen gave the scattering length aK−p [13]. However, the direct determination of the two scattering lengths in
the KN channelneedsfurther experimentalmeasurements. Unlike the pion-nucleoncase,thereis the possibility that
r
a measurements on kaonic deuterium are not enough to give the two KN scattering lengths [14].
There had been efforts to extract the scattering lengths from kaon-nucleon scattering data using various models
[15, 16, 17, 18]. Kaon-nucleon scattering lengths were calculated to order (p3) in the SU(3) HBχPT in Ref. [5].
O
There were large cancellations at the second and the third chiralorder. Recently two lattice simulations also tried to
extract the kaon-nucleon scattering lengths [19, 20].
ηN scattering is particularly interesting because the attractive ηN interaction may result in η-mesic nuclei which
have a long history [21]. The formation of η-mesic hypernuclei is also possible [22]. An early experiment gave a
negativeresultonthesearchforη-mesicboundstates[23]. However,arecentexperimentsignalstheexistenceofsuch
states [24]. The experimental situation is rather ambiguous.
On the theoretical side, the existence of η-3He quasibound state is also controversial, mostly because it is hard
to determine the η-nucleus scattering length. In the optical models, the scattering length can be obtained with ηN
scattering length a . However,theoretical predictions of a are different from various approaches(see overviewin
ηN ηN
[25]). It is worthwhile to calculate ηN scattering length in the SU(3) HBχPT.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
There are very few investigations of pion-hyperon, kaon-hyperon and eta-baryon scattering lengths in literature.
Experimentally these observables may be studied through the strangeness program at Japan Hadron Facility (JHF)
and Lan-Zhou Cooling Storage Ring (CSR). In this paper we will perform an extensive study of these scattering
lengths to the third chiral order in the framework of SU(3) HBχPT.
We present the basic notations and definitions in Section II. We present the chiral corrections to the threshold
T-matrices of meson-baryon interactions in Section III. This section contains our main results. We discuss useful
relations between these threshold T-matrices in IV. Then we discuss the determination of LECs in Section V. The
final section is a summary.
II. LAGRANGIAN
For a system involving pions and one octet baryon, the chirally invariant Lagrangianof HBχPT reads
= + , (1)
φφ φB
L L L
where φ represents the pseudoscalar octet mesons, B represents the octet baryons. The purely mesonic part
φφ
L
incorporates even chiral order terms while the terms in start from (p).
φB
L O
χ
(2) =f2tr(u uµ+ +), (2)
Lφφ µ 4
(1) =tr(B(i∂ B+[Γ ,B])) Dtr(B ~σ ~u,B ) Ftr(B[~σ ~u,B]), (3)
LφB 0 0 − { · } − ·
(2) = b tr(B χ ,B )+b tr(B[χ ,B])+b tr(BB)tr(χ )
LφB D { + } F + 0 +
D2 3F2 DF
+ 2d + − tr(B u2,B )+ 2d tr(B[u2,B])
(cid:16) D 2M0 (cid:17) { 0 } (cid:16) F − M0(cid:17) 0
F2 D2
+ 2d + − tr(BB)tr(u2)
0 2M 0
(cid:16) 0 (cid:17)
3F2 D2
+ 2d + − tr(Bu )tr(u B), (4)
1 0 0
3M
(cid:16) 0 (cid:17)
where
i i
Γ = [ξ ,∂ ξ], u = ξ ,∂ ξ , ξ =exp(iφ/2f), (5)
µ † µ µ † µ
2 2{ }
χ+ =ξ†χξ†+ξχξ, χ=diag(m2π, m2π, 2m2K −m2π), (6)
π0 + η π+ K+ Σ0 + Λ Σ+ p
√2 √6 √2 √6
φ=√2 π− −√π02 + √η6 K0 , B = Σ− −√Σ02 + √Λ6 n . (7)
 K− K0 −√26η   Ξ− Ξ0 −√26Λ
f is the meson decay constant in the chiral limit. Γ is chiral connection which contains even number meson fields.
µ
u contains odd number meson fields. D+F = g = 1.26 where g is the axial vector coupling constant. The first
µ A A
three terms in (2) are proportional to SU(3) symmetry breaking. Terms involving M are corrections produced by
LφB 0
finite baryon mass in the chiral limit. Others are proportional to LECs d , d , d and d .
D F 1 0
We calculate threshold T-matrices for meson-baryonscattering to the third order according to the power counting
rule [1, 2]. The leading and next leading order contributions can be read from the tree level Lagrangians (1) and
LφB
(2)
respectively.
LφB
At the third order, both loop diagrams and (p3) LECs from (3) contribute. In principle, divergence from the
O LφB
loop integration will be absorbed by these LECs. There are many unknown LECs at (p3) which may reduce the
O
predictive power of HBχPT. The contribution of these (p3) LECs to threshold T-matrix was carefully investigated
O
using the resonance saturation approach in the SU(2) HBχPT in Refs. [10]. Luckily, the counter-term contributions
at (p3) are much smaller than the loop corrections and the chiral corrections at this order mainly come from the
O
chiralloop[10]. Therefore,thesenegligiblecounter-termcontributionswereignoredinthecalculationofkaon-nucleon
scattering lengths in Ref. [5]. We follow the same approach in our crude numerical analysis because of the lack of
enough data to fix these small LECs.
3
FIG.1: Non-vanishingloopdiagramsinthecalculationofmeson-baryonscatteringlengthstothethirdchiralorderinHBχPT.
Dashed lines represent Goldstone bosons while solid lines represent octet baryons. The fourth diagram generates imaginary
parts for kaon-baryon and eta-baryon scattering lengths.
III. T-MATRICES FOR MESON BARYON SCATTERINGS
The threshold T-matrix T is related to scattering length a through T = 4π(1+ mP)a , where m and
PB PB PB MB PB P
m are the masses of the pseudoscalar meson and baryon, respectively.
B
There exist many diagrams for a general elastic meson-baryon scattering process. However, the calculation is
simpler at threshold due to ~σ ~q = 0 where ~q is the three momentum of the meson. The lowest order contribution
·
arises from the chiral connection term in (1). Only meson masses and decay constants appear in the expressions.
LφB
(2)
At the next leading order,the expressionsare proportionalto the combinations ofseveralLECs in . At the third
LφB
chiral order, the loop diagrams in Fig. 1 have non-vanishing contributions. The vertices come from (1) and (2)
LφB Lφφ
There are two types of diagrams. The first six contain vertices from the chiral connection, and thus contributions
fromthesediagramsdonotcontainthe axialcouplingconstantsDandF.Becausethemassofanintermediatemeson
is always larger than or equal to m , no diagrams generate imaginary parts for the pion-baryon scattering lengths.
π
In contrast, the fourth diagram generates imaginary parts for kaon-baryonor eta-baryon threshold T-matrices.
In the following subsections we list the expressions of threshold T-matrices order by order for every channel. This
is our main result, which may be helpful to the chiral extrapolations of the scattering lengths on the lattice.
A. Kaon-nucleon scattering
For kaon-nucleon scattering lengths, we reproduce the tree level expressions of Ref. [5]. At the third chiral order,
our expressions are slightly different [36].
At the leading order,
m m 3m
(1) K (0) (1) K (0) K
T = , T =0, T = , T = , (8)
KN − f2 KN KN 2f2 KN 2f2
K K K
4
where the superscripts represent total isospin and f is the renormalized kaon decay constant.
K
The second chiral order T-matrices are
m2 m2 m2 m2
T(1) = KC , T(0) = KC , T(1) = K C +C , T(0) = K 3C C , (9)
KN f2 1 KN f2 0 KN 2f2 (cid:18) 1 0(cid:19) KN 2f2 (cid:18) 1− 0(cid:19)
K K K K
where C are defined in Ref. [5]
1,0
D2+3F2
C = 2(d 2b )+2(d 2b )+d ,
1 0 0 D D 1
− − − 6M
0
D(D 3F)
C = 2(d 2b ) 2(d 2b ) d − . (10)
0 0 0 F F 1
− − − − − 3M
0
WewillexpresscombinationsofLECsintheotherchannelswithC andC . Thiscanreducethenumberofparameters
0 1
for a given channel.
At the third order, we have
m2 m m m
T(1) = K m 3+2ln π +ln| K| +3ln η
KN 16π2f4 (cid:26) K(cid:18)− λ λ λ (cid:19)
K
m + m2 m2 m
+2 m2 m2 ln K K − π 3 m2 m2 arccos K
q K − π pmπ − q η− K mη
π m2
(D 3F) 2(D+F) π +(D+5F)m , (11)
η
−6 − (cid:20) m +m (cid:21)(cid:27)
η π
3m2 m m m + m2 m2
T(0) = K m ln π ln| K| + m2 m2 ln K K − π
KN 16π2fK4 (cid:26) K(cid:18) λ − λ (cid:19) q K − π pmπ
π m2 1
+ (D 3F) (D+F) π + (7D+3F)m , (12)
η
3 − (cid:20) m +m 6 (cid:21)(cid:27)
η π
m2 m m m
T(1) = K m 3 5ln π +2ln| K| 3ln η
KN 32π2f4 (cid:26) K(cid:18) − λ λ − λ (cid:19)
K
m + m2 m2 m
+5 m2 m2 iπ ln K K − π 3 m2 m2 arccos− K
q K − π(cid:18) − pmπ (cid:19)− q η− K mη
π m2
+ (D 3F) 2(D+F) π +(3D F)m . (13)
η
3 − (cid:20) m +m − (cid:21)(cid:27)
η π
3m2 m m m
T(0) = K m 3 ln π 2ln| K| 3ln η
KN 32π2f4 (cid:26) K(cid:18) − λ − λ − λ (cid:19)
K
m + m2 m2 m
+ m2 m2 iπ ln K K − π 3 m2 m2 arccos− K
q K − π(cid:18) − pmπ (cid:19)− q η− K mη
π m2 1
(D 3F) 2(D+F) π + (5D+9F)m . (14)
η
−3 − (cid:20) m +m 3 (cid:21)(cid:27)
η π
In loop calculations, we have used dimensional regularization and minimal subtraction scheme. λ is the cutoff. In
deriving the expressions in square brackets, Gell-Mann Okubo relation was used. For the decay constants in the
−
loop expressions, we have used f in kaon processes in the above equations in order to make the expressions more
K
compact. One may also use f in π loops, f in kaon loops and f in η loops. The differences are of high order.
π K η
B. Pion-nucleon scattering
With the isospin amplitude, the tree level expressions are
m m
T(3/2) = π T(1/2) = π, (15)
πN −2f2 πN f2
π π
5
and
m2 m2
T(3/2) = π C +C +4C T(1/2) = π C +C +4C , (16)
πN 2f2(cid:18) 1 0 π(cid:19) πN 2f2(cid:18) 1 0 π(cid:19)
π π
where
DF
C =(d 2b ) . (17)
π F F
− − 2M
0
At the third order
m2 3 m m
T(3/2) = π m 2ln π ln K
πN 16π2f4(cid:26)− π(cid:18)2 − λ − λ (cid:19)
π
m π 1
m2 m2 π+arccos π + 3(D+F)2m (D 3F)2m , (18)
−q K − π(cid:18) mK(cid:19) 4(cid:20) π− 3 − η(cid:21)(cid:27)
m2 3 m m
T(1/2) = π 2m 2ln π ln K
πN 16π2f4(cid:26) π(cid:18)2 − λ − λ (cid:19)
π
3 m π 1
m2 m2 π+2arcsin π + 3(D+F)2m (D 3F)2m . (19)
−q K − π(cid:18)2 mK(cid:19) 4(cid:20) π− 3 − η(cid:21)(cid:27)
The isospin even and isospin odd amplitudes are also widely used in literature.
m m2 3 m m m
Tπ−N = 2fππ2 + 16π2πfπ4(cid:26)mπ(cid:18)2 −2ln λπ −ln λK(cid:19)−qm2K −m2πarcsinmKπ (cid:27), (20)
m2 1 3m2 1
T+ = π (C +C )+2C + π 2 m2 m2 +(D+F)2m (D 3F)2m . (21)
πN fπ2 (cid:26)2 1 0 π(cid:27) 64πfπ4(cid:26)− q K − π π− 9 − η(cid:27)
The above expressions agreewith those in Ref. [5]. The two conventions are related throughthe following equations:
Tπ(3N/2) =Tπ+N −Tπ−N, Tπ(1N/2) =Tπ+N +2Tπ−N. (22)
Sometimes a third convention with the isoscalar amplitude Tπ0N = Tπ+N and isovector amplitude Tπ1N = −Tπ−N are
used in the literature.
Most of the threshold T-matrices for pion-baryon and kaon-baryon scattering at the second order depend on the
above three combinations C , C and C . If C , C and C can be fixed from KN and πN scattering lengths, many
1 0 π 1 0 π
predictions can be made.
C. Pion-Sigma scattering
T-matrices at the leading order:
m m 2m
T(2) = π, T(1) = π, T(0) = π. (23)
πΣ − f2 πΣ f2 πΣ f2
π π π
At the next leading order we have
m2 m2 m2
T(2) = πC , T(1) = π C 2C , T(0) = π C +3C , (24)
πΣ f2 1 πΣ f2 (cid:18) 1− d(cid:19) πΣ f2 (cid:18) 1 d(cid:19)
π π π
where
D2 3F2
C =d − . (25)
d 1
− 6M
0
6
The extraction of C needs additional inputs besides pion-nucleon and kaon-nucleon scattering lengths. Due to the
d
lack of experimental data, one cannot determine it like the determination of C . We will estimate its value when
1,0,π
we discuss numerical results.
At the third order, we have
m2 3 m m m π 1
T(2) = π m 2ln π ln K m2 m2arccos π + 3F2m D2m , (26)
πΣ 8π2fπ4(cid:26)− π(cid:18)2 − λ − λ (cid:19)−q K − π mK 2(cid:20) π − 3 η(cid:21)(cid:27)
m2 3 m m m π 1
T(1) = π m 2ln π ln K m2 m2 arccos− π + (2D2 3F2)m D2m ,(27)
πΣ 8π2fπ4(cid:26) π(cid:18)2 − λ − λ (cid:19)−q K − π mK 2(cid:20) − π − 3 η(cid:21)(cid:27)
m2 3 m m 3 m
T(0) = π 2m 2ln π ln K m2 m2 π 2arccos π
πΣ 8π2fπ4(cid:26) π(cid:18)2 − λ − λ (cid:19)−q K − π(cid:18)2 − mK(cid:19)
π 1
3(D2 4F2)m + D2m . (28)
π η
−2(cid:20) − 3 (cid:21)(cid:27)
D. Pion-Cascade scattering
Because both nucleon and cascade are isospin-1/2 baryons, one expects similar expressions. At the leading order,
the expressions are the same as those for the πN scattering,
m m
T(3/2) = π, T(1/2) = π. (29)
πΞ −2f2 πΞ f2
π π
The next leading order contributions read
m2 m2
T(3/2) = π C +C , T(1/2) = π C +C . (30)
πΞ 2f2(cid:18) 1 0(cid:19) πΞ 2f2(cid:18) 1 0(cid:19)
π π
The expressions involve only combinations of LECs in kaon-nucleon scattering. This is the relic of isopin and U-spin
symmetry (see the relations in Section IV).
At the third order, we have
m2 3 m m
T(3/2) = π m 2ln π ln K
πΞ 16π2f4(cid:26)− π(cid:18)2 − λ − λ (cid:19)
π
m π 1
m2 m2 π+arccos π + 3(D F)2m (D+3F)2m , (31)
−q K − π(cid:18) mK(cid:19) 4(cid:20) − π− 3 η(cid:21)(cid:27)
m2 3 m m
T(1/2) = π 2m 2ln π ln K
πΞ 16π2f4(cid:26) π(cid:18)2 − λ − λ (cid:19)
π
1 m π 1
m2 m2 π+2arccos− π + 3(D F)2m (D+3F)2m . (32)
−q K − π(cid:18)2 mK (cid:19) 4(cid:20) − π− 3 η(cid:21)(cid:27)
The expressions are similar to πN T-matrices at this order. The difference lies in the factor F. One can get these
expressions from those for πN scattering through replacing F by F. In fact, we can also get the second order
−
expressions from those of T through replacing b by b , d by d and F by F.
πN F F F F
− − −
E. Pion-Lambda scattering
Theleadingordercontributionvanishes,whichcanbeunderstoodthroughthecrossingsymmetry: Tπ+Λ =Tπ−Λ =
[T ] . Thus T = T at the leading order. This analysis is also available for the following T-matrices,
Tπ+,ΛTmπ→a−nmdπT whosπeΛlead−ingπΛorder contributions also vanish.
KΛ KΛ ηB
7
The tree-level contribution appears only at the next leading order
m2
T = π C +2C +4C +C . (33)
πΛ 3f2(cid:18) 1 0 π d(cid:19)
π
At the third order, for a PΛ or ηB scattering, only diagram 4, 5, 10, 11 and 12 in Fig. 1 have non-vanishing
contributions. The expression for T at this order is
πΛ
m2 1
T = π 3 m2 m2 +D2 m m . (34)
πΛ 16πfπ4(cid:26)− q K − π (cid:18) π− 3 η(cid:19)(cid:27)
The above expression was first derived in Ref. [5].
F. Kaon-Sigma scattering
One expects the similarity between T and T because both K and K are isospin doublets. The leading order
KΣ KΣ
expressions are
m m m m
T(3/2) = K, T(1/2) = K, T(3/2) = K, T(1/2) = K. (35)
KΣ −2f2 KΣ f2 KΣ −2f2 KΣ f2
K K K K
At the next leading order, we have
T(3/2) = m2K C +C +4C , T(1/2) = m2K C +C 2C ,
KΣ 2fK2 (cid:18) 1 0 π(cid:19) KΣ 2fK2 (cid:18) 1 0− π(cid:19)
T(3/2) = m2K C +C , T(1/2) = m2K C +C +6C .. (36)
KΣ 2fK2 (cid:18) 1 0(cid:19) KΣ 2fK2 (cid:18) 1 0 π(cid:19)
At the third chiral order
m2 m m m
T(3/2) = K m 3+ln π 4ln| K| 3ln η
KΣ 32π2f4 (cid:26)− K(cid:18) λ − λ − λ (cid:19)
K
m + m2 m2 m
+ m2 m2 2iπ ln K K − π 3 m2 m2 arccos K
q K − π(cid:18) − pmπ (cid:19)− q η− K mη
4 m2
+ πD 2F π +(D+2F)m , (37)
η
3 (cid:20) m +m (cid:21)(cid:27)
η π
m2 m m m
T(1/2) = K m 3+ln π 4ln| K| 3ln η
KΣ 16π2f4 (cid:26) K(cid:18) λ − λ − λ (cid:19)
K
1 m + m2 m2 3 m
+ m2 m2 iπ+ln K K − π 3 m2 m2 π arccos K
q K − π(cid:18)4 pmπ (cid:19)− q η− K(cid:18)4 − mη (cid:19)
2 m2
+ πD 4F π +(D 4F)m , (38)
η
3 (cid:20)− m +m − (cid:21)(cid:27)
η π
m2 m m m
T(3/2) = K m 3+ln π 4ln| K| 3ln η
KΣ 32π2f4 (cid:26)− K(cid:18) λ − λ − λ (cid:19)
K
m + m2 m2 m
+ m2 m2 2iπ ln K K − π 3 m2 m2 arccos K
q K − π(cid:18) − pmπ (cid:19)− q η− K mη
4 m2
+ πD 2F π +(D 2F)m , (39)
η
3 (cid:20)− m +m − (cid:21)(cid:27)
η π
m2 m m m
T(1/2) = K m 3+ln π 4ln| K| 3ln η
KΣ 16π2f4 (cid:26) K(cid:18) λ − λ − λ (cid:19)
K
1 m + m2 m2 3 m
+ m2 m2 iπ+ln K K − π 3 m2 m2 π arccos K
q K − π(cid:18)4 pmπ (cid:19)− q η− K(cid:18)4 − mη (cid:19)
2 m2
+ πD 4F π +(D+4F)m . (40)
η
3 (cid:20) m +m (cid:21)(cid:27)
η π
8
T(3/2) differs from T(3/2) in the sign in front of F at the third order. The same property holds for T(1/2) and T(1/2).
KΣ KΣ KΣ KΣ
One can also testify that differences between T and T at the second order are the signs in front of b , d and
KΣ KΣ F F
F.
G. Kaon-Cascade scattering
The leading order contributions are
m 3m m
T(1) = K, T(0) = K, T(1) = K, T(0) =0. (41)
KΞ 2f2 KΞ 2f2 KΞ − f2 KΞ
K K K
At the second order, we have
T(1) = m2K C +C +4C , T(0) = m2K 3C C 4C ,
KΞ 2fK2 (cid:18) 1 0 π(cid:19) KΞ 2fK2 (cid:18) 1− 0− π(cid:19)
T(1) = m2KC , T(0) = m2K C +4C . (42)
KΞ fK2 1 KΞ fK2 (cid:18) 0 π(cid:19)
The third order expressions are
m2 m m m
T(1) = K m 3 5ln π +2ln| K| 3ln η
KΞ 32π2f4 (cid:26) K(cid:18) − λ λ − λ (cid:19)
K
m + m2 m2 m
+5 m2 m2 iπ ln K K − π 3 m2 m2 arccos− K
q K − π(cid:18) − pmπ (cid:19)− q η− K mη
π m2
+ (D+3F) 2(D F) π +(3D+F)m , (43)
η
3 (cid:20) − m +m (cid:21)(cid:27)
η π
3m2 m m m
T(0) = K m 3 ln π 2ln| K| 3ln η
KΞ 32π2f4 (cid:26) K(cid:18) − λ − λ − λ (cid:19)
K
m + m2 m2 m
+ m2 m2 iπ ln K K − π 3 m2 m2 arccos− K
q K − π(cid:18) − pmπ (cid:19)− q η− K mη
π m2 1
(D+3F) 2(D F) π + (5D 9F)m , (44)
η
−3 (cid:20) − m +m 3 − (cid:21)(cid:27)
η π
m2 m m m
T(1) = K m 3+2ln π +ln| K| +3ln η
KΞ 16π2f4 (cid:26) K(cid:18)− λ λ λ (cid:19)
K
m + m2 m2 m
+2 m2 m2ln K K − π 3 m2 m2 arccos K
q K − π pmπ − q η− K mη
π m2
(D+3F) 2(D F) π +(D 5F)m , (45)
η
−6 (cid:20) − m +m − (cid:21)(cid:27)
η π
3m2 m m m + m2 m2
T(0) = K m ln π ln| K| + m2 m2 ln K K − π
KΞ 16π2fK4 (cid:26) K(cid:18) λ − λ (cid:19) q K − π pmπ
π m2 1
+ (D+3F) (D F) π + (7D 3F)m . (46)
η
3 (cid:20) − m +m 6 − (cid:21)(cid:27)
η π
H. Kaon-Lambda scattering
Non-vanishing contributions start from the next leading order.
T =T = m2K 5C +C +2C 4C . (47)
KΛ KΛ 6fK2 (cid:20) 1 0 π − d(cid:21)
9
At the third chiral order loop corrections are
9m2 8
T =T = K i m2 m2 m2 m2 + D2m . (48)
KΛ KΛ 64πfK4 (cid:26) q K − π −q η− K 27 η(cid:27)
Similarly, one may get T from T through the replacement b b , d d and F F. However, b
KΛ KΛ F → − F F → − F → − F
and d disappear in the second order T-matrices and F disappears in the third order T-matrix. Hence T =T .
F KΛ KΛ
I. Eta-baryon scattering
The leading order contribution vanishes for every channel. With the Gell-Mann Okubo relation the next leading
−
order contributions can be written as:
1
T = (5C +C 4C 4C )m2 4(b 3b )(m2 m2) ,
ηN 6f2(cid:20) 1 0− π− d η− D− F η− π (cid:21)
η
1
T = (C +2C +4C +C )m2 +4b (m2 m2) ,
ηΣ 3f2(cid:20) 1 0 π d η D η− π (cid:21)
η
1
T = (5C +C +8C 4C )m2 4(b +3b )(m2 m2) ,
ηΞ 6f2(cid:20) 1 0 π− d η− D F η− π (cid:21)
η
1
T = 3(C +C )m2 4b (m2 m2) . (49)
ηΛ 3f2(cid:20) 1 d η− D η− π (cid:21)
η
These expressions rely on C as well as b and b . b and b are determined with the octet baryon masses.
1,0,π,d D F D F
At the third chiral order, we have
1 1
T = 9im2 m2 m2 (D 3F)2(4m2 m2)m
ηN 32πfη4(cid:26) ηq η− K − 6 − η− π η
3 2
(D+F)2m3 + (5D2+9F2 6DF)m3 , (50)
−2 π 3 − K(cid:27)
1 1
T = 3im2 m2 m2 D2(4m2 m2)m
ηΣ 16πfη4(cid:26) ηq η− K − 3 η− π η
1
(D2+6F2)m3 +2(D2+F2)m3 , (51)
−3 π K(cid:27)
1 1
T = 9im2 m2 m2 (D+3F)2(4m2 m2)m
ηΞ 32πfη4(cid:26) ηq η− K − 6 η− π η
3 2
(D F)2m3 + (5D2+9F2+6DF)m3 , (52)
−2 − π 3 K(cid:27)
1 1 2
T = 9im2 m2 m2 D2(4m2 m2)m D2m3 + (D2+9F2)m3 . (53)
ηΛ 16πfη4(cid:26) ηq η− K − 3 η− π η− π 3 K(cid:27)
Again there is the similarity between T and T .
ηN ηΞ
IV. THRESHOLD T-MATRIX RELATIONS
In this section, we list the relations between kaon-baryon and anti-kaon-baryonthreshold T-matrices according to
thecrossingsymmetry. WealsopresenttherelationsoftheseT-matricesintheSU(3)symmetrylimit. Theserelations
are used to cross-checkour calculations.
10
Crossing symmetry relates KB processes with KB processes. According to this symmetry, we have the following
relations:
1 1
(1) (1) (0) (0) (1) (0)
T = T +T , T = 3T T , (54)
KN 2h KN KNimK mK KN 2h KN − KNimK mK
→− →−
1 1
T(1) = T(1) +T(0) , T(0) = 3T(1) T(0) , (55)
KΞ 2h KΞ KΞimK mK KΞ 2h KΞ− KΞimK mK
→− →−
1 1
T(3/2) = T(3/2)+2T(1/2) , T(1/2) = 4T(3/2) T(1/2) , (56)
KΣ 3h KΣ KΣ imK mK KΣ 3h KΣ − KΣ imK mK
→− →−
and
T = T . (57)
KΛ KΛ
h imK mK
→−
In the SU(3) flavor symmetry limit, we derive the following relations using the isospin and U-spin symmetry:
(1) (2) (1)
T =T =T ,
KN πΣ KΞ
T(3/2) =T(3/2) = 1[T(1) +T(0)],
πN KΣ 2 KΞ KΞ
T(3/2) =T(3/2) = 1[T(1) +T(0) ],
πΞ KΣ 2 KN KN
T(1) = 1[T(3/2)+2T(1/2)]= 1[T(3/2)+2T(1/2)],
KΞ 3 πN πN 3 KΣ KΣ
T(1) = 1[T(3/2)+2T(1/2)]= 1[T(3/2)+2T(1/2)],
KN 3 πΞ πΞ 3 KΣ KΣ
T(1) +T(0) =T(1) +T(0) = 1[T(2)+3T(1)+2T(0)],
KN KN KΞ KΞ 3 πΣ πΣ πΣ
T =T ,
ηΣ πΛ
1[T(2)+T(1)]+T = 1[2T(3/2)+T(1/2)]+T = 1[2T(3/2)+T(1/2)]+T
2 πΣ πΣ πΛ 3 KΣ KΣ KΛ 3 KΣ KΣ KΛ
= 1[2T(3/2)+T(1/2)]+T = 1[2T(3/2)+T(1/2)]+T . (58)
3 πN πN ηN 3 πΞ πΞ ηΞ
V. DETERMINATION OF LOW-ENERGY-CONSTANTS
There aresevenunknownLECs in (2), b , d , andd . The thresholdT-matricesdepend onfour combina-
LφB D,F,0 D,F,0 1
tionsofthem: (d 2b ),(d 2b ), (d 2b )andd . Afterthe regrouping,wedefinedanotherfourcombinations:
F F D D 0 0 1
− − −
C , C , C and C . The values of C , C and C can be extracted from the experimental values of T and T .
1 0 π d 1 0 π KN πN
With C , C and C one can predict most of pion-baryonand kaon-baryonscattering lengths.
1 0 π
Threshold T-matrices T(1), T(0), T and T depend on additional LECs C and b . The values of b and
KΣ KΣ MΛ ηB d D,F D
b can be extracted from the octet baryon mass differences. In order to determine C =d D2 3F2, one has to fix
F d 1− 6−M0
both d and M .
1 0
For the sake of comparison, let’s recall the procedure of determining LECs in Ref. [5]. (1) First, Kaiser fixed the
energy scale parameter fromthe loopintegrationλ to be 0.95 GeV using the experimental value of Tπ−N; (2) Then he
determined the LECs C1,0 using the experimental values of TK(1N) and TK(0N) [15]: C1 =2.33GeV−1, C0 =0.36GeV−1.
Throughout his analysis he used D =0.8 and F =0.5; (3) The three LECs bD =0.042GeV−1, bF = 0.557GeV−1,
b0 = 0.789GeV−1 and M0 =918.4 MeV were obtained by fitting the octet baryon masses and the π−N sigma term
σπN =−45MeVfromRef. [26];(4)UsingtheexperimentalvalueofTπ+N,theextractedparametersdF =−0.968GeV−1,
2d0+dD = 1.562GeV−1, d1+dD =1.150GeV−1 and the value ofd0 1.0GeV−1 determined elsewhere[27], one
− ≃−
extracts d .
1
OurprocedureisslightlydifferentfromKaiser’s. Wetakeλ=4πf asthe chiralsymmetrybreakingscale,whichis
π
widely adoptedin the chiralperturbationtheory. The otherstepsaresomehowsimilar. The mesonmassesanddecay
constantsarefromPDG[28]: m =139.57MeV,m =493.68MeV,m =547.75MeV,f =92.4MeV,f =113MeV,
π K η π K
f =1.2f . D =0.8 and F =0.5.
η π
We first determine the combinations C and C . Using the experimental values of a(1) = 0.33fm and a(0) =
1 0 KN − KN
0.02fm [15], we extract
C1 =1.786GeV−1, C0 =0.413GeV−1. (59)