a r X i v :0711.1977v 2 [h e p -p h ] 14 N o v 2007
S-wave meson scattering up to
√
s
2GeV.Our framework is bad on Unitary Chiral Perturbation Theory.We include for I =0the channels:ππ,K
K ∗
,a 1(1260)πand π⋆(1300)π.In addition,and in order to constrain our fits,we also study the I =1/2,3/2channels given by Kπ,Kηand Kη′.We finally prent the resonant content of our fits with the σ,f 0(980),f 0(1310),f (1500),f 0(1710)and f 0(1790).
1Lagrangians.U (3)symmetry
Due to the spontaneous breakdown of chiral SU (3)symmetry the π,K and ηare the octet of pudo-Gol
dstone bosons.As it is well known,chiral symme-try strongly constrains the allowed interactions between the pudoscalars and it is a basic ingredient in any study of strong interactions involving tho mesons.If one considers higher energy regions,as it is our ca here where we study the I =0and 1/2S-waves up to about 2GeV,one also needs to take into account the ηη,ηη′and η′η′channels.Interestingly,in the large N c limit,the η1becomes the ninth Goldstone boson.This fact can be ud to ttle down chiral Lagrangians bad on U (3)chiral symmetry and to include the η1field.The η1-η8mixing angle is taken as sin θ=−1/3→θ≈−20◦.
leakM.Albaladejo et al.S-wave meson scattering in UChPT
K,(3)ηη,(4)σσ,(5)ηη′,(6)ρρ,(7)ωω,(8)ηη′,(9)ωφ,(10)φφ,(11)K∗
D(s1)D(s2)
.(1)
成都电脑The a→(σσ)0amplitude,N a→(σσ)
,is obtained from,
lim s i→sσT a→(ππ)
0(ππ)0
=lim
s i→sσ
哪个培训机构好T2+R
a→(ππ)0(ππ)0
(s1−sσ)(s2−sσ)
(2)
The subscript II means that we have to calculate the corresponding function on the cond Riemann sheet,where theσpole appears.Finally,calculating this limit with an appropriate Laurent expansion around sσ,
N a→(σσ)
0=T2+R
a→(ππ)0(ππ)0 α0gσππ
2≃9.1·10−3GeV2.(3)
2
sdcpM.Albaladejo et al.S-wave meson scattering in UChPT
annotate8π2 ∞s th,i d s′p i(s′)/√
(s′−s0)(s′−s+iǫ)(4) A remark is in order.The integrals involve the mass of the particles of the scattering states,but some of them,as theσ,ρ,a1(1260)andπ⋆(1300)have very large widths.To take the effects into account,we consider instead of eq.(4)an integral of this loop function times a mass distribution over a wide range of mass for each of the unstable particles.
piano怎么读3Results and spectroscopy
With all the amplitudes,one can construct the S-matrix and calculate obrvables;in our ca,the will be pha shifts and amplitude moduli.The curves resulting from ourfit are depicted in Fig.1.We u13parameters for about373experimental data,and a fair agreement with data is achieve
d.We have reduced the number of free parameters compared with other approaches in the literature which do not employ(chiral)Lagrangians.
Once the obrvables arefitted,we can explore the s−complex plane to find the relevant poles of the amplitudes,and discuss their resonance content. We prent in Table1the mass and widths of the resonances that wefind. The agreement with the ones in the PDG is remarkable.
Table1:Parameters of resonances.On the left columns we have the mass and widths that wefind.On the right ones,the values are given by the PDG or the BES Collaboration.
Resonance Mass(MeV)
475
44
350
100−170
≈160
≈390Mass(MeV)
600-1200
40-100
200-500
109±7
137±8
270+60
−30
3
M.Albaladejo et al.百威啤酒广告歌曲
S-wave meson scattering in UChPT
s (MeV)
|S 15|2
1850
1800
1750
170016501600
1550
1500
0.10.0750.05
0.025
0Binon et al.
allrightsrerved
√
s (MeV)
φ(I =1/2)
2000
1800
1600
14001200
1000
800
250200150100500
√
s (MeV)
|S 12|2
2000
180016001400
12001000
0.80.70.60.50.40.30.20.10Cohen et al.multiply是什么意思
Etkin et al.√
s (MeV)
|S 11|
1600
1400
床单 英文
1200
1000800
600
21.51
0.50BNL-E865
NA48/2400
350300
3020100
Prom.
Kaminski et al.
Ke4
√
K .Right panels,from up to down:modulus squared
of the S-matrix elements ππ→ηη′and ππ→ηη.The last two figures correspond to the pha and modulus of the K −π+→K −π+scattering.
References
[1]M.Jamin,J.A.Oller and A.Pich,Nucl.Phys.B 587(2000)331;
M.Jamin,J.A.Oller and A.Pich,Nucl.Phys.B 622,279(2002)[2]J.Gasr and H.Leutwyler,Nucl.Phys.B 250,465(1985);G.Ecker,
J.Gasr,A.Pich and E.de Rafael,Nucl.Phys.B 321,311(1989).[3]J.A.Oller and E.Ot,Nucl.Phys.A 620,438(1997)[Erratum-ibid.
A 652,407(1999)];J.A.Oller and E.Ot,Phys.Rev.D 60,074023(1999)
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