Does the light and broad sigma(500) exist

a r X i v :h e p -p h /9904346v 1 15 A p r 1999

Frascati Physics Series Vol.XXX (1997),pp.000-000Conference Title -Conference Town,Oct 3rd,1996

DOES THE LIGHT AND BROAD σ(500)EXIST?Nils A T¨o rnqvist

Physics Dept.University of Helsinki,PB9,Fin-00014Helsinki,Finland

ABSTRACT

The lightest scalar and pseudoscalar nonets are discussed within the framework of the broken U3×U3linear sigma model,and it is shown that already at the tree level this model works remarkably well predicting scalar masses and couplings not far from present experimental values,when all parameters are ?xed from the pseudoscalar masses and decay constants.The linear σmodel is the simplest way to implement chiral symmetry together with the broken SU3of the quark model,and its success in understanding experiment is comparable to that of the naive quark model for the heavier multiplets.It is argued that this strongly suggests that the light and very broad σresonance exists near 500MeV.

1Introduction

In this talk 1I shall discuss mainly the light and broad σ,which was picked by Matts Roos and myself 1)from the particle data group wastebasket 4years ago,having been there for over 20years.Today an increasing number of papers,many of which have been reported at this meeting 2),are quoting its parameters,with a pole position near 500-i250MeV (See the table 1).

Table1:σpole position. Reference

532±12?i(259±7) Locher et al.2)

≈500?i250 Lucio et al.2)

602±26?i(196±27) Kaminski et al.3)

469.5?i178.6 T¨o rnqvist et al.3)

1100?i300 Amsler et al.3)

387?i305 Achasov et al.3)

370?i356

Zou et al.3)

870?i370 Beveren et al.3)

750±50?i(450±50) Protopescu et al.3)

650?i370 Scadron et al.4)

600+200

?100?i350 Igi et al.6)

predictions for the whole nonet we strongly believe it is qˉq.

The same is true for theσ.No single analysis of theππS-wave,however re?ned,could ever decide on what is the nature of theσ.Even the decision as to whether it really exists,cannot be done using data on theππS-wave alone,since there are inherent,model dependent,ambiguities as to how to continue analytically to a pole which is far from the physical region,as is the case for the broadσ.

It is also obvious why the NQM fails for the scalars:Chiral symmetry is absent in the NQM,but is crucial for the scalars.Chiral symmery is believed to be broken in the vacuum,and two of the scalars(σand f0)have the same quantum numbers as the vacuum.Thus to understand the scalar nonet in the same way as we believe we understand the vectors,and to make a sensible comparison with experiment,one must include chiral symmetry in addition to?avour symmetry in the quark model.The simplest such chiral quark model is the linear U3×U3sigma model with3?avours.

Then we can treat both the scalar and pseudoscalar nonets simultaneously, and on the same footing,getting automatically small masses for the pseudoscalar octet,and symmetry breaking through the vacuum expectation values(VEV’s)of the scalar?elds.

As an extra bonus we have in principle a renormalizable theory,i.e.“uni-tarity corrections”are calculable.In fact,in the?avour symmetric limit the unitarity corrections can be thought to be already included into the mass parameters of the theory,once the original4-5parameters are replaced by the4physical masses for the singlet and octet0?+and0++masses and theσVEV.

Unfortunately this over30years old model7)has had very few phenomeno-logical applications.An important exception is the intensive e?orts of M.Scadron and collaborators.

2The Linear sigma model with3?avours

The well known linear sigma model7)generalized to3?avours with complete scalar (s a)and pseudoscalar(p a)nonets has at the tree-level the Lagrangian the same ?avour and chiral symmetries as massless QCD.The U3×U3Lagrangian with a symmetry breaking term L SB is

L=12μ2Tr[ΣΣ?]?λTr[ΣΣ?ΣΣ?]?λ′(Tr[ΣΣ?])2+L SB.(1)

HereΣis a3×3complex matrix,Σ=S+iP= 8a=0(s a+ip a)λa/√

the singletλ0=(2/N f)1/21is included.Each meson in Eq.(1)has a de?nite SU3f symmetry content,which in the quark model means that it has the same?avour structure as a qˉq meson.Thus the?elds s a and p a and potential terms in Eq.(1) can be given a conventional quark line structure8).

The symmetry breaking terms are most simply:

L SB=?σσuˉu+dˉd+?sˉsσsˉs+c[detΣ+detΣ?],(2) which give the pseudoscalars mass and break the?avour and U A(1)symmetries.The small parameters?i can be expressed in terms of the pion and kaon decay constants and masses:?σ=m2πfπ,?sˉs=(2m2K f K?m2πfπ)/√

√

2=fπ/

2:One?nds denoting the often occurring combinationμ2+4λ′(u2+d2+s2)byˉμ2,and expressing the?avourless mass matrices in the ideally mixed frame:

Table2:Predicted masses in MeV and mixing angles for two values of theλ′parameter.The asterix means that mπ,m K and m2η+m2η′are?xed by experiment together with fπand f K.

Quantity Modelλ′=3.75

137?)137

m K495?)

538?)547.3

mη′963?)

-5.0?(-16.5±6.5)?3)

1028

m a

11231430

mσ619

1229980

32.3?

Θσ?singlet

S

2(u+d)

?c√

2(u+d)

√

(4λ′s+c)

As can be seen from Table 2the predictions are not far from the experimen-tal values.Considering that one expects from our previous analysis 1)that unitarity corrections can easily be more than 20%,and should go in the right direction,one must conclude that these results are even better than expected.

The trilinear coupling constants follow from the Lagrangian,and are at

the tree level:

g σπ+π?=cos φid S (m 2σ?m 2π)/f πg σK +K ?=?

√

3cos(φid S ?35.26?)(m 2f 0?m 2

K )/(2f K )

(4)

g a 0πη=cos φid P (m 2a 0?m 2

η)/f π

g a 0πη′=sin φid P (m 2

a 0?m 2η′)/f π

g a 0K +K ?=(m 2a 0?m 2

K )/f K g κ+K 0π+=(m 2κ?m 2π)/(

√3sin(φid P ?35.26?)(m 2κ?m 2η)/(2f K )g κK +η′=

√

8π

k cm (m )

Table3:Predicted couplings i g2i

i g2i i g2i iΓi

in model model

7.22678

0.28

2.17574

0.160

1.67see text

6.540

2.29273see text

2.050

These results strongly favour the interpretation that theσ(500),a0(980), f0(980),K?0(1430)belong to the same nonet,and that they are the chiral partners of theπ,η,K,η′.If the latter are believed to be unitarized qˉq states,so are the light scalarsσ(500),a0(980),f0(980),K?0(1430),and the broadσ(500)should be interpreted as an existing resonance.

Theσis a very important hadron indeed,as is evident in the sigma model, because this is the boson which gives the constituent quarks most of their mass and thereby it gives also the light hadrons most of their mass.It is the Higgs boson of strong interactions.

References

1.N.A.T¨o rnqvist and M.Roos,Phys.Rev.Lett.761575(1996);N.A.T¨o rnqvist,

Z.Physik C68647(1995).

2.M.P.Locher,V.E.Markuskin and H.Q.Zheng,Eur.Phys.J.C4317(1998);R.

Kaminski,L.Lesniak and B.Loiseau,hep-ph-9810386;J.L.Lucio,G.Moreno and M.Napsusciale,hep-ph-9810299;J.L.Lucio,M.Napsusciale,hep-ph-9903234;M.Harada,F.Sannino and J.Schechter,Phys.Rev.D541996(1991).

3.C.Caso et al.(the Particle Data Group),Eur.Phys.J.C31(1998).

4.M.Scadron Eur.Phys.J.6131(1999);R.Delbourgo,D Liu,and M Scadron

Phys.Lett.B446332(1999).

5.J.L.Lucio,M.Napsusciale and M.Ruiz-Altaba,hep-ph/9903420.

6.K.Igi and K.Hikasa,Phys.Rev.D5*******(1999).

7.J.Schwinger,Ann.Phys.2(1957)407;M.Gell–Mann and M.Levy,Nuovo Cim.

XVI(1960)705;B.W.Lee,Nucl.Phys.B9(1969)649;S.Gasiorowicz and D.A.

Ge?en,Rev.Mod.Phys.41(1969)531;J.Schechter and Y.Ueda Phys.Rev.

D32874(1971).

8.N.A.T¨o rnqvist Physics Letters B426(1998)105;N.A.T¨o rnqvist,Physics

Letters B406(1997)70.

9.N.N.Achasov and G.N.Shestakov,Phys.Rev.D5805011-1.

10.N.N.Achasov et al.,hep-ex980910,hep-ex/980716,(V.Ivanchenko at this meet-

ing).