Radiative decays with scalar mesons a0(980) and f0(980) in Resonance Chiral Theory

0.14 MB·English

by  S. Ivashyn

9 Radiative decays with scalar mesons a (980) and f (980) in Resonance
0 0
0
0 Chiral Theory
2
S. Ivashyna and A. Korchina
n ∗ †
a
a Institute for Theoretical Physics, NSC “KharkovInstitute of Physics and Technology”,
J
1, Akademicheskaya str., Kharkov 61108,Ukraine
6
2
The φ(1020) radiative decays form a good playground for study of the low-lying a0(980) and f0(980) scalar

] mesons. Thecomplicated interplayof kaonloop mechanism,isoscalar meson mixingand momentumdependence
h of the effective couplings manifests itself in the invariant mass distributions in the φ(1020) → π0π0γ and πηγ
p
spectra. These distributions are fitted in framework of Resonance Chiral Theory.
-
p
e
h −
1. Introduction e γ
[ ∗
γ φ
1 Thee+e− experimentsinNovosibirsk[1,2]and
v Frascati [3,4] allow one to study the φ(1020)
5 ππγ (and πηγ) radiative decays. The invaria→nt + S
e
4
mass distributions of the pseudoscalar (P) pairs
0
in these decays are of considerable interest. The
4
. scalarmesons(S)f0(980)anda0(980)areimpor-
1
tant intermediate resonances in the ππ and πη
0 Figure 1. The e+e annihilation to ππγ (and
9 channel correspondingly, and thereby they show −
πηγ) via the φ vector resonance. S denotes
0 up in these spectra.
the intermediate scalar resonance. Blob shows
: Itis believedthat the kaonloop(KL) coupling
v schematically the kaon loop mechanism.
i of S to the vector meson φ is very important
X
(see, e.g. the data analysis [1,5,6]). Fig.1 shows
r schematically the processes with scalar mesons.
a
Although many authors relate the dominance of
the KL mechanism to a large KK¯ component sal: one can use either the analytical expression
in the a (980) and f (980) mesons and to the for I(a,b) [10], or calculate it numerically.
0 0
proximity of the KK¯ threshold to the scalar me- The φ decays are rather suitable for study of
son mass, in fact the KL mechanism is a feature the chiral dynamics, and the framework is out-
of the chiral dynamics and reflects the impor- linedinSection2. WeshouldnotethatRχTleads
tant role of the pseudoscalar mesons in low- and toanimportantfeature–momentumdependence
intermediate-energy interactions. It is shown in of the scalar meson effective couplings. Employ-
Refs. [7,8] that the kaon loops in the φ Sγ ing the RχTLagrangian of [9] we obtained [7] a
transitions naturally arise in the leading→order completesetof (p4)contributionstovariousra-
O
in Resonance Chiral Theory (RχT) [9], irrespec- diative decays with the scalar mesons. Later on
tivelyofthethresholdandmassposition,andthe this model was applied [8] to the study of π0π0
internalstructureofthe scalars. Theloopcontri- andπ0ηinvariantmassdistributions. InRef.[7,8]
butionis convergent,gauge invariantanduniver- we compared the model results with estimates of
other related approaches [11,12].
∗Partly supported by NASU grant for young researchers
The free parameters of the model are c , c ,
(contract 8.63/2008) and Joint NASU-RFFR scientific d m
c˜ and c˜ , which are the coupling constants in
projectN38/50-2008. d m
†SupportedbytheINTASgrant05-1000008-8328. the Lagrangian [9]. It is possible to fix their val-
1
2
1.2 themomentum-independentvertexicSPP/fπ2and
2eV] 1 .11 momentum-decpoennsdtaenntt t(hseeemToambleen1tu),mw-dheepreenfden=t9o2n.e4iMcˆSePVP.(Tp1h·epse2)v/efrπ2-
G 0.9 π
π4 [ 0.8 tices vanish in the chiral limit, as soon as cSPP
2)/ 0.7 is proportional to the pseudoscalar meson mass
p
(-K 0.6 squared. The f ππ tree-level decay width is
+a K0 00..45 then 0 →
2
G 0.3
0.2 Γ (p2) = Γ˜ (p2)Θ(p2 4m2),
0.5 0.6 0.7 0.8 0.9 1 f0→ππ f0→ππ − π
p2 [GeV2]
where Θ(x) is the Heaviside step function. For
4 the further convenience we employ the analytic
2] 3.5 momentum-dependent function of p2
2πp)/4 [GeV 2 .235 constant Γ˜f0→ππ(p2) = 23 21p2rp42 −m2πG2f0π4ππ(p2), (1)
(-K
+K 1.5 which is defined above the threshold and
2Gf 0 1 below the threshold with p2/4 m2 =
− π
0.5 i p2/4 m2 . Thefactor3/2accountsforboth
0.5 0.6 0.7 0.8 0.9 1 | − π| p
p2 [GeV2] πp+π− and π0π0 final states and Gf0ππ is a cou-
pling ofthef to π+π . The analogousformulae
0 −
holdforotherdecays. Wesupposethatthescalar
iso-singletf (980)=(Ssing cosθ Soct sinθ)isa
0 − 8
mixture of the singlet Ssing and the eights com-
Figure 2. Effective couplings G2 (p2)/4π
a0K+K− ponent of octet Soct with the angle θ.
(upper panel) and G2 (p2)/4π (lower 8
f0K+K− It is important to note that the scalar me-
panel). The values c = c = f /2
46.2MeV[13]arechosenm. Mixingdangleθπ=27.3≈8 son couplings G2SPP(p2)/4π are the essentially
momentum-dependent functions, see Table 1 for
istakenfromourfitwithFlatt´e-likepropagators.
explicit expressions. As an example, in Fig. 2
The horizontallines indicate the constantvalues,
we compare the functions G2 (p2)/4π and
see [4]. a0K+K−
G2 (p2)/4π to the corresponding constant
f0K+K−
values of these couplings, which are often dis-
cussed.
uesfromthelimitoflargenumberofquarkcolors The total widths Γ (p2) of the scalar
S,tot
(N ), and by imposing the constraints sug- mesons are
c
→∞
gested in Ref. [13]. In the present paper we use
the corresponding values in the description of in- Γf0,tot(p2) = Γf0→ππ(p2)+Γf0→KK¯(p2), (2)
avanrdiaπnηtγmdaescsaydsi.stributions in φ(1020) → π0π0γ Γa0,tot(p2) = Γa0→π0η(p2)+Γa0→KK¯(p2). (3)
The application of the model is presented in In Fig. 1 the mechanism of the scalar-meson
Section3andfutureprospectsareoutlinedinSec- production is depicted. The scalar meson propa-
tion 4. gatesanddecaystoapairofpseudoscalarmesons.
Thefiniteresonancewidtheffectsintheinvariant
2. Scalar meson vertices and propagators mass distributions for π0π0 and π0η in the φ ra-
in RχT diative decays are known to be important [14].
For the scalar meson propagator one may use
IntheRχTascalarmesonS(p)decaysintothe
pseudoscalar meson pair P (p )P (p ) through D (p2) = [p2 m2 +i p2Γ (p2)] 1, (4)
1 1 2 2 S − S S,tot −
p
3
Table 1
Numerical values of these couplings, in princi-
RχTeffective couplings (see Table 9 in Ref. [7]).
ple,canbedeterminedfromtheunderlyingQCD.
ccfππ == −mm22π((44c˜c˜mccoossθθ−+2 22//33ccmssiinnθθ)),. Fstrroamtedth[9e]athssautmthpetioocnteNtcan→dsi∞gletit(wwiaths “dteimldoe”n)-
fKK − K m p m chiral couplings obey the constraints
cˆ = 4c˜ cosθ 2 2/3pc sinθ,
fππ d d c c
− m d
cˆfKK = 4c˜dcosθ+ p2/3cdsinθ. c˜m = √3, c˜d = √3. (8)
caKK = −√2cmm2Kp, Inthislimitthe octetandsingletmesonsbecome
caπη = −2Z 2/3cmm2π . degenerate and have equal masses.
We use the latter constraintalthough one may
cˆ = √2c p,
aKK d
arguewhetherthelarge-N considerationisappli-
c
cˆ = 2 2/3c .
aπη Z d cable to the scalar mesons, especially in view of
G 1 pcˆ (m2 p2/2)+c , so-called Inverse Amplitude Method results [17].
Gf0fK0πKπ ≡≡ ff1ππ22 (cid:0)cˆff00KππK(m2πK−−p2/2)+cf0fπ0πKK, (cid:1) oTnhaenucensuwsuaaslrleacregnet-lNycsubmehmavairoizreodfitnhe[1s8c].alar res-
G 1 cˆ (m2 p2/2)+c ,
a0KK ≡ fπ2 (cid:0) a0KK K − a0K(cid:1)K In Ref. [13] based on the short-distance con-
G 1 cˆ (m2 +m2 p2)/2+c .
a0πη ≡ fπ2 (cid:0) aπη η π − aπ(cid:1)η straints on the flavor-changing Kπ, Kη and Kη′
The factor Z = co(cid:0)scθo0s−(θ√8−2θs0in)θ8 ≈ 1.53, where θ0(cid:1) socfaclamr faonrdm-cfdacctoourspliitngwsasinshtohwen t(hpa2t)trheesovnaalnucees
and θ are the η - η mixing angles. O
8 ′ chiral Lagrangiansatisfy the relation
f
π
c =c = 46.2MeV. (9)
m d
2 ≈
wherethescalarmesonmassisdenotedbym . A
S
This relation, in particular, allows us to reduce
more advanced form of the propagator including
the number of adjustable parameters in the fit.
bothrealandimaginarypartsofthescalar-meson
selfenergyΠ (p2)wasproposedrecently[15]. As The expressionfor the invariantmassdistribu-
S
tionintheKLmodelisobtainedinRef.[10],and
anoption,inthepresentworkweusetheso-called
its physical meaning is explained in terms of the
Flatt´e-likeform[16] the scalar-mesonpropagator
SPP andφPP couplings[6]. Takingintoaccount
DS(p2) = [p2−m2S +i p2Γ˜S,tot(p2)]−1, (5) the momentum-dependent vertices we obtain
where p dB =1 1 α p2 1−4m2π/p2 (10)
dm 2Γ 4 48π4m4
Γ˜f0,tot(p2) = Γ˜f0→ππ(p2)+Γ˜f0→KK¯(p2), (6) π0π0 φI,(tao,tb)p2 ×pM2 p2K3
Γ˜a0,tot(p2) = Γ˜a0→π0η(p2)+Γ˜a0→KK¯(p2), (7) × (cid:12)Df0(p2)(cid:12) φM−φ !
wyshisic,hseiess[6u]p.pTohrteedprboypathgeatpohreinnotmheenfoolromgiocafleaqn.(a5l)- G(cid:12)(cid:12)(cid:12) 2f0ππ(p2)(cid:12)(cid:12)(cid:12)G2f0KK(p2) √2GVMφ 2
accountsforacontributionoftheKK¯ channelto × 4π 4π fπ2 !
the imaginary part of the self energy above the
KK¯ threshold. It also approximately describes for the π0π0 invariant mass (mπ0π0 p2) dis-
≡
tribution, here a = M2/m2 , b = p2/m2 and
the real part of the self energy below the thresh- φ K p K
α 1/137. Thefactor1/2takesintoaccountthe
old via the analytic continuation.
≈
identity oftheneutralpions. Theformulaforthe
π0η case is analogous to (10).
3. Application to the radiative decays
The φ(1020) π0π0γ (and πηγ) spectra fit-
→
A priorithe scalar octet and singlethave inde- ted by varying only the masses of scalar mesons
pendent couplings c ,c and c˜ ,c˜ respectively. andthesinglet-octetmixing angleθ areshownin
d m d m
4
MM..NN..AAcchhaassoovv eett aall.. [[PPLL,, BB448855,, 334499 ((22000000))]] MM..NN..AAcchhaassoovv eett aall.. [[PPLL,, BB447799,, 5533 ((22000000))]]
AA..AAllooiissiioo eett aall.. ((KKLLOOEE)) [[PPLL,, BB553377,, 2211 ((22000022))]] CC..BBiinnii eett aall.. [[KKLLOOEE nnoottee 117733,, 0066//0022]]
FFRRllaaeettttΠΠee ((RRpp22ee))ΠΠ==00((pp,, 22MM)),,ff 00MM==ff((0011==00((77888888..1144..5522 77++ ++//--// --77 ..6644..6633))88 ))MM MMeeVVeeVV,, θθ,, ==θθ==227711..5533..8899oo22 oo++ ++//--// --11 ..1166..8855oo11,,oo χχ,, 22χχ//22dd//..ddoo....ooff....==ff..== 88 6655..9955..33336699////22229999 FFRRllaaeettttΠΠee ((RRpp22ee))ΠΠ==00((pp,, 22 )) ,, MM MMaaaa0000====((((1111000022885599..44..668811 ++ ++//--//-- 11 9911....00116655)))) MMMMeeeeVVVV,,,, χχχχ2222////dddd....oooo....ffff....==== 1188000066..00..00668844////33332222
9900 116600
8800 114400
7700
112200
-1-1MeV]MeV] 6600 -1-1MeV]MeV] 110000
88mm x 10 [ x 10 [0000ππππ 45450000 88mm x 10 [ x 10 [00ηπηπ 68680000
dB/ddB/d 3300 dB/ddB/d
4400
2200
1100 2200
00 00
770000 775500 880000 885500 990000 995500 11000000 770000 775500 880000 885500 990000 995500 11000000
mmππ00ππ00 [[MMeeVV]] mmηηππ00 [[MMeeVV]]
Figure 3. The invariant mass distributions in φ πoπoγ (left panel) and φ πoηγ (right panel).
→ →
Theoretical calculation is performed with the values of parameters c = c = f /2 46.2 MeV [13].
m d π
≈
The scalar resonance masses and the mixing angle θ obtained from the fit are shown in the legend. The
two variants of the scalar meson propagator are discussed in the text. Data: [1,3] (left) and [2,4] (right).
Fig. 3. We focus on the region near the f (980) Usually the experimental results are inter-
0
peak and correspondingly use the data above pretedintermsoftheconstantvaluesoftheSPP
700MeV.The σ =f (600)meson is not included couplings g2 /4π instead of G (p2) func-
0 SPP SPP
in the analysis. tions. Constantg couplingscanbejustifiedin
SPP
The figure also illustrates an effect from the thephenomenologicalapproaches,howeverthisis
Flatt´e-likemodificationofthescalar-mesonprop- not supported by chiral models. An importance
agator. The shape of the π0π0 and πη distribu- of the derivative-coupling interactions was em-
tions is well reproduced. For the both forms of phasized in Ref. [11] some time ago,and recently
the propagator(4) and (5) the value of χ2/d.o.f. discussed in [19]. The replacement of G (p2)
SPP
is relatively small. functions by the constantvalues may have a dra-
Themassofthef (980)andthemixingangleθ matic influence on the invariant mass distribu-
0
turn out to be strongly correlatedin the fit. The tions. In fact, in these distributions p2 varies
massesofthescalarresonancesobtainedfromthe within the wide region – from the PP thresh-
fitwiththepropagator(4)areoverestimated. We old up to the φ mass squared – and the scalar
observe that the account for the real part of the resonances may “feel” the variation of G (p2)
SPP
scalar-mesonself energy, even in a simple Flatt´e- alongthespectrum. Thisbecomesmoremanifest
like approximation, results in a considerable de- in the case of a broad σ meson, which is some-
creaseofthef massinthefit. Thereforethereis times included in the fit.
0
a strong motivation for the careful treatment of
theself-energyandtheeffectsrelatedtoaspecific
4. Prospects and conclusion
behavior of the scalar meson spectral function.
The importance ofthese aspects is discussedalso We fit the φ(1020) π0π0γ and πηγ spec-
in Ref. [14]. tra by varying only the→masses of scalar mesons
5
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