arXiv:0905.2017v3 [hep-ph] 2 Sep 2011 hoyadsmrlsd o oki hs channels. these in work mesons, not scalar perturbation light do both became rules GeV because sum 1 QCD 1 and to of up theory block region stumbling the in a channels scalar The Introduction 1. Outline † ∗ 1 References. .Appendix 8. .Ftr Trends. Future 7. summary. Preliminary 6. .Pouto fthe of Production 5. .Aayi fhg ttsisBledt ntereactions the on data Belle statistics high of Analysis 4. .Lgtsaa eosi h ih fpoo-htncolli- photon-photon of light hadron the the in mesons in scalar mesons Light scalar light 3. the of place Special 2. Introduction. 1. -al [email protected] E-mail: -al [email protected] E-mail: eoac akrud atclry ntecs ftesoli the of case the in in fro inseparable channels, scalar Particularly, large i.e., vector tary a by resonances, classic background. accompanied solitary to resonance not not are contrast which are in nances, there that, region this is point The φ o bevda l eas h ieeta rbblte o probabilities differential the because all at observed not ttesm ieteqeto ntentr fthe of nature the on question the time same the At → γa 8.1. eigand tering J/ψ in.73 erhsfrthe for Searches 7.3. sions. π . The 7.1 σ σ K γγ resonances. scalar Pro- of mechanisms 3.4. duction distributions. angular and contributions Born a iuto.33 yaiso h reactions the of Dynamics experimen- Current 3.3. 3.2. situation. tal investigations. of History 3.1. sions. structure. four-quark their of Evidences world. a 0 (600), 60 and (600) 0 + 0 η 90 and (980) (980) K . → γγ aoaoyo hoeia hsc,SL ooe Institute Sobolev S.L. Physics, Theoretical of Laboratory → − π niur oe rmteote.A rsn h otiiln nontrivial the present At outset. the from model antiquark ASnmes 23.x 34.f 36.e 13.75.Lb 13.60.Le, 13.40.-f, 12.39.-x, numbers: PACS data. experimental mec current new production of a two-photon entered analysis concerning it the Recently, results the mesons. review scalar We light substan a the made of has nature physics Two-photon anybody. practically → and f + → ρa f 0 h ih clrmsn,dsoee vrfryyasao be ago, years forty over discovered mesons, scalar light The π 0 90,and (980), ππ γπη 0 f 90 and (980) − f 90 eas . nlsiiyof Inelasticity 7.4 decays. (980) f 0 γγ 0 0 (980) 8.2. . and 90 resonances. (980) 90 eoacs h eoac ek nthe in peaks resonance the resonances, (980) and σ → a 0 (600), 90 eoac ntereaction the in resonance (980) K γγ − φ γγ 0 ih clrMsn nPoo-htnCollisions Photon-Photon in Mesons Scalar Light a → K a ¯ → 0 a → 0 90 mixing. (980) 0 0 γf κ 90 eoacsin resonances (980) 90 eoacsnear resonances (980) (800), ecintrsod.72 The 7.2. thresholds. reaction π π 0 (980) 0 0 π η 0 8.3. . aiettoso the of Manifestations . J/ψ → a 0 γππ 90,and (980), γγ ..Achasov N.N. → → easwudbe would decays ωf K 0 K 90 and (980) ¯ γγ γγ ππ . f ∗ 0 → γγ these f (980) γγ colli- scat- reso- ππ ∗ → → m - n ..Shestakov G.N. and : htthe the that prin- against of One reasons reviews. on cipal PDG information from way disappeared an states conclusive and these a disappointment in existence general their entailed prove to attempts ful nmn,adhnefrudrtnigteconfinement con- the the understanding of from for itself. mechanism arising hence the and realization, understanding finement, symmetry for chiral major the is 11], [10, 1,1,1]teewr icvrdtenro ih scalar light narrow the isovector discovered seventies the were the resonances, of there clear beginning 19] the made 18, at At been and QCD. [13, sixties of has the realization of it energy end low years the the ten be could after LSM surpris- that that The is contains thing bosons. ing and Goldstone as symmetry mesons chiral pseudoscalar of breaking the was The spontaneous mesons scalar [12–14]). for example, search linear for a for (see, ground time theoretical light had that reviews the at (PDG) on Group appeared Data information Particle in preliminary mesons a scalar and already ties S hr sangtv akrudpaei the in phase background negative a is there continued. the been had which themselves, in processes, investigation of theoretical and experimental Nevertheless, n,d o asoe 90 over pass not do ing, 2 structures: ftelgtsaa orqaksae 2] esgetdals suggested He [20]. incor- states which four-quark model, scalar bag that n light MIT the the the exists in of there phenomenologically, that confinement noted case. porates a Jaffe meson scalar than 1977 light more In the nothing in is exercise approximation academic resonance solitary The h rnia oeo h hrlbcgon nteft fthe of fate the in background chiral Secti the see [1–5], of invariance σ role gauge for principal region The photon e photon soft the a of in function cubic proportionally vanish decays time a ftelgtqakspectroscopy. quark light the of sfrthe for As utn h light the Hunting iuto hne hnw hwd[]ta nLSM in that [6] showed we when changes Situation aeapiuewt isospin with amplitude wave 0 − 60 eoac a eosrtdi h linear the in demonstrated was resonance (600) o ahmtc,609,Nvsbrk Russia Novosibirsk, 630090, Mathematics, for 90 = (980) a a σ 0 0 tr fteesae sn ogrdenied longer no is states these of ature 90 and (980) 90 and (980) oe LM 1–7,wihtksit account into takes which [15–17], (LSM) model tg fhg ttsismeasurements. statistics high of stage ilcnrbto oudrtnigthe understanding to contribution tial aim ftelgtsaas ae on based scalars, light the of hanisms aeacalnefrteniequark- naive the for challenge a came S aepaesit,both shifts, phase wave d a us ¯ σ 0 + s ¯ 90 = (980) and , and † f f 0 0 90 eoacscm noblvdchildren beloved into came resonances (980) 90 ih eteesae ihsymbolic with states these be might (980) σ κ f and 0 eos ogsadn unsuccess- long-standing mesons, 90 ( = (980) u 0 ds ¯ a κ σ tpttv eoac masses. resonance putative at 0 s ¯ 90 n isoscalar and (980) , σ eoshdbgni h six- the in begun had mesons and a and 0 0 us 90 ( = (980) I u ¯ κ ,wihhdsthe hides which 0, = s ¯ κ + eoswstefact the was mesons ππ ttscudreveal could states ds d and ¯ s us ¯ ) / u ¯ ππ √ s ¯ σ πK .Fo that From 2. − oe [6–9]. model scattering f ds 0 scatter- (980). d ¯ s ¯ ) nergy n2. on / onet √ 2, σ n 2 o 2
− − ′ meson with the result that the ππ S wave phase shift e (p1) e (p1) does not pass over 900 at a putative resonance mass. It has been made clear that shielding wide lightest scalar mesons in chiral dynamics is very natural. This idea γ was picked up and triggered new wave of theoretical and experimental searches for the σ and κ mesons, see, for Hadrons example, [21–28]. As a result the light σ resonance, since 1996, and the light κ resonance, since 2004, appeared in γ the PDG reviews [29, 30].
By now there is an impressive amount of data about + + ′ e p2 e p the light scalar mesons [10, 11, 31, 32]. The nontrivial ( ) ( 2) nature of these states is no longer denied practically any- + − body. In particular, there exist numerous evidences in Figure 1: The two-photon process of hadron formation at e e colliders; p , p′ and p , p′ are the 4-momenta of electrons and favour of their four-quark structure. These evidences are 1 1 2 2 positrons. widely covered in the literature [1, 4, 31–84]. They are presented also in Sections 2–6. One of them is the suppression of the a (980) and 0 signals of the f (980) resonance in the both charge chan- f (980) resonances in the γγ π0η and γγ ππ reac- 0 0 nels. The previous indications for the f (980) production tions, respectively, predicted→ in 1982 [85, 86]→ and con- 0 in the γγ collisions [94–100] were rather indefinite. firmed by experiment [10, 11]. The elucidations of the In the given paper there are presented the results mechanisms of the σ(600), f0(980), and a0(980) reso- + nance production in the γγ collision and their quark of the investigation the mechanisms of the γγ π π−, γγ π0π0, and γγ π0η reactions (see Sections→ 3–5) structure are intimately related. That is why the studies → → of the two-photon processes are the important part of the based on the analysis [9, 101–106] of the Belle data [89– light scalar meson physics. 93] and our previous investigations of the scalar meson physics in the γγ collisions [48, 85, 86, 107–110]. We It should be noted that the reactions of hadron pro- also briefly (sometimes critical) survey analyses of other duction in photon-photon collisions are measured at + authors. e e− colliders, i.e., the information on the transitions hadrons The joint analysis of the Belle high-statistics data on γγ is extracted from the data on the pro- + 0 0 → + + + the γγ π π− and γγ π π reactions is presented cesses e e− e e−γγ e e− hadrons (Fig. 1). The → → most statistics→ is obtained→ by the so-called “non tag” and the principal dynamical mechanisms of these pro- method when hadrons only are detected and the scat- cesses are elucidated in the energy region up to 1.5 GeV. The analysis of the Belle high-statistics data on tered leptons are not. In this case the main contribution 0 + + the reaction γγ π η is presented too. It is shown to the cross section of e e− e e− hadrons is provided → by photons with very small→ virtualities. Therefore, this that the two-photon decays of the light scalar resonances are the four-quark transitions caused by the rescatter- method allows to extract data on hadron production in + + + ings σ π π− γγ, f0(980) (K K− + π π−) γγ, collisions of almost real photons. The absolute majority → → 0 →0 → and a (980) (KK¯ + π η + π η′) γγ in contrast to of data on the inclusive channels γγ hadrons has been 0 → → obtained with the use of this method.→ If the scattered the two-photon decays of the classic P wave tensor electrons are detected (which leads to a loss of statistics), qq¯ mesons a2(1320), f2(1270) and f2′ (1525), which are 2 caused by the direct two-quark transitions qq¯ γγ in then one can investigate in addition the Q dependence → 2 the main. As for the direct coupling constants of of the hadron production cross sections in γγ∗(Q ) col- 2 the σ(600), f0(980), and a0(980) resonances with the lisions, where γ is a real photon and γ∗(Q ) is a photon 2 2 3 γγ system, they are small. It is obtained the two- with virtuality Q = (p1 p′ ) . − 1 photon widths averaged over resonance mass distribu- Recently a qualitative leap had place in the experimen- 0 tions Γf0 γγ ππ 0.19 keV, Γa0 γγ πη 0.4 keV and tal investigations of the γγ ππ and γγ π η processes h → i ≈ h → i ≈ → → Γσ γγ ππ 0.45 keV. [89–93] that proved the theoretical expectations based on h → i ≈ the four-quark nature of the light scalar mesons [85, 86]. In Section 7, we attend to the additional possibili- ties of the investigation of the a0(980) and f0(980) res- The Belle Collaboration published the data on the cross + 0 0 + 0 0 onances in the reactions γγ K K− and γγ K K¯ , sections for the γγ π π− [90, 91], γγ π π [92], and → → γγ π0η [93] reactions,→ statistics of which→ are hundreds which are as yet little studied experimentally, and also → to the promising possibility of investigating the nature of of times as large as statistics of all previous data. The 2 Belle Collaboration observed for the first time the clear the light scalars σ(600), f0(980), and a0(980) in γγ∗(Q ) collisions.
2. Special place of the light scalar mesons 3 Detailed formulae for experimental investigations of the reactions in the hadron world. Evidences of their e+e− e+e− hadrons may be found in the reviews [87, 88]. four-quark structure →
3
· ·
Even a cursory examination of PDG reviews gives an ·
à à idea of the four-quark structure of the light scalar meson Ã
4
a =f
a =f =f
nonet , σ(600), κ(800), a (980), and f (980), a
¼ ¼
¼ ¼ ¼
0 0 ¼
Ã
à Ã
¡
¡ ¡ 0 + a0− a0/f0 a0
+ − κ κ (1) Figure 2: The K K loop mechanism of the radiative decays φ(1020) γ(a0(980)/f0(980)). { }{ } → σ a considerable part of the strange pair ss¯ in the wave inverted [49] in comparison with the classical P wave qq¯ function of the f (980). tensor meson nonet f (1270), a (1320), K (1420), and 0 2 2 2∗ At the beginning of eighty it was demonstrated in a se- f2′ (1525) ries of papers [34, 123, 132–136] that data on the f0(980) and a (980) resonances, available at that time, can be f 0 2′ interpreted in favour of the q2q¯2 model, i.e., can be ex- plained by using coupling constants of the f0(980) and K2∗ K2∗ (2) { }{ } a0(980) states with pseudoscalar mesons superallowed by 0 + the Okubo-Zweig-Iizuka (OZI) rule as it is predicted by a2− a2/f2 a2 , the q2q¯2 model. In particular, in these papers there were obtained and specified formulae for scalar resonance or also in comparison with the classical S wave vector propagators with taking into account corrections for fi- meson nonet ρ(770), ω(782), K (892), and φ(1020). 5 In ∗ nite width in case of strong coupling with two-particle the naive quark model such a nonet cannot be understood decay channels. Late on, these formulae were used in as the P wave qq¯ nonet, but it can be easy understood fitting data of a series of experiments on the f (980) as the S wave q2q¯2 nonet, where σ(600) has no strange 0 and a (980) resonance production (see, for example, [90– quarks, κ(800) has the s quark, a (980) and f (980) have 0 0 0 92, 137–151]). Recently, it was shown that the above the ss¯ pair. scalar resonance propagators satisfy the K¨allen-Lehmann The scalar mesons a0(980) and f0(980), discovered representation in the domain of coupling constants usu- about forty years ago, became the hard problem for the 6 ally used [152]. naive qq¯ model from the outset. Really, on the one hand At the end of eighties it was shown that the study of the the almost exact degeneration of the masses of the isovec- radiative decays φ γa0 γπη and φ γf0 γππ tor a0(980) and isoscalar f0(980) states revealed seem- can shed light on the→ problem→ of a (980)→ and f→(980) + ¯ 0 ¯ √ 0 0 ingly the structure a0 (980)= ud, a0(980)=(uu¯ - dd)/ 2, mesons [1]. Over the next ten years before experiments ¯ a0−(980)= du¯ and f0(980)=(uu¯ + dd)/√2 similar to the (1998) the question was considered from different points structure of the vector ρ and ω or tensor a2(1320) and of view [153–165]. f2(1270) mesons, but on the other hand, the strong cou- Now these decays have been studied not only theo- ¯ pling of the f0(980) with the KK channel as if suggested retically but also experimentally with the help of the SND [137–140] and CMD-2 [141, 142] detectors at Bud- ker Institute of Nuclear Physics in Novosibirsk and the KLOE detector at the DAΦNE φ-factory in Frascati 4 To be on the safe side, notice that the linear σ model does [143, 144, 146–148, 151, 166–168]. not contradict to non-qq¯ nature of the low lying scalars because These experimental data called into being a series of Quantum Fields can contain different virtual particles in different regions of virtuality. theoretical investigations [2–5, 40, 169–172] in which ev- 5 In Eqs. (1) and (2) the mass and isotopic spin third component idences for the four-quark nature of the f0(980) and of states increase bottom-up and from left to right, respectively. a0(980) states were obtained. Note the clear qualitative 6 Note here a series of important experiments of seventies in which one. The isovector a0(980) resonance is produced in the the f0(980) and a0(980) resonances were investigated [111–116], as well as a few theoretical analyses of scalar meson proper- radiative φ meson decay as intensively as the isoscalar η′(958) meson containing 66% of ss¯, responsible for ties relevant to this period [20, 117–122]. In the last-named ≈ paper there was theoretically discovered the fine threshold phe- the φ ss¯ γss¯ γη′(958) decay. In the two-quark nomenon of the a0(980) f0(980) mixing which breaks the iso- ≈ 0 → → ¯ − model, a0(980)=(uu¯ dd)/√2, the φ ss¯ γa0(980) topic invariance (see also [123]). Now a rebirth of interest in the decay should be suppressed− by the OZI≈ rule.→ So, ex- a0(980) f0(980) mixing takes place and there appear new sug- gestions− on search for this phenomenon (see, for example, [124– periment, probably, indicates for the presence of the ss¯ 127] and references in these papers) as well as the first indications pair in the isovector a0(980) state, i.e., for its four-quark + − 0 for its manifestation in the f1(1285) π π π decay, which is → nature. measured with the help of the VES detecor at IHEP in Protvino When basing the experimental investiga- [128–130], and in the decays J/ψ φf0(980) φa0(980) φηπ and χ π0a (980) π0f (980)→ π+π→−π0, which are→ tions [1], it was suggested the kaon loop model c1 0 0 + 0 + → → → φ K K− γa (980) γπ η and φ K K− being investigated with the BESIII detector at BEPCII in Chine → → 0 → → → [131]. γf (980) γππ, see Fig. 2. This model is used 0 → 4
6 70 1
- 2 2 1 - 5 60 qq¯ states as well as between qq¯ and q q¯ states but their GeV , 4 50 dm, MeV
dm intensities depend strongly on a type of the transition. A / ) / ) γ 40 0 0 3 π π η 0 γ π 30 radiative four-quark transition between two qq¯ states re- --> --> φ 2 φ ( 20
dBR quires creation and annihilation of an additional qq¯ pair, x
4 1 x dBr ( 8
10 10
10 i.e., such a transition is forbidden according to the OZI 0 0 0.7 0.75 0.8 0.85 0.9 0.95 1 300 400 500 600 700 800 900 1000 rule, while a radiative four-quark transition between qq¯ m, GeV m, MeV and q2q¯2 states requires only creation of an additional qq¯ Figure 3: The left and right plots illustrate the fit to the KLOE pair, i.e., such a transition is allowed according to the data for the π0η and π0π0 mass spectra in the φ γπ0η [143] and OZI rule. The consideration of this question from the → φ γπ0π0 [144] decays, respectively. See for details [169–172] large NC expansion standpoint [4, 5] supports a suppres- → . sion of a radiative four-quark transition between two qq¯ states in comparison with a radiative four-quark transi- tion between qq¯ and q2q¯2 states. So, both intensity and 8 mechanism of the a0(980) and f0(980) production in the 2 |
) 6 radiative decays of the φ(1020) meson indicate for their m g( | four-quark nature. x
5 4 10 Note also that the absence of the decays J/ψ 2 γf (980), J/ψ a (980)ρ, J/ψ f (980)ω against→ 0 → 0 → 0 0 a background of the rather intensive decays into the 0.8 0.85 0.9 0.95 1 m, GeV corresponding classical P wave tensor qq¯ resonances J/ψ γf2(1270) (or even J/ψ γf2′ (1525)), J/ψ Figure 4: A new threshold phenomenon in φ K+K− γR → → → → → a2(1320)ρ, J/ψ f2(1270)ω intrigues against the P decays. The universal in the K+K− loop model function → 2 2 wave qq¯ structure of the a0(980) and f0(980) states [36– g(m) = g (m)/g + − is drawn with the solid line. The | | | R RK K | 38, 41]. contributions of the imaginary and real parts of g(m) are drawn with the dashed and dotted lines, respectively. 3. Light scalars in the light of two-photon collisions in the data treatment and is ratified by experiment 3.1. History of investigations [137–144, 146–148, 151, 166–168, 173–175], see Fig. Experimental investigations of light scalar mesons in the + 0 0 0 3. The key virtue of the kaon loop model has the γγ π π−, γγ π π and γγ π η reactions with the + → → → built-in nontrivial threshold phenomenon, see Fig. e e−-colliders began in eighties and have continued up 4. To describe the experimental mass distributions to now. In first decade many groups, DM1, DM1/2, dBR(φ γRγab; m)/dm g(m) 2ω(m), 7 the function PLUTO, TASSO, CELLO, JADE, Crystal Ball, MARK g(m) 2 →should be smooth∼ at |m |0.99 GeV. But gauge II, DELCO, and TPC/2γ, took part in that. Only Crys- 0 0 0 |invariance| requires that g(m)≤ is proportional to the tal Ball and JADE could studied the π π and π η chan- + photon energy ω(m). Stopping the impetuous increase nels, the others (and JADE) the π π− channel. For of the function (ω(m))3 at ω(990 MeV)=29 MeV is those, who wish to read more widely in the contribution + the crucial point in the data description. The K K− of this impressive period in light scalar meson physics, loop model solves this problem in the elegant way [2– one can recommend the following reviews and papers: 5, 39, 40], see Fig. 4. In truth this means that a0(980) [48, 85, 86, 88, 107, 108, 176–197]. and f0(980) resonances are seen in the radiative decays First results on the f0(980) resonance production are + of φ meson owing to the K K− intermediate state. collected in Tables I and II. So the mechanism of the a0(980) and f0(980) mesons It is reasonable that first conclusions had a qualita- production in the φ radiative decays is established at a tive character and data on the f0(980) γγ decay width → physical level of proof at least. had large errors or were upper bounds. Note as a guide that the TASSO and Crystal Ball results, see Table II, Both real and imaginary parts of the φ γR ampli- 1 + → based on the integral luminosity equals to 9.24 pb− and tude are caused by the K K− intermediate state. The 1 + 21 pb− , respectively. imaginary part is caused by the real K K− intermediate As for the a0(980) resonance, it was observed in the state while the real part is caused by the virtual compact 0 + γγ π η reaction only in three experiments. The Crys- K K− intermediate state, i.e., we are dealing here with tal→ Ball group [189] collected during two years the integral the four-quark transition [4, 5, 39, 40]. Needless to say, 1 luminosity of 110 pb− , selected at that 336 events rele- radiative four-quark transitions can happen between two 0 vant to the γγ π η reaction in the a0(980) and a2(1320) region, see Fig.→ 5, and published in 1986 the follow- 0 +0.10 ing result: Γa0 γγ B(a0 π η)=(0.19 0.07 0.07) keV, → → ± − 7 where Γa0 γγ is the width of the a0(980) γγ decay and Here m is the invariant mass of the ab-state, R = a0(980) or → 0 0 0 0 → 0 f0(980), ab = π η or π π , the function g(m) describes the φ B(a0 π η) is the branching ratio of the a0(980) π η → → → 0 γ[a0(m)/f0(m)] transition vertex. decay. The measured value of Γa0 γγB(a0 π η) → → 5
Table I: First conclusions on the f0(980) production in Table III: 1990–1992 data on the γγ width of the f0(980) (see γγ → ππ (see reviews [176–178]). the text).
Experiments Conclusions Experiments Γf0→γγ [keV]
Crystal Ball No significant f0(980) Crystal Ball (1990) 0.31 ± 0.14 ± 0.09
CELLO Hint of f0(980) MARK II (1990) 0.29 ± 0.07 ± 0.12 +0.08 JADE No evidence for f0(980) JADE (1990) 0.42 ±0.06−0.18 TASSO Good fit to data book values Karch (1991) 0.25 ± 0.10
for f2(1270) includes f0(980) Bienlein (1992) 0.20 ± 0.07 ± 0.04 (3 σ effect) ≤ 0.31 (90% CL)
MARK II No significant f0(980) signal
the earlier Crystal Ball and JADE experiments. The Table II: First results on the γγ width of the f (980) (see 0 detailed analysis of the new Belle data we present reviews [88, 176, 177, 179, 180, 184]). in Section 5. Here we only point out the value for +0.003+0.502 Experiments Γf0→γγ [keV] 0 Γa0 γγB(a0 π η)=(0.128 0.002 0.043) keV obtained + − → → − − TASSO (1.3 ± 0.4 ± 0.6)/B(f0 → π π ) by the authors of the experiment [93] and the average 0 +0.8 Crystal Ball < 0.8/B(f0 → ππ) (95% C.L.) value for Γa0 γγB(a0 π η)=(0.21 0.4) keV from the → → − JADE < 0.8 (95% C.L.) last PDG review [11]. Kolanoski (1988) [184] 0.27 ± 0.12 (average value) The JADE group [96] measured also the γγ π0π0 → cross section and having (60 8)-events in the f0(980) region (and, for comparison,± (2177 47) events in the ± characterizes the intensity of a0(980) production in f2(1270) region) obtained for the f0 γγ decay width 0 +0.08 → the channel γγ a0(980) π η. The prehistory of Γf0 γγ = (0.42 0.06 0.18) keV (that corresponds to → → → ± − this result see in Refs. [179, 185, 186]. After four Γf0 γγ < 0.6 keV at 95% C.L.). → years, the JADE group [96] (see also [184]) obtained In addition, in 1990 the MARK II group in experiment 0 + Γa0 γγ B(a0 π η)=(0.28 0.04 0.10) keV based on the γγ π π reaction with the integral luminosity → → ± 1± 0 − on the integral luminosity 149 pb− and 291 γγ π η 1 → → 209 pb− [95] and the Crystal Ball group in 1990–1992 events. The Crystal Ball [189] and JADE [96] data in experiments on the γγ π0π0 reaction with the in- on the a (980) γγ decay have aroused keen inter- 1→ 1 0 tegral luminosities 97 pb− [94] and 255 pb− [98, 200] est, see, for example,→ [108, 182–184, 187, 192, 195, obtained also similar results for Γf0 γγ. All data are 196, 198, 199]. Late on, need for high-statistic data listed together in Table III, and Figs.→ 6(a) and 6(b) il- arose. But until very recently, there are no new ex- lustrate the manifestations of the f0(980) and f2(1270) periments on the γγ π0η reaction. According to the → resonances observed by MARK II and Crystal Ball in the PDG reviews from 1992 to 2008, the average value for cross sections for γγ ππ. 0 +0.8 Γa0 γγ B(a0 π η)=(0.24 0.7) keV [10, 196]. Only → in 2009,→ the→ Belle Collaboration− obtained new high- Although the statistical significance of the f0(980) sig- statistics data on the reaction γγ π0η at the KEKB nal in the cross sections and the invariant ππ mass reso- + → lution left much to be desired, the existence of a shoulder e e− collider [93]. The statistics collected in the Belle experiment is 3 orders of magnitude higher than in in the f0(980) resonance region in the γγcollision might be thought as established, see Fig. 6 The experiments of eighties and beginning of nineties 60 showed that the two-photon widths of the scalar f0(980) 50 and a0(980) resonances are small in comparison with the L
nb 40 two-photon widths of the tensor f2(1270) and a2(1320) LH
Η resonances, for which there were obtained the following 0 30 values Γf2 γγ 2.6 3 keV [94–97] (see also [10, 11])
ΓΓ®Π 20 → H ≈ −
Σ and Γa2 γγ 1 keV [96, 189] (see also [10, 11]). This 10 → ≈ fact pointed to the four-quark nature of the f0(980) and 0 a0(980) states [94, 96, 182–184, 187, 189, 192, 195, 196, 0.6 0.8 1 1.2 1.4 1.6 1.8 2 !!!!s GeV 199, 201, 202]. H L As mentioned above, in the beginning of eighties it Figure 5: Cross section for γγ π0η as a function of √s for was predicted [85, 86] that, if the a (980) and f (980) → 0 0 cos θ 0.9, where √s is the invariant mass of π0η and θ is the mesons are taken as four-quark states, their production | | ≤ polar angle of the produced π0 (or η) meson in the γγ center-of- rates should be suppressed in photon-photon collisions mass system. The data are from the Crystal Ball Collaboration by a factor ten in relation to the a0(980) and f0(980) [189]. mesons taken as two-quark P wave states. The estimates 6
350 One dwells else on predictions of the molecule ì MARK II 1990 ; cosΘ £0.6 a H L È È H L model in which the a0(980) and f0(980) resonances 300 are non-relativistic bound states of the KK¯ system 2 2 250 [212, 213]. As the q q¯ model, the molecule one ex- L plains the state mass degeneracy and their strong cou- nb
LH 200 pling with the KK¯ channel. As in the four-quark - Π
+ f 980 model, in the molecular one no questions arise with the 0H L 150 ¯ small rates B[J/ψ a0(980)ρ]/B[J/ψ a2(1320)ρ] and ΓΓ®Π
H → →
Σ B[J/ψ f0(980)ω]/B[J/ψ f2(1270)ω] (see specialities 100 in Refs.→ [36, 38]). However,→ the predictions of this model 50 for the two-photon widths [187, 198], f 1270 2H L 0 Γa0(KK¯ ) γγ =Γf0(KK¯ ) γγ 0.6 keV, (5) 0.2 0.4 0.6 0.8 1 1.2 1.4 → → ≈ !!!!s GeV H L 175 are rather big, within two standard deviations contra- Crystal Ball 1990 ; cosΘ £0.8 b H L È È H L dict the experiment data from Table III. More than 150 æ Crystal Ball 1992 ; cosΘ £0.7 ¯ H L È È that, the widths of KK molecules must be smaller (strictly speaking, much smaller) than the binging en- 125 L ergy ǫ 10 MeV. Recent data [11], however, contra-
nb ≈ dict this, Γa0 (50 100)MeV and Γf0 (40 100)MeV.
LH 100 0 ¯ ∼ − ∼ − Π The KK molecule model predicted also [157, 162] that 0 5 B[φ γa (980)] B[φ γf (980)] 10− that contra- 75 → 0 ≈ → 0 ∼ ΓΓ®Π
H dicts experiment [11]. In addition, recently [214, 215] Σ f 980 50 0H L ¯ it was shown that the kaon loop model, ratified by ex- Shielded Σ 600 periment, describes production of a compact state and 25 H L ¯ not an extended molecule. Finally, experiments in which f 1270 2H L the a0(980) and f0(980) mesons were produced in the 0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 π−p π ηn [216, 217] and π−p π π n [218–220] reac- !!!!s GeV → → H L tions within a broad range of four-momentum transfer squared, 0 < t< 1 GeV2, have shown that these states − Figure 6: Cross sections for γγ π+π− (a) and γγ π0π0 (b) are compact, e.g. as two-quark ρ, ω, a (1320), f (1270) → → 2 2 as functions of the invariant mass √s of ππ. The data correspond and other mesons and not as extended molecule ones with to limited angular ranges of the registration of the final pions; θ is form factors determined by the wave functions. These the polar angle of the produced π meson in the γγ center-of-mass experiments have left no chances for the KK¯ molecule system. model. 8 As to four-quark states, they are as compact as two-quark states. 9 obtained for the four-quark model were [85, 86] The Particle Data Group gives information on an av- erage value of Γf0 γγ beginning from 1992. Note that → Γa0 γγ Γf0 γγ 0.27 keV, (3) no new experimental data on Γf0 γγ emerged from → ∼ → ∼ 1992 up to 2006, nevertheless, its→ average value, ad- which were supported by experiments. As for the qq¯ duced by PDG, evolved noticeably in this period. Based model, it predicted that on the data in Table III, the Γf0 γγ value would be (0.26 0.08)keV. In 1992 PDG [196]→ obtained the av- Γ0++ γγ 15 ± → = corrections 1.3 5.5 (4) erage value Γf0 γγ = (0.56 0.11) keV combining the → ± Γ2++ γγ 4 × ≈ − JADE result (1990) [96], see Table III, with the value → PC ++ ++ Γf0 γγ = (0.63 0.14)keV, which was found by Morgan for the P wave states with J =0 and 2 from and→ Pennington± (1990) [208] as a result of a theoret- the same family, see, for example, [36, 94, 108, 187, 15 ical analysis of the MARK II (1990) [95] and Crystal 192, 195, 198, 199, 203–211]. The factor 4 is obtained Ball (1990) [94] data. In 1999 Boglione and Pennington in the non-relativistic quark model according to which 2 2 4 Γ0++ γγ = (256/3)α R′(0) /M and Γ2++ γγ = → 2 2 4 | | → (1024/45)α R′(0) /M , where R′(0) is the derivative of | | the P state radial wave function with a mass M at the ori- 8 A KK¯ formation of unknown origin with the average relativistic gin. Γ2++ γγ differs from Γ0++ γγ by the product of the Euclidean momentum squared
350 175 350 a à Belle, ì Mark II, ò CELLO b à Belle, Crystal Ball L L a Belle data 180 b Belle data H L H L
H L nb H L 300 150 L L nb 300 ______ΣBorn=ΣBorn+ΣBorn nb LH
0 2 nb
LH Born LH 160 250 ...... Σ0 LH 125
250 0.6
0.6 Born < 0.6 --- Σ 0.8 È
< 2 È £ £ Θ È È Θ Θ 200 140 200 Θ 100 cos È cos cos cos È È È , ; ; ,
150 - - 150 0 75 - Π Π Π 120 Π 0 + + + 100 100 50 ΓΓ®Π 100 ΓΓ®Π H H ΓΓ®Π Σ Σ H ΓΓ®Π
H 50
Σ 50 25 Σ 0 80 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.85 0.9 0.95 1 1.05 1.1 1.15 0 0 !!!!s GeV 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 !!!!s GeV H L !!!!s GeV !!!!s GeV H L H L H L
− Figure 7: (a) The high statistics Belle data on the γγ π+π− Figure 8: (a) The data on the γγ π+π reaction cross sec- → → reaction cross section for cos θ 0.6 [91]. Plot (b) emphasizes tion from Mark II [94] and CELLO [97], for √s 0.85 GeV, | | ≤ ≤ the region of the f (980) peak. Errors shown include statistics and from Belle [91], for 0.8 √s 1.5 GeV. (b) The data on 0 ≤ ≤ only. They are approximately equal to 0.5%–1.5%. The √s bin the γγ π0π0 reaction cross section from Crystal Ball [95], for → size in the Belle experiment has been chosen to be 5 MeV, with the √s < 0.8 GeV, and from Belle [92], for 0.8 √s 1.5 GeV. Plots ≤ ≤ mass resolution of about 2 MeV. (a), for √s > 0.85 GeV, and (b), for √s > 0.8 GeV, show exclu- sively the Belle data to emphasize the discovered miniature signals from the f0(980) resonance. The theoretical curves, shown on plot (a), correspond to the cross sections for the process γγ π+π− carried out a new theoretical analysis [222] of the sit- → +0.09 for cos θ 0.6 caused by the electromagnetic Born contribution uation and halved value, Γf0 γγ = (0.28 0.13)keV (see | | ≤ also [223]). The Particle Data→ Group noted− that the from the elementary one pion exchange: the total integrated cross section σBorn = σBorn + σBorn and the integrated cross sections Boglione and Pennington (1999) result replaces the Mor- 0 2 σBorn with helicity λ = 0 and 2. gan and Pennington (1990) one but used both results λ coupled with the JADE (1990) one for calculation of the average f0 γγ decay width. In this way the value →+0.10 Table IV: The current data on the f0(980) → γγ decay width. Γf0 γγ = (0.39 0.13)keV emerged in the PDG review → − (2000) [224]. Experiments Γf0→γγ [keV] + − +0.095+0.147 In 2003 preliminary super-statistics Belle data on γγ → π π Belle (2007) [90] 0.205−0.083−0.117 + γγ π π− were reported. They contain a clear sig- γγ → π0π0 Belle (2008) [92] 0.286 ± 0.017+0.211 → −0.070 nal from the f0(980) resonance [89]. In 2005 there PDG average value [10, 11] 0.29+0.07 emerged our first response [101] to these data. It has −0.06 become clear that Γf0 γγ is bound to be small. In 2006 → PDG excluded the Morgan and Pennington (1990) re- 1 based on the integral luminosity 95 fb− [92]; see also sult, Γf0 γγ = (0.63 0.14) keV, from its sample and → [226–228]. Here also the clear signal of the f (980) reso- using only the JADE± (1990) data and the Boglione 0 nance was detected for the first time. Note that the back- and Pennington (1999) result obtained a new guide +0.08 ground conditions for the manifestation of the f0(980) in Γf0 γγ = (0.31 0.11) keV [225]. To the effect that hap- 0 0 → − the γγ π π channel are more favourable than in the pened later to the average value of Γf0 γγ and can else +→ → γγ π π− one. happens to the one, we are going to tell in the following → Figures 8(a) and 8(b) illustrate a general picture of subsections 3.2 and 3.4. + 0 0 data on the cross sections of the π π− and π π produc- 3.2. Current experimental situation tion in photon-photon collisions from the ππ threshold up In 2007 the Belle collaboration published the data on to 1.5 GeV after the Belle experiments. It is instructive + cross section of the γγ π π− reaction in the region to compare these results with a previous picture illus- + → of the π π− invariant mass, √s, from 0.8 up 1.5 GeV trated by Figs. 6(a) and 6(b). 1 based on the integral luminosity 85.9 fb− [90, 91]. These The current information about Γf0 γγ are adduced data are shown on Fig. 7. Thanks to the huge statistics → in Table IV. The Belle collaboration determined Γf0 γγ and high energy resolution in the Belle experiment, the (see Table IV) as a result of fitting the mass distributions→ clear signal of the f (980) resonance was detected for 0 (see Figs. 7(b) and 8(b)) taking into account the f0(980) the first time. Its value proved to be small that agrees and f2(1270) resonance contributions and smooth back- qualitatively with the four-quark model prediction [85, ground contributions, which are a source of large system- 86]. The visible height of the f0(980) peak amounts of atic errors in Γf0 γγ (see for details in Refs. [90–92]). about 15 nb over the smooth background near 100 nb. Its → visible (effective) width proved to be about 30–35 MeV, 3.3. Dynamics of the reactions γγ → ππ: Born see Fig. 7. contributions and angular distributions Then the Belle collaboration published the data on To feel the values of the cross sections measured by ex- cross section for the γγ π0π0 reaction in the region periment, in Fig. 8(a) the total Born cross section of the + → + Born Born Born of the π π− invariant mass, √s, from 0.6 to 1.6 GeV γγ π π− process, σ = σ + σ , and the par- → 0 2 8
Born + γ π+ γ π+ γ π+ γ π+ dσ (γγ π π−)/d cos θ , compare Figs. 10(a) and 10(b). The→ interference,| destructive| in the first half of π± π+ +π+ + Born = the cos θ 0.6 interval and constructive in the second one,| flattens|≤ out the θ angle distribution in this interval, γ π− γπ− γπ− γ π− so that this effect increases with increasing √s, see Fig. 10(a). Figure 9: The Born diagrams for γγ π+π−. G PC + ++ → Since the first resonance with I (J )= 0 (4 ) has the mass near 2 GeV [10, 11], then seemingly the
300 300 S and D wave contributions only should dominate at ______total a b ºº H L !!!!s GeV = 0.6, H L Λ=0 contribution L L √s 1.5 GeV and the differential cross section of the 250 --- Λ=2 contribution 250 ______Contribution nb nb ≤ + from Λ=2 D wave ÈH !!!! ÈH γγ π π− process could be represented as [91]
Θ s GeV = 0.6, Θ 200 200 → cos cos È È 0.8, d d
L L + 0 2 2 2
- 0.8, - dσ(γγ π π−)/dΩ= S + D Y + D Y , (6) Π 150 Π 150 0 2 2 2 + + 1.0, → | | | | 1.0, 1.2,
ΓΓ®Π 100 ΓΓ®Π 100 H 1.2, H 1.4 where S, D0, and D2 are the S and Dλ wave ampli-
Born 1.4 Born
Σ 50 Σ 50 m d d tudes with the helicity λ = 0 and 2, YJ are the spherical 12 0 0 harmonics. But, the above discussion shows that the 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 + cosΘ cosΘ smooth background contribution in the γγ π π− cross È È È È section contains the high partial wave due to→ the one pion exchange, so that the smooth background can imitate the Figure 10: Plots (a) and (b) show the γγ π+π− differential → large S wave at cos θ 0.6. cross section in the Born approximation (i.e., for the elementary | |≤ The one-pion exchange is absent in the γγ π0π0 one pion exchange mechanism) and its components for different → values of √s/GeV. The vertical straight lines cos θ = 0.6 show channel and the representation of the cross section of | | the upper boundary of the region available for measurements. this reaction similar to Eq. (6) is a good approximation at √s 1.5 GeV ≤
Born 0 0 0 2 2 2 tial helicity ones, σ , are adduced as a guide, where dσ(γγ π π )/dΩ= S + D0Y + D2Y , (7) λ → | 2 | | 2 | λ = 0 or 2 is the absolute value of the photon helicity dif- ference. These cross sections are caused by the elemen- where S, D0, and D2 aree theeS and Dλe wave ampli- tary one pion exchange mechanism, see Fig. 9. By the tudes with the helicity λ =0 and 2. Nevertheless, the Low theorem 10 and chiral symmetry 11, the Born con- partiale wavee analysise of the γγ π0π0 events, based on tributions should dominate near the threshold region of Eq. (7), is not prevented from difficulties→ for the relation + 2 0 0 the γγ π π− reaction. As shown in Fig. 8(a), this √6 Y = √5Y – Y , which gives no way of separating → 2 0 2 anticipation does not contradict the current data near the| partial| waves when using only the data on the differ- threshold, but, certainly, errors leave much to be desired. ential cross section [91, 92, 96]. So, the separation of the In additional, one can consider the Born contributions contributions with the different helicities requests some as an reasonable approximation of background (non- guesswork, for example, the domination of the helicity 2 + resonance) contributions in the γγ π π− amplitudes in the f (1270) resonance production [209, 240–242] that → 2 in all the resonance region, including the f2(1270) one. agrees rather well with the experimental angle distribu- The Born contributions are also the base for a construc- tion. tion of amplitudes, including strong interactions in final The dσ(γγ π0π0)/dΩ differential cross section in Eq. state, see, for example, [9, 101, 181, 190, 193, 194, 232– (7) is a polynomial→ of the second power of z =cos2 θ, 13 239]. which can be expressed in terms of its roots z1 and z1∗, The Born contributions have the following particular Born 0 0 qualities. First, σ has a maximum at √s 0.3 GeV, dσ(γγ π π )/dΩ= C(z z1)(z z1∗) , (8) Born Born Born ≈ → − − where σ σ0 , then σ0 falls with increasing ≈ Born Born √s, so that the σ2 contribution dominates in σ where C is a real quantity. So, from fitting experimen- at √s > 0.5 GeV, see Fig. 8(a). Second, although the tal data on the differential cross section one can deter- Born σ2 value is approximately 80% caused by the D wave mine only three independent parameters, for example, amplitude, its interference with the contribution of higher C, Rez1, and Imz1 up to the sign and not four ones, S , waves are considerable in the differential cross section | | e
12 Eq. (6) corresponds the situation “untagget” when the depen- 10 According to this theorem [229–231], the Born contributions give dence on the pion azimuth ϕ is not measured, that took place in the exact physical amplitude of the crossing reaction γπ± γπ± all above experiments. close to its threshold. → 13 Such a procedure is the base of the determination all solutions 11 Chiral symmetry guarantees weakness of the ππ interaction at when carrying out partial wave analyses, see, for example, [120, low energy. 217, 219, 220, 243–245]. 9
12 !!!!! 0.96£!!!!!s £0.98 GeV 0.8£ s £0.82 GeV 300 1.26£!!!!!s £1.28 GeV the effects of the η η′ mixing and the SU(3) symmetry L L 40 L nb nb 10 nb − 250 ÈH ÈH ÈH Θ Θ Θ breaking [183, 262, 265]. 8 30 cos cos cos 200 È È È d d 6 d L L L As for the tensor mesons, in the ideal mixing case, i.e., 0 0 0 150 Π Π 20 Π 0 0 4 0 ¯ √ 100 if f2 = (uu¯ + dd)/ 2 and f ′ = ss¯, the quark model 2 2 ΓΓ®Π ΓΓ®Π ΓΓ®Π H H 10 H Σ Σ Σ 50 d d 0 d predicts the following relations for the coupling constant
-2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 squared: cosΘ cosΘ cosΘ È È È È È È 2 2 2 g : g : g ′ =25:9:2 . (9) f2γγ a2γγ f2γγ Figure 11: The Belle data on the angular distributions for γγ π0π0 [92]. The solid lines are the approximations. The ver- → Though absolute values of the two-photon widths of the tical straight lines cos θ = 0.8 show the upper boundary of the | | tensor meson decays cannot be obtained from the first region available for measurements. principles [88, 183, 192, 203, 206, 266–268] (see also ref- erences herein), the qq¯ model prediction (9), underlying D , D , and cos δ (δ is a relative phase between the S the relations between the widths of the f2(1270) γγ, | 0| | 2| a (1320) γγ, and f (1525) γγ decays, are used→ with and D amplitudes), as one would like. 2 2′ 0 taking into→ account the effects→ of a deviation from eIn Fig.e 11 the Belle data on the angular distribu-e the ideal mixing and the SU(3) symmetry breaking tions in γγ π0π0 are adduced at three values of √s. e [11, 88, 183, 191, 269, 270]. Roughly speaking, the All of them→ are described very well by the simple two- qq¯ model prediction (9) is borne out by experiment. parameter expression a 2 + b Y 2 2 [103]. This suggests 2 Among other things, this implies that the final state in- that the γγ π0π0 cross| | section| is| saturated only by the → teraction effects are small, in particular, the contribu- S and D2 partial wave contributions at √s< 1.5 GeV. + tions of the f (1270) π π− γγ rescattering type are 2 → → 3.4. Production mechanisms of scalar resonances small in comparison with the contributions of the direct Expectatione e of the Belle data and their advent have qq¯(2++) γγ transitions. → called into being a whole series of theoretical papers The observed smallness of the a0(980) and f0(980) which study dynamics of the f0(980) and σ(600) pro- meson two-photon widths in comparison with the two- duction in the γγ ππ processes by various means and photon tensor meson ones and thus the failure of the qq¯ discuss the nature→ of these states [9, 45, 46, 65, 73, 77, model prediction of the relation (4) between the widths 78, 101–105, 211, 239, 246–258]. of the direct 0++ and 2++ γγ transitions point to that → The main lesson from the analysis of the production a0(980) and f0(980) are not the quark and antiquark mechanisms of the light scalars in γγ collisions is the bound states. If the qq¯ component is practically absent following [45, 46]. in the wave functions of the light scalars and in their q2q¯2 The classical P wave tensor qq¯ mesons f2(1270), component the white neutral vector meson pairs are prac- a2(1320), and f2′ (1525) are produced in γγ collisions due tically absent too, as in the MIT bag model [85, 86], then to the direct γγ qq¯ transitions in the main, whereas the σ(600) γγ, f0(980) γγ, and a0(980) γγ decays → → → → the light scalar mesons σ(600), f0(980), and a0(980) could be the four-quark transitions caused by the rescat- + + + are produced by the rescatterings γγ π π− σ, terings σ(600) π π− γγ, f0(980) K K− γγ, + + 0 → → → + →0 → → γγ K K f , γγ (K K , π η) a , and so and a0(980) (K K−, π η) γγ. Already in 1998 we − 0 − 0 → → on,→ i.e., due to→ the four quark→ transitions.→ As to the di- considered such a scenario extensively [108] analyzing rect transitions γγ σ, γγ f0, and γγ a0, they the Crystal Ball data [189] on the a0(980) resonance are strongly suppressed,→ as it→ is expected in→ four-quark production in the γγ π0η reaction; see also the dis- → model. cussion of the γγ KK¯ reaction mechanisms in Refs. → This conclusion introduces a new seminal view of the [109, 110]. Fifteen years later, when the preliminary high γγ ππ reaction dynamics at low energy. Let us dwell statics Belle data [89] on the f0(980) resonance produc- → + on this point. tion in the γγ π π− reaction were reported, we studied → Recall elementary ideas of interactions of C even what role the rescattering mechanisms, in particular, the + + mesons with photons based on the quark model [11, 88, γγ K K− f0(980) π π− mechanism, could play → → → 183, 184]. Coupling of the γγ system with the classical in this process [101]. As a result we showed that just this PC + qq¯ states, to which the light pseudoscalar (J = 0− ) mechanism gives a reasonable scale of the f0(980) mani- ++ + 0 0 and tensor (2 ) mesons belong, are proportional to four festation in the γγ π π− and γγ π π cross sections. → → power of charges of constituent quarks. Then in the SU(2)L SU(2)R linear σ model frame Only the width of the π0 γγ decay is evaluated from we showed that the σ ×field are described by its four- → the first principles [259–262]. Γπ0 γγ is determined com- quark component at least in the energy (virtuality) re- pletely by the Adler-Bell-Jackiw→ axial anomaly and in gion of the σ resonance and the σ(600) meson decay + this case the theory (QCD) is in excellent agreement into γγ is the four-quark transition σ(600) π π− γγ with the experiment [263, 264]. The relations between [9]. We also emphasized that the σ meson→ contribution→ 0 the widths of the π γγ, η γγ, and η′ γγ decays in the γγ ππ amplitudes is shielded due to its strong are obtained in the q→q¯ model→ with taking→ into account destructive→ interference with the background contribu- 10
tions as in the ππ ππ amplitudes, 14 i.e., the σ me- rect transitions of the resonances in two photons are son is produced in the→ γγ collisions accompanied by the taken into account (see, in addition, [9, 101–103, 108– great chiral background due to the rescattering mecha- 110, 164, 181, 247, 249, 253]), + nism γγ π π− (σ + background) ππ, that results + → → 0 0 → + Born π in the modest γγ π π cross section near (5–10)nb in M0(γγ π π−; s,θ)= M0 (s,θ)+ → → + the σ meson region, see Fig. 8(b). The details of this π − − +I + − (s) Tπ+π π+π (s)+ π π → shielding are given in the next Section. + K direct +I + − (s) TK+K− π+π− (s)+ Mres (s), (10) The above considerations about dynamics of the K K e → σ(600), f0(980), and f2(1270) resonance production were e + Born π+ developed in analyzing the final high-statistics Belle data M2(γγ π π−; s,θ)= M2 (s,θ)+ + 0 0 →2 [102, 103] on the γγ π π− and γγ π π reactions, + − → → +80πd20(θ)Mγγ f2(1270) π π (s), (11) to a discussion of which we proceed. → → M (γγ π0π0; s,θ)= M (γγ π0π0; s)= 4. Analysis of high statistics Belle data on 0 → 00 → + − 0 0 π+ the reactions γγ → π π and γγ → π π . = I + − (s) Tπ+π− π0π0 (s)+ π π → + Manifestations of the σ(600) and f0(980) K − direct +IK+K− (s) TK+K π0π0 (s)+ Mres (s), (12) resonances e → 0 0 As noted above, the S and Dλ=2 partial wave contri- e M2(γγ π π ; s,θ)= Born → butions dominate in the Born cross sections σ0 and =5d2 (θ)M (γγ π0π0; s)= Born 20 22 σ , respectively, in region of interest, √s < 1.5 GeV, 2 → 2 0 0 = 80πd20(θ)Mγγ f2(1270) π π (s) , (13) and the ππ interaction is strong also in the S and → → D waves only in this region, that is why the final- where d2 (θ) = (√6/4) sin2 θ. The diagrams of the above state strong interaction modifies these Born contribution 20 + 15 amplitudes are adduced in Figs. 9, 12, 13, and 14. in γγ π π− essentially. In addition, the inelastic →+ The first terms in the right sides of Eqs. (10) and γγ K K− ππ rescattering plays the important role + → → (11) are the Born helicity amplitudes γγ π π− cor- in the f0(980) resonance region (for the first time this responding to the elementary one pion exchange→ mech- process was noted in Refs. [85, 86]). anism (see Fig. 9). Their explicit forms are ad- So, we use the model for the helicity, Mλ, and par- duced in Appendix 8.1. The terms in Eqs. (10) tial, M , amplitudes of γγ ππ in which the Born − − 0 λJ and (12), containing the Tπ+π π+π (s) = [2T0 (s) + → → charged π and K exchanges modified by the strong final- 2 − 0 2 T0 (s)]/3, Tπ+π π0π0 (s) = 2[T0 (s) T0 (s)]/3, and → state interactions in the S and D waves and the di- − − − − 2 TK+K π+π (s)= TK+K π0π0 (s) amplitudes, take into account→ the strong final-state→ interactions in the S wave. Eqs. (10) and (12) imply that Tπ+π− ππ(s) + → and TK+K− ππ(s) in the loops of the γγ π π− ππ 14 As already noted in Introduction, the presence of the large back- →+ → → and γγ K K− ππ rescatterings (see Figs. 12 ground, which shields the σ resonance in ππ ππ, is a conse- → → → and 13) are on the mass shell. In so doing the quence of chiral symmetry. + + 15 π K It is reliably established by experiment that the S and D wave Iπ+π− (s) and IK+K− (s) functions are the amplitudes contribution dominate in the ππ scattering cross sections in the of the triangle loop diagrams describing the transi- isospin I = 0 and 2 channels at √s < 1.5 GeV (see, for exam- + tionse γγ π πe− (scalar state with a mass = √s) ple, data [112, 113, 115, 120, 218–220, 271, 272]). The ππ par- → + → I I I and γγ K K− (scalar state with a mass = √s), tial wave amplitudes TJ (s)= ηJ (s) exp[2iδJ (s)] 1 /[2iρπ+ (s)] → → + + { I I − } in which the meson pairs π π− and K K− are pro- with J = 0, 2 and I = 0 (where δJ (s) and ηJ (s) are the phase and inelasticity for the J wave in the ππ scattering channel with duced by the electromagnetic Born sources, see Figs. 2 1/2 the isospin I; ρ + (s) = (1 4m /s) ) reach their unitarity π π+ 9 and 14. Their explicit forms are adduced in Ap- − direct limits at some values of √s in the region of interest and demon- pendixes 8.1 and 8.3. The amplitude Mres (s) in strate both the smooth energy dependence and the sharp res- 0 Eqs. (10) and (12) caused by the direct coupling con- onance oscillations. The T2 (s) amplitude is dominated by the 0 stants of the σ0(600) and f0(980) with photons, and the f2(1270) resonance contribution. The T0 (s) amplitude contains + − the σ (600) and f (980) resonance contributions. The σ (600) f2(1270) production amplitude Mγγ f2(1270) π π (s)= 0 0 0 → → 0 0 resonance contribution is compensated strongly by the chiral Mγγ f2(1270) π π (s) in Eqs. (11) and (13) are specified background near the ππ threshold to provide for the observed below.→ → 0 smallness of the ππ scattering length a0 and the Adler zero in 0 2 0 Let us show by the example of the S wave T0 (s) at s mπ/2 [6, 9, 171, 172]. T0 (s) reaches the unitary + 0 0 ≈ | | amplitudes M (γγ π π−; s) and M (γγ π π ; s) limit in the 0.85–0.9 GeV region and has the narrow deep (practi- 00 → 00 → cally up to zero) right under the KK¯ threshold caused by the de- that the unitary condition requirement or the Wat- structive interference of the f0(980) resonance contribution with son theorem [273] about interaction in final-state the large smooth background. It is established also that the ππ holds in the model under consideration. First of ¯ scattering in the I = 0 channel is elastic up to the KK channel all note that the 4π and 6π channel contribu- threshold in the very good approximation, but directly above this 0 tions are small for √s< 1GeV [112–114] and conse- the inelasticity η0 (s) shows the sharp jump due to the production 0 ¯ − − iδ0 (s) − − of the f0(980) resonance coupled strongly with the KKchannel. quently Tπ+π K+K (s)= e Tπ+π K+K (s) and → | → | 11
16 γ π+ γ π+ γ π+ π+ where A(s) and B(s) are the real functions. Eqs. (14) π± π± and (15) show that at one with the Watson theorem the λ =0 = Born + Born S wave λ =0 λ =0 phases of the S wave amplitudes γγ ππ with I = 0 and γ π− γ π− γ π− π− → 0 2 coincide with the phases of the ππ scattering δ0(s) and γ K+ π+ γ π+ 2 ¯ ± δ0(s), respectively, in the elastic region (below the KK K direct + Born S wave + , threshold). λ =0 res γ K− π− γ (σ, f0) π− We use the following notations and normalizations for the γγ ππ cross sections: γ π+ γ π+ γ π+ → ± f2 + + π (1270) σ(γγ π π ; cos θ 0.6) σ = σ0 + σ2, (16) λ =2 = Born + → | |≤ ≡ λ =2 0 0 σ(γγ π π ; cos θ 0.8) σ˜ =σ ˜0 +˜σ2, (17) γ π− γ π− γ π− → | |≤ ≡
Figure 12: The diagrams corresponding to the helicity amplitudes ρ + (s) 0.6 σ = π M (γγ π+π ; s,θ) 2d cos θ, (18) (10) and (11) for the γγ π+π− reaction. λ 64πs 0.6 λ − → − | → | ρπ+ (s) 0.8 0 0 2 σ˜λ = 128πs R 0.8 Mλ(γγ π π ; s,θ) d cos θ. (19) − | → | γ π0 γ π+ π0 π± Hereinafter theR corresponding partial cross sections will λ =0 = Born S wave λ =0 be denoted as σλJ andσ ˜λJ . γ π0 γ π− π0 Before fitting data it is helpful to center on a simplified 0 0 γ K+ π γ π (qualitative) scheme of their description. K± direct + Born S wave + , In Fig. 15(a) from Ref. [102] are adduced the the- λ =0 res Born oretical curves for the cross section σ = σ0 + σ − 0 (σ, f0) 0 2 γ K π γ π Born and its components σ0 and σ2 corresponding to the γ π0 γ π0 simplest variant of the above model in which only the + + f2(1270) S wave Born amplitudes γγ π π− and γγ K K− λ =2 = are modified by the pion and→ kaon strong final-state→ in- γ π0 γ π0 teractions. As for all higher partial waves with λ =0 and 2, they are taken in the Born point-like approxima- Figure 13: The diagrams corresponding to the helicity amplitudes tion [101, 102]. This modification results in appearing (12) and (13) for the γγ π0π0 reaction. → the f0(980) resonance signal in σ0, the value and shape of which agree very well with the Belle data, see Fig. 0 15(a). From comparing the corresponding curves in Figs. M direct(s) = eiδ0 (s) M direct(s) for 4m2 s 4m2 [9, res ± | res | π ≤ ≤ K 15(a) and 8(a) it follows that the S wave contribution 101, 102, 171, 172]. Taking into account that + π+ Born π+ to σ(γγ π π−; cos θ 0.6) is small for √s > 0.5 ImI + − (s)= ρπ+ (s)M00 (s) one finds → | | ≤ π π GeV. It is clear that the f2(1270) resonance contribution
+ Born π+ is the main element required for the description of the e M00(γγ π π−; s)= M (s)+ + → 00 Belle data on γγ π π− in the √s region from 0.8 up π+ → +I + − (s)Tπ+π− π+π− (s)+ to 1.5 GeV. For describing data only near the f0(980) res- π π → K+ direct onance one can the large non-coherent background under +I + − (s)TK+K− π+π− (s)+ Mres (s)= K K e → the resonance, caused by σ2, approximate by a polyno- = (for 2mπ √s 2mK)= mial of √s. The result of a such fit is shown in Figs. e 0 ≤ ≤ 2 = 2 eiδ0 (s)A(s)+ 1 eiδ0 (s)B(s), (14) 15(c) and 15(d) [102]. 3 3 By Fig. 13 and Eq. (12) taking into account + the final-state interactions in the Born γγ π π− and + + → 0 0 π − γγ K K amplitudes leads to the prediction of the S M00(γγ π π ; s)= I + − (s)Tπ+π π0π0 (s)+ − π π → → 0 0 +→ wave amplitude of the γγ π π reaction [9, 101–103]. K direct − + − 0 0 0→0 +IK+K (s)TK K π π (s)+ Mres (s)= In Fig. 15(b), the γγ π π cross section, evaluated in →e → = (for 2mπ √s 2mK)= the outlined above manner, are compared with the Crys- ≤ ≤ 17 e 2 0 2 2 tal Ball and Belle data. In view of the fact that no = eiδ0 (s)A(s) eiδ0 (s)B(s), (15) 3 − 3
γ K+ γ K+ γ K+ γ K+ 16 Born π+ 0 A(s)= M00 (s) cos δ0 (s) + ± + + K + + π 0 3 K = K +K + (1/ρ + (s))Re[I (s)] sin δ (s)+ I (s) T + − + − (s) Born π π+π− 0 2 K+K− | K K →π π |± 3 direct Born π+ 2 2 Mres (s) and B(s)= M00 (s) cos δ0 (s) + γ − γ− γ− γ − | | K K K K π+ 2 (1/ρ + (s))Re[Ie (s)] sin δ (s). e π π+π− 0 17 + − Note that the step of √s for the Crystal Ball and Belle data, Figure 14: The Born diagrams for γγ K K . shown in Fig. 15(b), is 50 MeV and 20 MeV respectively. → e 12
350 175 25 25 a à Belle, ì Mark II, ò CELLO b à Belle, Crystal Ball L a L b H L H L H L H L 300 150 nb nb L L Born 20 20
LH LH ______Σ=Σ0+Σ ______Σ0 nb
nb 2 0.6 0.6
LH 250 ...... Σ0 LH125 £ £
Born È È Θ Θ 0.6 --- Σ 2 0.8 15 15 £ £ È È
Θ 200 100 Θ cos cos È È , , cos cos È È - -
; 10 10 ; Π Π
- 150 75 0 + + Π Π + 0 100 50
ΓΓ®Π 5 ΓΓ®Π 5 ΓΓ®Π ΓΓ®Π H H H H 0 0 Σ Σ 50 25 Σ Σ 0 0 0 0 0.85 0.9 0.95 1 1.05 1.1 0.85 0.9 0.95 1 1.05 1.1 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 !!!!s GeV !!!!s GeV !!!!s GeV !!!!s GeV H L H L H L H L 115 200 c Belle data d Belle data H L H L Figure 16: The structure of the f0(980) signal in σ0. (a) The L L 180 110 ______+ − + − ______Σ=Σ +Σ Σ=Σ0+Σ2 nb nb 0 2 contributions from the γγ K K π π (dashed line), ...... Σ2 → → LH LH ...... Σ + − + − 160 2 105 γγ π π π π (dotted line) rescattering amplitudes, and 0.6 0.6 → → £ £ È È Θ Θ their sum (solid line). (b) The dashed line is identical to the solid 140 100 cos cos Born È È one in (a), the dotted and dot-dashed lines show the σ and , , 0 - - Π Π 120 Born Born Born
+ 95 + σ00 cross sections, respectively (σ0 < σ00 because of the destructive interference between the S and higher partial waves), ΓΓ®Π ΓΓ®Π 100 H H 90 Σ Σ and the solid line corresponds to the resulting f0(980) signal in σ0. 80 85 0.85 0.9 0.95 1 1.05 1.1 1.15 0.92 0.94 0.96 0.98 1 1.02 !!!!s GeV H L !!!!s GeV H L rescattering in its turn transfers the narrow deep under the KK¯ threshold from the Tππ ππ(s) amplitude, see Figure 15: Theoretical curves in plots (a) and (b) correspond to → the simplest model which incorporates only the Born contributions the footnote 18, to the γγ ππ one. The interference + → + γγ π+π−, from π exchange, and γγ K+K−, from K exchange, of the resonance γγ K K− π π− amplitude with → → + +→ → 19 modified for strong final-state interactions in the S wave. Plot (c) the γγ π π− π π− amplitude exerts the essen- → → illustrates the description of the Belle data in the f0(980) region. tial effect on the resulting shape of the f0(980) signal, as (d) The fragment of (c). indicated by Fig. 16(a). As for the Born contributions, their influence on the resulting shape of the f0(980) signal is small, see Fig. 16(b).
fitting parameters are used for the construction ofσ ˜0, Once more notable fact lies in the drastic change of the + one should accept that the agreement with the data is f (980) production amplitude γγ K K− f (980) 0 → → 0 rather well at √s 0.8 GeV, i.e., in the σ(600) reso- in the f0(980) peak region [101, 102] just as the ≤ + nance region. It is clear also that at √s> 0.8 GeV the γγ K K− a (980) amplitude in the a (980) region → → 0 0 f2(1270) resonance responsibility region begins. [108], see Section 5. In the cross section its contribution is K+ 2 So, already at this stage it emerges the following. First, proportional to IK+K− (s) , see Eqs. (10) and (12). The K+ | 2 | if the direct coupling constants of σ(600) and f0(980) − function IK+K (s) decreases drastically immediately | + | with γγ are included in fitting their role will be negli- under the K K−ethreshold, i.e., in the f0(980) resonance 20 gible in agreement with the four-quark model prediction region, seee Fig. 17. Such a behavior of the f0(980) [85, 86]. Second, by Eqs. (10) and (12) the σ(600) γγ two-photon production amplitude reduces strongly the → and f0(980) γγ decays are described by the tri- left slope of the the f0(980) peak defined by the reso- → + angle loop rescattering diagrams Resonance (π π−, nance amplitude TK+K− ππ(s). That is why one cannot + → → K K−) γγ and, consequently, are the four-quark tran- approximate the f (980) γγ decay width by a constant → 0 sitions [9, 101–103]. even in the region m –Γ→ /2 √s m +Γ /2 [101]. f0 f0 ≤ ≤ f0 f0 The interesting and important feature of the f0(980) So, the above consideration teaches us that all simplest + signal in γγ π π− is its complicated structure which is approximations of the f0(980) signal shape observed in → + + 0 0 shown by Figs. 16(a) and 16(b). The γγ K K− ππ the γγ π π− and γγ π π cross sections can give rescattering amplitude plays the determinant→ role trans-→ → → ferring the f0(980) peak from the TK+K− ππ(s) ampli- 18 →+ + tude to the γγ ππ one. The γγ π π− π π− → → → + − + − Its estimate gives σ(γγ K K f0(980) π π ; cos θ 0.6) 0.6 0.62α2R /m→2 8 nb→ R , where→ α = 1/137| and| ≤ ≈ × f0 f0 ≈ × f0 mf0 is the f0(980) mass. 18 19 It provides the natural scale of the f0(980) production cross Note that the relative sign between these amplitudes is fixed section in γγ collisions [101]. The maximum of the cross sec- surely [101, 102]. + tion σ(γγ K+K− f (980) π+π−) is controlled by the 20 The function IK (s) 2 decreases relatively its maximum at 0 K+K− product of→ the ratio→ of the squares→ of the coupling constants | | √s = 2mK+ 0.9873 GeV by 1.66, 2.23, 2.75, 3.27, and 6.33 2 2 K+ 2 2 ≈ Rf = g + − /g + − and the value I + − (4m + ) . times at √s = 0.98, 0.97, 0.96, 0.95, and 0.9 GeV, respectively. 0 f0K K f0π π | K K K | e
e 13
where δππ(s) and δKK¯ (s) are the phase of the elastic S 0.007 B B wave background in the ππ and KK¯ channels with I = 0; 0.006 the ππ scattering phase 0.005 2 LÉ s
H 0 ππ 0.004 - δ (s)= δ (s)+ δ (s). (24) K res 0 B + K I
É 0.003
0.002 The amplitudes of the σ(600)– f0(980) resonance com- plex in Eqs. (10), (12), (20), (22), and (23) are [171, 172] 0.001
+ − + − 0.8 0.9 1 1.1 1.2 1.3 1.4 ππ gσπ π ∆f0 (s)+ gf0π π ∆σ(s) !!!!s GeV Tres (s)=3 2 , (25) H L 32π[Dσ(s)Df (s) Π (s)] 0 − f0σ K+ 2 Figure 17: The solid curve shows I − (s) as a function of | K+K | √s (see Appendix 8.3); the dashed and dotted curves above the ¯ g + − ∆f (s)+ g + − ∆σ(s) T KK ππ(s)= σK K 0 f0K K , (26) K+K− threshold correspond to the contributions of the real and res → 2 + e 16π[Dσ(s)Df0 (s) Πf0σ(s)] imaginary parts of IK (s), respectively. − K+K−
(0) (0) e ππ gσγγ∆f0 (s)+ g ∆σ(s) direct iδB (s) f0γγ only a rather relative information on the f0(980) state Mres (s)= se 2 , (27) Dσ(s)Df (s) Π (s) two-photon production mechanism and the f0(980) pa- 0 − f0σ rameters. where ∆ (s)= D (s)g + − + Π (s)g + − and Fortunately, the already current knowledge of the dy- f0 f0 σπ π f0σ f0π π (0) − 0 2 − ∆ (s) = D (s)g + − + Π (s)g + − , g and namics of Tπ+π ππ(s)[T0 (s), T0 (s)] and TK+K ππ(s) σ σ f0π π f0σ σπ π σγγ strong interaction→ amplitudes allows to advance in→ under- (0) gf0γγ are the direct coupling constants of the σ and f0 standing the signals about the light scalar mesons which resonances with the photons. We use the expressions for ππ KK¯ the data on the γγ ππ reaction send to us. In fitting the δB (s) and δB (s) phases, the propagators of the → 0 − data we use the model for the T0 (s) and TK+K ππ(s) σ(600) and f0(980) resonances 1/Dσ(s) and 1/Df (s), → 0 amplitudes which was suggested and used for the joint and the polarization operator matrix element Πf0σ(s) 0 0 analysis of the data on the π π mass spectrum in the from [171] (see also Appendix). The mf0 was free, the 0 0 φ π π γ decay, the ππ scattering at 2mπ < √s < 1.6 other parameters in the strong amplitudes (mσ, gσπ+π− , → 0 GeV, and the ππ KK¯ reaction [171, 172]. The T (s) + − 0 gf0K K , etc.) correspond to variant 1 from Table 1 → 21 2 model takes into account the contributions of the σ(600) from this paper. We also put η0 (s) = 1 for all √s 2 and f0(980) resonances, their mixing, and the chiral under consideration and take δ0(s) from [274]. background with the large negative phase which shields The amplitudes of the f2(1270) resonance production the σ(600) resonance (see additionally [6, 9, 45, 46]). Eqs. in Eqs. (11) and (13) are (10) and (12) transfer the effect of the chiral shielding + − 0 0 of the σ(600) resonance from the ππ scattering into the Mγγ f2(1270) π π (s)= Mγγ f2(1270) π π (s)= γγ ππ amplitudes. This effect are demonstrated by → → → → + √s G2(s) (2/3)Γf2 ππ(s)/ρπ (s) → → Fig. 18(a) with the help of the ππ scattering phases = 2 tot . (28) ππ 0 mf s i√sΓf (s) δres(s), δB (s), and δ0(s) [see Eqs. (20)–(22) and (24)], p2 − − 2 and by Figs. 18(b) and 18(c) with the help the corre- tot 0 0 The main contribution in its total width Γ (s) = sponding cross sections of the ππ ππ and γγ π π f2 → → Γf2 ππ(s)+Γf2 KK¯ (s)+Γf2 4π(s) is given by the ππ reactions. As seen from Fig. 18(c), if it were not for → → → such a shielding the γγ π0π0 cross section nearby the partial decay width → threshold would be not (5 10) nb but approximately 100 tot 2 + − Γf2 ππ(s)=Γf (mf )B(f2 ππ) nb due to the π π− loop mechanism of the σ(600) γγ → 2 2 → × 2 5 2 → m D (q + (m )r ) decay [9]. The decay width corresponding to this mech- f2 qπ+ (s) 2 π f2 f2 − 5 2 , (29) anism, Γσ π+π γγ(s), is shown in Fig. 18(d); see also + × s q + (mf ) D2(qπ (s)rf2 ) Eq. (64) in→ Appendix→ 8.1. π 2 2 4 According Refs. [171, 172], we write + + where D2(x)=9+3x + x , qπ (s)= √sρπ (s)/2, rf2 ππ is the interaction range and B(f2 ππ) = 0.848 [10]. 0 ππ 2iδB (s) ππ T0 (s)= TB (s)+ e Tres (s) , (20) →
ππ ππ TB (s)= exp[2iδB (s)] 1 /[2iρπ+ (s)] , (21) { − } 21 Removing the misprint in the sign of the constant C C we ≡ f0σ ππ 0 use Cf0σ = 0.047 GeV. Notice that our principal conclusions T (s)= η (s) exp[2iδ (s)] 1 /[2iρ + (s)] , (22) − res { 0 res − } π [the insignificance of the direct transition γγ Light Scalar and the dominant role of the four-quark transition→ γγ + + + − → ππ KK¯ ¯ (π π ,K K ) Light Scalar] are independent on a specific i[δB (s)+δB (s)] KK ππ → TK+K− π+π− (s)= e Tres → (s) , (23) variant from [171, 172]. → 14
theorem requirement for the amplitude γγ ππ with a 5000 ______0 0 2 →
L 150 H L Σ0=32Π T0 s s 10 b
∆res L @ È H LÈ D H L λ = J = 2 and I =0 under the first inelastic threshold. ΠΠ 2 --- Σres=32Π T s s mb È resH LÈ H 4000 ΠΠ 2 This term gives a small contribution, less then 6%, in 100 ...... ΣB=32Π T s s degrees B
H È H LÈ 2 23 Γf2 γγ (m ). 0 f2
∆ ΠΠ®ΠΠ 3000 → 50 0
ΠΠ®ΠΠ The simplest approximation (32) of the main con- 2000 (0) 0 tribution, Γf2 γγ(s), in the f2(1270) γγ decay → → 2 1000 width is completely adequate to the current state ∆0 Cross sections Phase shifts -50 ΠΠ of both theory and experiment. The parameter ∆B 0 (0) 2 1 2 3 0.2 0.4 0.6 0.8 1 Γ (m )= [g /(16π)]m in Eq. (32) accumu- 0.2 0.4 0.6 0.8 1 f2 γγ f2 5 f2γγ f2 !!!!s GeV !!!!s GeV → H L H L lates effectively lack of knowledge of the values of the amplitudes responsible for the f2(1270) γγ decay. By 100 17.5 c → ______+ H L d Σ00= ΡΠ s 80Πs H L the above reasons, see Section 3, it is generally agreed L @ H LH LD 2 15 --- Re IΠ+ Π- s contribution
nb + - @ H LD 80 IΠ Π s TΠ+ Π-®Π0 Π0 s È H L H LÈ ºº + - that the direct quark-antiquark transition qq¯ γγ dom- L Im IΠ Π s contribution LH --- TΠΠ s contribution 12.5 @ H LD resH L → keV 0.8 ΠΠ inates in the f2(1270) γγ decay and its amplitude is £ ºº È 60 T s contribution B LH Θ H L →
s 10 H characterized by the gf2γγ coupling constant. As shown cos È ®ΓΓ - ; 7.5 0 40 Π in [9, 101–106, 108] and as we state here step by step, the + Π 0
Σ®Π 5 situation is quite different in the case of the light scalar G ΓΓ®Π
H 20
Σ 2.5 mesons.
0 0 Now everything is ready to come to the discussion of 0.2 0.4 0.6 0.8 1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 + 0 0 fitting the Belle data on the γγ π π− and γγ π π !!!!s GeV !!!!s GeV H L H L cross sections which was carried→ out in [102, 103].→ One considers firstly fitting the γγ π0π0 cross Figure 18: The figure demonstrates the chiral shielding effect in section only, see Fig. 19(b), which has→ the smaller the reactions ππ ππ and γγ π0π0. All the plots have been → → background contributions under the f0(980) and taken from Ref. [9] dedicated the lightest scalar in the SU(2) + L × f2(1270) resonances then the γγ π π− cross section, SU(2)R linear σ model. compare Figs. 19(a) and 19(b).→ The solid curve in Fig. 19(b), describing these data rather well, corresponds the following parameters of the model: The small contributions of Γf2 KK¯ (s) and Γf2 4π(s) tot 2 → → mf2 =1.269 GeV, Γf2 (mf2 ) = 0.182 GeV, rf2 = 8.2 are the same ones as in [102]. The parameter rf2 1 (0) [90, 92, 94, 95, 97, 102, 103] controls the relative form GeV− , Γf2 γγ(mf2 ) = 3.62 keV [Γf2 γγ(mf2 ) = 3.43 → → of the f2(1270) resonance wings and is very important (0) 1 keV], mf0 = 0.969 GeV, gσγγ = 0.536 GeV− and especially for fitting data with small errors. (0) 1 24 g = 0.652 GeV− . The fitting indicates small- The amplitude G2(s) in Eq. (28) describes the cou- f0γγ (0) pling of the f2(1270) resonance with the photons, ness of the direct coupling constants gσγγ and (0) (0) 2 2 (0) 2 + g : Γσ γγ (mσ)= mσgσγγ /(16πmσ) = 0.012 Born π f0γγ → (0) M22 (s) 2 | | + (0) 2 2 (0) 2 G2(s)= Γf γγ (s)+i ρπ (s)Γf2 ππ(s) . keV and Γ (m )= m g /(16πm ) = 0.008 2 → f0 γγ f0 f0 f0γγ f0 → 16π 3 → | | q r (30) keV, in accordance with the prediction [85, 86] 25 Born The explicit form of the M22 (s) amplitude is in Ap- The dominant rescattering mechanisms give the + pendix 8.1, Eq. (53). The f (1270) γγ decay width σ(600) π π− γγ width (1–1.75) keV averaged in 2 → → ≈ is → the region 0.4 < √s < 0.5 GeV [9], see Fig. 18(d), and + 2 the f0(980) K K− γγ width (0.15 0.2) keV Γf2 γγ(s)= G2(s) (31) → → ≈ − → | | and 2 m s 23 + − Γ(0) (s)= f2 Γ(0) (m2 ) (32) As for a real part of the f2(1270) π π γγ amplitude, its f2 γγ f2 γγ f2 4 modulus is far less than the one of the→ direct transition→ amplitude → √s → mf 2 as different estimations show. [here the s2 factor, and also the s factor in Eq. (27), 24 The formally calculated errors in the significant parameters of is caused by the gauge invariance requirement]. The the model are negligible due to the high statistical accuracy of the Belle data. The model dependence of adjustable parameter second term in G2(s) corresponds to the rescattering values is the main source of their ambiguity. + f2(1270) π π− γγ with the real pions in interme- 25 One notes that the small values of these coupling constants are →22 → direct diate state and ensure the fulfillment of the Watson grasped in fitting due to the interference of the Mres (s) am- plitude, see Eqs. (10), (12), and (27), with the contributions of the dominant rescattering mechanisms. In such a case not the specific above values of g(0) and g(0) are important, but the σγγ f0γγ (0) 22 fact of their relative smallness, corresponding to Γ (m2 ) and That is, it corresponds to the imaginary part of the f2(1270) σ→γγ σ π+π− γγ amplitude. → Γ(0) (m2 ) both 0.1 keV. → f0→γγ f0 ≪ 15
350 175 350 à Belle, ì Mark II, ò CELLO a à Belle, Crystal Ball b H L H L à Belle, ì Mark II, ò CELLO a H L 300 ______Σ=Σ0+Σ2 150 ______Σ = Σ + Σ L 0 2 ______L 300 ______Σ=Σ +Σ
nb 0 2 Σ0 nb --- Σ 0 L LH 250 --- Σ2 LH 125 ºº ______Σ0 Σ nb Born Born Born 2 0.6 ××-×× Σ =Σ0 +Σ2 0.8 Born £ £ È È Born LH 250 ×××××× Σ
Θ 2 200 Θ 100 -×- Σ0 Born Born cos 0.6 ×××××× Σ cos -×- Σ x
È Σ 1 2 È 2
; H L ; £ È - 150 0 75 Θ Π Σ Π 200
2 0 +
100 50 cos È ΓΓ®Π ΓΓ®Π H ; H Σ Σ - 150
50 25 Π + Σ0 Σ 0 0 100 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 ΓΓ®Π !!!!s GeV !!!!s GeV H H L H L Σ 50 Σ0 Figure 19: Cross sections for the γγ π+π− and γγ π0π0 → → 0 reactions. Only statistical errors are shown for the Belle data [91, 0.2 0.4 0.6 0.8 1 1.2 1.4 !!!!s GeV 92]. The curves in plot (a) are described in the text and on the H L figure. The curves in plot (b) are the result of the fit to the data 175 on the γγ π0π0 reaction. à Belle, Crystal Ball b H L → 150 ______Σ = Σ0 + Σ2 L
nb --- Σ0
LH 125 averaged over the resonance mass distribution [101]. ºº Σ2
0 0 0.8 £
But such a fitting of the γγ π π cross section comes È
Θ 100 → + into conflict with the data on γγ π π−, see solid curve cos → È ;
for σ = σ0 + σ2 in Fig. 19(a). This is connected with 0 75 Π the large Born contribution in σ2 and its strong con- 0 structive (destructive) interference with the f (1270) res- 50
2 ΓΓ®Π H
onance contribution at √s
26 Such a natural modification of the point-like Born contribution was discussed in connection with the data on the γγ π+π− [95, 27 At the same time we emphasize that the considerable systematic 97, 191, 193, 194] and γγ K+K−(K0K¯ 0) reactions→ [109, 110]. errors, the sources of which are described in detail in Refs. [90– But only the problem of→ the consistent description of the Belle 92], do not depreciate the role of the high statistics of the data, data on γγ π+π− and γγ π0π0 indicates the modification which allows to resolve the small local effects connected with the → → need of the Born sector of the model unambiguously [102, 103]. f0(980) resonance manifestation. 16
curves in Fig. 20 correspond the following values of the 3 3s 8Π2 Σ ΓΓ®Π+Π-;s tot 2 @ H LD resH L parameters: mf2 =1.272GeV, Γf2 (mf2 ) = 0.196 GeV, 2.5 1 (0) rf2 = 8.2 GeV− , Γf2 γγ(mf2 ) = 3.83 keV [Γf2 γγ(mf2 ) → → 2 = 3.76 keV], m = 0.969 GeV, g(0) = 0.049 GeV 1 f0 σγγ − units 6 (0) 2 (0) − 1 (0) - 1.5 [Γσ γγ (mσ) miserable], gf0γγ = 0.718 GeV− [Γf0 γγ
→ → In 10 (m2 ) 0.01 keV], x = 0.9 GeV and x =1.75 GeV. It 1 f0 ≈ 1 2 is clear from comparison of Figs. 19(b) and 20(b) that 0.5 the form factor effect on the γγ π0π0 cross section is +→ weak in contrast to the γγ π π− one [compare Figs. 0 → 0.2 0.4 0.6 0.8 1 19(a) and 20(a)] in which the σ2 contribution is modified !!!!s GeV H L mainly. One emphasizes that all our conclusions about the mechanisms of the two-photon decays (productions) Figure 21: The integrand in Eq. (35) corresponding to the joint 28 − of the σ(600) and f0(980) resonances are in force. fit to the γγ π+π and γγ π0π0 data (Fig. 20) is shown by → → Thus the physics of the two-photon decays of the light the solid curve. The dotted and dashed curves show the contribu- tions from the resonant elastic γγ π+π− π+π− and inelastic scalar mesons acquires the rather clear outline. The → → γγ K+K− π+π− rescatterings, respectively. mechanism of their decays into γγ does not look like the → → mechanism of the classic tensor qq¯ meson decays, which is the direct annihilation qq¯ γγ. The light scalar me- → [see Eqs. (10) and (25)–(27)], where the functions son decays into γγ are suppressed in comparison with + + π K the the tensor meson ones. They are caused by the Iπ+π− (s; x1) and IK+K− (s; x2) are the analogs of the π+ K+ rescattering mechanisms, i.e., by the four-quark tran- Iπ+π− (s) and IK+K− (s) functions constructed with tak- + + sitions σ(600) π π− γγ, f0(980) K K− γγ, inge the form factorse into account, see Appendixes 8.1 + → → → → a0(980) K K− γγ, etc. Such a picture is suggested ande 8.3. Fig.e 21 shows the dependence of the → → 2 2 + by experiment and supports the q q¯ nature of the light σ (γγ π π−; s) cross section, multiplied by the fac- res → scalars. It is significant that in the scalar meson case the tor 3s/(8π2), on the energy. In and around 1 GeV there longing for the exhaustive characteristic of their coupling is the impressive peak from the f0(980) resonance due 2 + + with photons via the constant values Γ0++ γγ(m0++ ) by to the inelastic γγ K K− π π− rescattering in the analogy with the tensor mesons cannot be→ realized for main. Following Refs.→ [101, 103,→ 108], one determines the quite a number reasons. f0(980) γγ decay width averaged over the resonance First of all it is clear that when we deal with reso- mass distribution→ in the ππ channel nances accompanied by fundamental background, when 1.1 GeV two-photon decay widths change sharply in the resonance 3s + Γf0 γγ ππ = σres(γγ π π−; s)d√s. (35) region for close inelastic thresholds, then there is no point h → i 8π2 → in discussing the two-photon width in the resonance peak. 0.8Z GeV In this connection it is interesting to consider the cross This value is the adequate functional characteristic of the + section γγ π π− caused only by the resonance contri- coupling of the f0(980) with γγ. For the presented com- butions, i.e.,→ bined fitting, Γf0 γγ ππ 0.19 keV [103]. Taking into h → i ≈ + account that the wide σ(600) resonance dominates in re- σres(γγ π π−; s)= → gion 2mπ < √s < 0.8 GeV one obtains by analogy with + ππ π 2iδB (s) ππ = [ρπ+ (s)/(32πs)] Iπ+π− (s; x1) e Tres (s) (35) Γσ γγ ππ 0.45 keV [103]. Note that the cusp near h → i ≈ 2 + the ππ threshold in the [3s/(8π )]σres(γγ π π−; s) ex- + 2 → K − − direct + I + − (s; x2) TK+eK π+π (s)+ Mres (s) (34) pression, shown in Fig. 21, is the manifestation of the K K → correction for the finite width in the propagator of the scalar resonance. In Appendix 8.1 there are adduced the e transparent explanation of this phenomenon. In the total 28 S wave amplitude γγ ππ such a threshold enhancement Notice that the point-like ω and a2(1320) exchanges in the → γγ π0π0 and γγ π+π− amplitude, respectively, which give is absent due to shielding the resonance contribution in → → 0 the contributions (mainly the S wave one) in the cross section, the amplitude T0 (s) by the chiral background one, see, runaway with increasing the energy and comparable with the for example, Fig. 20(b). f2(1270) resonance contribution even in its energy region, are not observed experimentally. This was puzzled out in our pa- The above examples, each in its manner give to feel per [109] with the γγ π0η example (the details are discussed clear the nontriviality of accessing information about the → bellow in Section 5). The proper Reggeization of the point-like σ(600) and f0(980) decays into γγ. For instance, to de- exchanges with the high spins reduces the dangerous contribu- 2 termine Γσ γγ (mσ) directly from data is impossible for tions greatly. In addition, the partial cancelations between the → 0 0 the cross section in the σ region is formed by both res- ω and h1(1170) exchanges in γγ π π and the a2(1320) and + →− onance and compensating background. The specific dy- a1(1260) exchanges in γγ π π take place. As to the ρ ex- change in γγ ππ, its contribution→ is small for g2 g2 /9 namic model of the total amplitude needs to their sepa- → ρπγ ≈ ωπγ and canceled additionally by the b1(1235) one. ration. The simple Breit-Wigner is not enough here. 17
60 æ Belle cosΘ £0.8 As for the f0(980) resonance, experimenters began to HÈ È L Crystal Ball cosΘ £0.9 take into consideration two from three important circum- HÈ È L 50 stances [90–92] (see also [10, 11]), for which we drew at-
L 40
tention in Ref. [101]. Firstly, there was taken account nb LH Η
the correction for the finite width due to the coupling 0 of f (980) with KK¯ channel in the f (980) resonance 30
0 0 ΓΓ®Π H Σ propagator, which effects essentially on the shape of the 20 f0(980) peak in the ππ channel. Secondly, there was 10 taken into account the interference of the f0(980) reso-
nance with the background though in the simplest form. 0 0.8 1 1.2 1.4 1.6 But no model was constructed for the f0(980) γγ de- !!!!s GeV cay amplitude which was approximated simply→ by a con- H L stant [90–92]. Fitting data in this way the Belle collab- Figure 22: The Belle [93] and Crystal Ball [189] data for the γγ 2 → oration extracted the values for Γf0 γγ(m ) presented 0 → f0 π η cross section. The average statistical error of the Belle data in Table IV. But the discussion needs how to under- is approximately 0.4 nb, the shaded band shows the size of their ± stand these values. First and foremost, one cannot use systematic error. The solid, dashed, and dotted lines correspond 0 them for determining a coupling constant gf γγ in a ef- to the total, helicity 0, and S wave γγ π η cross sections caused 0 → fective lagrangian, i.e., a constant of the direct transition by the elementary ρ and ω exchanges for cos θ 0.8. | |≤ f0(980) γγ, because such a constant is small and does not determine→ the f (980) γγ decay, as shown above. 0 → Until the model of the f0(980) γγ decay amplitude is → 2 of the first experiments Crystal Ball [189] (see Fig. 5) not specified, a meaning of the Γ (m ) values, ex- 0 f0 γγ f0 and JADE [96] on the γγ π η reaction. Unfortunately, tracted with the help of the simplified→ parametrization, → 29 2 the large statistical errors in these data and the rather is rather vague. In principle the Γf0 γγ (mf ) values in 0 → 0 rough step of the π η invariant mass distribution (equal Table IV can be taken as the preliminary estimations of 40 MeV in the Crystal Ball experiment and 60 MeV in Γf0 γγ , i.e., as the f0(980) γγ decay width averaged the JADE one) left many uncertainties. overh → thei hadron mass distribution→ [101, 103, 108]. As we have mentioned in Subsection 3.2, recently, the In the dispersion approach there are introduced usually Belle Collaboration obtained new data on the γγ π0η the pole two-photon widths ΓR γγ(pole), R = σ, f0 to + → → reaction at the KEKB e e− collider [93], with statistics characterize the coupling σ(600) and f0(980) resonances three orders of magnitude higher than those in the pre- with photons (see, for example, [194, 208, 222, 239, 247]). ceding Crystal Ball (336 evens) and JADE (291 events) These widths are determined through the moduli of the experiments. complex pole residues of the γγ ππ and ππ ππ par- The experiments revealed a specific feature of the tial amplitudes constructed theoretically.→ Basing→ on our γγ π0η cross section. It turned out sizable in the investigation [9] we would like to note the following. The region→ between the a (980) and a (1320) resonances (see residues of the above amplitudes are essentially complex 0 2 Fig. 22), 30 which certainly indicates the presence of ad- and cannot be used as any coupling constants in a hermi- ditional contributions. These contributions must be co- tian effective lagrangian. These residues are “dressed” by herent with the resonance ones, because only two lowest the background for they relate to the total amplitudes. S and D partial waves dominate in the γγ π0η am- As our analysis in the SU(2) SU(2) linear σ-model 2 L R plitude at the invariant mass of the π0η system→√s< 1.4 [9] indicated, the background× effects essentially on the GeV [93]. The authors of Ref. [93] performed the phe- values and phases of the residues. Thus the focus on the nomenological fitting of the γγ π0η data taking into values of the Γ (pole) type in dispersion approach R γγ account interference between resonance→ and background does not help to reveal→ the mechanism of the two-photon contributions. It was found that the description of the decays of the scalar mesons and so cannot shed light on S wave requires not only contributions from the a (980) the nature of the light scalars. 0 resonance and the possible heavy a0(Y ) resonance, but also a smooth background whose amplitude is compara- 5. Production of the a0(980) resonance in ble with the amplitude of the a0(980) resonance at the the reaction γγ → π0η maximum and has a large imaginary part [93]. As a re- sult, the background results in almost the quadrupling of + Our conclusions about the important role of the K K− the cross section near the a0(980) peak and in the filling loop mechanism in the two-photon production of the of the dip between the a0(980) and a2(1320) resonances. a0(980) resonance and its possible four-quark nature The origin of such a significant background in the S wave [48, 108, 109] were based on the analysis of the results
30 The JADE data [96] on γγ π0η are nonnormalized and there- 29 The above comments are true also in the σ(600) resonance case. fore are not shown in Fig. 22.→ 18
0 0 0 0 is unknown. Meanwhile, the imaginary part of the back- γ π γ Born π γ Born π π ground amplitude is due to the contributions from real ρ, ω ρ, ω λ =0 = + S wave intermediate states, πη, KK¯ , and πη′, and, naturally, λ =0 λ =0 ′ requires the distinct dynamical decoding. γ η γ η γ η, η η 0 0 In Refs. [105, 106] we shown that the observed ex- γ Born K π γ π ∗ direct perimental pattern is a result of the interplay of many + K,K S wave + , λ =0 ′ a0, a0 dynamical factors. To analyze the data, we signifi- γ K¯ η γ η cantly developed a model previously discussed in Refs. [104, 108, 109]. The basis for this model is an idea of γ π0 γ Born π0 γ π0 a2 ρ, ω (1320) what the a0(980) resonance can be as a suitable can- λ =2 = + didate in four-quark states. There exists a number of λ =2 γ η γ η γ η significant indications in favor of the four-quark nature of the a0(980); see, for example, Refs. [1, 4, 34, 36, 45, Figure 23: The diagrams corresponding to the helicity amplitudes 48, 85, 86, 133, 169]. The solution obtained by us for γγ π0η, see Eqs. (36) and (37). the γγ π0η amplitude is in agreement with the expec- → tations→ of the chiral theory for the πη scattering length, with the strong coupling of the a0(980) resonance with ¯ γ π0,K γ π0,K γ π0,K the πη, KK, and πη′ channels, and with the key role Born ¯ 0 0 of the a0(980) (KK + π η + π η′) γγ rescattering ρ,ω, ρ,ω, ρ,ω, → → (a) ∗ = ∗ + ∗ mechanisms in the a0(980) γγ decay. This picture is K K K 2 2 → much in favor of the q q¯ nature of a0(980) resonance and ′ ′ ′ γ η,η , K¯ γη,η , K¯ γ η,η , K¯ is consistent with the properties of its partners, σ0(600) γ K+ γ K+ γ K+ γ K+ and f0(980) resonances, in particular, with those mani- Born fested in the γγ ππ reactions. The important role of ± + + vector exchanges→ in the formation of the non-resonant (b) K = K +K + background in the γγ π0η reaction has been revealed γ − γ− γ− γ − and preliminary information→ on the π0η π0η reaction K K K K → has been obtained also in Refs. [105, 106]. Figure 24: The Born ρ, ω, K∗, and K exchange diagrams for To analyze the Belle data, we constructed the helicity γγ π0η, γγ π0η′, and γγ KK¯ . → → → amplitudes Mλ and the corresponding partial amplitudes M of the γγ π0η reaction, where the electromagnetic λJ → Born contributions from ρ, ω, K∗, and K exchanges mod- ified by the form factors and strong elastic and inelas- Born V 0 M2 (γγ π η; s,θ)= tic final-state interactions in the π0η, π0η , K+K , and → ′ − m2 m2 tu G (s,t) G (s,u) K0K¯ 0 channels, and the contributions from the direct =2g g π η − ω + ω ,(39) ωπγ ωηγ 4 t m2 u m2 interaction of the resonances with photons are taken into − ω − ω account:
0 Born V 0 1 2 2 M0(γγ π η; s,θ)= M0 (γγ π η; s,θ)+ where gωηγ = 3 gωπγ sin(θi θP ), gωπγ =12πΓω πγ[(mω → → 2 3 − 2 → − V V ′ mπ)/(2mω)]− 0.519 GeV− [10, 11], the “ideal” +Iπ0η(s) Tπ0η π0η(s)+ Iπ0η′ (s) Tπ0η π0η(s)+ ≈ → → mixing angle θi = 35.3◦, θP is the mixing angle in the ∗ ∗ K + K 0 K+ + I + − (s) I (s)+ I + − (s; x ) pseudoscalar nonet, which is a free parameter; t and e K K K0K¯ 0 e K K 2 − × u are the Mandelstam variables for the reaction γγ direct 0 → TK+K− π0η(s)+ Mres (s), (36) π η, Gω(s,t) and Gω(s,u) are the t and u channel e × e→ e 0 Born V 0 form factors [for the elementary ρ and ω exchanges M2(γγ π η; s,θ)= M (γγ π η; s,θ)+ 2 f Gω(s,t)= Gω(s,u)=1]. In the corresponding Born am- → 2 → 0 0 ′ 1 +80πd20(θ)Mγγ a2(1320) π η(s), (37) plitudes for γγ π η′, gωη γ = gωπγ cos(θi θP ), and → → 3 ¯ → 2 − 0 for γγ KK with the K∗ exchange, gK∗+K+γ 0.064 where θ is the polar angle of the produced π (or η) meson 2→ 2 2 ≈ GeV− and g ∗0 0 0.151 GeV− [10, 11]. in the γγ center-of-mass system. Figs. 23 and 24 show K K γ ≈ the diagrams corresponding to these amplitudes. Note that information on the bare Born sources of the The first terms in the right-hand parts of Eqs. (36) and γγ π0η reaction corresponding to the exchanges with (37) represent the real Born helicity amplitudes, which the→ quantum numbers of the ρ and ω mesons (as well are the sums of the ρ and ω exchange contributions equal as the b1(1235) and h1(1170) mesons) is very scarce in magnitude and are written in the form [108, 109] in the nonasymptotic energy range of interest. It is
Born V 0 known certainly only that the elementary ρ and ω ex- M0 (γγ π η; s,θ)= changes, whose contributions to the γγ π0η cross → → s tGω(s,t) uGω(s,u) section (primarily to the S wave) increase very rapidly =2g g + , (38) ωπγ ωηγ 4 t m2 u m2 with the energy, are not observed experimentally (see − ω − ω 19
31 Fig. 22). This fact was explained in Ref. [109] by where ∆ ′ (s)= D ′ (s)g 0 + Π ′ (s)g ′ 0 a0 a0 a0π η a0a0 a0π η the Reggeization of the elementary exchanges, which sup- and ∆ (s) = D (s)g ′ 0 + Π ′ (s)g 0 ; a0 a0 a0π η a0a0 a0π η presses dangerous contributions even in the range of 1- ′ ga0ab and ga0ab are the coupling constants; and 1.5 GeV. For this reason, we use the Regge type form 1/D (s)=1/(m2 s + [ReΠab (m2 ) Πab (s)]) 2 a0 a0 − ab a0 a0 − a0 factors Gω(s,t) = exp[(t mω)bω(s)], Gω(s,u) = exp[(u is the propagator for the a resonance (and similar 2 − 0 4− 0 mω)bω(s)], where bω(s)= bω + (αω′ /4) ln[1 + (s/s0) ], for the a resonance), whereP ReΠab (s) is deter- 0 2 2 0′ a0 bω = 0, αω′ = 0.8 GeV− and s0 = 1 GeV (and similar for mined by a singly subtracted dispersion integral the K∗ exchange). ab 2 of ImΠa0 (s) = √sΓa0 ab(s) = ga abρab(s)/(16π), → 0 As for the b1(1235) and h1(1170) exchanges, their am- ′ ′ ′ ab ′ Πa0a0 (s)= Ca0a0 + ab(ga0ab/ga0ab)Πa0 (s), and Ca0a0 is plitudes have the form similar to Eqs (38) and (39) ex- the resonance mixing parameter; the explicit form of the cept for the common sign in the amplitude with helicity P ab polarization operators Πa0 (s) [104, 132, 169, 170] see in 0. The estimates show that the axial-vector exchange Appendix 8.2. The amplitude amplitudes are at least five times smaller than the cor- (0) (0) responding vector exchange amplitudes and we neglect g ∆ ′ (s)+ g ′ ∆ (s) a0γγ a0 a γγ a0 bg their contributions. 32 M direct(s)= s 0 eiδπη(s) (44) res ′ 2 Da0 (s)Da (s) Π ′ (s) The terms in Eq. (36) proportional to the S 0 − a0a0 wave hadron amplitudes Tπ0η π0η(s), Tπ0η′ π0η(s), and f 0 → → in Eq. (36) describes the γγ π η transition caused by TK+K− π0η(s) are attributed to the rescattering mech- → (0) (0) → the direct coupling constants ga0γγ and g ′ of the a0 anisms. In these amplitudes, we take into account the a0γγ and a resonances with the photons; the factor s appears contribution from the mixed a0(980) and heavy a0(Y ) 0′ due to the gauge invariance. resonances (bellow, for brevity, they are denoted as a0 Equation (36) implies that the amplitudes T 0 (s) and a0′ , respectively) and the background contributions: ab π η in the γγ ab π0η rescattering loops (see Fig.→ 23) 1 → → V 1 2iδ0 (s) are on the mass shell. In so doing, the functions Iπ0η(s), 1 η0(s)e 1 ∗ T 0 0 (s)= T (s)= − = V K π η π η 0 ′ → 2iρπη(s) Iπ0η (s), IKK¯ (s), and the above mentioned function + bg K e bg 2iδπη (s) res IK+K− (s; x2), are the amplitudes of the triangle loop di- = Tπη(s)+ e Tπ0η π0η(s) , (40) → agramse describinge the transitions γγ ab (scalar state bg bg res i[δ ′ (s)+δπη (s)] → →0 0 ′ πη with a mass = √s), where the meson pairs π η, π η , and Tπ0η π0η(s)= Tπ0η′ π0η(s) e , (41) e ′ → → KK¯ are created by electromagnetic Born sources (see i[δbg (s)+δbg (s)] − res KK¯ πη TK+K π0η(s)= TK+K− π0η(s) e , (42) Fig. 24); corresponding formulae see in Appendixes 8.2 → → 0 and 8.3. The constructed amplitude M0(γγ π η; s,θ) bg → bg 2iδπη (s) res satisfies the Watson theorem in the elastic region. where Tπη(s) = (e 1)/(2iρπη(s)), Tπ0η π0η(s)= res − → For the a2(1320) production amplitude in (37), we use (η1(s)e2iδπη (s) 1)/(2iρ (s)), δ1(s)= δbg (s)+ δres(s), 0 − πη 0 πη πη the parametrization similar to (28) and (29): (+) 2 ( ) 2 ( ) ρ (s)= s m s m − s, m ± = m m , ab ab ab ab b a 0 − − ± Mγγ a2(1320) π η(s)= q+ 0 q¯ 0 bg bg bg → → ab = πη, K K−, K K , πη′; δπη(.s), δπη′ (s) and δ ¯ (s) tot KK sΓa2 γγ(s)Γa (s)B(a2 πη)/ρπη(s) → 2 → are the phase shifts of the elastic background contribu- = 2 tot , (45) ¯ ma2 s i√sΓa2 (s) tions in the channels πη, πη′, and KK with isospin I = 1, p − − respectively (see Appendix 8.2). where The amplitudes of the a0 – a0′ resonance complex in 2 5 2 m q (s) D2(qπη(m )ra ) Eqs. (40)–(42) have the form analogous to Eqs. (25), Γtot(s)=Γtot a2 πη a2 2 , (46) a2 a2 s q5 (m2 ) D (q (s)r ) (26) [105, 106, 171, 275] πη a2 2 πη a2 2 4 qπη(s)= √sρπη(s)/2, D2(x)=9+3x + x , ra2 is the in- g ∆ ′ (s)+ g ′ ∆ (s) res a0ab a0 a0ab a0 √s 3 T 0 (s)= , (43) teraction radius, and Γa2 γγ(s) = ( m ) Γa2 γγ. Recall ab π η ′ 2 → a2 → → 16π[Da0 (s)Da (s) Π ′ (s)] 0 − a0a0 that the f (1270) γγ and a (1320) γγ decays widths 2 → 2 → rather well satisfy the relation Γf2 γγ/Γa2 γγ =25/9 [10, 11, 197], which is valid in the→ naive qq¯→model for
31 the direct transitions qq¯ γγ. These contributions are weakly sensible to the θP values under → 0 ◦ The results of our fit to the Belle data on the γγ π η discussion [11]. The curves in Fig. 22 correspond to θP = 22 . → 32 The exchanges with high spins in the γγ π0η reaction− are reaction cross section are shown in Figs. 25 and 26. The the correction against the background of the→ K exchange con- corresponding values of the model parameters are quoted tribution. This correction is required to describe the data for in Appendix 8.2. The good agreement with the experi- 0 γγ π η, as shown bellow. As to the γγ ππ reactions, the mental data, see Fig. 25, allows for definite conclusions corrections→ from the high spin exchanges prove→ to be less signifi- on the main dynamical constituents of the γγ π0η reac- cant against the background of the summary contribution of the → π and K exchanges, and we are unable to catch them at this tion mechanism whose contributions are shown in detail stage. in Figs. 26(a) and 26(b). 20
c a 50 H L 50 H L L L
nb 40 nb 40 LH LH 0.8 0.8 < < È È
Θ 30 Θ 30 cos cos È È ; ; Η Η 0 20 0 20 ¬1 ΓΓ®Π ΓΓ®Π
H H 2® Σ 10 Σ 10 ¬3
¬4 0 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 !!!!s GeV !!!!s GeV H L H L
0 b Figure 25: The fit to the Belle data on the γγ π η reaction H L → 50 cross section. The resulting solid line corresponds to the solid line 1 in Fig. 26(a) (or in Fig 26(b)), folded with a Gaussian with L nb 40
σ = 10 MeV mass resolution; the shaded band shows the size of the LH
systematic error of the data. 0.8 < È
Θ 30 cos È ; Η Let us begin with the contribution from the in- 0 20 ¬1 + 0 ΓΓ®Π elastic rescattering γγ K K− π η, where the in- H 2® + → → Σ 10 termediate K K− pair is create due to the charge one kaon exchange (see Fig. 24(b)). This mech- anism, as in the case of the f0(980) production in 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 the γγ ππ reactions [102, 103], specifies the natu- !!!!s GeV → H L ral scale for the a0(980) resonance production cross section in γγ π0η, and leads also to the narrowing → Figure 26: The fit to the Belle data. (a) Solid line 1 is the total a0(980) peak in this channel [104, 108]. The maximum + 0 γγ π0η cross section, solid line 2 and the dotted line are the of the cross section σ(γγ K K− a0(980) π η) is → → → → helicity 0 and 2 components of the cross section, solid line 3 is the controlled by the product of the ratio of the squares + − 0 2 2 contribution from the γγ K K π η rescattering with the of the coupling constants Ra = g + − /g and → → 0 a0K K a0πη intermediate K + K pair created due to the Born K exchange, + K 2 2 − ¯ 0 the value IK+K− (4mK+ ; x2) . Its estimate gives the dashed line is the contribution from the γγ KK π η with + | 0| ¯ → → ∗ σ(γγ K K− a0(980) π η; cos θ 0.8) 0.8 the intermediate KK pairs created due to the Born K and K 2→ 2 → → | | ≤ ≈ × exchanges, the dash-dotted line is the contribution from the Born ρ 1.4α Ra0 /mae0 24 nb Ra0 (here we neglect the heavy ≈ × + and ω exchanges with λ =0, and solid line 4 is the joint contribution a′ resonance contribution). Bellow the K K− thresh- 0 0 K+ 2 from these exchanges and the S wave rescattering γγ (π η + old, the function I + − (s; x ) decreases sharply, re- → K K 2 π0η′) π0η. (b) Solid lines 1 and 2 are the same as in panel | | → sulting in the narrowing of the a0(980) peak in the (a), the short-dashed line corresponds to the contribution of the + 0 direct ′ γγ K K− a0(980)e π η cross section [104, 108]. amplitude M (s) caused by the direct decays of the a0 and a → +→ 0 → res 0 The γγ K K− π η rescattering contribution to the resonances into photons, the dotted line is the total contribution →0 → ′ γγ π η cross section is shown by solid line 3 in Fig. from the a0 a resonance complex, and the long-dashed line is → e− 0 26(a). The K∗ exchange also slightly narrows the a0(980) the helicity 0 cross section without the contribution of the direct direct peak (see the dashed line under solid line 3 in Fig. 26(a)). transition amplitude Mres (s). One γγ KK¯ π0η rescattering mechanism is evi- dently insufficient→ → to describe the data in the region of e the a0(980) resonance. The addition of the Born con- but their coherent sum with the contribution from the tribution from the ρ and ω exchanges, which is modi- γγ KK¯ π0η inelastic rescattering (see the diagrams 0 0 0 → → fied by the S wave γγ (π η + π η′) π η rescattering, for the amplitude with λ = 0 in Fig. 23) results in the →direct → considerable enhancement of the a (980) resonance (see and the amplitude Mres (s), which is due to the di- 0 rect transitions of the a0 and a′ resonances into photons, solid line 2 in Fig. 26(a)). Recall that all the S wave con- 0 0 + tributions to the γγ π η amplitude below the K K− makes it possible tof obtain the observed cross section → magnitude. The contributions of these two mechanisms threshold have the same phase according to the Watson themselves are small in the region of the a0(980) reso- theorem. nance (see solid line 4 in Fig. 26(a) for the first of them Note that, as a by-product, we extracted from the fit- and the short-dashed line in Fig. 26(b) for the second), ting of the γγ π0η data the preliminary information on → 21
1 33 0 0 300 tails). a Π Η®Π Η b 0 0 H L H L Π Η®Π Η Let us consider now the γγ π0η cross section due
0.8 L → 200 ∆res s only to the resonance contributions and, by analogy with ΠΗ H L L s H
1 0 Eq. (35), determine the width of the a0(980) γγ decay
0.6 degrees Η 1 H 1 s → Η0 s ® ∆0 , H L 100 H L averaged over the resonance mass distribution in the πη LÈ s H 1 1 0 0.4 T0 s channel [104, 108]:
T È H LÈ È 0 0.2 Phase shifts 1.1 GeV bg ∆ΠΗ s s H L 0 √ -100 Γa0 γγ πη = 2 σres(γγ π η; s)d s (47) 0 0.6 0.8 1 1.2 1.4 h → i 4π → 0.6 0.8 1 1.2 1.4 Z !!!!s GeV !!!!s GeV 0.9 GeV H L H L (the integral is calculated over the region of the 0 0 1 Figure 27: The S π η π η amplitude. (a) T0 (s) and inelas- a0(980) resonance). Taking into account the contri- 1 → 1 | | ticity η0 (s); (b) phase shifts (a0 = 0.0098). butions from all of rescattering processes and direct decays into γγ to the σres cross section, we obtain ′ Γa0 (KK¯ +πη+πη +direct) γγ πη 0.4 keV. Taking into haccount→ the contributions→ fromi only≈ the rescattering pro- 0 0 ′ the S wave amplitude of the π η π η reaction, which cesses, Γa0 (KK¯ +πη+πη ) γγ πη 0.23 keV, and, taking h → → i ≈ is important for the low-energy physics→ of pseudoscalar into account the contributions from only the direct de- direct mesons. The characteristics for the S wave amplitude cays, Γa0 γγ πη 0.028 keV. h → i ≈ π0η π0η are represented in Fig. 27. Here worth not- The performed analysis indicates that the → ¯ 0 0 ing is the important role of the background π0η elastic a0(980) (KK + π η + π η′) γγ rescattering bg mechanisms,→ i.e., the four-quark transitions,→ dominate amplitude Tπη(s), see Eq. (40). First, the choice of the negative background phase δbg (s) (see Fig. 27(b)) in in the a0(980) γγ decay. This picture is evidence πη of the q2q¯2 nature→ of the a (980) resonance and is in T bg (s) makes it possible to fit the πη scattering length 0 πη agreement with the properties of the σ (600) and f (980) in the model under consideration to the estimates based 0 0 resonances, which are its partners. As to the ideal qq¯ on the current algebra [276, 277] and chiral perturba- 1 model prediction for the two-photon decay widths of the tion theory [278, 279], according to which a0 (in units of 1 f0(980) and a0(980) mesons, Γf0 γγ /Γa0 γγ = 25/9, it mπ− ) 0.005 0.01. The resonance contribution ( 0.3) 34 → → 1 ≈ − ≈ is excluded by experiment. to a0 is compensated by the background contribution. bg Second, the significant negative value of δπη(s) near 1 GeV ensures the resonance-like behavior of the cross sec- tion shown by solid line 4 in Fig. 26(a). 33 Recall, that in the previous Section it was not required to in- troduce any heavy scalar isoscalar resonance for the theoretical We now turn to Fig. 26(b) and discuss the contribution description of the γγ π+π− and γγ π0π0 processes, as → → from the possibly existing heavy a0′ resonance [11]) with well as in Refs. [91, 92] for the phenomenological treatment of the mass m ′ 1.4 GeV. The cross section corresponding the experimental data. In principle, it could be the f0(1370) a0 ≈ direct resonance [11]. As a matter of fact, situation with the heavy to the amplitude Mres (s) (see the short-dashed line) > scalar resonances with the masses ∼ 1.3 GeV has been strongly exhibits a pronounced enhancement near 1.4 GeV. In tangled for a long time. For example, the authors of the re- the cross section correspondingf to the total contribution view [75] seriously doubt in the existence of such a state as the from the resonances (see the dotted line), i.e., from the f0(1370) (in this connection see also Refs. [280, 281]). It is pos- sible that the wish to see the scalar resonances with the masses amplitude M direct (s) and rescattering amplitudes pro- res of (1.3–1.4) GeV as the partners of the well established b1(1235), portional to the amplitudes of the resonance transitions h1(1170), a1(1260), f1(1285), a2(1320), and f2(1270) states, be- 0 + 0 ¯ 0 ab π η (fab = πη, K K−, K K , πη′), this enhance- longing to the lower P wave qq¯ multiplet, is not realized in the ment→ is transferred to a shoulder. Finally, in the total naive way. In any case, this question remains open and requires further experimental and theoretical investigations. cross section σ0 (see solid line 2) additionally including 34 0 0 0 As already mentioned in Ref. [104], the model of nonrela- the Born γγ π η contribution and the γγ π η π η tivistic KK¯ molecules is unjustified, because the momenta in → 0 → 0 → + − rescattering caused by the background π η π η elastic the kaon loops describing the φ K K γ(f0/a0) and → + − → → amplitude, any resonance attributes near 1.4 GeV are f0/a0 K K γγ decays are high [45, 214, 215]. Our anal- → → absent. Thus, a strong destructive interference exists be- ysis gives an additional reason against the molecular model. The point is that the a0(980) resonance is strongly coupled with the tween different contributions and masks the a0′ resonance KK¯ and πη channels, which are equivalent in the q2q¯2 model. A in the γγ π0η cross section. Nevertheless, in many re- ¯ → weakly bound KK + πη molecule seems to be impossible. More- spects owing to the a0′ , we succeed in modeling a signifi- over, the widths of the two-photon decays of the scalar resonances cant smooth background under the a2(1320) and between in the molecular model are calculated at the resonance point [282, 283], but this is insufficient for describing the γγ π+π−, a0(980) and a2(1320) resonances, which is required by the γγ π0π0, and γγ π0η reactions. Attempts of→ the de- Belle data [93]. Note that due to the resulting compen- scription→ of the data on→ these processes in the framework of the sations, the wide interval of (1.28–1.42)GeV is allowed molecular model are absent and, therefore, the results obtained for the mass of the a0′ resonance (see Ref. [105] for de- in this model have the academic character. 22
120 80 50 6. Preliminary summary a ARGUS b L3 c ARGUS H L H L H L 100 40 60 L 30 MeV 20 MeV 80 nb L Results of the theoretical analysis of the experimental L LH 0 S - 30 - K K K + 0 S
60 40 + K K achievements in the low energy region, up to 1 GeV, can K 20 ΓΓ® 40 ΓΓ® ΓΓ® H H H
be formulated in the following way. 20 Σ 20 10 Events Events 0 0 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1 1.1 1.2 1.3 1.4 1.5 1. Naive consideration of the mass spectrum of the !!!!!s GeV !!!!!s GeV !!!!!s GeV H L H L H L light scalar mesons, σ(600), κ(800), f0(980), and 2 2 a0(980), gives an idea of their q q¯ structure. + − 0 0 Figure 28: The K K (a) and KSKS (b) mass spectra mea- sured by ARGUS [269] and L3 [291], respectively. (c) This plot 2. Both intensity and mechanism of the illustrates the scale of the KK¯ production cross section in γγ a0(980)/f0(980) production in the φ(1020) collisions. The experimental points show the cross section for meson radiative decays, the four-quark transitions γγ K+K− with allowed contributions from λJ = [22, 02, 00] + → φ(1020) K K− γ[a0(980)/f0(980)], indicate [269]. The upper dashed, dot-dashed, and solid curves correspond the q2q¯2 →nature of the→ a (980) and f (980) states. to the γγ K+K− Born cross section with λJ =00, λJ = [00, 22], 0 0 → and the total one, respectively (the λJ =02 contribution is negligi- 3. Intensities and mechanisms of the two-photon pro- ble). The lower dashed, dot-dashed, and solid curves correspond to duction of the light scalars, the four-quark transi- the same cross sections but modified by the form factor (see Sec. + 0 4 and Appendix 8.3). The dotted curve shows the estimate of the tions γγ π π− σ(600), γγ π η a0(980), + − → + → → → the S wave γγ K K cross section in our model. and γγ K K− f0(980)/a0(980), also indi- → cate their→q2q¯2 nature.→
f2(1270), a2(1320), and f ′ (1525). But near the thresh- 4. In addition, the absence of the J/ψ γf0(980), 2 → old, 2mη = 1.0957 GeV < √s< 1.2 GeV, there is a no- ρa0(980), ωf0(980) decays in contrast to the in- ticeable S wave contribution, (1.5 0.15 0.7) nb, tensive J/ψ γf2(1270), γf ′ (1525), ρa2(1320), → 2 which indicates the presence of≈ some subthreshold± ± reso- ωf2(1270) decays intrigues against the P wave qq¯ nance strongly coupled with the ηη channel. Such a res- structure of the a0(980) and f0(980) resonances. 2 2 onance in the q q¯ model is the f0(980). Unfortunately, 5. It seems also undisputed that in all respects the the γγ ηη reaction is not so good for its investigation, because→ here only the end of the tail of this resonance a0(980) and f0(980) mesons are strangers in the can be seen. company of the well established b1(1235), h1(1170), a1(1260), f1(1285), a2(1320), and f2(1270) mesons, High statistics information is still lacking for the reac- + 0 0 tions γγ K K− and γγ K K¯ in the 1 GeV region. which are the members of the lower P wave qq¯ mul- → → tiplet. It is expected that the four-quark nature of the a0(980) and f0(980) resonances shows itself in these channels very originally [109, 110]. 7. Future Trends As the experiments show [100, 190, 269, 286–292], the + 0 0 γγ K K− and γγ KSKS cross sections in the region 7.1. The f0(980) and a0(980) resonances near 1 <→√s< 1.7 GeV are→ saturated in fact with the contri- → + − → 0 ¯ 0 γγ K K and γγ K K reaction thresh- butions of the classical tensor f2(1270), a2(1320), and olds f2′ (1525) resonances, creating in the helicity 2 states, see + The Belle Collaboration investigated the γγ π π−, Fig. 28. The constructive and destructive interference 0 0 0 → γγ π π , and γγ π η reactions with the highest between the f2(1270) and a2(1320) resonance contribu- → 35 → + 0 0 statistics. In July 2010, the Belle Collaboration re- tions is observed in γγ K K− and γγ K K¯ , re- ported also the first data on the γγ ηη reaction [285]. spectively, in agreement→ with the qq¯ model→ [293]. Notice → The γγ ηη cross section for √s> 1.2 GeV is domi- that the region of the KK¯ thresholds, 2m < √s< 1.1 → K nated by the contributions from the tensor resonances GeV, sensitive to the S wave contributions is not investi- gated in fact. The sensitivity of the ARGUS experiment + to the K K− events for 2mK+ < √s < 1.1 GeV was negligible [269], see Fig. 28(a), and the total statistics of 35 Note that high precision measurements of the γγ π+π− cross → the L3 [291], see Fig. 28(b), and CLEO [100] experiments section for 0.28 GeV < √s< 0.45 GeV are planned for the KLOE- 0 0 on γγ K K for 2m 0 < √s< 1.1 GeV are within 60 2 detector at upgraded DAΦNE φ factory in Frascati [175, 284]; → S S K the existing MARK II data [95] have in this region very large events. error-bars, see Fig. 6(a). Measurements of the integral and dif- The absence of the considerable non-resonance back- ferential cross sections for γγ π+π− and γγ π0π0 in the √s + → → ground in the γγ K K− cross section seems at first region from 0.45 GeV to 1.1 GeV [175, 284], which will complete sight rather surprising→ for the one kaon exchange Born the information from previous experiments on the σ(600) and contribution comparable with the tensor resonance con- f0(980) resonance production, are also planned. In particular, the statistical uncertainty in the γγ π0π0 cross section in the tributions, see Fig. 28(c). As seen from this figure, the σ(600) meson region (see Fig. 6(b))→ will be reduced to 2%. S wave contribution dominates in the Born cross section 23 at √s < 1.5 GeV. One would think that the large non- experimental investigations of the f0(980) and a0(980) coherent background should be under the tensor meson mesons in various reactions. Nevertheless, it turns out + peaks in the K K− channel. But taking into account that equally good descriptions of the available data can + of the resonance interaction between the K and K− be obtained for appreciably different sets of the coupling + − + − mesons in the final state results in the cancelation of the constants gf0K K , gf0π π , etc. (see, for example, Refs. considerable part of this background [109, 110]. The prin- [34, 132, 136, 171, 172]). Certainly, it would be highly de- + cipal point is that the S wave Born γγ K K− ampli- sirable to fix their values. In respect of the coupling con- → + + − + − + − + − tude acquires the ξ(s) = [1 + iρK (s)TK K K K (s)] stants gf0K K and gf0π π , this question could be elu- + + → factor due to the γγ K K− K K− rescattering cidated by precise data on the inelasticity of ππ scatter- amplitude with the real→ kaons in→ the intermediate state. ing near the KK¯ threshold, that have not been updated The a0(980) and f0(980) resonance contributions domi- since 1975 [112–115]. It is very likely that such data in + nate in the TK+K− K+K− (s) amplitude near the K K− the raw form are in hand of the VES Collaboration, which → + threshold and provide it with the considerable imaginary was performing measurements of the π−p π π−n re- part for the strong coupling with the KK¯ channels in action at IHEP (Protvino). Moreover, the product→ of the 2 + − + − the four-quark scheme. As a result the ξ(s) factor is coupling constants ga0K K gf0K K may be fixed from | + | considerably less than 1 just above the K K− thresh- data on the f0(980) a0(980) mixing, that are expected old and the seed S wave Born contribution is consider- from the BESIII detector− [131]. + − + − ably reduced in the wide region of √s. The dotted curve Exclusive information on ga0K K gf0K K can result in Fig. 28(c) represents the estimation of the S wave from investigations of the spin asymmetry jump, due to + γγ K K− cross section obtained in the model under the f0(980) a0(980) mixing, in the π−p f0(980)n → −0 → → consideration (see details in Appendix 8.3). This estima- a0(980)n π ηn reaction [124]. tion agrees with those obtained earlier [109, 110]. → This work was supported in part by the RFFI Grant So one can hope to detect in the partial wave analysis of + No. 10-02-00016 from the Russian Foundation for Basic the γγ K K reaction at 2m + < √s< 1.1 GeV the − K Research. scalar contributions→ at the rate of about 5–10 nb. As for the γγ K0K¯ 0 reaction, its amplitude has not the Born → 8. Appendix contribution and the a0(980) resonance contribution has + the opposite sign in comparison with the γγ K K− 8.1. γγ → ππ channel. As a result the contributions of the→S wave Below there is the list of the expressions for the Born he- + 0 0 γγ K K− K K¯ rescattering amplitudes with the licity amplitudes corresponding to the charged one pion isotopic→ spin I→=0 and 1 practically cancel each other and exchange mechanism and for the triangle loop integrals π+ π+ the the corresponding cross section should be at the rate − − Iπ+π (s) and Iπ+π (s; x1) used in Section 4. In addition, of about < 1 nb. a few useful auxiliary formulae for the solitary scalar res- ∼ onance are adduced. 7.2. The σ(600), f0(980), and a0(980) resonances e e ∗ The Born helicity amplitudes for the elementary one in γγ collisions + 2 pion exchange in the γγ π π− reaction have the form The investigations of light scalar mesons in the γγ∗(Q ) → 2 collisions are promising. If σ(600), f0(980), and a0(980) + 4m + 8πα M Born π (s,θ)= π , (48) resonances are four-quark states, their contributions 0 2 2 s 1 ρ + (s)cos θ 2 0 0 2 0 π to the γγ∗(Q ) π π and γγ∗(Q ) π η cross sec- − → →2 tions should decrease with increasing Q more rapidly 2 2 Born π+ 8παρπ+ (s) sin θ than the contributions from the classical tensor mesons M2 (s,θ)= 2 2 , (49) 1 ρ + (s)cos θ f2(1270) and a2(1320). A similar behavior of the con- − π tribution from the exotic q2q¯2 resonance state with + 2 G PC + ++ 0 0 (ρπ (s) = 1 4mπ+ /s). Their partial wave expan- I (J )=2 (2 ) [48, 85, 86] to the γγ∗ ρ ρ and − + → sions are γγ∗ ρ ρ− cross sections was recently observed by the q L3 Collaboration→ [294–297]. Born π+ Born π+ J Mλ (s,θ)= (2J + 1)MλJ (s)dλ0(θ) , (50) → → J λ 7.3. Searches for the J/ψ ωf0(980) and J/ψ X≥ ρa0(980) decays J where dλ0(θ) are usual d-functions (see, for example, [10, These decays are important to elucidate the nature of the 11]). Three lower partial waves have the form f (980) and a (980) resonances [36, 38, 41]. The J/ψ 0 0 2 → + 1 ρ + (s) 1+ ρ + (s) ρa0(980) decay has not been discovered as yet, B(J/ψ Born π π π 4 → M00 (s)=4πα − ln , (51) ρa0(980)) < 4.4 10− [36]. As for the information on ρπ+ (s) 1 ρπ+ (s) × 4 − B(J/ψ ωf0(980)) = (1.4 0.5) 10− [11], it would be → ± × 2 2 more correctly replaced by a suitable upper limit [38, 41]. Born π+ 1 ρπ+ (s) 3 ρπ+ (s) M02 (s)=4πα −2 − ρ + (s) 2ρ + (s) × 7.4. Inelasticity of ππ scattering and f0(980) − π π a (980) mixing 1+ ρ + (s) 0 ln π 3 , (52) By now considerable progress has been made in the × 1 ρ + (s) − − π 24
2 2 Born π+ 3 (1 ρπ+ (s)) 1+ ρπ+ (s) In this case the numerical integration needs certainly. M22 (s)=4πα − 3 ln 2 2ρ + (s) 1 ρ + (s) To make easy understanding the structure and normal- r π π − − ization of the sufficiently complicated expressions used 1 5 in fitting data, one adduces the formulae for the pro- 2 + . (53) −ρ + (s) 3 π duction cross section of the σ resonance and for its two- The amplitude of the triangle loop diagram, describ- photon decay width due to the rescattering mechanism, + + + γγ π π− σ π π−, in the imaginary case of the ing the transition γγ π π− (scalar state with a → → → mass = √s), is defined by→ → solitary scalar σ resonance coupled only to the ππ chan- nel. ∞ Born π+ The corresponding resonance cross section has the fa- π+ s ρπ+ (s′)M00 (s′) Iπ+π− (s)= ds′ . (54) miliar form π s′(s′ s iε) 4mZ2 − − π+ + e σres(γγ π π−; s)= + → π + − + − The behavior I + − (s) s, when s 0, is the gauge in- 8π √sΓσ π π γγ(s) √sΓσ π π (s) π π → → → ∝ 2 → = 2 , (63) variance consequence. For s 4m + s D (s) ≥ π | σ | e 2 π+ mπ+ 2 where I + − (s)=8α( [π 2 arctan ρ + (s) ] 1) , (55) π π s − | π | − 1 2 2 Γσ π+π− γγ(s)= Mσ π+π− γγ (s) = fore s 4m + → → 16π√s| → → | ≥ π 2 2 2 2 1 + g + − + m + 1+ ρπ+ (s) π σπ π π π = Iπ+π− (s) . (64) Iπ+π− (s)=8α π + i ln 1 . 16π 16π√s s 1 ρπ+ (s) − ( − ) (56) e e If the σ can else directly transit into γγ with the ampli- The form factor, see Eq. ((33), (0) tude sgσγγ then the width Γσ π+π− γγ(s) in Eq. (63) → → 2 2 should be replaced by 1 mπ+ t mπ+ u Gπ+ (t,u)= −2 2 + −2 2 s 1 (u mπ+ )/x1 1 (t mπ+ )/x1 1 2 − − − − Γσ γγ (s)= Mσ γγ(s) , (65) 2 2 → 16π√s| → | (here t = m + s[1 ρ + (s)cos θ]/2 and u = m + s[1 π − − π π − + ρπ+ (s)cos θ]/2) modifies the Born partial wave ampli- where tudes. Let us introduce the notations: (0) 2 + − + + Mσ γγ(s)= Mσ π π γγ(s)+ sgσγγ . (66) Born π 1 ρπ+ (s) Born π → → → M0J (s)= − F0J (ρπ+ (s)) , (57) ρ + (s) π The propagator of the σ resonance with the mσ Breit- Wigner mass in Eq. (63) has the form Born π+ Born π+ M (s)= ρ + (s)F (ρ + (s)) . (58) 2J π 2J π 1 1 = , (67) Then the modified amplitudes can be represented in the 2 ππ 2 ππ Dσ(s) mσ s + ReΠσ (mσ) Πσ (s) form − − ππ 2 where Πσ (s) is the polarization operator of the σ reso- Born π+ 1 ρπ+ (s) + 0 0 M0J (s; x1)= − nance for the contribution of the π π− and π π inter- ρπ+ (s) × 2 2 mediate states. For s 4mπ+ (= 4mπ0 ) Born π+ Born π+ ≥ + + F0J (ρπ (s)) F0J (ρπ (s; x1)) , (59) 2 × − ππ 3 gσπ+π− 1 1+ ρπ+ (s) h i Πσ (s)= ρπ+ (s) i ln . (68) + + 2 16π − π 1 ρπ (s) Born π − M2J (s; x1)= ρπ+ (s) + + × 2 Born π Born π If 0 π+ The σ ππ decay width is The function Iπ+π− (s), see Eqs. (54)– (56), is replaced, → 2 with taking into account the form factor, by 1 ππ 3 gσπ+π− ρπ+ (s) Γσ ππ(s)= ImΠσ (s)= . (70) e → ∞ Born π+ √s 2 16π √s π+ s ρπ+ (s′)M00 (s′; x1) Iπ+π− (s; x1)= ds′ . ππ 2 ππ π s′(s′ s iε) The function Re[Πσ (mσ) Πσ (s)] in the denomi- 4mZ2 − − − π+ nator of Eq. (67) is the correction for the finite width e (62) of the resonance. In Fig. 29 the real and imaginary 25 ( ) 2 (+) 2 0.4 m − L ab ab 2 mΣ=0.6 GeV, GΣ=0.45 GeV GeV 2 (+) ( ) g − ab a0 ab mab mab ma 0.2 → LD< H Πa0 (s)= ln s H 16π πs mb − Σ " D @ Im (+) 2 - 0 2 mab s ρab(s) 1 arctan − , (72) and − − π q ( ) 2 LD s m − s ab H -0.2 Σ − D @ q Re (+) 2 ( ) 2 - 8 where ρab(s)= m s s m − s, and for s -0.4 ab − − ab ≤ 0 0.1 0.2 0.3 0.4 0.5 ( ) 2 2 q q . s GeV m − H L ab 2 (+) ( ) g m m − m Figure 29: Demonstration of the finite width correction with an ab a0 ab ab ab a Πa0 (s)= → ln example of the single σ resonance. The curves are described in the 16π " πs mb − text. (+) 2 ( ) 2 1 mab s + mab− s ρab(s) ln − − , (73) − π q (+) 2 q ( ) 2 parts of the inverse propagator Dσ(s) (taken with m s m − s ab − − ab − the sign minus) are shown by the solid and dashed q q curves in the case of the resonance with the mass (+) 2 ( ) 2 2 where ρab(s)= mab s mab− s s. mσ = 0.6 GeV and the width Γσ =Γσ ππ(mσ) = 0.45 → − − GeV. As may be inferred from this figure, Re[Dσ(s)] The triangleq loop integralq in Eq. (36). is 2 can be close to 0 at s =4mπ+ due to the correction for the finite width in the case of the large one. Then ∞ Born V 0 V s ρπη(s′)M00 (γγ π η; s′) this results in the threshold cusp in the amplitudes Iπη(s)= → ds′ , 36 π s′(s′ s iε) proportional to 1/D (s) . For reference, in Fig. 29 Z 2 − − | σ | (mπ+mη) the real and imaginary parts of the inverse propagator e (74) 2 2 2 2 D (s)= m s im Γ (s 4m + )/(m 4m + ) where σ σ − − σ σ − π σ − π without the correction forq the finite width [118] (also M Born V (γγ π0η; s) = (75) taken with the sign minus) are shown by the dotted and 00 → 1 dot-dashed curves, respectively, at the same values of 1 = M Born V (γγ π0η; s,θ)d cos θ , mσ and Γσ. 2 0 → 0 Z1 8.2. γγ → π η − The polarization operators of the a resonance Πab (s) 0 a0 is the S wave Born amplitude, and the amplitude + 0 ¯ 0 (ab = πη, K K−, K K , πη′), introduced in Section 4 M Born V (γγ π0η; s,θ) is defined by Eq. (38). The 0 ∗ (see the paragraph with Eqs. (43) and (44)), have the V → K functions I 0 ′ (s) and I (s) in Eq. (36) are calcu- (+) 2 ( ) π η KK¯ + following form: for s mab (mab± = mb ma, mb K lated similarly and the function I + − (s; x ) is calcu- m ) ≥ ± ≥ K K 2 a lated withe Eq. (92) in Appendixe 8.3. 2 (+) ( ) For the background phase shifts we use the simplest g − e ab a0 ab mab mab ma Πa0 (s)= → ln + ρab(s) parametrizations, which are suitable in the physical re- 16π " πs mb × gion of the γγ π0η reaction: → ( ) 2 (+) 2 s m − + s m iδbg (s) 1/2 1 ab ab e ab = [(1 + iF (s))/(1 iF (s))] , (76) i ln − − , (71) ab − ab × − π q ( ) 2 q (+) 2 s m − s m (+) 2 (+) 2 − ab − − ab 1 mπη /s c0 + c1 s mπη − − q q Fπη(s)= q 2 , (77) (+) 2 ( ) 2 (+) 2 where ρ (s)= s m s m − s, for 1+ c2 s mπη ab − ab − ab − q q . − FKK¯ (s)= fKK¯ √s (ρK+K (s)+ ρK0K¯ 0 (s)) /2, (78) (+) 2 ′ ′ Fπη (s)= fπη s mπη′ . (79) 36 − The references to papers, in which the the finite width corrections q and the analytic properties of the propagators of the realistic The curves in Figs. 25, 26, and 27 cor- f0(980), a0(980), and σ(600) resonances have been investigated, are pointed out in Section 2. In connection with the γγ π0η respond to the following model parameters: → + − ′ and γγ ππ reactions these corrections are discussed also in the (ma0 , ga0πη, ga0K K , ga0πη ) = (0.9845, 4.23, 3.79, → papers [101, 108]. 2.13) GeV; (ma′ , ga′ πη, g ′ + − , ga′ πη′ ) = (1.4, 3.3, − 0 0 a0K K 0 26 3 2 2 + 0.28, 2.91) GeV; (ga0γγ, ga0γγ) = (1.77, 11.5) 10− (here t = mK+ s[1 ρK (s)cos θ]/2 and u = mK+ s[1 1 2 − × − − −+ GeV− ; Ca a′ = 0.06 GeV , c0 = 0603, c1 = 6.48 Born K 0 0 + ρK+ (s)cos θ]/2), the partial amplitudes MλJ (s) 2 4 − − + ¯ ′ Born K GeV− , c2 = 0.121 GeV− ; (fKK , fπη ) = ( 0.37, are replaced by M (s; x ). Substituting ρ + (s) 1 tot − λJ 2 K 0.28) GeV− ; (ma , Γ ) = (1.322, 0.116) GeV; 2 2 a2 instead ρπ+ (s) and ρK+ (s; x2)= ρK+ (s)/(1+2x2/s) in- (0) 1 + Γa2 γγ = 1.053 keV, ra2 = 1.9 GeV− , θP = 24◦ (see stead ρπ (s; x1) in Eqs. (57)–(60), one gets Ref.→ [105] for details). − 2 + 1 ρ + (s) + → ¯ Born K K Born K 8.3. γγ KK M0J (s)= − F0J (ρK+ (s)), (88) + ρK+ (s) The Born amplitude of the reaction γγ K K−, caused → Born K+ by the elementary one kaon exchange, Mλ (s,θ) Born K+ Born K+ Born K+ and MλJ (s) result from the corresponding M2J (s)= ρK+ (s)F2J (ρK+ (s)), (89) + Born π+ γγ π π− Born amplitudes Mλ (s,θ) and Born→ π+ + + 2 MλJ (s) by the substitution of mK for mπ and of + + Born K 1 ρK+ (s) Born K M (s; x2)= − F (ρ + (s)) + 2 + 0J 0J K ρK (s) = 1 4mK+ /s for ρπ (s) in Eqs. (48), (49), ρK+ (s) − − h and (51)–(53):q Born K+ F (ρ + (s; x )) , (90) − 0J K 2 2 Born K+ 4mK+ 8πα i M0 (s,θ)= 2 2 , (80) s 1 ρK+ (s)cos θ Born K+ Born K+ − M (s; x )= ρ + (s) F (ρ + (s)) 2J 2 K 2J K − Born K+ h 2 2 + + 8παρ + (s) sin θ F (ρK (s; x2)) . (91) Born K K − 2J M2 (s,θ)= 2 2 , (81) 1 ρ + (s)cos θ i − K Correspondingly, with taking into account the form fac- K+ tor, the function I + − (s) is replaced by 2 K K Born K+ 1 ρK+ (s) 1+ ρK+ (s) M00 (s)=4πα − ln , (82) + ρK+ (s) 1 ρK+ (s) e ∞ Born K − K+ s ρK+ (s′)M00 (s′; x2) IK+K− (s; x2)= ds′. π s′(s′ s iε) Z2 − − 2 2 4m + + 1 ρ + (s) 3 ρ + (s) K Born K K K e M02 (s)=4πα − − (92) 2 + ρ + (s) 2ρK (s) × + K K 2 Note that 0.68 IK+K− (s) coincides with 1+ ρ + (s) + × | | ln K 3 , (83) K 2 IK+K− (s; x2) within an accuracy better than 3% × 1 ρK+ (s) − | | − in the range 0.8 GeV < √s 2 2 + + m + 1+ ρ + (s) + ρK (s) + 2 K K K σ (γγ K K−)= M (γγ K K−; s) , IK+K− (s)=8α π + i ln 1 . 00 00 s 1 ρ + (s) − → 32πs | → | ( − K ) (95) e (86) Taking account of the form factor 0 0 0 ρK (s) 0 ¯ 0 2 σ00(γγ KSKS)= M00(γγ K K ; s) . 2 2 → 64πs | → | 1 mK+ t mK+ u (96) GK+ (t,u)= −2 2 + −2 2 s 1 (u m + )/x 1 (t m + )/x The amplitudes of the ππ KK¯ reactions, − − K 2 − − K 2 → (87) Tπ+π− K+K− (s) = Tπ+π− K0K¯ 0 (s)= TK+K− π+π− (s), → → → 27 + + are defined by Eqs. (23) and (26). The K K− K K− + 0 0 → and K K− K K¯ reaction amplitudes are given by → 1 1 ga K+K− ∆ ′ (s)+ ga′ K+K− ∆ (s) 0 1 KK¯ 0 a0 0 a0 T + − + − (s) = [t (s)+ t (s)]/2, (97) T (s)= , (102) K K K K 0 0 res;1 ′ 2 → 8π[Da0 (s)Da (s) Π ′ (s)] 0 − a0a0 0 1 TK+K− K0K¯ 0 (s) = [t0(s) t0(s)]/2, (98) where → − 0 I ∆ (s)= D (s)g + − + Π (s)g + − , where t (s) are the S wave KK¯ KK¯ reaction ampli- f0 f0 σK K f0σ f0K K 0 → 0 + − + − tudes with isospin I = 0 and 1; ∆σ(s)= Dσ(s)gf0K K + Πf0 σ(s)gσK K , 1 KK¯ ∆ ′ (s)= D ′ (s)g + − + Π ′ (s)g ′ + − , 2iδ (s) a a0 a0K K a0a0 a0K K e B 1 KK¯ ¯ 0 t0(s)= − + e2iδB (s)T KK (s) , (99) 1 0 res;0 ∆ (s)= Da (s)g ′ + − + Π ′ (s)g + − . 2iρK+ (s) a0 0 a0K K a0a0 a0K K The amplitudes of the direct resonance transitions into photons are given by 2iδbg (s) e KK¯ 1 bg ¯ 1 2iδ ¯ (s) KK t0(s)= − + e KK Tres;1(s) , (100) (0) 0 (0) 0 + ¯ 2iρK (s) KK gσγγ∆f0 (s)+ g ∆σ(s) direct iδB (s) f0γγ Mres; (s)= se 2 ± D (s)D (s) Π (s) KK¯ bg σ f0 f0σ where δB (s) and δ ¯ (s) are the phases in the channels − KK (0) 1 (0) 1 ′ with I = 0 and 1, respectively. bg ga0γγ∆a (s)+ g ′ ∆a (s) iδ (s) 0 a0γγ 0 se KK¯ . 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