Strong isospin breaking at production of light scalars

0.16 MB·

by  N. N. Achasov

Strong isospin breaking at production of light scalars
N.N. Achasov and G.N. Shestakov
Laboratory of Theoretical Physics, S.L. Sobolev Institute for Mathematics, 630090 Novosibirsk, Russia
It isdiscussed breakingtheisotopic symmetryasthetool of studyingtheproduction and nature
of light scalar mesons.
I. INTRODUCTION
K+ K0
a00 f0 a00 f0
The thirty seven years ago we discovered theoret- +
ically a threshold phenomenon known as the mix-
K− K¯0
7 ing of a0(980) and f (980) resonances that breaks
0 0
01 the isotopic invariance considerably, since the effect ∼ FIG. 1. The KK¯ loop mechanism of the a00(980)−f0(980)
2(M M )/M 0,13inthemoduleoftheam-
2 K0 − K+ K0 ≈ mixing.
plitude [1]; see also Ref. [2]. This effect appears as the
n nparrow, 2(M M ) 8 MeV, resonant structure
K0 K+
9 Ja KpbreKot¯wpoe→esanfls0t(hw9e8e0rKe)+aa−nKpdp−veaiacreneddv,e≈Krcsao0.Kn¯cS0eirntnchienretghshabotoltdtihsm,etahm00e(a9sn8ey0a)nrce→hw- −ρK0K¯0(m)(cid:17)− ρK+Kπ−(m)ln11+−ρρKK++KK−−((mm))
] ing it and estimating the effects related with this phe- +ρK0K¯0(m)ln1+ρK0K¯0(m)
h nomenon [3-29]. π 1−ρK0K¯0(m) #
p-p peNrimoweandtaalylys,atnhdisstpuhdeineodmweinthonthheahseblpeeonfddeistceocvtoerresdVeExS- ≈ ga00K+K1−6gπf0K+K− i ρK+K−(m)−ρK0K¯0(m) ,
e in Protvino [30, 31] and BESIII in Beijing [32–34] in the (cid:16) (cid:17)
h processes where m (invariant virtual mass of scalar resonances)
≥
v1 [ (a) π→−πN−→π+ππ−−fπ10(N128[53)0N, 3→1],π−f0(980)π0N → Tm2mhe≤K0m2aomndKdul,ρuKρsKK¯aKn(¯md(m)t)h=esphpoh1ual−dse4bmoef2KrΠe/pam00laf20c;(emidn)bthyaereir|ρesgKhioKo¯wn(mn0)i≤n|.
2 (b) J/ψ φf (980) φa (980) φηπ0 [32], Fig. 2. In the region between the K+K− and K0K¯0
2 → 0 → 0 →
1 (c) χ a (980)π0 f (980)π0 π+π−π0 [32],
c1 0 0
2 → → → 30
0 (d) J/ψ γη(1405) γf0(980)π0 γ3π [33], HaL Les 100 HbL
1701. ((fe)) JJ//ψψ→→→φφff10((1928805)π)→0→→φfφ03(π980[3)π4]0,→→φ3π [34] 23-LÈHLm10GeV 122505 HLHmdegreP0fa -00 468000
iv: Iimntgihxaeinffsgeb,cetcbcouamnteaacpllsepoaerafr[o3nr5o,ta3no6yn]tlyhmadetuctehhtaeonsitishmmeilaao00rf(i9s8toh0se)p−inpfrb0o(r9de8au0kc)-- HÈP0fa -00150 Phaseof 200
X
tion of the KK¯ pairs in the S wave, X KK¯ 00.97 0.98 0.99 1 1.01 0.97 0.98 0.99 1 1.01
ar f0(980)/a00(980).1 Thus anew toolto study →the produ→c- m HGeVL m HGeVL
tion mechanism and nature of light scalars is emerged.
FIG. 2. (a) An example of the modulus of the a0(980) −
0
f0(980) mixing amplitude. (b) The phase of the a00(980)−
II. THE a00(980)−f0(980) MIXING f0(980) mixing amplitude.
Themaincontributiontothea0(980) f (980)mixing thresholds, which is the 8MeV wide,
0 − 0
amplitude, caused by the diagrams shown in Fig. 1, has
Πtha00ef0fo(mrm)= ga00K+K1−6gπf0K+K−"i ρK+K−(m) |Πa000.1f02(7m|g)a|0≈K+|Kga−00Kgf+0KK1−+6Kgπf−0K| +K0−.0|3sG2e(mV2Km0 −K0mK+)
(cid:16) ≈ 16π ≃
m m2 m2 m3/2√m m .
≈ K K0 − K+ ≈ K d− u
1 Eachsuchmechanismreproducesboththenarrowresonantpeak q
and the sharp jump of the phase of the amplitude between the Note that Πρ0ω Ππ0η 0.003 GeV2 (md mu)
K+K− andK0K¯0 thresholds. 1 GeV. | |≈| |≈ ≈ − ×
2
The branching ratios of the isospin-breaking decays
1
f0(980) → ηπ0 and a00(980) → π+π−, caused by the 0£-t£0.025GeV2
a0(980) f (980) mixing, are [36] 0.75
0 − 0
BR(f (980) KK¯ a0(980) ηπ0) 0.5
0 → → 0 →
2
=Z (cid:12)(cid:12)Da00(m)DΠf0a(00mf0)(m−)Π2a00f0(m)(cid:12)(cid:12) mmetry 0.250
2(cid:12)(cid:12)m2Γa0→ηπ0(m) (cid:12)(cid:12) Asy -0.25
(cid:12) 0 dm 0.3%,(cid:12)
× π ≈
-0.5
BR(a0(980) KK¯ f (980) ππ)
0 → → 0 → -0.75
2
Π (m)
= a00f0
Z (cid:12)(cid:12)Da00(m)Df0(m)−Π2a00f0(m)(cid:12)(cid:12) 0.92 0.94 0.9m6 HG0.e9V8L 1 1.02 1.04
(cid:12)(cid:12)(cid:12)2m2Γf0→ππ(m)dm 0.2%,(cid:12)(cid:12)(cid:12)
× π ≈ FIG. 4. Manifestation of the a0(980)−f0(980) mixing effect
aw0h(e9r8e0)Daan00(dmf) (a9n8d0)Drfe0s(omna)nacrees,trheesppercotpivaeglay.torFsigoufrteh3e itnargthetearteaPcltπai−bon=π1−8p.3↑ G→eVa00in(9t8h0e)nρ2→anηdππ0nexocnhaangpeomlaroidzeedl.
0 0 The solid (dotted) curves show the spin asymmetry A(0 ≤
t ≤ 0.025 GeV2,m) as a function of ηπ0 invariant mass, m
1-LV 0.3 (smoothed with 10 MeV mass resolution).
Ge f0H980L®ΗΠ0
H
cays0.25 a00H980L®Π+Π- configurationisdominatedbythea0(980) f (980)mix-
de 0 − 0
g 0.2 ing then the spin asymmetry of the cross section jumps
kin near the KK¯ thresholds. An example is the reaction
a
bre-0.15 π−p↑ → a00(980)+f0(980) n → a00(980)n → ηπ0n on
spin apolarize(cid:0)dprotontarget. T(cid:1)hecorrespondingdifferential
o cross section has the form
is 0.1
of d3σ 1
a = M 2+ M 2
ectr0.05 dtdmdψ 2π | ++| | +−|
p
s +2 (M(cid:2) M∗ )P cosψ ,
ss ℑ ++ +−
Ma 0 0.94 0.96 0.98 1 1.02 1.04 and the dimensionless normalized spin as(cid:3)ymmetry is
m HGeVL A(t,m) = 2 (M M∗ )/( M 2 + M 2), 1
A(t,m) 1.ℑ3 Fi+g+ure+4−illus|tra+te+s|the|str+o−ng| asy−mme≤-
≤
FIG. 3. Mass spectra in the isospin-violating decays try jump which is straightforward manifestation of the
f0(980)→ηπ0anda00(980)→π+π−,causedbythea00(980)− a0(980) f0(980) mixing amplitude interfering with the
f0(980)mixing. Thesolidanddashedlinesaregenerallysim- isospin p−reserving one in the ρ and π Regge exchange
2
ilar each other. The dotted vertical lines show the locations
model. Details and various variants may be found in
of the K+K− and K0K¯0 thresholds.
Refs. [17, 18].
These effects are still in waiting for their studies.
shows the mass spectra correspond to the integrands in
the above equations.2
IV. THE DECAY f1(1285) →f0(980)π0 →3π
III. POLARIZATION PHENOMENA
Estimated are the contributions of the following
mechanisms responsible for the decay f (1285)
The phase jump (see Fig. 2(b)) suggests the idea to 1
study the a0(980) f (980) mixing in polarization phe- f0(980)π0 π+π−π0 [36]: →
0 − 0 →
nomena [17, 18]. If the process amplitude with the spin
3 Here M+− and M++ arethe s-channel helicity amplitude with
2 Herewe usethe values of the coupling constants of the f0(980) ana without nucleon helicity flip, ψ is the angle between the
and a0(980) resonances with the ππ, KK¯, and ηπ channels ob- normal to the reaction plainformedby the momenta of the π−
0
tainedinRef. [36]fromtheBESIIIdataontheintensitiesofthe and ηπ0 system, and the transverse (to the π− beam axis) po-
f0(980)→a00(980)anda00(980)→f0(980) transitionsmeasured larization of the the proton target, and P is a degree of this
inthereactions (b)and(c)[32]. polarization.
3
: (1) the contribution of the a0(980) f (980) mixing, and, as a result, the inconvenient coupling constants of
f (1285) a (980)π0 0(K+K−− +0 K0K¯0)π0 the scalar mesons with the pseudo-scalar mesons in the
1 0
f (980)π0→ π+π−π0, → → many cases
0
→
: (2) t(hKe+cKon−t+ribKu0tKi¯on0)πo0f thfe0(9tr8a0n)sπi0tionπf+1π(1−2π805,)ar→is- gf204ππ+π− =1.2 GeV2, gf20K4π+K− =5.7 GeV2,
ing due to the pointl→ike decay f1(1→285) KK¯π0, g2 g2
→ a00ηπ0 =1.9 GeV2, a00K+K− =9.9 GeV2.
: (3) the contribution of the transition f (1285) 4π 4π
1
(K∗K¯ + K¯∗K) (K+K− + K0K¯0)π0 →
f0(980)π0 →π+π−→π0, where K∗ =K∗(892), and→ wFoitrhetxhaemKplKe¯,cdhuaentnoelt,htehevewriydtshtroofntghecoau00p(9li8n0g)orfesao00n(a9n80ce)
: (4) the contribution of the transition f (1285) in the ηπ0 mass spectrum turns out to be near 15 MeV.
fa(K0n(d09∗K8K¯0)∗+(→14K3¯π00∗+)K.π)−π0→, wh(eKre+KK0∗−=+KK0∗(018K¯000))π(0or →→κ) (B2)RT(fh1e(1t2h8e5p)o→intfl0ik(9e8d0e)cπa0y→f1π(1+2π8−5)π→0) K=K0¯.π0002g2ives
0 BR(f (1285) KK¯π)
1
→
These mechanisms break the conservation of the isospin
instead of the experimental value
due to the nonzero mass difference of the K+ and K0
mesons. They result in the appearance of the narrow BR(f (1285) f (980)π0 π+π−π0)
1 0
resonance structure in the π+π− mass spectrum in the BR(f→(1285) KK→¯π)
region of the KK¯ thresholds, with the width ≈2mK0 − =0.0133 0.0→10.
2mK+ 8 MeV. The observation of such a structure ±
in expe≈riment is the direct indication on the KK¯ loop The π+π− mass spectrum in the decay f (1285)
1
mechanism of the breaking of the isotopic invariance. (K+K−+K0K¯0)π0 f (980)π0 π+π−π0 lookssim→i-
0
Wepointoutthatexistingdatashouldbemoreprecise, → →
lartothe curvesinFig. 3forthe a (980)-f (980)mixing
0 0
anditisdifficulttoexplainthemusingthesinglespecific
case. However,itisclearthatthepointlikemechanismof
mechanism from those listed above. Taking the decay the decay f (1285) KK¯π cannot by itself provide the
fd1is(c1u2s8s5t)he→gefn0e(r9a8l0a)pπp0ro→achπt+oπt−hπe0deasscrtiphteioenxaomftphlee,KwK¯e cπo+nπs−idπe0ratbrlae1npsirtoiobna.b→ility of the f1(1285)→f0(980)π0 →
loop mechanism of the breaking of isotopic invariance. (3) The isospin-breaking decay f (1285) (K∗K¯ +
1
(1) The matter is that the J/ψ φf0(980) K¯∗K) (K+K− + K0K¯0)π0 f (980)π0→ π+π−π0
φa (980) φηπ0 [32] and χ a→(980)π0 → → 0 →
0 c1 0 is induced by the diagram shown in Fig. 5, because
→f (980)π0 →π+π−π0 [32] decays are →described b→y
0
→
the a0(980) f (980) mixing well enough:
0 − 0 K∗(K¯∗) π0, p
3
BR(J/ψ φf (980) φa0(980) φηπ0)
BR→(J/ψ0 φf0→(980)0 φππ→) f1(1285), p1 K(K¯) π+
→ →
=(0.60 0.20(stat.) 0.12(sys.) 0.26(para.))%
BR(f±(980) KK¯± a0(980)± ηπ0) K¯(K) f0(980), p2
0 → → 0 → ,
≈ BR(f0(980) ππ) π−
→
BR(χc1 →a00(980)π0 →f0(980)π0 →π+π−π0) Fπ+IGπ.−5π.0TvihaetdhieagKra∗mK¯ o+fKt¯h∗eKdeincateyrmf1e(d1i2a8t5e)s→tatefs0.(980)π0 →
BR(χ a0(980)π0 ηπ0π0)
c1 → 0 →
=(0.31 0.16(stat.) 0.14(sys.) 0.03(para.))% the contributions from the K+K− and K0K¯0 pair pro-
± ± ±
BR(a0(980) KK¯ f (980) π+π−) duction are not compensated entirely. The transition
0 → → 0 → . f (1285) (K∗K¯ +K¯∗K) (K+K− +K0K¯0)π0
≈ BR(a00(980)→ηπ0) f01(980)π0→ π+π−π0 gives th→e shape of the π+π− spe→c-
→
Asforthef (1285) f (980)π0 3π decay[31],itsde- trum practically coincides with the corresponding spec-
scriptionreq1uiresth→e“te0rrible”a00→(980)−f0(980)mixing: trum caused by the a00(980)−f0(980) mixing, but its
BR(f (1285) a0(980)π0 f (980)π0 π+π−π0) BR(f1(1285) f0(980)π0 π+π−π0) 0.0255%
1 → 0 → 0 → → → ≈
BR(f (1285) a0(980)π0 ηπ0π0)
1 → 0 → is much less then the experimental value
=(2.5 0.9)%
±
BR(a00(980)→KK¯ →f0(980)→π+π−), BR(f1(1285)→f0(980)π0 →π+π−π0)
≈ BR(a0(980) ηπ0) =(0.30 0.09)%.
0 → ±
4
So, the f (1285) (K∗K¯ + K¯∗K) (K+K− + estimate [36]
1
K0K¯0)π0 f (98→0)π0 π+π−π0 tra→nsition mecha-
0
nism alone→is also insuffic→ient to understand the experi- Γf1(1285)→f0(980)π0→π+π−π0
mental data. = A(2m )22.59 10−6GeV5.
(4) The variant f (1285) (K∗(800)K¯ + | K+ | ×
K¯∗(800)K) (K+K−1+ K0K¯0)π→0 f0(980)π0 Thus its comparison with the data on the decay
π+0π−π0 is →rejected by the shapes →of th0e Kπ an→d f1(1285) π+π−π0 permits one to verify their consis-
KAsK¯fomrafss(1s2p8e5c)tra in(tKhe∗(1d4e3c0a)yK¯f+1(1K2¯8∗5()143→0)KK)K¯π. tweinthcethweitihd→etahoefdtahteaborenatkhinegdoefciasyotfo1p(i1c2i8n5v)ar→ianKceK¯cπauasnedd
(K+K− +1 K0K¯0)π→0 f0(980)π0 π+0π−π0, it pr→o- by the mass difference of K+ and K0 mesons.
0
videstheresultssimila→rtof (1285) →(K∗K¯+K¯∗K)
1
(K+K− + K0K¯0)π0 f (980)π→0 π+π−π0 an→d
0
consequently cannot de→scribe the data→alone. VI. THE DECAY
J/ψ→γη(1405)→γf0(980)π0 →γπ+π−π0
V. THE CONSISTENCY CONDITION According to BESIII [33], the mass and the width of
theη(1405)peakintheπ+π−π0 channelare1409.0 1.7
±
The isospin breaking amplitude MeVand48.3 5.2MeV,respectively,while thebranch-
±
(m) can be expanded near the KK¯ ing ratio is
Mf1(1285)→f0(980)π0
threshold into the series in ρKK¯(m)= 1−4m2K/m2: BR(J/ψ γη(1405) γf0(980)π0 γπ+π−π0)
→ → →
Mf1(1285)→f0(980)π0(m)=gf0K+K−{Ap(m) =(1.50±0.11±0.11)·10−5.
×i[ρK+K−(m)−ρK0K¯0(m)]+B(m)[ρ2K+K−(m) In addition, the BESIII gives the ratio
ρ2 (m)]+O[ρ3 (m) ρ3 (m)]+ .
− K0K¯0 K+K− − K0K¯0 ···} BR(η(1405) f (980)π0 π+π−π0)
0
→ →
With a good accuracy BR(η(1405) a0(980)π0 ηπ0π0)
→ 0 →
=(17.9 4.2)%,
Mf1(1285)→f0(980)π0(m)=gf0K+K−A(m) ±
×i[ρK+K−(m)−ρK0K¯0(m)]. that rules out practically the explanation of the discov-
ered effect by means of the a (980) f (980) mixing.
0 0
The amplitude A(m) contains the information about all −
possible mechanisms of production of the KK¯ system
KwiKt¯hπi.sospin I = 1 in S wave in the process f1(1285)→ K∗(K¯∗) π0, p3
η(1405), p
π+Fπr−omπ0thoenedcaatanoenxttrhacetdtehceayinffo1r(m12a8t5io)n→abfo0u(9t80A)(πm0)→2 1 K(K¯) π+
in the region of the K+K− and K0K¯0 threshold|s, | K¯(K) f0(980), p2
dΓf1(1285)→f0(980)π0→π+π−π0(m) π−
dm
1 FIG. 6. The diagram of the decay η(1405) → (K∗K¯ +
= (m)2
16π|Mf1(1285)→f0(980)π0 | K¯∗K)→KK¯π→π+π−π0. In theregion of theη(1405) res-
p3(m)2m2Γf0→π+π−(m), odniaagnrcaemacllaninltieeromnedthiaetirempaarstsicslheesllisn. Tthheatloios,pinofththeihsytproiatnhgelte-
× π|Df0(m)|2 ical case of the stable K∗ meson the logarithmic singularity
appears in theimaginary part of thetriangle diagram.
where p(m)=[m4 2m2 (m2+m2)+(m2 m2)2]1/2
f1 − f1 π − π
/(2m ). Moreover, the information about A(m)2 at
f1 | | We discuss the possibility of the theoretical explana-
m > 2m can be obtained from the data on the
K tion of the large breaking of isotopic invariance in the
KKKK¯¯πm. aFsosrsipnescttarnacem,etahseuKred+Kin−thspeedctercuamysifn1(t1h2e8d5e)ca→y decay η(1405)→f0(980)π0→π+π−π0 by means of the
anomalous Landau thresholds (the logarithmic triangle
f (1285) K+K−π0 can be represented in the form
1 → singularities), which are in the transition η(1405)
dΓf1(1285)→K+K−π0 π(K+π∗K−¯π+0 K(¯s∗eKe )F→ig.(K6)+,Ka−nd+sKho0wK¯0t)hπa0t→thef0a(9c8co0u)πn0t →→of
dm
= 2πmρK+K−(m)p3(m)|A(m)|2. sthmeoofitnhietes twhiedtlhogoafritthhme iKc ∗si(n8g9u2l)ar(iΓtiKes∗→inKπthe≈a5m0plMiteuVde)
and results in the suppression of the calculated width of
Fvaitlutieng|At(h2emdKa+ta)|2onanddΓfo1b→tKai+nKt−hπe0f/odllmow,ionngeacpapnrofixnidmtahtee 6th−e 8deicnacyomη(p1a4r0i5s)on→wift0h(9th80e)cπa0se→of3ΓπKb∗→yKthπe=fa0ct[o3r5]o.f
5
TheaccountingofthefinitewidthoftheK∗resonance, that reasonably agrees with experiment.
i.e., the averaging of the amplitude over the resonance
Breit–Wigner distribution in accord with the spectral
Conclusion
Ka¨ll´en–Lehmannrepresentationforthepropagatorofthe VIa.
unstable K∗ meson, smoothes the logarithmic singular-
ities of the amplitude and hence makes the compensa- We also analyze the difficulties related with the as-
tion of the contributions of the K∗+K−+K∗−K+ and sumption of the dominance of the η(1405) (K∗K¯ +
K∗0K¯0+K¯∗0K0 intermediate states more strong. This K¯∗K) KK¯π decay mechanism and discu→ss the pos-
→
results in both the diminishing of the calculated width sible dynamics of the decay η(1405) ηππ [35]. The
→
of the decay η(1405) π+π−π0 by a number of times decisive improvement of the experimental data on the
icnonccoemntpraartiisoonnowfitthhethm→eacinaseeffoefctΓKof∗→thKeπis=os0p,inanbdreianktinhge rKesK¯on,aKncπe,sηtπru,catnudreπηπ(1m40a5s/s1s4p7e5c)ttroaKinK¯thπeadnedcaηyππo,fatnhde
in the domain of the π+π− invariant mass between the on the shape of the resonance peaks themselves in the
KK¯ thresholds. KK¯π and ηππ decay channels is necessary for the fur-
ther establishing the η(1405) 3π decay mechanism.
Assuming the dominance of the η(1405) (K∗K¯ + →
K¯∗K) KK¯π0 decay, one obtains → ACKNOWLEDGMENTS
→
ThepresentworkispartiallysupportedbytheRussian
Foundation for Basic Research Grant No. 16-02-00065
BR(J/ψ γη(1405) γf (980)π0 γ3π)
0 and the Presidium of the Russian Academy of Sciences
→ → →
1.12 10−5, project No. 0314-2015-0011.
≈ ·
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