CENTRAL INSTITUTE CF PHYSICS ' ' '""' INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING Cucharett, P.O.Box JliG-6,ROMANIA
department ofa fundamental Phytic*
ÎFÎM ~ FT-2J2-J98Z June The Quark Model Predictions for the Pseudoscalar Meson Decays into Lepton n i o + - +-.,o + - Pairs; » •• e e , n -»• u v , KL •* y y
M.0.SCADR0N ' and M.l/ISINESCU
Abstract The p*eudo*calar decay* P •* i SL ane driven by second order electromagnetic loop diagram* which are rendered finite uiing appropriate o^-ihell P •* yy ^orm factor* determined in the quark model. The. theoretical prediction* are in good agreement -)ith thz experimen tal rate* and branching ratio* and with the. unitarity bound* lor each decay ir •*• e e, r\ •*• \i p and K. •* y y .
* Permanent address : Physics Department, University of Arizona Tucson, Arizona 85721, USA. - 2 -
It ie well known that the pseuxloscalar nes on decay into lepton pairs P •* £• A (e.g. it -*• e e", r\ •*• \i*\x~ and K° -»• y+v") involving "leadirg" second order electromagnetic loop diagrams ae depicted in Flg.l are primaţively divergent. Before these small branching ratios were measured, the scale ambiguity of Fig.l meant that many alternative dynamical models for the two-photon loops should be given serious consideration /1,2/. In the case of the weak transition K° •+ U+P~, this divergent, scale amblgiiity also meant .that a two-W exchange loop as calculated in the GSW electro- weak model /3/ could be taken as a .dynamical candidate /4j5/. Now, however, these three P -* £ £*" branching ratios are
v 0 „ + reasonably well understood experimentally /6/ : B(1T • -*• e e /all) = ' (2,2 *i'*> x 10"7; B(n -• y%~/all) • (2.2 ± 0.8)x IO-5 ; B(K? •• y%~/all) = (9.1 + 1.9)x 10* . Furthermore, the three cor- id • — responding branching ratios scaled to the radiative decays P-+ YY are between approximately 1 and 5 timet the unita-rtty lovrer bounds determined by the absorptive V -*• YY intermediate state, as listed in columns A and C of table I (with associated references /6-14./) • The compatibility between these six entries strongly suggests that the two-rphoton loop graph of Fig.l atone, drives all three P •*• Z I decays.We attempt to show that this is indeed the case. By now the obvi-ous model for the off-shell P •*• YY form factor in Flg.l is the color - quark loop of Fig.2, where it is
AB known /IB/ that the analogous 3-Js ^ triangle graph correctly predicts the observed TT • YY decay rate. For convenience in the subsequent P •+ I i" calculation we employ the pseudoscalar irqq coupling *„_ 4Y-4 in Fig-2 with g__. obeying the Goldberger - Trelman relation at* the quark level, f g„. * m, «hero fi is the 2 — Cp -independent) C0n6tA.tae.nt nonstrange quark mass • % 340 MeV. 2 P v 4 4 Then for SflOr° - YY> = -* ^ (k ,q*> epv(kq) E* e* (2TT) 6 (Pf±y with e ,(kq) H e kaq^, Flg.2 recovers /16/ the ABJ soft pion \iv |i v op result for on-jhell pnotons, F (o,o)= -a/irff , predicting _3 " T T ™ r = 7 6 eV Moreover for vYY 64n" 'FTYY'< ^ ' " off-shell photons 2 2 k ? 0 (but still q = 0), the ITYY PCAC form factor obtained from the quark loop of Fig.2 is
2 2am xdx **TYCk •« '" 0) " TT-» ' „I", "-I (1) ' ' ir o k x(l-x)-m
2 + - The k dependence ln*(l) renders the two-photon P -*• % I graph of Fig.l finite and calculable in all three cases, as is de picted in the quark-triangle, photon-loop graph of Fig.3. Before proceeding on with the cbaputataon, however, we note that just a3 the ABJ quark graph Cup to a factor of g ) was preceded by* the
Stelnberger nucleon triangle calculation
/2/ In a dispersion-theoretic framework. Thus we need only borrow the Pratap - Seith result but with m replaced by m to find for 8 fi " »«.. "V CM>V(Pfl> «« fw * -3 MeV, pj = 1-4.J/.J ,
w • • /W~ 1-6 2m , W IT IT
2 • O2 -Î -7 2l Ţ7^- »e In —j- ^ a.59 x 10 . <2b) ir m
Since (2) involves three rather Involved dispersion theory intermediate state computations, we should test It in a simplified field theory context, at least to check factors of 2 •>» 2,11, etc. We again do this in the q ^ 0 plon PCAC limit, keeping "22 2 only p -» = - • /4 / 0 in the resulting infrared logarithm factor. o + - Then the IT •* e e Feynman amplitude is
4 2 2 P V 2 d k *wvv
2 4»2 I F»ee lp AC * 7TT % Since (3b) has.the sane structure as the Pratap - Smith (PS) ampli tude (2b) and is within the usual PCAC error (now 28% due to the logarithm factor), we henceforth apply the PS formula (2a) to all P •*• £*£~ decays. 2 First of all, employing r Q » (p/4ir)|F | and 11 * e" -7 l p: 17 MeV,(2) predicts T • 3.66 x 10 eV within l£ standard T e- e deviations of experiment and virtually identical with the unitari*:y Halt. In table I, column B, the theoretical branching ratio scaled to IT •*• yy is given .but it is nonetheless important to remember that not only the branching ratio but scale of TT * e+e" is reasonably predicted in the quark model. To extend this analysis to r\ -* p ţi decay, we must first properly account for the different nonstrange and strange quark •ass traversing the quark loop of Fig.3 as manifested by the mixing angle 9 'v» 42°, measured relative to the quark basis /IS/, with Iri > - cos 9 In»- > - sin ţ |n > (This corresponds to a singlet- no s - 5 - o b* octet nixing angle of 8 «v» -13 which la known to consistent with experlnent). The on-shell n •+ YY current algebra quark no del aapli- tude Is then /18/ f = -— (5cose ~/2sine) £-2.30 —- x IO-11 eV_1 (4) nTT 3irf fn r which predicts the observed width nYY = 323 +. 46 eV for t ft - • 1.18 _+ 0.09, the latter also an expected result . Likewise the quark no del prediction for the- n •* y y" amplitude, containing es sentially the analog of the V, f forafactor CD can be extracted froa C2a) and (4), again with B* - 1 - 4a2 / a* and â 3, 340 MeV, a y 510 MeV: 2 -2 a lfg .1 a 2 n «n /Bn i-en 2a .-1 a2 a - /I sin? Csin -) ~rf } (5a) - 2aa a * ' > 1.99 x 10~5. (5b) \ 2 47I p na 253 MftV Given T * (P/ >l nţJVl * P " » W predicts _ o ^ 0.80 x 10 MeV,between the two experlaental values of /13/ I\ • (1.87 + 0.68) x 10"8 MeV and /14/ T - CO.55 • 0.1»)xl0-8 nyy - nwy — MeV, given T ^ £0.85 keV. * P.,to At ^ Finally we turn to the weak decay Ko •* y +y - md appeal to L the w pole graphs of Figs.4a and 4b as realistically describing K? -* YY "d *? "*" M%~ dscays. In the «,YY case, it is known that it la \t Fig.4a iaplles (see, e.g. ref /10/) 0 .-6 _2 < ir 1 Hw \%l > |* (.J - .J) I t^yy / Fm| 1 * 1.6x10" .tYY ' mrY ' n (6) whereaa an Independent current algebra analyals of the Al = 1/2 B system flnda /16/ the experimental scale |< TT°| H^ , JK°>| ^ -6 2 2.0 z 10 a . Corrections In Pig.4a due to T\ and n' poles, with <ţ ^ 4a0 again, decrease the scale of | £h* kaon mass shell, renders the K, -*• ţi ]x" amplitude finite,with id the scale .for 8^ • 1 - 4n2/«? 0 2 , , »• l»„!r-.";° -» -«• i'« hW .? . «* ,i »» /B" K TT IT K K * 1-rp -i m 2 _ In (fin -*-> | % 2.53 x 10 <7) 1-PK 2. a 22 Given p = 225 MeV, C7) predicts T - Cp/4ir)!r , | % 1.14x10 KLyp KLPP * MeV almost identical to experiment I* « CI.16 i 0.24)x 10 HeV. If we also include the n and n' poles in Fig.4b,. the | P_ | seal* is Increased by 8% Coablnlng this with the 11% decrease In |< ir°|oiH_ |Ki_®o > | then overall lowers our predlctloa to T £1.03x10 •> I tnen overall lowers our preaictioa to i MeV, still very close to experiment. V* fonclud* then that «11 three decay rates ir •*••",• t) •*• u y" and E"f -»• y p~ and branching ratios as given ia tabl* I are accurately predicted by th* colored quark aod*l theory. This 2 Includes th* usual color factor of 3 In all caees. Not* that while th* theoretical ir«* rat* is close to th* ualtarlty bound, tbs theore K tical n and Luu rates are respectively )2.4 and 1.4 tla*s greater than th« Clower) ualtarlty bound.Pr**uaablj th* diff*renc* ll*s In 7 the values of m , m , m and m , m relative to the constituent quark mass scale of m 'v- 340 lleV. The effect is therefore kinematic in nature and creates no dynamical problem. Finally we suggest that the second'order weak graphs for K. -*• V V~ are therefore much smaller than -the above first order weak, second order electromagnetic amplitude (T) and have no bearing on the observed rate. The upper and lower bounds of the six-quark model KM weak mixing angles as calculated in ref. /5/ a:s £-asequently far too large and should be drastically reduced. Acknowledgement* One o{ u4 (M.P.S.I g*ate.iully acknowledges the. hospitality of the ln.6titu.te. $oi Phytic* and Hu.cJLe.ax Engineeiing, 'Buckaiett, and ioi the iuppoit 4tom the International PAogiam* Division c& the. U.S. National Science foundation ioi scitntitic exchange voith Romania unde.1 gitttt INT 8026521 and alio ioi ijuppo/it item the U.S. Vepaitment of .Eneigy undei contiact VE-AC 02 - fO ER 10663. - 8 - REFERENCES /l/ S.D.Drell, Nuovo Cira.-, 1_1 (1959) 692 Bf.Barman and D.A.Geffen, Nuovo Cin., 1JJ (1960) 1192 L.M.««hgal, Nuovo Clns., 45_ (1966) 785 B.-L.Young, Phys.Rev., 161 (1967-) 1620 C.Qulgg and J.D.Jackson, UCBL Report No.18487 (unpublished) I.K.Litskevitch and V.A.Franke, Yadern.Fiz., _10 (1970) 815 (Soviet J.Nucl .Phys . , ^9. (1970) 471) ; Yadern.Fiz. 11 (1970) 1078 (Soviet J.Nucl.Phys.. 1_1 (1970) 599) V.a.Dubovik, J.Vaklev and M.Visinescu, Rev.Roum.Phys. 27 (1982) 3 /2/ M.Pratap and J.Smith, Phys.Rev., D5_ (1972) 2020 /3/ S.L.Glashow, Nucl.Phys., 22 (1961) 579 A.Salam in "Elementary Particle Theory" ed. H.Svarthola (Stockholm : Almqvist, Forlag AB) (1968) p.367 S.Weinberg, Phys . 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Branching Onltarity Quark model Exjjwriinental bounds predictions mefiurements J ratios o + - t T * ee +2 4 -7 7 -7 (2.2 _**])xlO ' > 0.47 x 10~ 0.47 x IO 0 ir •+ YY (Ref./7/) (Ref./ll/) , (1.8 + 0.6)x 10"7 (Ref./12/) •n -" y y" (5.8 + 2.1)x IO-5 5 2.47 x 10"5 > 1.07 x 10~ Ref./13/) n * YY (Ref./8/) 5 (1.7 + 0.7)x 10,' CRef./14/) K° . y\" 2: 1.17 x 10~5 1.67 x IO-5 (1.86 £ 0.39)x IO"5 K° - YY. (Ref./9/) (Ref./6/) 2: 0.96 x IO-5 (Ref./10/) FIGURE CAPTIONS Figure 1 General second order electromagnetic decay of a pseudoscalar i«ion Into lepton pairs P •*• I I Figure 2 Quark model loop graph for the Pyy form factor. The dark circles represent dressed constituent quarks with nonetrange and strange mass m, • , respectively. Figure 3 Quark model two-photon loop graph for IT -* e e and n "*• y •»• V - decays. Figure 4 Pion pole amplitude for K? -+ YY decay (Fig.4a) Li and for K° •* y+p~ decay (Fig.4b>. y Table I Unitarity bouads (column A), quark model predictions (column B) and experimental measurements (column C) of the branching ratios B(P * 9+%~/yy ). -p~crf^*l >— ^>»>J <— I* t ^1 ^ + 3lr— —»— (— ,---e^---€K'L f * ----e----C^: