C. Aydın*, M. Bayar and A. H. Yılmaz Physics Department, Karadeniz Technical University, 61080, Trabzon, Turkey.

Transcription

1 γ nd γ ouplin onstnts in three point QCD sum rules C. Aydın*,. Byr nd A. H. Yılmz Physis Deprtment, Krdeniz Tehnil University, 68, Trbzon, Turkey * oskun@ktu.edu.tr hkny@ktu.edu.tr The ouplin onstnt o γ nd γ deys re lulted usin -point QCD sum rules. We estimte the ouplin onstnt nd γ whih re n essentil inredient in the γ nlysis o physil proesses involvin isoslr (98) nd isovetor (98) mesons. Keywords: ouplin onstnt, isovetor (98), isoslr (98), vetor mesons, -point sum rule PACS Nos.:.8.L,..H,..A,

2 I. Introdution One o the very importnt ols o studyin nuler physis is to understnd the behvior o hdrons nd hdroni intertions on the bsis o the untum hromodynmis (QCD). One o the powerul tools or this im is the QCD sum rule invented by Shimn, Vinshtein, nd Zkhrov [] provides us with wy to relte the physil untities o the hdrons to the mtrix elements o the urk-luon omposite opertors by mens o the opertor produt expnsion (OPE) [, ]. The ield o pplition o the sum rules hs been extended remrkbly sine 98 s. The QCD sum rule method hs been utilized to nlyze mny hdroni properties, nd it yields n eetive rmework to investite the hdroni observbles suh s dey onstnts nd orm tors within the nonperturbtive ontributions proprotionl to the urk nd luon ondenstes []. With inresin experimentl inormtion bout the dierent members o the meson spetrum it beomes very importnt to develop onsistent understndin o the observed mesons rom theoretil point o view. For the low-lyin pseudoslr, vetor, nd tensor mesons this hs been done uite suessully within the rmework o the simple urk model ssumin the mesons to be urk-ntiurk ( ) sttes rouped toether into nonets. The dey hnnels o (98 ) nd (98 ) mesons n be nlyzed in the ontext o QCD sum rules. The lvor SU() orms pproximte lobl symmetry o hdron spetrum ordin to whih mesons re lssiied s bound sttes o urk nd ntiurk ( ) nd they re pled in nonet representtions o SU() roup. However, whether liht slr mesons orm slr nonet is still n open uestion. From the experimentl point o view, the isoslr (98) nd isovetor (98 ) re well estblished, but the nture nd the urk substruture o these slr mesons, the uestion whether they re onventionl sttes, K K moleules, or multiurk exoti sttes hs been subjet o ontroversy. Understndin the nture nd the urk substruture o the slr mesons is still n open problem in hdron physis. Rditive trnsitions between pseudoslr (P) nd vetor (V) mesons hve been n importnt subjet in low-enery hdron physis or more thn three dedes. These trnsitions hve been rerded s phenomenoloil urk models, potentil models, b models, nd eetive Lrnin methods [,5]. Amon the hrteristis o the eletromneti

3 intertion proesses VPγ ouplin onstnt plys one o the most importnt roles, sine they determine the strenth o the hdron intertions. Low-enery hdron intertions re overned by nonperturbtive QCD so tht it is very diiult to et the numeril vlues o the ouplin onstnts rom the irst priniples. For tht reson semiphenomenoloil method o QCD sum rules n be used, whih nowdys is the stndrt tool or studyin o vrious hrteristis o hdron intertions. On the other hnd, vetor meson-pseudoslr meson-photon VPγ vertex lso plys role in photoprodution retions o vetor mesons on nuleons. It should be notble tht in these deys (V Pγ) the our-momentum o the pseudoslr meson P is time-like, p >, wheres in the pseudoslr exhne mplitude ontributin to the photoprodution o vetor mesons it is spe-like p <. Thereore, it is o interest to study the eetive ouplin onstnt VPγ rom nother point o view s well. In this work, we studied γ nd γ dey in the rmework o three-point QCD sum rules nd we obtined the ouplin onstnt nd γ. γ II. Clultion Aordin to the enerl strtey o QCD sum rules method, the ouplin onstnts n be lulted by eutin the representtions o suitble orreltor lulted in terms o hdroni nd urk-luon derees o reedom. In order to do this we onsider the ollowin orreltion untion by usin the ppropritely hosen urrents ip'. y ip. x γ ω Π ( p, p )' = d x d y e e < T{ J () J ( x) J ( y )} µ > () We hoose the interpoltin urrent or the nd S mesons s = ( uγ u d d ) J S (or b b b b, J = [ u u d d ] nd or j µ γ µ µ b b b b, J [ u u + d d ] sinθ s osθ, nd = [6] + s γ respetively. -meson onsist o u nd d -urks. J = e ( uγ u) + e ( d d) is the eletromneti urrent with e u nd e d bein the urk hres. µ u µ d γ µ The theoretil prt o the sum rule in terms o the urk-luon derees o reedom or the ouplin onstnt nd γ re lulted by onsiderin the perturbtive γ ontribution nd the power orretions rom opertors o dierent dimensions to the three-

4 point orreltion untion Π. For the perturbtive ontribution we study the lowest order bre-loop dirm. oreover, the power orretions rom the opertors o dierent dimensions < >, < η. G >, nd < ( ) > re onsidered in this work. Sine it is estimted to be neliible or liht urk systems, we did not onsider the luon ondenste ontribution proportionl to < G >. We perorm the lultions o the power orretions in the ixed point ue [7]. We lso work in the limit m = nd in this limit the perturbtive bre-loop dirm does not mke ny ontribution. In t, by onsiderin this limit only opertors o dimensions d= nd d=5 mke ontributions whih re proportionl to < > nd < η. G >, respetively. The relevnt Feynmn dirms or power orretions re iven in Fi. On the other hnd, in order to lulte the phenomenoloil prt o the sum rule in terms o hdroni derees o reedom, double dispersion reltion stisied by the vertex untion Π is onsidered [,, 8]: Π ( p, p )' = π ds ds ( p ( s, s ) s )( p' s ) where we inore possible substrution terms sine they will not mke ny ontributions ter Borel trnsormtion. For our purpose we hoose the vetor nd pseudoslr hnnels nd sturtin this dispersion reltion by the lowest lyin meson sttes in these hnnels the physil prt o the sum rule is obtined s Π < J p, p )' = S >< S( p) J ( p )' >< J > γ ( µ s + ( p m )( p' m S ) ()..., () where the ontributions rom the hiher sttes nd the ontinuum re iven by dots. The overlp mplitudes or vetor nd pseudslr mesons re < J >=, where ε is µ ε µ µ the polriztion vetor o the vetor meson nd < S J s >= S, respetively, where S = or. The mtrix element o the eletromneti urrent is iven by γ e < S( p) J µ ( p )' >= i Sγ K( )( p. u µ u. pµ ) () m where = p p' nd K ( ) is orm tor with K ( ) =. This mtrix element deines the ouplin onstnt SVγ by mens o the eetive Lrnin = e m ( A S (5) S γ µ ν ν β βa ) ν

5 desribin the Sγ vertex [9]. We perorm the lultions o the power orretions in the ixed-point ue x A µ µ =. The enerl orms o the ontributions orrespondin to Feynmn dirms re derived with respet to their dimensions. For the irst dirms in Fi., there re two ontributions with dierent dimensions or d 5 s F (d) = N ψ ψ TrΓ Γ Γ / (6) p - m - m F (5d) N ψ Γ Γ Γ (7) - m - m = () ψ (x) x x Tr + N N F (5d) = 6 m 6 ψψ ψg ' p p ( /) σ ψ ' TrΓ p - m p Γ TrΓ - m - m Γ Γ - m Γ (8) For the seond nd third dirms in Fi. b nd, there re ontributions with dimensions or d 5 s F (5d) = in 9 G σ () TrΓ Γ Γσ (9) k - m - k - m - m k = F (5d) = in 9 G σ () TrΓ Γ Γσ () k - m - k - m - m k = For, i i ( ) γ dey the vertex untions re Γ = γ ν, Γ = ie γ µ, nd Γ = or i Γ = sinθ or. Usin these vertex untions in Es.(6-) we then et the nonvnishin ontributions rom power orretions to the orreltion untion s Π = C N ψψ [ p p m + ( p p + p p 6 p p p p )]( p p ν µ p p ) () 5

6 where C = or nd C = sinθ or. The lowest order perturbtive urk loop dirms do not mke ny ontributions. The struture p p p p ) is hosen to ompere to theoretil nd ( ν µ phenomenoloil prts nd to obtin the ouplin onstnt prt o the invrint untion T or S γ dey s Sγ. We then ind theoretil T 5 ψ ψ + m + m () p p p p 8 p p = C N Ater perormin the double Borel trnsorm with respet to the vribles Q = p nd Q =, nd by onsiderin the ue-invrint struture p p p p ), we ' p' ( ν µ obtin the sum rule or the ouplin onstnts, m γ = 8 8 u ( ) / m / 5 m m m e e e e < uu > sinθ () nd m / m / 5 m = ( + ) m m < > γ eu e e e uu () 8 8 where nd re Borel msses orrespondin to ( ) nd mesons, respetively. III. Numeril Results nd Disussion For the numeril evlution o sum rule we use the vlues < uu >=. GeV, m =.98 GeV, m =. 98 GeV, =.8. GeV [], =.. 5 GeV [], m =.77 GeV. We note tht neletin the eletron mss the e + e dey width o meson is iven s Γ( e + πα e ) =. Then usin the vlue rom the experimentl leptoni dey width Γ( e + e ) = 7.. kev or [, ], we obtin the vlue = (.7.) GeV or the overlp mplitude meson. In order to exmine the 6

7 dependene o nd γ on the Borel msses γ nd, we hoose.,. = nd. GeV or γ nd.7,. 8 nd. 9 GeV or γ. Sine the Borel mss = is n uxiliry prmeter nd the physil untitites should not depend on it, one must look or the reion where nd γ re prtilly independent o γ. We irst determined tht this ondition is stisied in the intervl. GeV. 6 GeV or γ nd. 8 GeV. GeV or γ. The vrition o the ouplin onstnt γ s untion o Borel prmeters while or or dierent vlues o re shown in Fis. nd γ in Fis. nd 5. Exmintion o these iures show tht the sum rule is rther stble with these resonble vritions o = onstnt o nd. We then hoose the middle vlue. GeV or the Borel prmeter in its intervl o vrition nd obtin the ouplin s between γ =.96. nd γ = nd lso or γ γ we ind the ouplin onstnt or the middle vlue. GeV t θ = ' s γ = between = nd =.., where only the error risin rom the γ γ numeril nlysis o the sum rule is onsidered. When we use the mixin nle or θ = 68' [] we hve γ s..97 γ.7. 6 t =. GeV. In the previous work [] ouplin onstnts o nd γ were ound s γ = nd γ = in the rmework o liht-one QCD sum rules. The ouplin onstnt γ ws lso lulted [] s.. 5 nd.. in QCD sum rules. γ Aknowledments This work prtly supported by the Reserh Fund o Krdeniz Tehnil University, under rnt ontt no... 7

8 Reerenes [].A. Shimn, A.I. Vinshtein nd V.I. Zkhrov, Nul. Phys. B 7, 85 (979); Nul.Phys. B 7, 8 (979). [] L.J. Reinders, H.R. Rubinstein, nd S. Yzki, Phys.Rep. 7, (985). [] P. Colnelo nd A. Khodjmirin A, in Boris Ioe Festshrit, At the Frontier o Prtile Physis, Hndbook o the QCD, edited by. Shimn World Sientii, Sinpore. [] A. Göklp nd O. Yılmz, Eur. Phys. J. C, 7 (). [5] P.J. O Donnell, Rev.od.Phys. 5, 67 (98). [6] A. Göklp, Y. Srç, nd O. Yılmz, Phys.Lett. B 69, 9 (5) (nd reerenes therein). [7] A.V. Smil, Sov.J.Nul.Phys. 5, 7 (98). [8] L.J. Reinders, S. Yzki, nd H.R. Rubinstein, Nul.Phys. B96, 5 (98). [9] A.I. Titov, T-S.H. Lee, H. Toki, nd O. Streltsov, Phys.Rev. C 6, 55 (999). [] A. Göklp nd O. Yılmz, Eur. Phys. J C, (). [] F. De Fzio nd.r. Penninton, Phys.Lett. B 5,5 (). [] A. Göklp nd O. Yılmz, Phys.Rev. D 6, (); Phys.Lett. B 55, 7 (). [] C. Aydın,. Byr, nd A.H. Yılmz, hep-ph/67. [] V.V. Anisovih, hep-ph/8; Phys. Usp. 7, 5 (). 8

9 Fiure. Feynmn dirms or the Sγ -vertex: ) bre loop dirm, b) d= opertor orretions, nd ) d=5 opertor orretions. The dotted lines denote luons. 9

10 ,,8,6, Μ =. GeV Μ =. GeV Μ =. GeV, γ,,8,6,,,,,,,,,5,6 (GeV ) Fiure. The ouplin onstnt γ s untion o the Borel prmeter or dierent vlues o.

11 ,,5 Μ =.7 GeV Μ =.8 GeV Μ =.9 GeV, γ,5,,5,,8,9,,, (GeV ) Fiure. The ouplin onstnt γ s untion o the Borel prmeter or dierent vlues o t θ = '.

12 γ,8,, (GeV ),6,,,6 (GeV ) Fiure. Couplin γ s untion o the Borel prmeters or dierent vlues o.

13 5 γ, (GeV ),6,,8,,,,6 (GeV ),8, Fiure 5. Couplin γ s untion o the Borel prmeters or dierent vlues o.