Higher Order Moments of Momentum Spectra

Higher Order Moments ofMomentum SpectraNick Brook Ian SkillicornUniversity of Bristol•Introduction• MLLA predictions• Comparisons with e + e - & DIS data

Fragmentation FunctionsD(ξ,Q,Λ) not calculable in pQCDxp pp= =2p QmaxBUT know how to evolve in Q 2⎛ξ = log⎜1⎝x p⎟⎠⎞For e + e -: Q = √sFor DIS: Q is the virtuality of the exchanged photon andp is measured in the current region of the Breit frame(where p max is Q/2)

The Breit Frame`Brickwall’ framespacea)Electron-positron Annihilatione +e -Z/ γqqDIS in the Breit Framee - e -Z/ γqqqtimePhase space for e + e -annihilation evolveswith Q/2 = √s/2pTb)qe -e +qqqqqe -e -pLCurrent hemisphereof Breit frameevolves as Q/2c)-Q/2pTQ/2pLpT-Q/2 Q/2(1-x)Q/2xpLCurrent region ≡ e + e -annihilationCURRENTTARGET

Moments of ξ dist bnMLLA theory assumesenergy spectra - exptmeasure momentaKhoze, Lupia & Ochssuggested (Phys LettB386 1996 451) the cutoffin evolution can berelated to an effectivemass, m eff , of the hadrons1 1 1( ξ Q,Λ) ∝ exp k − sδ− ( 2 + k)MLLA predictions forthe Q evolution ofmoments (Dokshitzer etal, Int J Mod Phys A71992, 1875.)Fong & Webbersuggested region aroundpeak can be described bya distorted GaussianExtract moments fromfits to MLLA theoreticalspectra and data in aconsistent manner(PLB 479 2000 173 & PLB 497 2001 55)[ ]2 1 3 1 4δ + sδkD , + δwhere8δ =2( ξ −ξ)σ4624

Scaled Momentum Distributions, ξ pFollowing the approach of Khoze et al., can relate a cutoff,Q 0 , in the parton evolution to the mass of hadrons,m eff , and the MLLA spectra to expt. spectra:1Ndnhdξp∝pEhhD(ξ,Q,Λ)whereξ =⎛log⎜⎜⎝Q2eQ−2ξ p 2+ Q0⎞⎟⎟⎠&E2h= ph+Q20

MLLA Spectra•Typical hump back shapewith dependence on energy,Q 0 & ΛIntroduce m eff•Changes in RHS side ofdistorted Gaussian byintroducing m eff term (i.e.momentum getting smaller)•No longer displays usualtruncation at large ξE h = p hE h ≠ p h

MLLA Moments•Marked differencebetween Dokshitzer et al &fit to MLLA spectra(calculating moments overwhole range of spectra givesame answer as analyticcalculation)•Important moments arecalculated/extracted inconsistent manner!!•Different assumption infor small ξ account fordifference between F&Wand Dokshitzer et al ??•As Q↑ all predictionsconvergefit to MLLA spectra; Dokshitzer et alanalytic; Fong & Webber analytic

Moments of Spectra &e + e - DataMoments extracted from fitof distorted Gaussian to dataBUT also to theory•MLLA-0 gives reasonabledescription of mean &skewness•MLLA-0: σ is smaller & moreplatykurtic than data•MLLA-M gives poorerdescription of all varaibles atlow E CM•MLLA-M approaches MLLA-0& data at high E CMMLLA-0 (no m eff term - Q 0 = Λ)--- MLLA-M (with m eff term - m eff = Q 0 = Λ)

Moments of Spectra &DIS DataMoments extracted from fitof distorted Gaussian to data& theory curves•m eff allowed to be freeparameter in fits to data•MLLA-M & MLLA-0 bracketthe data. At high Q bothapproach the data•Fit to DIS data Λ=280 MeV &m eff = 230 MeV. Reasonabledescription of all variablesexcept mean ( and peakposition)ξσsmax= ξ + 1( ( ) )12 21−+ sQ 0 = Λ ≠ m eff

Summary•Care needs to be taken in choosing the range of ξ when comparing data& theory•Wide discrepancy in higher order moments (skewness & kurtosis)between MLLA-M & MLLA-0 and data. Predictions converge as Q↑•Limiting spectra preferred over truncated spectra (Q 0 ≠ Λ) incomparison with data•m eff has a large effect at all but highest Q•MLLA gives reasonable agreement with higher moments of data forfitted values of Λ and m eff