Higher Order Moments of Momentum Spectra
Higher Order Moments of Momentum Spectra
Higher Order Moments of Momentum Spectra
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<strong>Higher</strong> <strong>Order</strong> <strong>Moments</strong> <strong>of</strong><strong>Momentum</strong> <strong>Spectra</strong>Nick Brook Ian SkillicornUniversity <strong>of</strong> 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 <strong>of</strong> the exchanged photon andp is measured in the current region <strong>of</strong> 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 hemisphere<strong>of</strong> Breit frameevolves as Q/2c)-Q/2pTQ/2pLpT-Q/2 Q/2(1-x)Q/2xpLCurrent region ≡ e + e -annihilationCURRENTTARGET
<strong>Moments</strong> <strong>of</strong> ξ dist bnMLLA theory assumesenergy spectra - exptmeasure momentaKhoze, Lupia & Ochssuggested (Phys LettB386 1996 451) the cut<strong>of</strong>fin evolution can berelated to an effectivemass, m eff , <strong>of</strong> the hadrons1 1 1( ξ Q,Λ) ∝ exp k − sδ− ( 2 + k)MLLA predictions forthe Q evolution <strong>of</strong>moments (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 <strong>Momentum</strong> Distributions, ξ pFollowing the approach <strong>of</strong> Khoze et al., can relate a cut<strong>of</strong>f,Q 0 , in the parton evolution to the mass <strong>of</strong> 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 <strong>Spectra</strong>•Typical hump back shapewith dependence on energy,Q 0 & ΛIntroduce m eff•Changes in RHS side <strong>of</strong>distorted Gaussian byintroducing m eff term (i.e.momentum getting smaller)•No longer displays usualtruncation at large ξE h = p hE h ≠ p h
MLLA <strong>Moments</strong>•Marked differencebetween Dokshitzer et al &fit to MLLA spectra(calculating moments overwhole range <strong>of</strong> 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
<strong>Moments</strong> <strong>of</strong> <strong>Spectra</strong> &e + e - Data<strong>Moments</strong> extracted from fit<strong>of</strong> distorted Gaussian to dataBUT also to theory•MLLA-0 gives reasonabledescription <strong>of</strong> mean &skewness•MLLA-0: σ is smaller & moreplatykurtic than data•MLLA-M gives poorerdescription <strong>of</strong> 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)
<strong>Moments</strong> <strong>of</strong> <strong>Spectra</strong> &DIS Data<strong>Moments</strong> extracted from fit<strong>of</strong> 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 <strong>of</strong> 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 <strong>of</strong> 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 <strong>of</strong> data forfitted values <strong>of</strong> Λ and m eff