The Soft-Collinear Effective Field Theory

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1 The Soft-Collinear Effective Field Theory Iain W. Stewart University of Washington Flavor Physics and CP Violation University of Pennsylvania ay Iain Stewart p.1

2 Outline otivation Soft-Collinear Effective Theory Power counting Symmetries and Factorization Factorization for and Sudakov Logarithms Reparameterization Invariance Large Conclusions ased on work with C. auer D. Pirjol versus Large S. Fleming I. Rothstein T. ehen A. anohar. Grinstein Factorization Iain Stewart p.2

3 ! # # # $ # # # otivation To understand hadronic uncertainties we need to separate short ( ) and long ( QCD) distance contributions this is Factorization Effective Field Theory Useful for separating physics at different momentum scales HQET QCD systematic expansion in new symmetries for QCD universal non-perturbative parameters What about processes with energetic particles DIS Drell Yan " QCD? Iain Stewart p.3

4 otivation To understand hadronic uncertainties we need to separate short ( ) and long ( QCD) distance contributions this is Factorization Effective Field Theory Useful for separating physics at different momentum scales HQET QCD systematic expansion in new symmetries for QCD universal non-perturbative parameters For QCD the SCET helps to simplify power counting in make symmetries explicit sum IR logarithms %'& QCD understand factorization in a universal way Iain Stewart p.3

5 ( # 7 # & *? > > C A % Kinematics eg & )+* Pion has 9 ) & ) ; <= then pion constituents! & & Let <= 9! D % 9% Iain Stewart p.4

6 9G N ) % % % N ) R % 9 % " % % ] \$ ^ ] \$ [ Soft-Collinear Effective Theory Introduce fields for infrared degrees of freedom modes collinear soft usoft ) & FE 7 9% 9% %P %P % % auer Fleming Luke.F.P.S..S..P.S. fields HJILK Q K Q " O ILK N ) O K Typically %'& or W SUTV % & Z SUXY and is soft or usoft Offshell modes with are integrated out SCET operators and diagrams are homogeneous in and leading order results Iain Stewart p.

7 ? >? > Collinear Gauge Invariance integrate out offshell quarks u O N ILK Q`_ b Consider matching for heavy-to-light current d c?ba no gluons fhg c? HeILK Iain Stewart p.6

8 ? >? >? > c Q? > Collinear Gauge Invariance integrate out offshell quarks u O N ILK Q`_ b Consider matching for heavy-to-light current d c?ba one-gluon 9 i 7? HJILK O N ILK f g Iain Stewart p.6

9 ? >? > j Collinear Gauge Invariance integrate out offshell quarks u O N ILK Q`_ b Consider matching for heavy-to-light current d c?ba all orders f g c j? HeILK a Wilson line form of the current is protected by Collinear Gauge Invariance Iain Stewart p.6

10 ? >? > j k Collinear Gauge Invariance integrate out offshell quarks u O N ILK Q`_ b Consider matching for heavy-to-light current d c?ba all orders f g c j? HeILK a Wilson line form of the current is protected by Collinear Gauge Invariance also constrains form of hard loop corrections f g c j? HeILK?ml 9 a label operator? l Iain Stewart p.6

11 O? >? j Ž Œ Œ ƒ ƒ q Collinear Gauge Invariance u integrate out offshell quarks N ILK Q`_ > b d c?ba Consider matching for heavy-to-light current a Wilson line f g c j? HeILK all orders form of the current is protected by Collinear Gauge Invariance and the form of the collinear Lagrangian x r t v qz Š hš ˆ Š hš ˆ t v z { ƒ ~ } z { y x qsrutwv npo y t v t v Iain Stewart p.6

12 œ _ š H 9 0 d a * Factorization 9 š H 9 H 9 & DIS žh ] ž Ÿ [ 9 š H 9«š H Hª $ = &?b? ] Politzer-Wise..N.S. Ÿ ±³² [ Iain Stewart p.7

13 œ _ š H 9 0 d a * Factorization 9 š H 9 H 9 & DIS žh ] ž Ÿ [ 9 š H 9«š H Hª $ = &?b? ] Politzer-Wise..N.S. Ÿ ±³² [ CollinsSoperSterman Traditional proofs analyze diagrams Iain Stewart p.7

14 œ _ š H 9 0 d a * µ I O i ¹ ¾ ½ % Factorization 9 š H 9 H 9 & DIS žh ] ž Ÿ [ 9 š H 9«š H Hª $ = &?b? ] Politzer-Wise..N.S. Ÿ ±³² [ In SCET replaced by power counting and operator analysis eg. decoupling of ultrasoft gluons by field redefinitions ]\$ N µ I ILK & O N ILK ]\$ H ILK µ I & HeILK 9 N" > ¾! º¼» 2 µ & I where Â I modes gives soft Wilson line ÁÀ integrating out Iain Stewart p.7

15 ÆÇ ÆÇ Ä Å Ä Ã Ä Ä Æ É Í Î Í Ë Æ Ì Ê Ê Ë É È Æ È Ã È Applications Hard Scattering Å Ç Factorization proofs for classic processes are simplified.f.p.r.s. É Ê É Ê Ñ Ð Ï Å Ê Iain Stewart p.8

16 Ò near endpoint Ù9 ) 0 _! # â ñ ê ð É 9Ø ) Ø ª OÖ R ÓÕÔ c Optical Thm 9 Ý - ÜÛ & Ú q q standard OPE endpoint region resonance region s <= b b p P X <= Z àá SUXY % & usoft collinear 132 ß- Þ ÞÝ For endpoint Î ìø Î ù øæ ó ±öõ ô ã æ ò ñ ð çþè ý ïü í Îpû ìø òú ò ñ ï;ð ìîí ë æçéè ãåä Korchemsky Sterman ( 94) SCET gives direct all orders proof of this result (.P.S.) Iain Stewart p.9

17 Ò near endpoint â ñ ê ð É Î ìø Î ù øæ ó ±³õ ô ã æ ò ñ ð çþè ý ïü í Îpû ìø òú ò ñ ï;ð ìîí ë æç è ãÿä CLEO Data Spectator odel Useful for extracting 40 " and using spectrums 0 Weights 100 ev Neubert igi et al. Leibovich Low Rothstein E (GeV) Iain Stewart p.9

18 * H > = * * Factorization for History Naive Factorization jorken ( 89) Dugan & Grinstein ( 91) Generalized Factorization Politzer & Wise ( 91) eneke et al. ( 00) auer Pirjol I.S. ( 01) & = Color Transparency LEET & [ &? H» Proposal Two loop proof All orders proof 9«ª 9 π <= * & are soft collinear D Iain Stewart p.10

19 Ã Ã Æ Æ Ä Ë Æ Other Applications Reparameterization Invariance small changes in Ì Chay and Kim anohar ehen Pirjol Stewart constrains power suppressed collinear operators Sudakov Logarithms in SCET solving RGE s sums Sudakov double logarithms e.g. Ä Ë and È Ë.F.L. and.f.p.s. auerchiangflemingleibovichlow end of SCET part Iain Stewart p.11

20 !! Large vs. Large Proof Applies to? $! $ (and (and (and ) ) ) agreement is at the 0-30% level these decays also factor for large Iain Stewart p.12

21 d d a a Large vs. Large Factorization Large modes e.g. large strong phase disfavored by? 0? &?? in general gives ) (without $!! and good for small and large! Ligeti Luke Wise Û Iain Stewart p.12

22 a a $ 7 $ ~ Large vs. Large Large Factorization disfavored by in general gives! and good for small and large What about inclusive decays? no dn dx in SV limit modes e.g. large strong phase? d? &!! Û d? (without )? 0 Ligeti Luke Wise auergrinsteinpirjolstewart (in progress) see also Calmet annel Schwarze (CLEO 97) agrees ok for large model dependence from?b¾ "! would be cleaner x Iain Stewart p.12

23 Ã Ã Ã ÆÇ Ã È Conclusions Factorization emerges from an effective theory proofs are simplified Universal same steps for many inclusiveexclusive processes New framework for investigating power corrections Factorization in ÆÇ Ç work in progress Iain Stewart p.13

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