The Renormalization Scale Problem
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1 The Renormliztion Sle Prolem How does one set sle Q? with M. Binger LoopFest Stn Brodsky SLAC
2 Eletron-Eletron Sttering in QED No renormliztion sle miguity Two seprte physil sles. Guge Invrint. Dressed photon propgtor Sums ll vuum polriztion non-zero et terms into running oupling. p p t If one hooses different sle one must sum n infinite numer of grphs -- ut then reover sme result Numer of tive leptons orretly set Anlyti: reprodues orret ehvior t lepton mss thresholds LoopFest 006 Stn Brodsky SLAC
3 e " e " # $ $ es s µ s es Sle of α QED µ unique The QED Effetive Chrge Im s ' Im & s e %% Complex Anlyti through mss thresholds Distinguishes etween timelike nd spelike moment Anlytiity essentil: See C. Berger nd L. Dixon tlks 9 LoopFest Stn Brodsky SLAC
4 The Renormliztion Sle Prolem No renormliztion sle miguity in QED Gell Mnn-Low-Dyson QED Coupling defined from physil oservle; Sums ll Vuum Polriztion Contriutions Renormliztion Sle in QED sheme: Identil to Photon Virtulity Anlyti: Reprodues lepton-pir thresholds Exmples: muoni toms g- Lm Shift Time-like nd Spe-like QED Coupling relted y nlytiity Uses Dressed Skeleton Expnsion M. Binger sj LoopFest Stn Brodsky SLAC
5 Lessons from QED : Summry Effetive ouplings re omplex nlyti funtions with the orret threshold struture expeted from unitrity Multiple renormliztion sles pper The sles re unmiguous sine they re physil kinemti invrints Optiml improvement of perturtion theory LoopFest Stn Brodsky SLAC
6 Fetures of BLM Sle Setting On The Elimintion Of Sle Amiguities In Perturtive Quntum Chromodynmis. Lepge Mkenzie sj Phys.Rev.D8:81983 All terms ssoited with nonzero et funtion summed into running oupling Resulting series identil to onforml series Renormlon n growth of PQCD oeffiients from et funtion eliminted In generl BLM sle depends on ll invrints LoopFest Stn Brodsky SLAC
7 BLM Sle Setting Use n f dependene t NLO to identify A VP Conforml Coeffiient LoopFest Use skeleton expnsion Grdi Rthsmn sj Stn Brodsky SLAC
8 LoopFest Stn Brodsky SLAC
9 . BLM sles for DIS moments LoopFest Stn Brodsky SLAC
10 Three-Jet Rte Krmer & Lmpe Other Jet Oservles: LoopFest Rthsmn Stn Brodsky SLAC
11 Applition of BLM to Multi-Sle Threshold Prodution Hong Kuhn Tuener SJB Phys.Lett.B359: LoopFest Stn Brodsky SLAC
12 Fetures of BLM Sle Setting All terms ssoited with nonzero et funtion summed into running oupling Conforml series preserved BLM Sle Q* sets the numer of tive flvors Corret nlyti dependene in the qurk mss Only n f dependene required to determine renormliztion sle t NLO Result is sheme independent: Q* hs extly the orret dependene to ompenste for hnge of sheme Corret Aelin limit LoopFest Stn Brodsky SLAC
13 lim N C 0 t fixed α = C F α s n l = n F /C F QCD Aelin Guge Theory Anlyti Feture of SUN Guge Theory Huet sj LoopFest Stn Brodsky SLAC
14 Relte Oservles to Eh Other Eliminte intermedite sheme No sle miguity Trnsitive Commensurte Sle Reltions Exmple: Generlized Crewther Reltion LoopFest Stn Brodsky SLAC
15 LoopFest 006 Apply BLM Eliminte MSr Find Amzing Simplifition 15 Stn Brodsky SLAC
16 Geometri Series in Conforml QCD Generlized Crewther Reltion dd Light-y-Light Lu Ktev Gddze Sj LoopFest Stn Brodsky SLAC
17 Generlized Crewther Reltion [1 + α Rs π ][1 α g 1 q π ] = 1 s 0.5Q Conforml reltion true to ll orders in perturtion theory LoopFest Stn Brodsky SLAC
18 LoopFest Stn Brodsky SLAC
19 Leding Order Commensurte Sles p Trnslte etween shemes t LO LoopFest Stn Brodsky SLAC
20 Unifition in Physil Shemes * i Q0 i Q % 1$ & ˆ ˆ i Q + & i Q0 & ˆ * i % ' p i Q i s p p 4 * i=13 log-like funtion: L PHYSICAL RENORMALIZATION SCHEMES AND GRAND UNIFICATION M.B. nd Stnley J. Brodsky. Phys.Rev.D69: p - s p log e p $ Q / mp L Q / m $ ###" - p % 8/3 5/3 40/1 For spin sp = 0 nd 1 Elegnt nd nturl formlism for ll threshold effets LoopFest Stn Brodsky SLAC
21 The Pinh Tehnique Cornwll Ppvssiliou q V selfenergylike projetion q # V p k " S 1 p S 1 k selfenergylike projetion Guge-dependent PT = + + Guge-invrint gluon self-energy nturl generliztion of QED hrge 13 LoopFest Stn Brodsky SLAC
22 " 1 1 " 1 " 1 3 LoopFest Binger sj Stn Brodsky SLAC
23 Anlytiity nd Mss Thresholds MS does not hve utomti deoupling of hevy prtiles Must define set of shemes in eh desert region nd mth # f s M Q " # f 1 s M The oupling hs disontinuous derivtive t the mthing point At higher orders the oupling itself eomes disontinuous Does not distinguish etween spelike nd timelike moment Q AN ANALYTIC EXTENSION OF THE MS-BAR RENORMALIZATION SCHEME S. Brodsky M. Gill M. Melles J. Rthsmn. Phys.Rev.D58: LoopFest Stn Brodsky SLAC
24 Unifition in Physil Shemes Smooth nlyti threshold ehvior with utomti deoupling More diretly reflets the unifition of the fores Higher unifition sle thn usul LoopFest Stn Brodsky SLAC
25 BLM nd Non-Aelin QCD Generl Struture of the Three-Gluon Vertex THE FORM-FACTORS OF THE GAUGE-INVARIANT THREE-GLUON VERTEX M.B. nd Stnley J. Brodsky. hep-ph/ Sumitted to PRD Binger sj p 1 ˆ " 1 3 $ 1 Full lultion generl msses spin p 3 index tensor " ˆ uilt out of nd 1 # 3 with p % p % p $ p 3 3 g p1 p p3 14 sis tensors nd form ftors LoopFest Stn Brodsky SLAC
26 The Guge Invrint Three Gluon Vertex Cornwll nd Ppvssiliou performed the PT onstrution : The pinhed prts re dded to the regulr 3 gluon vertex PT = + pinhed prts guge invrint guge dependent Lter shown to = BFMFG Integrls were not evluted 1 LoopFest Stn Brodsky SLAC
27 Form Ftors : Supersymmetri $ QG Reltions Mssless.ut ertin liner sums re simple : d " F # FQ F G 0 for 7 of the 13 FF s in physil sis Simple N=1 SUSY ontriution in d=4 F 4 F 10 " d F % 0 For ll FF s G Q S N=4 SUSY in d=4 gives 0 These re off-shell generliztions of reltions found in SUSY sttering mplitudes y Z. Bern L.J. Dixon D.C. Dunr nd D.A. Kosower NPB LoopFest Stn Brodsky SLAC
28 Summry of Supersymmetri Reltions Mssless Mssive F G # Q S 4 F # 10 " d F 0 F MG # MQ MS 4 F # 9 " d F 0 % QG d " F $ FQ # F G % MQG d " 1 F $ FMQ # F MG = simple = simple LoopFest Stn Brodsky SLAC
29 Multi-sle Renormliztion of the Three-Gluon Vertex p 1 guge-invrint suset of rd. or. p 3 p ~ 3 g p1 p p g p 1 oupling t eh vertex sor the rd. or. g g p p 3 36 LoopFest Stn Brodsky SLAC
30 Stn Brodsky SLAC LoopFest Sle Effetive Chrge " 4 ~ ~ g # First suggested y H.J. Lu $ % & ' + *** + -. / " " ~ 1 0 L re ~ 1 ~ L L + - / " " L = 3-sle log-like funtion 0 0 L = log 38
31 Stn Brodsky SLAC LoopFest Sle Effetive Sle " Im log L i Q L eff # $ Governs strength of the three-gluon vertex % & 4 1 ~ 1 ~ L L ' # * + + ~ ˆ Generliztion of BLM Sle to 3-Gluon Vertex
32 Stn Brodsky SLAC LoopFest Properties of the Effetive Sle Q Q eff eff " Q Q eff eff # # # # " Q eff " 5.54 Q eff $ for 3.08 Q eff %% $ for.8 Q eff %% $ for.8 Q eff %% $ 41 Surprising dependene on Invrints
33 The Effetive Sle 10 GeV 10 GeV p " Q eff Q eff # 10 GeV # 10 GeV p " Q eff 10 GeV p p " Q eff # 10 GeV p p " 4 LoopFest Stn Brodsky SLAC
34 Stn Brodsky SLAC LoopFest Effetive Numer of Flvors " # $ % & ' " # $ % & log M M M L M d d M M M N MQ F " # $ $ % & ' ' ' M Q M Q M Q N F 3 5/ 1/ 1 1 log e Q M M Q L M d d M Q n f * " # $ $ % & ' " # $ $ % & 51
35 Hevy Qurk Hdro-prodution proton proton jet jet p T p T = 0 = 0 C Q C Q Preliminry lultion using mssless results for tree level form ftor Very low effetive sle muh lrger ross setion thn MS with sle " R M or QQ M Q where = + + rossed Future : repet nlysis using the full mssdependent results nd inlude ll form ftors Expet tht this pproh ounts for most of the one-loop orretions LoopFest 006 Stn Brodsky SLAC 35 5
36 Future Diretions Guge-invrint four gluon vertex PT L p1 p p3 4 4 p Q4 eff p p1 p p3 4 Hundreds of form ftors LoopFest Stn Brodsky SLAC
37 Summry nd Future Multi-sle nlyti renormliztion sed on physil guge-invrint Green s funtions Optiml improvement of perturtion theory with no sle-miguity sine physil kinemti invrints re the rguments of the multi-sle ouplings LoopFest Stn Brodsky SLAC
38 Conventionl renormliztion sle-setting method: Guess ritrry renormliztion sle nd tke ritrry rnge. Wrong for QED nd Preision Eletrowek. Predition depends on hoie of renormliztion sheme Vrition of result with respet to renormliztion sle only sensitive to nononforml terms; no informtion on genuine onforml higher order terms Conventionl proedure hs no sientifi sis. FAC nd PMS give unphysil results. Renormliztion sle not ritrry: Anlyti onstrint from flvor thresholds LoopFest Stn Brodsky SLAC
39 Use Physil Sheme to Chrterize QCD Coupling Use Oservle to define QCD oupling or Pinh Sheme Anlyti: Smooth ehvior s one rosses new qurk threshold New perspetive on grnd unifition Binger Sj LoopFest Stn Brodsky SLAC
40 Ftoriztion sle µ ftoriztion µ renormliztion Aritrry seprtion of soft nd hrd physis Dependene on ftoriztion sle not ssoited with et funtion - present even in onforml theory Keep ftoriztion sle seprte from renormliztion sle do dµ ftoriztion = 0 Residul dependene when one works in fixed order in perturtion theory. LoopFest Stn Brodsky SLAC
41 Use BLM Stisfies Trnsitivity ll spets of Renormliztion Group; sheme independent Anlyti t Flvor Thresholds Preserves Underlying Conforml Templte Physil Interprettion of Sles; Multiple Sles Corret Aelin Limit NC =0 Elimintes unneessry soure of impreision of PQCD preditions Commensurte Sle Reltions: Fundmentl Tests of QCD free of renormliztion sle nd sheme miguities BLM used in mny pplitions QED LGTH BFKL... LoopFest Stn Brodsky SLAC
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