Proposal No QWORK

Transcription

1 Sction 1: Quantum Chromodynamics at WORK Introduction to partons and Parton Distribution Functions Quantum Chromodynamics (QCD) plays a cntral rol in this proposal. It is wll-stablishd ortical framwork for strong nuclar forc which binds togr quarks and gluons into protons and nutrons which in turn constitut building blocks of atomic nucli. Th ory of QCD dscribs intractions among quarks anti-quarks and gluons carrying colour chargs collctivly rfrrd to as partons. Bcaus nrgy grows with sparation btwn colour chargs quarks and gluons cannot xist in isolation but only in colour nutral combinations. This is known as confinmnt. Th simplst nontrivial combinations ar quark anti-quark bound stats (msons) and thr-quark bound stats (baryons). Th collctiv nam for strongly intracting particls mad up of quarks and gluons is hadrons. Evn if quarks and anti-quarks hav only tiny masss that ar of ordr of twnty tims that of lctron bound stats bcom vry massiv in comparison. Th proton is almost two thousand tims havir than lctron. This is consqunc of strong binding. Th siz of hadrons is confinmnt scal about 1 fm or m. At distancs smallr than confinmnt scal quarks and gluons in ssnc start bhaving as fr particls (known as asymptotic frdom Nobl Priz 2004). Basic kywords: Hadrons: strongly intracting particls lik protons and nutrons built from partons. Partons: quarks and gluons fundamntal particls intracting via ir colour chargs. In collisions btwn particls at high nrgis on probs distancs of ordr of corrsponding quantum mchanical wavlngth which in collidrs with nrgis considrably abov GV-scal is much shortr than confinmnt scal and on can dscrib scattring dirctly in trms of collisions btwn (quasi-fr) quarks and gluons. This dscription uss stablishd prturbativ quantum fild ortical mthods and allows comparison of cross sctions (counting rats) for various scattring procsss at various nrgis. To account for initial and final stat hadrons for xampl in proton-proton scattring at Larg Hadron Collidr (LHC) on in ssnc nds to know probability of finding quarks and gluons insid protons. Ths probabilitis ar known as Parton Distribution Functions (PDF s) functions f h i (x) with x bing parton s momntum as fraction of momntum of original hadron (a fraction which must li btwn 0 and 1). Thr ar such functions for any hadron h and any kind of parton i (i can b a quark or anti-quark of any flavour or a gluon). In a similar way on nds to know how many and which hadrons h a particular parton i can produc in final stat. This is dscribd by Parton Fragmntation Functions (PFF s) functions D i h (z); hr z is hadron s momntum fraction. Pictorial dscription of a high-nrgy collision with (considr figur from lft to right) two colliding hadrons (thick lins on lft) producing partons with probabilitis dscribd by Parton Distribution Functions (PDF s f). Ths partons collid with ach or a hard procss that can b calculatd in QCD and finally y fragmnt into jts of hadrons in final stat (black lins on right) dscribd by Parton Fragmntation Functions (PFF s D). Th intuitiv confirmation for abov pictur of scattring procss is apparanc of jts bunchs of hadrons all moving roughly paralll to ach or. In a worldwid ffort ovr last fw dcnnia this intuitiv dscription of PDFs and PFFs has rigorously bn incorporatd in QCD framwork with imprssiv prcision rsults. Th functions hav bn shown to b univrsal apparing in factorizd xprssions. Thy can b xtndd to includ additional dgrs of frdom namly spins of partons as wll as spins of hadrons dscribing transfr of polarization btwn hadrons and partons (spinspin corrlations). Brakthrough from intgratd PDF s to transvrs momntum dpndnt (TMD) PDF s A brakthrough in which my group playd initiating rol (during scond half of nintis [1]) and to which my studnts and myslf hav mad sminal contributions sinc n is considration of rol 1

2 of transvrs momnta in PDF s and PFF s which mans looking at f i (xk T ) whr x and k T charactriz parton s momntum in a hadron with momntum P ( parton momntum is writtn as k = x P + k T ) x bing fraction of hadron s momntum k T (transvrs momntum) componnt orthogonal to this momntum. This may at first sight sm vry trivial but problm is that transvrs momntum is small (hundrds of MV scal) quantum mchanically (through k ~ h/λ) corrsponding to wavlngths around confinmnt scal. In that domain forcs btwn quarks and gluons ar larg prohibiting us of (prturbativ) QCD. Prturbativ mthods in QCD can only dal with functions f i (x) intgratd ovr k T or y can b and hav bn usd in calculations that study limit in which transvrs momntum bcoms vry larg. In ordr to incorporat small transvrs momnta on nds transvrs momntum dpndnt (TMD) functions f i (xk T ). Ths ar a st of nw functions which constitut part of complx structur of hadrons and which hav spcific intrinsic k T -dpndnc. Our brakthrough in introducing s nw TMD functions howvr cam whn w found that y can incorporat spcific angular corrlations btwn (transvrs) momntum and spin of quarks. Som of s corrlations had bn studid bfor ors wr nw. Stting up a systmatic tratmnt w introducd tru corrlations that disappar upon intgration and ar absnt at high k T. Morovr it turnd out to b vry important to charactriz natur of corrlation functions according to ir bhaviour undr timrvrsal symmtry (T) with spcial focus on T-odd corrlation functions. Bcaus QCD rspcts tim rvrsal symmtry r is a uniqu xprimntal signatur for s T-odd corrlations. Thy show up in singl spin asymmtris (SSA) non-vanishing diffrncs btwn cross sctions of procsss in which spin of only on hadron is rvrsd. This singl spin ffct is also rflctd in T-odd corrlation functions. Instad of usual spin transfr btwn hadrons and partons y corrlat transvrs spin with a particular transvrs momntum configuration (momntum-spin corrlations). Th impact of concpt of transvrs momntum dpndnt PDFs on particl physics Two spcial (T-odd) transvrs momntum dpndnt (TMD) PDFs hav had major impact: on function corrlats transvrs momntum of quarks with transvrs spin of hadron it blongs to (Sivrs function) and a scond function dscribs a spcific corrlation btwn transvrs polarization and momntum of quarks in an unpolarizd hadron (Bor-Muldrs function). Both of s TMD functions hav gnratd trmndous ortical and xprimntal activity in last tn yars. On ortical sid y shd nw light on spin structur of hadrons. Our group among svral or groups in world ar working on ortical aspcts of s nw functions. On xprimntal sid our initiativs hav givn an normous boost to fild of transvrs spin physics for xampl RHIC Spin Physics program at Brookhavn National Laboratory (.S.A.). Also at DESY (Hamburg) and CERN (Gnva) xprimnts on SSA hav bn and ar bing prformd. If on sarchs using for instanc Googl with combination Bor-Muldrs on gts of ordr of tn thousand hits indicating that this work has a wid impact. Th nxt stp forward ambition of this ERC proposal Having stablishd potntial of TMD PDFs tim has com to tak this concpt a crucial stp forward. I want to brak with rstrictions of collinar approximation for partons in high-nrgy procsss and dvlop full QCD dynamics undrlying novl corrlations and mak m into workabl tools. Ths tools will nabl a full manipulation of spins and momnta of partons for undrstanding xprimntal rsults at all frontirs nrgy and prcision. If succssful rsults of projct proposd hr will crat a nw lvl of undrstanding in high-nrgy physics and nuclar physics providing unifying links btwn modls and computational tools that ar currntly disjoint. Th timly dvlopmnt of this nxt-gnration ortical toolst idally coincids with running of LHC and fasibility studis of futur ddicatd xprimntal facilitis such as Elctron-Ion Collidr (EIC) in.s. and Larg Hadron Elctron Collidr (LHC) in Europ so that y can b combind towards ffctivly rvaling intrplay of mchanisms in and byond standard modl. It is byond qustion that issus in our proposal will nd to b addrssd in valuating xprimntal rsults xpctd from s facilitis. Objctivs (I) Th first objctiv is to rach sam lvl of sophistication for TMD distribution and fragmntation functions as that for collinar approach in which no TMD corrlations ar considrd. This rquirs 2

3 propr idntification of rlvant quantum filds of QCD for instanc through idntification of k T - wightd obsrvabls as xpctation valus of spcific combinations of quark and gluon filds. It is ssntial that w clarly distinguish tratmnt within rigorous QCD framwork propr idntification of gaug-invariant matrix lmnts and rlvant quark and gluon oprators from or oftn intuitivly appaling approachs that ar basd on modl assumptions for nonprturbativ (confining) aspcts of QCD. This first objctiv gos far byond a mr xtnsion of work that has bn don so far [2]. It rquirs a nw rsarch lin aiming for a full undrstanding of quark-gluon dynamics that nds to b accountd for. In tratmnt all aspcts of QCD bing a non-ablian gaug fild ory play a rol and on nds to combin prturbativ and nonprturbativ aspcts. (II) Th scond objctiv is xploitation of corrlations as tools in high-nrgy scattring procsss. Following ground-braking work outlind in prvious paragraph studis on TMD corrlations ar now includd worldwid in rsarch programms of many xisting or futur facilitis that hav programms on hadron physics ( RHIC-Spin programm at Brookhavn National Laboratory and upgrad of Jffrson Laboratory in.s.a. COMPAS xprimnt at CERN plans for FAIR at GSI in Darmstadt J-PARC programm at KEK in Japan). Ongoing xprimnts confirm apparanc of novl phnomna such as spcific singl spin asymmtris. Although polarization is usful TMD corrlations also could b mployd in ddicatd LHC xprimnts. Thy may b usful to invstigat Higgs sctor through spcific ffcts in final stat. Furrmor TMD functions ar going to play a rol in or long-trm plans among m ambitious nw larg scal facilitis such as proposd LHC in Europ or EIC projct in S (w will giv dtails blow). sing rsults of invstigations that ar part of first objctiv it is possibl to critically assss which novl aspcts of hadron structur can b addrssd in such futur facilitis. Highlighting objctivs: I I want to grasp fundamntal novl aspcts of quark and gluon dynamics that ar ndd for TMD corrlations in particular T-odd ons and giv a nw maning to concpt of parton in high nrgy collisions. II I want to critically assss which TMD corrlations can rliably b usd in tagging of vry spcific partonic initial stats or in analysing spcific asymmtris in partonic final stats. Knowing ffort that ovr dcads wnt into stablishing collinar approach s objctivs ar crtainly vry ambitious. Incorporating small k T from start avoids problms (collinar singularitis) but dosn t com for fr. I am confidnt that by a combination of xisting worldwid knowldg bas for collinar tratmnt and our xprtis and knowldg bas on TMD functions I can dvlop a succssful nw rsarch lin with a tam of ddicatd Ph.D. studnts postdocs and visitors. I am in a uniqu position hr not only bcaus of xprtis but also bcaus of xcllnt connctions with physicists activ in fild of prturbativ QCD as wll as with physicist working in QCD phnomnology. And vn if w do not solv all issus initiativ will gnrat nw most probably unxpctd brakthroughs and will lad to nw ways of rigorously mploying QCD in hard scattring procsss byond collinar approach. Mthodology and rsourcs Th scal ambition and impact of projct proposd hr is largr than national individual programms allow and in lin with high-risk/high-impact charactr of ERC Advancd Grant schm. Th chosn mthodology is such that it in a natural way is dividd into parts that by mslvs may yild important rsults. Th ovrall strngth and chancs of producing ground-braking rsults howvr li in cohrnc of diffrnt aspcts in full proposal which will run for fiv yars as major projct in my group. Within proposal I nvision following intrrlatd topics: (A) Fundamntal studis of TMD functions (towards objctiv I): Whn on intgrats ovr transvrs momnta ( collinar approach) transition hadron-to-parton and parton-to-hadron involvs gluon dynamics but accounting for s ffcts is rlativly straightforward in form of quantum mchanical phass. In collinar cas only on dirction is probd and phass do not affct obsrvabls i.. y do not lad to intrfrnc; on can intrprt distribution and fragmntation functions as probabilitis and dcay functions rspctivly. Including transvrs momnta gluon dynamics producs phass that do affct obsrvabls. To b prcis y track flow of colour charg 3

4 in high-nrgy procss and can hav obsrvabl consquncs. Tchnically on ncountrs basic non-local contributions that go byond usual standard oprator product xpansion. In last yar T.C. Rogrs and M. Aybat (postdocs in Amstrdam) hav obtaind som intrsting nw rsults [3] that constitut a first stp to calculat scal dpndnc (volution quations) for TMD functions combining aspcts of TMD physics and collinar approachs ( Collins-Sopr-Strman formalism). Th dpndnc on colour flow can b compard with famous Aharonov-Bohm phas in quantum lctrodynamics (QED). Th phas of lctron bcoms visibl in intrfrnc xprimnts. If an lctron can travl to a scrn via two diffrnt paths passing two sids of a currnt-carrying long solnoid (s lft half of figur abov) an intrfrnc pattrn mrgs vn if it only passs through spac whr r ar no lctromagntic filds. Similarly a high-nrgy scattring procss is a suddn procss (s right half of figur abov) in which a parton is rmovd from a hadron. If two hadrons ar involvd phass in wav functions of colourd rmnants produc physical ffcts which ar charactrizd as T-odd (odd undr tim rvrsal). Thy show up as non-vanishing singl spin asymmtris combind with azimuthal asymmtris of producd particls. It is on of ffcts ncodd in TMD distribution and fragmntation functions. As mphasizd alrady it is rigorous mbdding of potntially vry rich TMD phnomnology as portabl ffcts in QCD framwork (stablish factorisation) that is bing pursud. I am confidnt that such a dscription can b formulatd but it constituts a nw rsarch lin rquiring combination of our strngth with that of xprts on volution of PDFs. (B) Phnomnology of TMD s (towards objctiv II): High-nrgy scattring procsss hav bn and ar bing studid at many acclrators around world. Highr nrgis allow to prob vr smallr distancs. Dtction of spcific particls allows focus on gluons or on various flavours of quarks (up down strang charm bottom and top). Polarization of bams and targts and polarimtry in final stat allows to compar diffrnt quantum stats of particls involvd hnc incrasing our undrstanding of dynamics of scattring procsss and nhancing our knowldg of dtaild innr structur of proton. As such TMD s clarly play a dual rol. Firstly functions mslvs ncod many aspcts of structur of proton which will b challnging input for lattic gaug calculations. Scondly ir undrstanding within QCD framwork maks m into tools for dtction of physics byond Standard Modl. In particular signs of asymmtris in various procsss dpnd on colour flow in hard part of procss. This maks it possibl to focus on particular final stats containing gluon jts or havy quark pairs or no colour at all. In particular study of TMD s for gluon distributions is far lss dvlopd as compard to that for quark distributions whil gluon distributions ar much mor important for applications at highst nrgis. (C) Involvmnt in long-rang planning: Th dual rol of TMD s in nhancing our undrstanding of QCD as a fundamntal cornrston of Standard Modl and providing tools in study of high nrgy scattring procsss givs m a wid applicability in QCD-rlatd invstigations. For instanc I xpct m to hav impact in dfining and clarifying physics cass for for instanc Elctron-Ion Collidr (EIC) in.s. or Larg Hadron Elctron Collidr (LHC) in Europ. I intnd to activly participat in such studis. 4

5 Larg Hadron Elctron Collidr (LHC) Th LHC is a proposd colliding bam facility at CERN which will xploit nw world of nrgy and intnsity providd by LHC for lpton-nuclon scattring. An xisting 7 TV LHC proton or havy ion bam will collid with a nw lctron bam running simultanously with proton-proton or havy ion collisions at LHC. It will push frontir rachd at HERA acclrator at DESY. Accss to vry low-x valus which ar linkd to study of transvrs momnta may mak it idal laboratory to study strong intractions in an nvironmnt of vry high parton dnsitis but with small nough strong coupling to apply prturbativ mthods. Elctron-Ion Collidr (EIC) In.S. Nuclar Physics Long-Rang Plan EIC has bn proposd as a nxt-gnration high luminosity lctron-ion collidr facility addrssing complling physics qustions ssntial for undrstanding fundamntal structur of mattr. Polarizd bams in EIC will giv unprcdntd accss to spatial (3-dimnsional) and spin structur of gluons in proton. Organization and prsonnl As PI I plan to spnd 60% of my tim on proposal. I hav prviously pionrd brakthrough that is undrlying currnt proposal so I am wll positiond to achiv its objctivs. I plan to b working at V nivrsity and Nikhf Institut with a group of about 10 ortical particl physicists of which on avrag 5-6 prsons ar fundd from this proposal. This group will involv thr snior physicists. Th PI at full profssor lvl will lad projct; snior rsarchrs with xprinc in sam fild ar availabl in immdiat nvironmnt in Amstrdam. This assurs an xcllnt cor group of physicists working togr and guiding postdocs and graduat studnts. I plan to hir two Ph.D. studnts and thr or four postdocs (in total 12 postdoc yars) and I will hav frquntly visitors as part of programm. I xpct to profit gratly from my xtnsiv ntwork of physicists in Europ built ovr yars among ors in prvious E Framwork Programms. This ntwork involvs orists and xprimntalists which is an ssntial ingrdint for making progrss on lins st out in this proposal. Also within Nrlands r is an xcllnt mbdding of my group. Th group is locatd at a nivrsity with on campus many facilitis and possibilitis to attract studnts. Furrmor my group is part of Ntwork for Thortical High Enrgy Physics with orists covring a broad rang of filds as wll as Nikhf collaboration providing a stimulating nvironmnt for livly intractions with xprimntalists. Rfrncs 1 Brakthrough paprs from our group includ two rnownd paprs (500 citations) P.J. Muldrs and R.D. Tangrman Th complt tr lvl rsult up to ordr 1/Q for polarizd dp inlastic lctroproduction Nucl. Phys. B 461: and Danil Bor and P.J. Muldrs Tim-rvrsal odd distribution functions in lptoproduction Phys. Rv. D 57: An xcllnt ovrviw of us of QCD mthods in high-nrgy scattring procsss including collinar approach and status of transvrs momntum dpndnt functions is found in book of J.C. Collins Foundations of Prturbativ QCD Cambridg nivrsity Prss Following mthods outlind in Rf. [2] and building on formalism of J.C. Collins D.E. Sopr and G. Strman Nucl. Phys. B250 (1985) 199 intrsting progrss has bn rportd in S.M. Aybat and T.C. Rogrs TMD Parton Distribution and Fragmntation Functions with QCD Evolution Phys. Rv. D83 (2011) including availability of volution programms via 5

6 Sction 2: Quantum Chromodynamics at Work a. Stat-of--art and objctivs Introduction Quantum Chromodynamics (QCD) plays a cntral rol in this proposal. It is wll-stablishd ortical framwork for strong nuclar forc which binds togr quarks and gluons into protons and nutrons which in turn constitut building blocks of atomic nucli. Th ory of QCD dscribs intractions among quarks anti-quarks and gluons carrying colour chargs collctivly rfrrd to as partons. Ths partons do not xist in isolation but only confind in colour nutral combinations known as hadrons. Th siz of hadrons is confinmnt scal about 1 fm or m. At distancs smallr than confinmnt scal quarks and gluons in ssnc start bhaving as fr particls (known as asymptotic frdom). In collisions btwn particls at high nrgis on probs distancs of ordr of corrsponding quantum mchanical wavlngth which in collidrs with nrgis considrably abov GV-scal is much shortr than confinmnt scal and on can dscrib scattring dirctly in trms of collisions btwn (quasi-fr) quarks and gluons. This dscription uss stablishd prturbativ quantum fild ortical mthods and allows comparison of cross sctions (counting rats) for various scattring procsss at various nrgis. To account for initial and final stat hadrons for xampl in proton-proton scattring at Larg Hadron Collidr (LHC) on in ssnc nds to know probability of finding quarks and gluons insid protons. Ths probabilitis ar known as Parton Distribution Functions (PDF s) functions f h i (x) with x bing parton s momntum as fraction of momntum of original hadron (a fraction which must li btwn 0 and 1). Thr ar such functions for any hadron h and any kind of parton i (i can b a quark or anti-quark of any flavour or a gluon). In a similar way on nds to know how many and which hadrons h a particular parton i can produc in final stat. This is dscribd by Parton Fragmntation Functions (PFF s) functions D i h (z); hr z is hadron s momntum fraction. Th intuitiv confirmation for abov pictur of scattring procss is apparanc of jts bunchs of hadrons all moving roughly paralll to ach or. In a worldwid ffort ovr last fw dcnnia this intuitiv dscription of PDFs and PFFs has rigorously bn incorporatd in QCD framwork with imprssiv prcision rsults. Th functions hav bn shown to b univrsal apparing in factorizd xprssions. Thy can b xtndd to includ additional dgrs of frdom namly spin carrid by partons as wll as hadrons n dscribing transfr of polarization btwn hadrons and partons (spinspin corrlations). Th fild ortical languag for PDF s and PFF s involvs matrix lmnts of quark and gluon oprators. In particular for PDFs on nds forward matrix lmnts lik Φ q ij = P ψ j (0) ψ i (0) P Pictorial dscription of a high-nrgy collision with (considr figur from lft to right) two colliding hadrons (thick lins on lft) producing partons with probabilitis dscribd by Parton Distribution Functions (PDF s f). Ths partons collid with ach or a hard procss that can b calculatd in QCD and finally y fragmnt into jts of hadrons in final stat (black lins on right) dscribd by Parton Fragmntation Functions (PFF s D). involving for quark distribution functions quark filds. Local matrix lmnts can b incorporatd into a fild ortical framwork. Th dscription of a PDF howvr rquirs a towr of local oprators including also (covariant) drivativs Dµ which can b rcast as a Fourir transform 6

7 Φ q ij (x) = d(ξ.p) i p.ξ P ψ (2π ) j (0) ψ i (ξ) P ξ.n=ξt =0 which only dpnds on momntum fraction x (which in a full xpansion of momntum of a quark is so-calld light-con componnt x = p.n). Involving only on componnt on has a non-local matrix lmnt with a light-lik sparation ξ btwn filds. This collinar tratmnt is usful at high nrgis whr or componnts of quark momnta ar oftn irrlvant. Th corrlator abov can b rprsntd in a diagrammatic way as conncting hadrons with partons (in this cas quarks). Sinc tim-ordring is not important in highnrgy limit (qual light-con tim) diagram can b considrd as a forward anti-parton hadron amplitud. This diagram is n in nxt stp combind with diagrams rprsnting squard hard amplitud of scattring procss involving quarks gluons or or particls for which on uss ory of Standard Modl of particl physics in this way incorporating hadron confinmnt into a powrful fild ortical dscription of high-nrgy scattring procsss. Transvrs momntum dpndnt (TMD) PDF s Nxt I want to xplain brakthrough in which my group playd initiating rol starting with now classic work of Muldrs and Tangrman [1]. W considrd rol of transvrs momntum dpndnc in PDF s and PFF s which mans looking at f i (xp T ) whr x and p T charactriz parton s momntum in a hadron with momntum P ( parton momntum is writtn as p = x P + p T ) x bing fraction of hadron s momntum p T (transvrs momntum) componnt orthogonal to this momntum. Th fild ortical matrix lmnts involving Fourir transform Φ ij q (x p T ) = d(ξ.p)d 2 ξ T i p.ξ P ψ (2π ) 3 j (0) ψ i (ξ) P ξ.n=0 now includ non-local combinations of parton filds whr sparation ξ is no longr light-lik but also involvs a transvrs sparation. This may at first sight sm vry trivial but problm is that transvrs momntum is small (hundrds of MV) quantum mchanically corrsponding to wavlngth of ordr of confinmnt scal. In that domain forcs btwn quarks and gluons ar larg prohibiting us of (prturbativ) QCD. Prturbativ mthods in QCD can only dal with functions f i (x) intgratd ovr p T or with bhaviour whn transvrs momntum bcoms vry larg a limit which indd has bn studid in dtail. Th lattr can b don using prturbation ory within QCD framwork. For xampl a quark can acquir a larg momntum by splitting off a gluon which can b calculatd and lads to a 1/p T 2 bhaviour. Th non-intgrability at larg p T lads to a logarithmic nrgy dpndnc that is xprimntally vrifid with grat prcision. For small-p T on finds in calculation unphysical collinar divrgncs which hav to (and can) b carfully dalt with in prturbation ory. In ordr to incorporat small transvrs momnta on can also turn to transvrs momntum dpndnt (TMD) functions f i (xp T ) in which cas on dosn t hav to worry about collinar problms. Instad on obtains a numbr of nw functions which constitut part of complx structur of hadrons and which hav spcific intrinsic p T -dpndnc. All of m hav vry natural intrprtations as momntum distributions (s box blow). Our brakthrough in introducing s nw TMD functions howvr cam whn w found that y can incorporat spcific angular corrlations btwn (transvrs) momntum and spin of quarks. Som of s corrlations had bn studid bfor ors wr nw. Stting up a systmatic tratmnt w introducd tru corrlations that disappar upon intgration and ar absnt at high p T. Morovr it turnd out to b vry important to charactriz natur of corrlation functions according to ir bhaviour undr tim-rvrsal symmtry (T) with spcial focus on T-odd ons in work of 7

8 Bor and Muldrs [2]. Bcaus QCD rspcts tim rvrsal symmtry r is a uniqu xprimntal signatur of such T-odd corrlations. Thy show up in singl spin asymmtris (SSA) non-vanishing diffrncs btwn cross sctions of procsss in which spin of only on hadron is rvrsd. This singl spin ffct is also rflctd in T-odd corrlation functions. Instad of usual spin transfr btwn hadrons and partons y corrlat transvrs spin with a particular transvrs momntum configuration (momntum-spin corrlations). Two spcial transvrs momntum dpndnt (TMD) PDFs ar on that corrlats transvrs momntum of quarks with transvrs spin of hadron it blongs to (Sivrs function) or on that dscribs a spcific corrlation btwn transvrs polarization and momntum of quarks in an unpolarizd hadron (Bor-Muldrs function). Both of s TMD functions hav gnratd trmndous ortical and xprimntal activity in last tn yars. On ortical sid y shd nw light on spin structur of hadrons. Our group among svral or groups in world ar working on ortical aspcts of s nw functions. On xprimntal sid dvlopmnts hav givn an normous boost to fild of transvrs spin physics for xampl RHIC Spin Physics program at Brookhavn National Lab. (.S.A.). Also at DESY (Hamburg) and CERN (Gnva) xprimnts on SSA hav bn and ar bing prformd. A usful rviw on phnomnology of TMD s is givn in Rf. [3]. An xcllnt ovrviw of fild ortical mthods for QCD in high-nrgy physics can b found in rcntly publishd book of Collins [4]. Th quark production matrix in a polarizd nuclon Th intrprtation as probability distributions of corrlator Φ is nicly illustratd by translating it into a quark production matrix. As basis stats on can us lft- and right-handd quarks in nuclon hlicity ignstats (pictorially givn abov matrix). On finds a production matrix of form R R L L f 1 + g 1L p T M iφ g 1T p T M iφ h 1L 2 h 1 p T M iφ g1t p f 1 g T 2 1L M 2iφ h 2 1T p T M iφ h 1L p T M iφ h 1L p T 2 M 2iφ h 2 1T f 1 g 1L p T M iφ g1t 2 h 1 p T M iφ h 1L p T M iφ g 1T f 1 + g 1L. Th p T -intgratd function f 1 q (xp T ) for any quark of flavour q givs collinar quark distribution usually dnotd q(x). For polarizd nuclons on has longitudinal spin distributions g 1L q (xp T ) intgratd usually dnotd as Δq(x) and transvrs spin distributions h 1T q (xp T ) intgratd usually dnotd as δq(x). Including transvrs momntum dpndnc s distributions fill full spin-spin corrlation matrix but ach with a charactristic azimuthal bhaviour. Th T-odd functions show up as imaginary parts of off-diagonal ntris. Objctivs of proposal (I) Th first objctiv is to rach sam lvl of sophistication for TMD distribution and fragmntation functions as that for collinar approach in which no TMD corrlations ar considrd. This rquirs propr idntification of rlvant quantum filds of QCD for instanc through idntification of p T - wightd obsrvabls as xpctation valus of spcific combinations of quark and gluon filds. It is ssntial that w clarly distinguish tratmnt within rigorous QCD framwork (factorization) study of gaug-invariant matrix lmnts and or oftn intuitivly appaling approachs that ar basd on modl assumptions for nonprturbativ (confining) aspcts of QCD. This first objctiv gos far byond a mr xtnsion of work that has bn don sofar. It rquirs a nw rsarch lin aiming for a full undrstanding of quark-gluon dynamics that nds to b accountd for using all aspcts of QCD bing a non-ablian gaug fild ory and combining prturbativ and nonprturbativ aspcts. 8

9 (II) Th scond objctiv is xploitation of corrlations as tools in high-nrgy scattring procsss. Following ground-braking work outlind in prvious paragraph studis on TMD corrlations ar now includd worldwid in rsarch programms of many xisting or futur facilitis that hav programms on hadron physics (upgrad of Jffrson Laboratory in.s.a. plans for FAIR at GSI in Darmstadt RHIC programm at Brookhavn National Laboratory J-PARC programm at KEK in Japan). Early xprimnts confirm apparanc of novl phnomna. Th TMD corrlations howvr also offr possibilitis for ddicatd LHC xprimnts that may b usful to invstigat Higgs sctor. Furrmor TMD s ar oftn mntiond in or long-trm plans among m ambitious nw larg scal facilitis such as proposd LHC in Europ or EIC projct in S (w will giv dtails blow). sing findings of study mntiond undr first objctiv I want to critically assss which novl aspcts of hadron structur can b addrssd in s facilitis. Highlighting objctivs: I I want to grasp fundamntal novl aspcts of quark and gluon dynamics for TMD corrlations in particular T-odd ons and giv a nw maning to concpt of parton in high nrgy collisions. II I want to critically assss which TMD corrlations can rliably b usd in tagging of vry spcific partonic initial stats or in analysing spcific asymmtris in partonic final stats. Knowing ffort that wnt into stablishing collinar approach first objctiv is crtainly vry ambitious. Incorporating small p T from start avoids problms (collinar singularitis) but dosn t com for fr. I am confidnt that givn xisting worldwid knowldg bas for collinar tratmnt and our xprtis on TMD functions I can dvlop a succssful nw rsarch lin with a tam of ddicatd Ph.D. studnts postdocs and visitors. And vn if w do not solv all issus initiativ will gnrat nw brakthroughs and may lad to nw ways of rigorously mploying QCD in hard scattring procsss byond collinar approach. b. Mthodology In gnral I hav bn quit succssful in obtaining funds for Ph.D. and/or postdoc position. An ERC Advancd Grant howvr will offr m possibility to start this ambitious ntrpris of which scal is largr than national individual programms allow. Evn if it is a larg scal ntrpris chosn mthodology is such that it in a natural way is dividd into parts that by mslvs will yild important rsults. Th ovrall strngth and chancs of producing ground-braking rsults howvr li in cohrnc of full proposal which will run for fiv yars starting on January and which will b major projct in my group in thos yars. Within proposal I nvision following intrrlatd topics: (I) Fundamntal studis of TMD s (towards objctiv I): starting point and projcts It has bcom clar that taking partonic dscription of nuclons byond collinar approach would b xtrmly nic to dpn our undrstanding of QCD and bcaus it hlps us in undrstanding of xprimnts. A systmatic formalism using prturbation ory known as Collins Sopr Strman (CSS) formalism [5] has bn st up alrady in 1985 and has bn usd in a numbr of high-nrgy scattring procsss. As compard to this formalism introduction of TMD functions is still in its infancy. Nvrlss promis of s TMD functions is grat bcaus of ir intuitiv simplicity and ir ability to provid in a natural way candidats for T-odd functions that can xplain singl spin asymmtris apparing at lading ordr rar than at sublading (highr twist) ordr. A first stp towards a full incorporation of TMD functions into fild ortical framwork of QCD has bn study of matrix lmnts and rol of gluon filds rin. Whn on intgrats ovr transvrs momnta ( collinar approach) transition hadron-to-parton and parton-to-hadron involvs gluon dynamics but accounting for s ffcts is rlativly straightforward in form of quantum mchanical phass. In collinar cas only on dirction is probd and phass do not affct obsrvabls i.. y do not lad to intrfrnc; on can asily intrprt distribution and fragmntation functions as probabilitis and dcay functions rspctivly. Including transvrs momnta gluon dynamics producs phass that do track flow of colour charg in high-nrgy procss and can hav obsrvabl consquncs. Tchnically on ncountrs basic non-local contributions that go byond usual standard oprator product xpansion. 9

10 In B1 w compard situation with famous Aharonov-Bohm phas in quantum lctrodynamics (QED). Th phas of lctron bcoms visibl in intrfrnc xprimnts. In a high nrgy scattring procss on has a suddn procss in which a parton is rmovd from a hadron. If two hadrons ar involvd phass in wav functions of rmnants produc physical ffcts which ar charactrizd as T- odd (odd undr tim rvrsal). Thy show up as non-vanishing singl spin asymmtris but only in combination with azimuthal asymmtris of producd particls. It is rigorous mbdding of such TMD ffcts as portabl ffcts in QCD framwork (stablish factorisation) that is bing pursud. At that point a numbr of problms hav bn ncountrd that hampr TMD-factorization. Nxt I want to outlin spcific studis mphasizing novl aspcts and discussing fasibility. (I.a) Propr fild ortical tratmnt of TMD s Th propr fild ortical dfinitions of PDFs both in collinar as wll as in TMD cas is a highly nontrivial issu [6]. In Fynman diagram languag on may hav additional partons participating in hard procss. Ths ar mostly supprssd at high nrgis but gluons with polarizations paralll to quark momntum must b rsummd modifying most naiv dfinition of PDFs. Two partons originating from on hadron ( lin-with-arrow rprsnting a quark and curld-up lin rprsnting a gluon) participat in hard scattring procss. Th addition of gluons is what opns Pandora s box! Th inclusion of gluons with polarization paralll to parton momntum givs ris to corrlators including appropriat futur/past pointing gaug links (s box on nxt pag). In fild ory languag s gluons provid gaug link rfrrd to as phass in introduction abov which connct nonlocal quark or gluon filds in dfining matrix lmnts. For TMD s on obtains corrlators with diffrnt gaug links in particular futur- or past-pointing links discussd in som dtail in box blow (possibilitis a and b for quarks rspctivly). Th sum of s is T-vn diffrnc T-odd. Sinc it dpnds on hard scattring procss (which absorbs additional gluons) which corrlator to us. As a consqunc on gts diffrncs for azimuthal asymmtris associatd with T-odd TMD s. Th simplst cass ar dp inlastic scattring (DIS) lctron-positron annihilation and lpton-pair production (Drll-Yan or DY). In s cass flow of colour charg is uniqu bing ir a simpl annihilation or cration of colour chargs or a simpl flow of colour from initial to final stat. Th fact that in DIS hadron corrlator contains a futur-pointing gaug link whil in DY on nds a past-pointing gaug link producs a crucial sign chang in asymmtris xpctd for DIS vrsus DY (for xprts known as Collins and Sivrs asymmtris rspctivly). W want to undrstand what happns byond s simplst procsss as wll as what happns for gluons. Th xtnsion to gluons involvs a mor complx gaug link structur (s box on gaug links) and is actually discussd in mor dtail as scond projct (I.b). Although procss dpndnc coming from gaug invarianc rquirmnts points to a factorization braking r ar situations in which full ffct can b cast in form of modifid cross sctions (s undr II). But full solution is as yt unknown rquiring inclusion of full QCD dynamics which includs bsids appropriat gaug links study of volution of distribution functions. A numbr of problms has bn ncountrd hampring TMD-factorization with rcntly som promising stps towards implmnting QCD-volution [7]. I am confidnt that a full dscription can b formulatd but it rquirs combining our strngth with that of svral xprts on volution of PDFs. This projct dals with basic ortical issus and is cntral to proposal. It is naturally linkd and compltd by or ortical parts in I and phnomnological studis in II and III. For this part (I.a) I plan to attract at last two xprincd postdoc with complmntary xprtis to myslf and with rlvant (diffrnt) backgrounds. Thy work togr with two Ph.D s on of m bing a Ph.D in this proposal. I intnd to spnd around 20% of my tim (on third of my commitmnt of 60%) on this part. 10

11 2(P P nfild combinations P considrd v v) Th corrlators ar not color gaug-invariant sinc y involv d( sofar P2) in i p n (x;important n) Tr Gnonlocal (0) [0 ] Gn ( ) [ 0] P iat ach spcific (1.50) H(p...= ; v) v yhp A-filds andvmor bcaus involv fild combinations. ordr + (2 ) pt H(p... ; v) LC 2 in Q on of cours xpcts gaug-invariant combinations. Along light-con lading combinations P v 2x (P v) {z } whil for light-front Workplan involv TMD parton filds corrlators O(1/Qd 2 ) n 2( ) = n n / / ( ) and G ( ) d( P ) d T i p + [nc] (xnpt ; n C) = A gluons (putting n H hp= 0) [0 ] 4 ati ( ) P i ordr + (1.51) ij j (0) mans that for whil soft part omitting implis lading 3 (2 ) A+ = A = n A noprators appar in gaug links along light-con LF ( = n = T = 0) omission of v A gluons (v H = 0).! d( P ) d2 T i p n [nc] n [nc 0 ] quark and gluon Gaug links for (x pt ; n C C 0 ) = TMD s hp Tr G (0) G ( ) P i (1.52) [0 ] [ 0] 3 [0 ](2 ) = P xp i d( P ) n A( ) (1.48) LF r gaug invarianc 0 For quark distribution functions fild ortical xprssion in trms of a non-local product of quark whr w in passing mntion that path dpndnc (indicatd by argumnts C and C 0 ) will aris which ar sofar ndd to connct colord parton filds. Which n appars in a corrlator is fixd by hard ld combinations considrd in corrlators arpic(s) not color filds ψ is givn by bcaus of (ncssary) transvrs in gaug-invariant gaug link. sinc y involv procss although sominvolv frdom in n may rmain. W not that xponnt filds and mor important bcaus y nonlocal fild combinations. At ach spcific ordr in gaug link is in 2 T [nc] ssnc built from combinations. oprators (nd( P )1dn A which ar O(1). Actually gaug-invariant corrlators will i p n of cours xpcts gaug-invariant light-con lading combinations hp j (0) [0 ] µ i ( ) P i quark (x pt ; n C) = 3 (2 ) parton momntum p (p; P ) n=0 tc. which implis a drivativ parton filds in som cass appar multiplid with µ in matrix which lmnts involving gluon filds Gµ. Th n ( ) lmnt / =n / + ( ) and Gn )standard ( ) d( P d2 T i p in matrix [nc] n [nc 0 ] link 0 n whr gaug (x p ; n light-con C C ) = corrlators hp Tr gluons G (0) [0 ] G ( ) [ 0] P i color gaug-invariant gluon 3 for quarks A+ = An = n A oprators appar int gaug links along ( +and = n = ar T = 0) (2 )light-con n=0! d( P ) 4 Hr condition on n) = of a possiblivp hp j (0)nbcoms i (1.49) ij (x; smallnss important. T in dfining [0 ] i ( ) P [0 ] = P xp i d( P(2 ) ) n A( ) LC (1.48) 0 d( P ) i p n n (x; n) = dpnds on hp Tr Gthrough (0) [0 ] Gpath ( )Crunning P ifrom point(1.50) [ 0] is a path-ordrd xponntial which procss 0 to ar ndd to connct colord parton filds. Which n (2 ) appars in a corrlator is fixd by hard LC s although som frdom in possibilitis n may rmain. not that xponnt in gaug link is in ξ. Th simplst ar W momnta of partons in high-nrgy scattring procsss 35 whil for Transvrs TMD light-front corrlators built from oprators 1 n A which aro(1). Actually gaug-invariant corrlators will 2 d( Pp)µd (p; T Pi) [nc] cass appar multiplid with parton momntum implis aξ Tdrivativ p tc. hp which i (1.51) ij (x pt ;ξn i ( ) P T C) = j (0) [0 ] 3 (2 ) matrix lmnt which is.g. standard in matrix lmnts involving gluon filds Gµ.LFTh aug-invariant light-con corrlators for quarks and gluons d( Par ) d2 T i p [nc] n [nc 0 ] n (x pt ; n C C 0 ) = hp Tr G (0) G ( ) P i (1.52) ξ ξ [0 ] [ 0] d( (2 )3 (a) P ) i p (b) LF hp j (0) [0 ] i ( ) P i (1.49) ij (x; n) = (2 ) LCquark-quark whr wfig. in passing that path dpndnc (indicatd by argumnts anddy C 0(b) ) will aris 8. Thmntion gaug link structur in corrlator in SIDIS (a)cand d( P ) bcaus of for (ncssary) transvrs pic(s) in scattring gaug link.or Drll-Yan procss (b)rspctivly. rspctivly which ar rlvant smi-inclusiv inlastic (a) i p ndp n (x; n) = hp Tr G (0) [0 ] G ( ) [ 0] P i (1.50) (2 )ortical xprssion For gluons fild of a non-local[nc] product of gluon fild strngths G is givn LC d( P in ) d2trms T i p hp j (0) [0 ] i ( ) P i quark (x pt ; n C) = by or TMD light-front corrlators (2 )3 n=0 d( P ) d2 T i p [nc] ) d2 T d( P [nc](1.51) [nc 0 ] n (x p ; n C) = ; n ChP 0 ) j=(0) [0 ] i i p hp Tr Gnfunctions ij T i ( ) P (x p C (0) G ( ) P ion T Th transvrs momntum dpndnt distribution howvr do dpnd 3 gluon [0 ] [ 0] (2 ) (2 )3 LF n=0 gaug link structur which is still tractabl in simpl procsss lik DIS or DY with d( P ) d2on Ti smallnss [nc 0 ] p n n condition of color agpossibl v[nc] dfining n bcoms T in G (x pt with ; n Calrady C40Hr ) = hp Tr (0) ( ) Pimportant. i (possibilitis (1.52) ntangld a simpl color flow but structur of various corrlators bcom as simplst possibilitis four diffrnt gaug link connctions a difin figur [0 ] [ 0] (2 )3 LF color flow is mor complicatd [ ]. blow)38sinc on nds for TMD s two gaug links [C] and [C ] to dal withp.j. colour of gluons w in passing mntion that path by C and C 0 )ar will aris Muldrs Evndpndnc if color (indicatd flow is simpl likargumnts SIDIS or DY r cts. Most notabl is of (ncssary) transvrs pic(s) in gaug link. ξ Tin singl spin asymmtris going from ξ T SIDIS to DY. To undrstand this a sign chang d( P ) d2 T on [nc] ar no longr constraind by tim-rvrsal as sign chang i p nots that TMDs hp j (0) [0 ] i ( ) P i[+] rmquark (x pt ; n C) = 3 tim (2 ) rvrsal opration intrchangs n=0 and [ ] links lading to apparanc of T-odd functions ξ and also occuring 0in ξ 121. On has.g. Eq. d( P ) d2 T i p in Eqs [nc] n [nc ] 0 n hp Tr G (0) [0 ] G ( ) [ 0] P i gluon (x pt ; n C C ) = (2 )3 (a) (b) n=0 ) ( ξt 1 important. i [/ pt n /+] condition on smallnss of a possibl vξ n bcoms [±] T Tin dfining (135) f1 (x pt ) n / + ± h?. 1 (x pt ) O (x pt ) = 2 2M ξ ξ Looking back at our xampl of an azimuthal asymmtry w hav sn sin( `h (c) (d) `? S ) asymmtry proportional to f1t D1. Th corrsponding asymmtry in Drll-Yan? proportional to f1t f 1 would gt an additional minus sign. Actually for fragmntation Fig. 9. corrlator Th gaug links TMDs such for as gluon O in Eq. 122 r is no such dpndnc on gaug link [ ]. Th function H1? howvr is nonzro bcaus stats Ph Xi in cas of fragmntation ar out-stats and tim rvrsal (changing out- into in-stats) simply g (oftncannot also rfrrd to as as a constraint. ) b usd Th asymmtry in Eq. 125 thus will not chang sign going from SIDIS to corrsponding DY asymmtry. In gnral situations in which (I.b) Gluon TMD s 1 of TMD d( P distribution ) d2 T ip functions of two hadrons in initial stat on0 ]has [ µ a convolution (145) (xpt ) = 2 (p n)is (2 )3 hamprd is alrady at transvrs tr-lvl bydpndnc ntanglmnt of Wilson lins [41 Th mphasis factorization in many discussions on momntum of quark distributions but at high nµ n 0 42].ofntanglmnt Wilson lins. Atthat lvl of F wightd implis Tr hp S Fof (0) [0 ] ( ) [ 0] P Si. ofitunivrsality nrgis rol quarks is lssof prominnt than gluons. For many ofasymmtris issus and LF mor complicatd factors than just a sign chang in apparanc of wightd volution it is natural and also simplr to start with quarks as y hav smallst possibl nonzro functions. fild-oprators ar on writtn in color-triplt rquirs colour Hr charg. Putting mphasis gluons howvr bringsrprsntation in nw subtltis.g. inclucolour flow can b 0 sion of two Wilson lins and. Thy again aris from rsummation of quit diffrnt. Dpnding on gluon charg can split into a colour gluon anti-colour charg [0 ]hard procss [ 0] and intractions. In gnral this will lad two links unrlatd Wilsonusgaug lins Gluonic matrix lvl Th inclusion ofto gaug alsodiffrnt allows to study which initial rquirs forfinal-stat TMD spol vn at lmnts. simplst considration of four linksin(s box 0 and som. on Inmor particular that structur 0 = quarks gluon corrlator also b novl writtn dtail pag). cas oprator of T-odd parts incan svral corrlators. Givn on gaug links prvious Lik for r ar also momntum-spin [±] as for product of two gluon filds corrlator Wilson collinar in a adjoint rprsntation ofparamtriz fullgluons oprator structur for with a matrix TMD including gaug links naturally on can xpliccorrlations (s production blow).lin Th functions S (N ). This is for instanc cas for gluon corrlators which acquir gauglinks itly calculat transvrs momnts (x) of which w hav givn paramtrization circular gluon polarizations in nuclon hlicity stats. Including TMD s on has for instanc as in Figs. 9a and but finds not for y gluondpnding corrlators Figurs. andcan d. b rlatd to color in Eq. 121.bOn that gaug 9c links 11 onin In pt -intgratd corrlator on lightcon procss dpndnc of TMD gluon corrlator disappars

12 intrsting option of paramtrizing longitudinally polarizd gluons through charactristic azimuthal asymmtris (discussd undr II). Such asymmtris can vn show up in scattring procsss with unpolarizd protons. Anor rason to considr gluon TMD s is my xpctation that y can play an important rol in diffractiv phnomna at this stag mrly a conjctur. In s phnomna r is no colour xchang btwn scattrd hadron and hard part (This could naturally occur whn two partons originating from a singl hadron ar both gluons). This projct will start by xtnding work on quarks rlatd to non-trivial gaug links. Togr with incorporation of aspcts of volution (projct I.a) it will provid ssntial background for phnomnology for LHC physics (s II) as wll as for stablishing physics cas for lctron-hadron collidrs (s III). This projct (I.b) is plannd to b focus of scond Ph.D. projcts. I also want to attract a postdoc with xprtis in this fild. Th gluon production matrix in a polarizd nuclon Th intrprtation as probability distributions of gluon corrlator can also b translatd into a production matrix similar as for quarks. As basis stats on can us circularly polarizd quarks in nuclon hlicity ignstats (pictorially givn abov matrix). On finds a production matrix of form Th imaginary (T-odd) functions ar xplicitly shown in matrix. Th function f 1 g (xp T ) is unpolarizd gluon distribution in p T -intgratd (collinar) cas usually dnotd g(x). For a polarizd nuclon on has longitudinal spin distributions g 1L g (xp T ) in collinar cas usually dnotd as Δg(x). Ths ar only functions that surviv in collinar limit. Th TMD s ar diffrnt with transvrs momnta naturally corrlatd with linar gluon polarizations. (I.c) Modl calculations With appropriat fild ortical dfinitions for TMD s on can turn to lattic gaug oris [8] ffctiv fild oris or or modls. Ths modl approachs offr intrsting possibilitis whn on wants to study matrix lmnts.g. ir dpndnc on gaug link structur. This is bing pursud by som of lattic gaug collaborations. I considr modl calculations as usful but y ar not main purpos. Modl calculations crtainly will provid guidanc whn on turns to phnomnology. W plan to do such calculations in many cass in collaboration with or groups. I do not plan to attract postdocs spcifically for modl calculations but in procss of attracting postdocs during runtim of projct nd for xprtis in spcific aras undr (I.c) may hav com up. 12

13 An intrmzzo: QCD-ntanglmnt in lab In proton-proton scattring a possibl subprocss is quark-quark scattring two quarks in two quarks out dscribd by a quantum mchanical amplitud. This amplitud can b calculatd with hlp of Fynman diagrams. In lowst ordr on has xchang of a colour forc particl (gluon) btwn two quarks. On has two contributions in amplitud (A 1 and A 2 ) and in squard amplitud (which is masurd probability) on has four contributions as Fynman diagrams pictorially rprsntd as Each of s diagrams also has two colour flow possibilitis. Th sum of s four (including colour flow ight) contributions givs rsult of hard procss and is multiplid with corrsponding PDF s rprsnting quark probabilitis in initial stat and dcay into final stats. In figur blow this rsult is shown as qqà qq contribution. For tim-rvrsal odd PDF s (dpnding on transvrs momnta) on gts a modifid combination of s ight contributions ach with appropriat sign. This lads to calculabl factors C G [(D)] whr D rfrs to diagrammatic contributions ach of which involvs a spcific gaug link structur (D) dtrmind by colour flow. In figur this rsult is shown as [q]qà qq contribution. Thr is a pronouncd diffrnc whn cross sctions (summd contributions) ar plottd (on a rlativ scal) as functiosn of scattring angl in cntr of mass fram of hard scattring procss. It should b notd that to singl out dpndnc on on of quarks on nds ddicatd xprimnts (as will b pointd out undr (II) whn w discuss for proton-proton scattring nonalignmnt of jts in transvrs plan). Thortically intrsting is fact that r xists mor than on colour gaug-invariant combination of squard amplituds. Th abov xampl [takn from papr: A. Bacchtta C.J. Bomhof P.J. Muldrs and F. Pijlman Singl spin asymmtris in hadron-hadron collisions Phys. Rv. D 72: ] illustrats possibilitis to idntify from hard cross sction spcific (T-odd) momntum spin corrlations in initial (or final) stat. 13

14 (II) Phnomnology of TMD s (towards objctiv II) Considration of intrinsic transvrs momntum only maks sns if it can b proprly masurd. At high nrgis that is not immdiatly obvious. Masurmnts of transvrs momntum indd ar possibl but y oftn rquir that colliding particls ar polarizd in a spcific dirction or that on masurs polarization of producd particls (polarimtry) or that on masurs angular distributions of producd particls (azimuthal asymmtris). An xampl of fasibility to accss intrinsic transvrs momntum in high nrgy procsss is for proton-proton scattring shown in this figur. Th non-collinarity (angl δφ) of projctions of two jts onto transvrs plan is a masur of transvrs momntum. By taking spcific wights and corrlations with azimuthal dirction st by polarizd proton on can isolat corrlations du to intrinsic transvrs momntum in thos cass whr prturbativ calculations giv zro as rsult in lading ordr [9]. Excpt dpndnc on δφ dpndnc on or variabls lik y = t/s follows bhaviour as xpctd for an ntangld situation (s box on prvious pag). Th importanc of symmtris as link btwn ory and xprimnt for TMD s has alrady bn mphasizd. Th strong intractions ar invariant undr spac invrsion (P) charg conjugation (C) and tim-rvrsal (T). Spac invrsion allows for a distinction of parity vn and odd phnomna. In sam way tim-rvrsal invarianc allows a distinction btwn tim-rvrsal vn (T-vn) and tim-rvrsal odd (T-odd) phnomna. This can b usd as a powrful tool to idntify spcific parts in intraction btwn quarks and gluons and rlat m to suitabl (xprimntally accssibl) obsrvabls as outlind abov. In particular singl spin asymmtris mntiond abov ar xampls of T-odd obsrvabls. For phnomnology of TMD s I distinguish thr kinds of analyss first lvl would b assuming TMD-factorization i.. just working with probabilitis f i (xp T ) and lmntary hard cross sctions irrspctiv of this factorization has bn stablishd or not. It is clar that trying to dscrib data on azimuthal asymmtris in this most naiv way is usful bcaus if only to point out discrpancis. At a scond lvl of analyss on would tak into account fact that naiv factorization has to b rplacd by a gnralizd factorization that accounts for colour flow. For simpl procsss lik ons mntiond also in (I.a) dp inlastic scattring lctron-positron annihilation or lpton-pair production colour flow is uniqu and on xpcts prdictabl sign changs. Mor complx procsss lik proton-proton scattring imply mor complx but calculabl ffcts (s box on QCD-ntanglmnt). Som applications at this scond lvl for quark and gluon TMD s hav bn workd out with collaborators worldwid [10] indicating intrsting options for masurmnts at LHC or futur high-luminosity lctron-hadron collidrs. At third lvl of analyss w want to includ full gluon dynamics using rsults of projcts undr (I) in particular (I.a). It rquirs incorporation of byond lading ordr (NLO NNLO) rsults not only to dal with rgim of larg p T and study volution of TMD s. At sam tim w want to us rgulating proprtis of TMD functions at small p T and includ full st of TMD s including T-odd ons. Implmntation of this third lvl is still unxplord trritory. Nxt I outlin a fw spcific phnomnological studis. (II.a) Analysis of obsrvabls Transvrs momnta of partons ar by dfinition hiddn variabls that only show up by constructing appropriat quantitis (cf. a-collinarity of jts discussd abov). In particular situations hiddn variabls may giv away ir origin bcaus y can only appar in combination with.g. particular polarization dirctions (usually transvrs polarization) or partons may giv away ir idntity through a quark mass (c- 14