Effective Field Theory

Eﬀective Field Theory

Iain Stewart MIT

Nuclear Physics 2007 Long Range Plan Joint Town Meeting on QCD Jan, 2007 Eﬀective Field Theories of QCD

ChPT HTL NRQCD cc states pions large T large HDET unstable density particle top quark QCD energetic EFT hadrons SCET jets nuclear SCET single nucleon forces bd states NNEFT Heavy Hadron HQET ChPT Arrows: Eﬀective theories describe infrared limits of QCD m ChPT π 1 π Λ ! talks by D. Kaplan, D. Phillips, p H. Griesshammer 1 Λ ! Eﬀective Field Theories of QCD

Q Soft-Collinear Eﬀective hard probes Theory (SCET) Λ 1 • Q ! Eﬀective Field Theories of QCD

g L QCD, π Nuclei 1 mπ− q a, L, QCD Q perturbative 1 F (Q2) = dxdy H(x, y) φ(x) φ(y) Q2 SCET Λ !

QCD

non-perturbative

2 ChPT mπ f µ (4) = tr ∂ Σ∂ Σ† + v tr m†Σ+m Σ† + (L ) L 8 µ q q L i ! " ! " Lattice & EFT - extrapolations Symanzik’s Effective Field Theory Symanzik’s EffeχctiSBvewFieithldWTilsheonoQryuarks ! −1 Near continuum a ΛQCD, lattice action described by a continuum effective fieldFtehremorioy n Discretization ! W−1ilson solves the doubling radically with second Near continuum a Λ2 QCD, lattice a1 ction describ1ed by a 2 2 0 1 2 Sy−manzik’s Effective Field Theory Symanzik Action: SSymanzik = S + a S + a Sder+iv.a.t.ive Go (m = 0) = iγµ a sin(pµa) + a sin (pµa/2) FecormniotinnuDumLisactetricefefetizdctiscriavtioeetizafientilodnt(hsheoortrryange) in χPT? Operators in EFT respect symmetries of laCttihceir−a1al ctsymionmetry1 recovered only in the continuum limit Naive discretization G (m = 0) = iγµ sin(p2µa) Symanzik Action: SSymanziko = S0 + a S1a + a S2 + . . . e.g. gauge invariance, hypercubic invarianLceeading Symanzik action contains!the−P1auli term Lattice discrSoelutitzaiontisonto(shferomrtiornandgoeu)bilninχgPgTe?neraNlley aleracod tonchtiniruaul m a ΛQCD, lattice action described by a OperatorssyminmEetFryTbreeaspkinegctat finsymite am, weitthriems=o0f lattice action S0 is the QCD action: S0 = ψ(D/ + m)ψ −1 S1 = a cS1W ψσµνcoGµnνtψinuum effective field theory Naive discretization Go (m = 0) = iγµ a sin(pµa) e.g. gaugBeuiildnvχaPTriafonr cethe,Shymypaenzircuk abctiiconinvariance 2 Discretization effects enter as higher dimenNsioonn-perturbative O(a) improvement renders cSW = 0 Solutions Shtoarfpee ramnd iSionngledtoonu’9b8,lLineegangdeShnaerprea’9ll9y lead toSchymiraalnzik Actireonvie: wSSedym byan zik = S0 + a S1 + a S2 + . . . operators, e.g., O S40sym=isc mth4eeψtryDQbCrDeDakiDnagctγaitoψfinn∈:iteS(aP20,ew=rithuψrm(bD/a=+t0ivme )imψprovement schemes deplete size of ci’s) H H Gµoldstµ onµe µmode propagation sensitOivepteoreaxptolircist iχnSEB FTTiburziresp ate ct symmetries of lattice action Build χPT for the Sym2 anzikFaoctriounnimproved actions mChiralust de Dynamicsal with O’06(a) errors, Hidden log 1/a-depeDniscrdenecetiza: Ectaiioch=nlcaeit(tfgicefe)ctfesrmeionnthearsadsiffehringghχeEreF.dTgi.mgeanusigoeninvariance, hypercubic invariance Sharpe and SinWgletiolnso’98n, ,LeTewanidstSheadrpem’i9an9 ssstrWuctilsoivne, Stotagcogenrestdr,uct χPT. opGeorldasttoornse,meo.dge.,pOropHa4ga=tiocnHse4 ψnsiDtivµeDtoµDexpµγliciµψt χ∈SBS2 Ginsparg-Wilson (Domain Wall), HybSr0idis(MthixedQACctiDon)action: S0 = ψ(D/ + m)ψ Each lattice fermion has difχfeSrinBg fχrEoFmT Pauli2term has the same form as the quark Hidden log 1/a-dependence: ci = ci(g ) Wilson, Twisted mass Wilsomn,aSsstagtgeerrmend., Discretization effects enter as higher dimension Partially O(a ) in χPT – p.3 n Ginsparg-Wilson (Domain Wall), Hybrid (MixeodpAectriaotno)rs, eO(.ag) i.n χ,PTO– p.4 H4 = cH4 ψDµDµDµγµψ ∈ S2 Quenched n O(a 2) in χPT – p.5 Hidden log 1/a-dependence: ci = ci(g ) O(an) in χPT – p.3 n valence O(a ) in χPT – p.4

sea n O(a ) in χPT – p.3 ﬁts F and G which are the same as ﬁt E except the argument of the log2 contribution is chosen to be M = mπ and M = mK , respectively. These choices lead to a larger variation in L˜5 and consequently in fK /fπ.

C. Discussion

To determine fK /fπ at the physical point and its associated uncertainty we synthesize the results of the NLO and NNLO fits. Fitting the lowest two mass points at NLO gives fK/fπ = 1.215 0.002, while fitting the three data points with pion masses below m 500 MeV ± π ∼ gives fK/fπ = 1.218 0.002. The difference between them is within statistical errors (1.5σ) but there appears to±be a systematic trend in the data which can be attributed to higher orders in the chiral expansion. As we are unable to fit the full NNLO expression to our small data set we can estimate the systematic uncertainty in this calculation by looking at the range of values of fK /fπ that result from the two types of NNLO extrapolation, both with and without the log2 contribution, including variation in the argument of the NNLO logarithm and including statistical errors. The range of variation in the NNLO estimate is an order of magnitude larger than the statistical error found at NLO. We take this NNLO +0.011 uncertainty, ∆(fK/fπ) =−0.022, to be an estimate of the systematic error in our calculation due to the truncation of the chiral expansion. We also assign a systematic error due to fitting procedures, obtained by varying the fitting ranges displayed in fig. 1, which gives +0.000 ∆(fK /fπ) =−0.010. Therefore, our final number is:

fK +0.011 = 1.218 0.002 −0.024 , (17) fπ ± wfihtesreFthaendfirGst werhriocrhisarsetatthiseticsaml aendasthfiet sEeceoxncdepetrrothreisasrygsutmemenatico,fwthiteh ltohge2 ecxotnratrpiobluattiioonn is error and fitting error added in quadrature. The error in this lattice QCD determination of chosen to be M = mπ and M = mK , respectively. These choices lead to a larger variation f /f˜ is clearly dominated by the systematics. iKn Lπ5 and consequently in fK /fπ. Decay constants fK , fπ Using a similar procedure, we arrive at a value for L5: Calculation of fπ, fK phy +0.18 −3 L (f ) = 5.65 0.02 − 10 , (18) PrCe.limDisicnusasiroyn Res5uStalπtgsgered: De cFaermions±y Co0n.54st×ants MILC from C.Bernard where the first error is statistical and the second is an estimate of the systematic error To determine fK /fπ at the physical point and its Casosmopcaiarteewdituhncertainty we synthepshiyze the due to omitted higher ordersunquenched,in the chiral expfaanst,s•ionligh. Thtismthπen scales to give L5(mη ) = results of th+e0.N18LO and−3NNLO fits. Fitting the lowepehsyxpt tewroimmenats.s points a+t0.N18LO giv−e3s fK/fπ = 2.22 0f.π02=−128.6 100.4 at 3th.0e Mη-emVass an[129d L.5(m 0.)9 = 31.452MeV0.]02 − 10 at the 1.215 0.0020.,54whi±le fittin±g the three data po5int±sρwith±pion masses be0l.o54w m 500 MeV ρ-ma±ss. As stated×previouslybut, th 4is “tais stanese” ffforect eachive L flaavors it, includ±es the higher×orπder strange gives f±K/f= 155= 1..2318 0.04.0023..1ThMeedVifferen[c156e b.e6twe5e1n.0them3.1i6s/4MwietVhi]n statistical e∼rrors (1.5σ) quark coKntriπbution. ± ±Pissue:relimdet(iD/n+amry) Rd±etes(D/u+l±tms): Decay Constants but there appears to±be a syste+ma7tic trend i→n the data which can be attributed to higher The resultfsKf/forπf=/f1.208(have2)a(n−1a4d)dition[1a.l210(syste4)m(a13)tic]error due to the non-zero lattice orders in the chiraKl exπpanslotsion. ofA paramets we arerse u nina bChPTle to fit the full NNLO expression to our spacing which we expect to be O((m m )b2). In principle one can reduce this error small data set we can estimate the sysst−emautic uncertainty in this calculation by looking at bOy lfidttriensug tltostihneraepdpraorperifartoemχPMTIfLπfCo=r,mP128uRlaDs.6t7h0at(02i.n40c0lu4d3)e.10t1hM4e5eOV0(1g.2b2)[129effe.c5ts d0u.e9to fl3a.5voMr- eV] ± ± ± ± sythmemreatnrgyeborefavkailnugesinotfhfeKs/efaπ-qtuhaarkt rsescutoltr [f2ro4m]. Hthoewetwveor,toyupresdaotfaNfiNt LwOelletxotrtahpeocloanttioinnu,ubmoth with and without the log2 nuclearcontfriKbu beta=tio155n,-decayin.3clud 0igivesn.4g var3i.a1tiMoneVin the [a156rgu.m6 ent1o.0f the3.N6NMLOeV] χPT fEorxpmuelarismaendnthaelnrcaetewefodroπnot eµxνpe+ctVtuhdatf±ruosme o(fsu±thpeeexratellnodwededχ)PT formu±las of±Ref. [24] log•arithm and including statist→ical errors. The range of v+ari7ation in the NNLO estimate is wouldnsuigclneifiacranbtelytaimdpercaovyse our refsπul=tsf130aKt /fth.7πis =st0a1g.1.e208(. O0u2).r4fi(M−na1elV4)r.esult[1is.210(consi4)st(e13)nt w]ith the an order of magnitude large⇒r than the statis±tical e±rror found at NLO. We take this NNLO MILC number [1] +0.011 uncerRtaeisuntyl,t f∆o(rffKK/f/fπ)π=, p−lus V, utod baendanKestimµaνte(eoxpf tth)e, gsyivsetesmaVtuics e.rror in our calculation O0.f0ld2K2 results in red are from MILC, PRD 70 (2004) 114501. du•e to the truncation of the chiral =ex1p.a2n1→0sion0. .0W04e al0s.o01a3ss,ign| a s|ystematic error d(u19e)to Following Marciano, fPR!L 93 (2004)±231803±, get fitting procedures, obtained πb!yMvILaCrying the fitting ranges displayed in fig. 1, which gives Expe+ri2m6 ental rate for π µν + V from (superallowed) where the firs+t0.e00r0ror Vis sta=tis0t.i2223(c!al and th)e sec[o0n.2219(d is th26)e t]otal systeumdatic error estimated ∆(fK /fπ) =− . Thuesrefore•, o!ur fin−a1l4number is: → Chiral Dynamics, Sept. 19, 2006 – p.16 0.010 | | Domainnucle War albel tvalenca decaeys NPLQCDfπ = 130.7 (‘060.1) 0.4 MeV. by MIPLDCG. S(i2n0ce06o)u,rfrvoamlenKce quπaµrkνs (aerxpe dt)omanadin-(wnaolnl -f⇒leartmticeion)s,thienocroy,n±trast w±ith the KS f • →ReKsult for fK /fπ, pl+u0s.01V1ud and K µν (expt), gives Vus . gives Vus =0.2257(21)•. = 1.218 0.002 −0.024 , → (1|7) | Foflπlowing 1M1a±rciano, PRL 93 (2004) 231803, get Could improve Vus determination, even on current data set, wh•ere the first error is statistical and the second error is syste+m2a6tic, with the extrapolation by setting scale with fπ scale erroVruseli=mi0n.a2223(ted. − ) [0.2219(26)] error and fitting error added in q⇒uadrature. T|he e|rror in this la1t4tice QCD determination of f /f is clearly dominated byPthDeGsy(s2te0m0a6t)i,csf.rom K πµν C(hireal Dxpynamtic)s, Saeptn. 19d, 200(6n– po.20n-lattice) theory, K π • → Using a similar procedure, wgeivaersrivVeusat=a0v.2257(alue fo21)r L.5:

phy +0.18 −3 L (fCo)ul=d i5m.6p5rov0e.0V2u−s determ10inatio, n, even on current (d1a8t)a set, 5• π ± 0.54 × by setting scale with fπ scale error eliminated. where the first error is statistical and the second is an ⇒estimate of the systematic error phy Chiral Dynamics, Sept. 19, 2006 – p.20 due to omitted higher orders in the chiral expansion. This then scales to give L5(mη ) = +0.18 −3 phy +0.18 −3 2.22 0.02 − 10 at the η-mass and L (m ) = 1.42 0.02 − 10 at the ± 0.54 × 5 ρ ± 0.54 × ρ-mass. As stated previously, this is an effective L5 as it includes the higher order strange quark contribution. The results for fK /fπ have an additional systematic error due to the non-zero lattice 2 spacing which we expect to be O((ms mu)b ). In principle one can reduce this error by fitting to the appropriate χPT formu−las that include the O(g2b2) effects due to flavor- symmetry breaking in the sea-quark sector [24]. However, our data fit well to the continuum χPT formulas and hence we do not expect that use of the extended χPT formulas of Ref. [24] would significantly improve our results at this stage. Our final result is consistent with the MILC number [1] f K = 1.210 0.004 0.013 , (19) f ! ± ± π !MILC ! where the first error is statistic!al and the second is the total systematic error estimated by MILC. Since our valence quarks are domain-wall fermions, in contrast with the KS

11 Preliminary Results: Low Energy Constants

Preliminary ResAndults: LoPwreElnimeri−gn3yaCryonRsteasnutslts:(preliminarLow Eneyr) gy Constants Also get (in units of 10 , at chiral scale mη): • −3 Also get (in units of 10 , at chiral scale mη): −3 • 2L6 AlsoL4g=et0(i.n5(u1)ni(t2)s of 10[0.5(, a2)t ch(4)ir]al scale mη): • − 2L8 L5 = 0.1(1)(1) [ 0.2(1)(2)] − 3 2L6 L4 = 0.5(1)(2)− [0.5(−2)(4)] − units 10 − 2L6 L4 = 0.5(1)(2) [0.5(2)(4)] 2L8 L5 = 0.1(1)(1) L[ 40.=2(1)0.(1(2)2)] (3) [0.2(3)(3)] − − − − 2L8 L5 = 0.1(1)(1) [ 0.2(1)(2)] L4 = 0.1(2)(3) L[05.2(=3)(23).0(] 3)(−2) [1−.9(3)(3)] − L5 = 2.0(3)(2) [1.9(3)(3)] L4 = 0.1(2)(3) [0.2(3)(3)] where errors are statistical & systematic. where errors are statistical & systematic. L5 = 2.0(3)(2) [1.9(3)(3)] Also look at various chiral limit quantities: Also look at various chiral limit quantities: • • where errors are statistical & systematic. Define fχ,2 as dDecaefiny coe nfstχ,a2nat isn dtheecatwyo-coflavnostr achnirtailn the two-flavor chiral • • Also look at various chiral limit quantities: limit: mu, md li0m; itm: smfixu,emd adt physi0;caml vaslufixe.ed at physical value. → • → Similarly fχ,3 is decay constant in thDeetfinhreee-fflaχ,v2oar schdireacal y constant in the two-flavor chiral • Similarly fχ,3 is• decay constant in the three-flavor chiral limit: mu, md•, ms 0. limit: mu, md 0; ms fixed at physical value. lim→it: mu, md, ms 0. → fχ,3 is more poorly determined sinceS→iimt rielaqruliyrefsχl,o3nigsedr ecay constant in the three-flavor chiral • f" is more poorly determined since it requires longer extrapolation m χ,3 0. • • s → " limit: mu, md, ms 0. extrapolation ms 0. Chiral Dynamics, Se→pt. 19, 2006 – p.21 f→χ,3 is more poorly determined since it requires longer • " Chiral Dynamics, Sept. 19, 2006 – p.21 extrapolation m 0. s → Chiral Dynamics, Sept. 19, 2006 – p.21 sector. In this paper we compute the leading finite-volume dependence of the axial-vector charge of the nucleon in heavy-baryon χPT (HBχPT), including the ∆-resonance as an explicit degree of freedom. The finite-volume corrections to the axial-vector charge of the nucleon depend on the ∆-nucleon mass splitting and on the chiral-limit values of the nucleon, ∆- nucleon and ∆ axial-vector charges. Traditionally, the nucleon and ∆ axial couplings have been estimated using the spin-flavor SU(4) symmetry of the quark model, and in recent work [34] the authors have conjectured the chiral-limit values of these couplings. We point sector. out that lattice QCD measurements of finite-volume effects in the axial-vector charge (and In this paper we compute the leading finite-volume dependence of the axial-vector charge mass) of the nucleon will provide a clean determination of the nucleon and ∆-resonance axial-vecotforthcoeunpulicnlgeso.n in heavy-baryon χPT (HBχPT), including the ∆-resonance as an explicit degree of freedom. The finite-volume corrections to the axial-vector charge of the nucleon depend on the ∆-nucleon mass splitting and on the chiral-limit values of the nucleon, ∆- II. THnEucNleUonCLanEdON∆ AaxXiaIlA-vLecCtoHrAchRaGrgEesI.NTrAadFitIiNonITalEly,VtOheLUnuMclEeon and ∆ axial couplings have been estimated using the spin-flavor SU(4) symmetry of the quark model, and in recent At onwe-olrokop[3l4e]vethl,etahuetmhoartsrihxaevleemcoentjescotuf rtehde tahxeiaclh-vieractl-olrimcuitrrveanltuebsetowf etehnesneuccoleuopnlsinogfs. We point flavor “ao”uatntdha“tb”lamttaicyebQeCwDritmteenaasus rements of finite-volume effects in the axial-vector charge (and mass) of the nucleon will provide a clean determination of the nucleon and ∆-resonance Nb jµ,5 Na =Nucleon[ Γab + cab ] 2 UaxialbSµUa ,charge g (1) axial-vector cou!plin| gs. | " A where cab represents a counterterm with a single insertion of the light-quark mass matrix. 3 The leadIiIn.g-oTrdHerELNagUraCnLgeEdOeNnsiAtyXdIeAscLribCinHgAtRheGiEnteIrNactAionFsINbeItTwEeenVtOhLe UpiMonEs and the 1.4 low-lying baStaryonsgisgered Sea quarks 1.4

At one-loµop level, the matarbicx,ν eldements of the axial-vector cνurrµent between nucleons of = 2gA NS AµN + g∆N T A Nb "cd + h.c. + 2g∆∆ T S AµTν . (2) 1.2 LflavDomainor “a” and “b” Wmayalble valencwritten aa,νs e ! " 1.3 LHPC ‘05 This Lagrange density gives rise to the diagrams in Fig. 1, whexpt.ich are the leading one-loop 1 (see talks by Negele,Nb jOrginosµ,5 Na =) [ Γab + cab ] 2U bSµ•Ua , (1) contributions to the axial-current!ma|trix e|leme"nts. In the isospin limit one finds [35] 0.8 A 1.2 A g g where cab rep4resent3s a counterterm with2 a single25in2sertion of the light-quark mass matrix. 0.6 LHPC/MILC ΓNTNhe=legaAdingi-or2de4rgLAaJg1r(amnπg,e0d, µen)si+ty 4degsc∆rNibgiAng+theg∆inNtegr∆a∆ctioJn1s(mbeπt,w∆e,eµn) the pions and the LHPC/SESAM − 3f # $ 81 % RBCK 1.1 0.4 QCDSF/UKQCD low-lying baryons is 3 32 2 L = + gA R1(mπ, µ) g∆N gA N1(mπ, ∆, µ) (3) Experiment ∞ 0.2 µ 2 − abc9,ν d & ν µ = 2gA NS AµN + g∆N T Aa,ν Nb "cd + h.c. + 2g∆∆ T S AµTν . (2) 1 L 1 1 1 0 where J (m, ∆, µ), R (m, µ) and N (m, ∆, µ) a!re loop integrals defined i0n"the A0.1ppendi0.2x and 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 2 2 2 2 ∆ is theT∆hChiral-insuLcalegornanm gExtrap.aessdesnpsliitttyin ggati.veA sfilrlnitiΓse["]teop otvolumehlees dhiaavgerabmeesninsuFbtirga.c1te, dw. hTichheyar—e athned ltehaemdirin g(GeVone-)loop m (GeV ) ! ! associatecdonctoruibntuetritoenrsmtocNtNh—e anxeiaeld-cnuortrecnotncmerantruixs ehleerme eansttsh. eInfintihte-visoolsupmine cliomrrietcotinoensfinddos [35] not depend on the ultraviolet behavior of the theory at leading one-loop order. All of the FIG. 1: Nucleon axial charge gA as a function of the pion FIG. 2: Comparison of all full QCD calculations of gA, as couplings (includLing f) in eq. (43) take3 their chiral-limit value2s. 25 2 ΓNN = gA i 2 4gA J1(mπ, 0, µ) + 4 g∆N gAm+ass. Lga∆tNticge∆d∆ata aJr1e(mdenπo,t∆ed, µby) squares (smaller volume) described in the text. The solid line and error band denote Using the notation δL (ϑ−) 3fϑ(L# ) ϑ( ) to denote th$e finite-vaonldu8ma1etrciaonrgrleect(%ilaorngsertovotluhmee), the lowest smaller volume the infinite volume χPT fit of Fig. 1 and its continuation to ≡ − ∞ N N N N quantity ϑ, and the results obtained in th3e Appendix, the fin3it2e-v2 oplouimnteiscodrirspelcatcieodnsslitgohNtΓlyNtNo theNrightN for clarityN, and ex- higher masses is indicated by the dotted line. + gA R1(mπ, µ) g∆ pegrAimNen1t(ims dπe,n∆ot,eµd )by the circle. The (h3ea)vy solid line and are N N N N N 2 − 9 shaded error band show& the (a)χPT fit to(a)the fin(b)ite volume d(b)ata 2 m 25 evaluated in the infinite volume limit,(a)and the lines below(b)it 1 π 1 3 2 1 2 2 δL (ΓNwNh)ere Jδg(Am=, ∆, µ), R (gmF, µ1)+andg∆Ng(Am+, ∆, µg∆) agre∆∆looFp 2in+stheoggwArFatlh3se+dbeegfih∆anveiodgrAioFnf4thhNies,(cA4h)ipraplefintNdiinxbaNonxeds of finiNte volume + 2 2 A N N 3 N N N N N ∆ is≡the ∆-nu3clπeofn m# ass split$ting. All Γ81["] poles %have Lbe,eans Lsuibs trredauccteeddt.o T3&.h5,e2y.5—, anadnd1.6thfmeirespectively. that the corrections applied in correcting our data from (c) (d) 2.5 or 3.5 fm to infinity in the χPT fit are quite small. At associated counterterm c — need not concern us here as the finite-volume correNction(c)s∆do N N (d) N where NN heavier masses, although the truncated χPT expansion is not depend on the ultraviolet behavior of the theory at leading one-loop order. All of(c)the (d) N N not quantitatively reliable, the finite volume effects are coupBeanelings (inc l&ud iSang fva) ige,n eq .‘K04(31()mtaLkne t)heiDetmoldr chiral-limit v&al uLin,es. ‘05 +w.fn. F1(m, L) = K0(mL n ) | | ; tween two nucleons, two Deltas, or a nucleon and Delta, physically suppressed. The order of magnitude of the (e) N N Using th=e n(otation δ|L|(ϑ−) mϑ(LL)n ϑ() ) to denote rtehsepefictniivteely-v),oalunmd ea ccoournrteecrttieornms Cto. the corrections from 1.6 to 2.5 fm is also consistent with the n'! 0 ≡ | −| ∞ quantity ϑ, and the results obtained in the Appendix, theFfiignuirtee-1voshluowmsethceorlraettcitcieondastatoanΓdNoNur fit(e)to it usNing fact that quenched calculations[6] have shown that in- are 3 finite volume χPT. TNhe χPT fuNnctiNon was fitN to each creasing the box length from 1.2 fm to 2.4 fm increases (e) data point at the correspon(f)ding mass and finite volume, gA by the order of 10% for pion masses ranging from 550 m2 25 (g) π 3 2 2 and the parameters of the2fit wNere then usedNto deNtermine tNo 870 MeV, and unquenched calculations for 770 MeV δL (ΓNN ) δgA = 2 2 gAF1 + g∆N gA + g∆N g∆∆ F2 + gAF3 + g∆N gAF4 ,(4) the inFIfiGn. 1i:teOnve-ololoup mgraephsatxhaitaclonctrhibautregtoet.heImnattrihx eeleambenstseonfcthee aoxfial acutrtreinctein the npucileoon.s[7] have shown that increasing L from 1.1 to 2.2 fm ≡ 3π f # $ 81 % & A solid, thick-solid and dashed line denote Na nucleon(f), a ∆-resonNance, andNa pion,(g)respectively. NThe calculations of gA at still lower pion masses, it is not increases gA by 20%. solid-squares denote an axial coupling given in eq.(2), while the crossed circle denotes an insertion where preseonf thleyaxpialo-vsecstiobr cluerretnot opderoatoar. Dcioagmrampsle(at)eto(f)e(ex) tareavperotelxacotriroecntionfsr,owmh(g)ile diagramsI(nf) addition to the statistical error arising from fitting and (g) give rise to wavefunction renormalization. lattice mFeIaGs.u1r: eOmnee-nlotosp garlaopnhset.haHt ceonntcreib,utfeotlolotwheinmgatrRixeefl.em[2en0t]s,of thetahxeiapl caurraremnteitnerthsegnAuc,lego∆n.∆, and C, several systematic errors K1(mL n ) we perforAmsoeldid,athcicokn-ssotlirdaaindeddasfihtedalninde dtehneothe eaanvuyclesoon,liad∆c-urersvoenance,manadyabpieone,srteismpeacttiveedly.. TThee three constrained parameters can F1(m, L) = K0(mL n ) | | ; soFlIiGd-.sq1u: aOrensed-leonooptegraanphasxitahlatcocuopnlt2irnigbugt2ievetno itnheeqm.(a2t)r,ixwehlielme ethnetscorofstsheed acxiriaclecduernreontetsinanthiensneurctileoonn. is determined by setting1f , nm − m , and g to their n=! 0 ( | | − mL n ) K (mπL ) ∆ ∆ Nm N∆ be calculated directly on the lattice and the linear re- F2o(mAf t,hs∆oe,liLad)x,i=athl-ivcekc-tsoorlidcuarnredndt|ao|sphee+rdatloinr.e−Dd2einaogKrte0a(mmasLn(unac))letoon(,ea) ∆ar-erevseornteaxncceo,rraencdtioanpsi,ownh, irlesdpeiacgtirvaemlys. (Tf)he ' | | − = " mL n m | | physical vsaolliud-essqu[2a0re]sn!ad! en0ndotepeanr|fao|xrimal icnougplainglegaivsetn sinqueqa.(r2e)s, wfihtilefotrhe crossspedocnirsceleodfenoouters cahn iirnaselrfitiotnto varying each of of them shows and (g) give rise to wavefunction renormaliza2tion.2 ∆ ∞ 2β K0(βL n ) + (∆ m )L n K1(βL n ) gA, g∆∆, oafnthde aCxi.al-Tvehcteorecrurrorrenbt aopnedrataorr.isDiniaggrafrmosm(a)thtois(et)hareeevertex corrections, while diagrams (f) 2 dβ | | 2 −2 2| | | | ; very weak dependence. Calculation of fπ on our lattices − m m √β + ∆ m $ 3 parametearnfid t(gi)sgisvhe orwisentoinw# aFveifgu.nc1tioannrdenotrhmealrizeastuio−lnt.ing value 3 K1(mL n ) yields 92.4 MeV ± 3%, corresponding to an error in gA F3(m, L) = | | ; 2 2 for the axial charg−e2an=!t0thmeLpnhysical pionn mass is gA(mπ = of 0.10%, and a rough calculation of m∆ − mN , (which ! | | K1(mL ) ∆ m F2(m, ∆, L) = −mL|n|| | +2 2 − K0(mL n ) 140 MeV) = 1.2128±0.0K814(m. LGni)venπethe sm∆oothm be∞2havioβ rKo0(fβL n ) − n=! 0 " mL n m | | can be improved) with 18% uncertainty corresponds to F4(m, ∆, L) = | | −2 2 dβ 2 2 2 | | 2 ,(5) 9 " mL!n −K21∆(Lm|nL| n− ) m ∆ ∆ m m√β + ∆ m $ the chiral fit and thn=! e0 small magnitu∞de of the c#hiral logs2, 2an error of 0.29%. Although we have not yet calculated F2(m, ∆,!L) = | | ∆ | | | |2β +K0(βL−n2 ) +K(0∆(mL−nm))L n K1(βL n ) − " mL n m | | it is clear that the extraponl=! a0tion2 fordβthe axial c| h|arge2 is−2 2| | | | ; and Kα(z) is a modified Bessel fu!nctiomn of thme s|eco|nd kind. The exte√nsion of this regsuNlt ∆to, it should be calculable to 20%, corresponding to − # β + ∆2 m2 $ quitePQbQeCnDig, innc,luadinngdstrtohnag itsostphineviloalatttioi∆nc,eis sdt∞raaitgahtf2oisrβwaeKrxd0tt(orβaeLxptnroacl)ta+ftroi(nm∆gRef. [−3m6] u)sLingn K1(βL n ) 3 K1(mL n ) an error in gA of 0.13 %. Thus the total error from the the results of this paper. We do not give an asympdtβotic expression |for|δgA as we−do not fin|d | | | ; F3(m, L) = 2 | | ; 2 2 2 conviitnucsienfugl lfoyr Lto

4 Extrapolations for Chen, O’Connell, Van De Water, Walker-Loud ‘06 Soft - Collinear Eﬀective Theory

Λ power counting parameter λ = 1 Q !

Bauer, Fleming, Luke, Pirjol, Stewart 7

VII. NRQCD 7

VII. NRQCD e+e− positronium (NRQED) 7 → 7 pe− Hydrogen (NRQED) → VII. NRQCD b¯b, cc¯ VeI+Ie. − NΥR, JQ/CΨD positr(oNnRiuQmCD)(NRQED) 7 → → 7 tt¯ pe− e+e− tt¯ Hydr(oNgeRnQCD) (NRQED) → V→I→I. NRQCD VIIb.¯b, cNc¯RQCD Υ, J/Ψ (NRQCD) NN e+e− deute→ropnositronium(few(nNuRcQleEoDn) EFT) e+e− →positro¯n→ium (NRQE+D−) ¯ − tt e e tt (NRQCD) (66) − → pe →Hydrogen → (NRQED) pe ¯ HyNdrNog→en (NRQdEeuDt)eron (few nucleon EFT) →e+eb−b, cc¯ p→osiΥtr,oJn/iuΨm (NRQ(NERDQ) CD) e+b¯be,−cc¯ poΥsi,tJro/nΨiu→m (NR(NQREQDC) D) − ¯ → + − ¯ (66) →→ pe tt+ − Hyderoegen tt (NRQ(NERDQ) CD) pe−tt¯ Hyedreog→en→tt¯ (NR(NQREQ→DC) D) →→b¯b, cNc¯ N → Υ, Jd/eΨutiepr·oxn (NR(QfeCwDn)ucleon EFT) b¯bN, ccN¯ Υ,dJe/uΨt→eψro(→xn) = (N(RfeeQwCDnu)ψcple(oxn)EFT) (67) ¯ + − ¯ (66) →→ tt + − e !ep tt (NRQCD) tt¯ e e →tt¯ (NR→QCD) ip·x (66) → NN → deuteronψ(x) = (fewenuclψeopn(xE)FT) (67) µ 2 NN deuter→on (few nucleon E!FpT) i∂ ψp(x) (mv )ψp(x) → (66) ∼ ip·x (66) µ 2 ψ(x) = e ψp(x) (67) i∂ ψp(x) (mv )ψp(x) ip·x ∼ ψ(x) = e ψp(x) !p (67) NRQCD !=p ultrasoft +ip·xpotential + soft (68) µ 2 ip·ψx(x) = e ψp(x) (67) i∂ ψp(x) (mv )ψp(x) Lψ(x) = e L ψ (x) L L (67) µ 2 NpRQC!Dp ultrasoft potential soft i∂ ψp(x) (mv )ψp(x) ∼ = + + (68) ∼ !p L L L L µ 2 i∂ ψp(x) (mv )ψp(x) µ 2 NRQCD ultrasoft potential soft i∂ ψp(x) (mv )ψp(x) ∼ VIII=. SCET+ + (68) ∼ NRQCD = ulLtrasoft + potLential +VIsIoIfLt. SCET L (68) L L L L NRQCD = ultrasoft + potential + soft (68) NRQCD Process Non-Pert. functions= UuLlttrialsiotfty+ poLtentiValI+II. soLfSt CET L (68) 0 + − Process L Non-PerLt. functionLs UtilityL B¯ D π , . . . ξ(w), φπ stVuIdIIy. QSCCDET → B¯0 D+π−, . . . ξ(w), φ study QCD ¯0 0 0 Process + Non-Pert. fuπnctions UtiliVtyIII. SCET B D π , . . . ¯S0 (→kj )0, φ0π + VsItIuId. y SQCCETD Proc→ess ¯0NBon-P+erDt−. fπun,c.ti.o.ns S(Ukjti)l,ityφπ study QCD endpt B D→+π , . . . ξ(w), φπ study QCD B0 X+s − γProc→esfs (k )endptNon-Pert. f+unctionnesw pUhtiylistyics, measure f B¯ D π , . . .¯0ξ(Bw), φ0Xπ0s γ +f(kstu)dy QCD new physics, measure f Pro→cess endpt ¯B0Non-→DP+erπ+−t. ,fu. .n.ctioSn(skj )U, tφilπity study QCD B¯0 → X0 0 'Bν →Df+(kπ ,)e.n.d.ptξ(w), φπ + meassutruedyVQuCbD B 0 D+uπ −, . . . S→B(k e)n,Xdφpπut 'ν +f(kstu)dy QCD measure Vub B¯ → D π , . . .B0ξ(w)Xj, 0sφ 0 γ f(+k )study QCD ne|w ph|ysics, measure f → B¯ D→ππ , .+. . S(k ), φπ + study QCD | | B → πe'nνdp,t. . . → φ+Ben(dkpt ), φπ(jx)+, ζπ(E) measure Vub , study QCD B¯0 Xs0 0 γ f→B(k+ ) π'ν, . . . φBn(ekw )p,hφyπsi(cxs),,mζeπa(sEu)remfeasure Vub , study QCD B → D π , . . . BS(kjXe)nu,dpφtπ 'ν f(+k )study QCD m|easur|e Vub → endpt +B −→X+s→ γ f+(k−) new physics,|mea|sur|e f | B → γe'nνdp,t γ' '→B+φ γ'ν, γ' ' φ+ measure φ , mneawsuprheyφsic,snew physics BB Xu γ'ν Bff((kkπe)'Bn)νd,pt. . . φB+(k nB)em,wφeaπpsh(uxyr)s,eicζsπV, (umEbe)asmureeaBsfure Vub , stuBdy QCD →→ s B Xu→ 'ν f(k ) | m| easure Vub → → + + − | | BB ππ'eνπn,d,.p.tK. π,B.→φ. B.B+(φγk'ν,,)π,γφπφ',π,K('xζπ),,(φζE.π.()+.Eφ)Bm, eφaπs,uζrneπe(wEVu)bph, ysmtsueidcasyu,|QnrCeeCwPφ|Dpv,hinoyelswaictspio,hnCy,sPicsviolation, B Xu 'ν B f(kπ'ν)B, . . . π φπB(Bk ),mφeπa(sxu)r,eζπV(uEb) measure Vub ,Bstudy QCD →→→ + − →→ → | | | | | | BB γπ'ν,,.γ. '. ' BφφB(πk+π),,Kφ+π(,−x.).,.ζφ(E, )φφm¯,me,ζaeζsauK(sEru(e)EreV)φB,,sntuenwdeywpQhpCyhsDyiscstsiucsd,yCQPCvDiolation, → B Bγφ'νK¯, γ, 'ζπK' (Eφ)BπB πK π sutbudmyeaQsuCreDφB, new physics → + − →→ ∗ | ⊥ | BB πγ'πν,,∗Kγ'π,'. . . φφBB, φπK, ζπγ(E, ρ)γ⊥φ ¯ , ζφm(ne,Eaeφws)urpe,hζφysi∗,c(snE,eCw) PpshvtyiuosdilcaymstiQeoanCs,Dure V /V B→ K γ, ργB Bπφπ,BK, πφ,K. .,. ζφB∗,K(φEπ,)KζBπ(EK) mKBeasnuerwe pVhytdsi/cVs,tCs P violattdion,ts → → →∗ K ⊥ | | → φ ¯ , ζ (E) stu∗dy QCD | | B ππ, Kπ, . .B. φBK,KφπK,γζ, π⊥ρ(γE) φK¯φ,Bζ,Kφφ(KnEρe,,)wζKρph((EyEs)i)cs, CPstvmuidoelyaastQuioCrneD, Vtd/Vts → ∗ → φ∗ρ, ζρ ⊥(E) ⊥ ⊥ | | B K γ, ργ B φφ¯BK,,ζφKγK,(Eρ, γζ) +∗ (E−φ)φ, φ, ζ , (ζsmEt+u∗e)(daEysu)QreCDVtdm/Veatssure V /V → + − KB X+sK' ' B ρ Kfρ (kK ) | | ntedw ptshysics B X∗ s' ' → f→(⊥k+ )⊥− ⊥+ new physics | | B K γ, ργ BφφBρ,−,XφζρKs'(,Eζ'K)−∗ (Eφ)ρf, (ζkρ ()Em)easure Vtd/Vts new physics −→→ − e p e X fi/p(ξ), f|g/p(ξ) | study QCD , measure p.d.f’s + − −→ +⊥+→− − + Be p Xs'e 'X Be φfpρ(,Xkfζsρi'e)/(pE'(X)ξ)+, f−gf/(fpki(/ξp)()ξ),nfegw/pp(ξhs)ytsuicdsyneQwsCtpuDhdyys,iQcmsCeDas,umreapsu.rde.fp’.sd.f’s →pp¯ X' ' fi/p(ξ), fg/p(ξ) study QCD Electroma−→ →gnetic−++− −Probes− →+ − + − epBp¯p XXse' 'X' ' epp¯pff(ik/−pfXe()→iξ/')Xp,('fξg)/−,p(fξ0gf)/i/fppi((/ξp)(),ξn)f,ges/wftpug(/pdξph)y(yξQss)itcCusDdy,stmQudseCtyausDQduyCreDQpC,.dDm.fe’assure p.d.f’s ? → →e γ e π φπ measure φπ − →→ −+ − − → +−− 0 pep¯−p Xe' X−' 0 pep¯ ffγi/pX∗((ξ'ξ)e),→',fπfg/p(ξ($)ξf)i/φp(ξ), sftgsu/tpdu(ydξ)yQCQDCD, msteuadmsuyeraQesCpu.rDde.fφ’s e γ→ e π →i/pφπ g/p π "measure φπ π − → −+ − −∗ γ→M− 0$ M φM , φM study QCD pp¯∗ →X' ' 0 $ e γfi/p(eξ),π→fg/p(ξ)φΥπ → stu"dy QCD measure φπ mW e γ→ e π γ φM→π+ −M " φM , XφMγmeasure φπ study QCD γ− M −M0 ∗ e φ→eM ,$ φMjets " study QCD study universality e∗ γ → e π$ γ +Mφ − M φM , φMmeasure φ study QCD γ M →M e φeπM , φMje→"ts study QCπ D study universality + −→ − →+ − e∗ e → j$ets + e →e " J/ΨX study universsatulidty QCD eγ+Me− Mjets ee+eφeM−, φjMeJt→s/ΨX stsutduydyQCunDiversstauldsittyyuudnyivQerCsDality ++ −−→→ + − → e+ ee− → jetJs/ΨeXe →J/ΨX study unsivteurdsayslittQuydCy DQCD mb Energetice e →pionJ/ (ΨprotX on) with→ collinear constituentsstudy QCD e+e− →→J/ΨX study QCD E → nµ ! mc

energetic ! ! n-collinear n-collinear ΛQCD jets jet jet

usoft particles m s B B - decays by weak interactions b mu,d Basic Building Blocks in SCET are collinear “parton” ﬁelds (can be derived starting from QCD and removing hard ﬂuctuations) matrix elements of these building blocks probe properties of hadrons

eg. Fourier Transform of standard twist-2 quark p.d.f Q

p (ξ¯nW )n¯/δ(w− Pˆ−)(W †ξn) p = fq(ω−/p−) − ! ! " " # " χn " • forward" m.elt. " parton ﬁeld with get GPD from: χ n,ω− momentum ω−

p χ¯n,ωn¯/χn,ω! p! p tr µ p = f (ω /p ) ! " " # gluon p.d.f. µ⊥ ,ω g − − " " B B⊥ µ " " ! " " # n " " ! twist-2 pion " " ω− π(pπ) χ¯n n¯/γ5 χn,ω 0 = φπ distribution pπ− ! " " # $ % hard interaction " " twist-2 proton " " creates a pion j ω− ω"− distribution p χ¯i χ¯ χ¯k Γijk 0 = φ , n,ω n,ω! n p or proton p− p− ! " " # $ % " " " " At leading order SCET reproduces well known factorization theorems.

Systematically improvable. So we can attack power corrections in a whole host of processes.

= (0) + (1) + (2) + . . . L L L L A ﬁeld theory for Soft & Collinear interactions

Qnµ + (Λ ) O QCD

ΛQCD Form Factors 3 !

proton

F Amendolia !+e elastics 2 , + GPD Regge Q 0.6 Previous p(e,e ! )n JLab 1997 pion

p 2

JLab 2003 1 F

Dyson - Schwinger / p

0.4 QCD Sum Rules 2 GPD modified Gaussian F 2

Constituent Quark Model Q 1 JLab data Expected error bars for JLab E04-108 0.2 JLab upgrade with SHMS

JLab @ 12 GeV (projected errors)

0 0 0 1 2 3 4 5 6 7 0 5 10 15 2 2 Q2 (GeV/c)2 Q (GeV )

2 2 2 f 1 2 p 2 fp p p Figure 11: Projected measurements of the pion electromagneticπ form factor,FigureFπ(Q6:),ThemadeJLabpossible6 GeV data on the ratio of proton form factors, F /F , and the projected Fπ(Q ) = Q2 0dxdy H(x, y) φπ(x)φπ(y) F1 (Q ) = 2 dxidyj H(xi, yj) φp(xi)φp(yj) 2 1 by the proposed 12 GeV Upgrade. Also shown are various model predictionsmeasuremenfor its behatsviorat 12in GeV.the PerturbativQ e QCD implies that F p /F p 1/Q2 for Q2 ∞. The graph 2 2 2 2 ! 2 1 region Q few GeV . Perturbative QCD predicts!Fπ 1/Q for Q illustrates∞. the potential of form factor measurements at 12 GeV to discriminate between GPD + . . . models. Shown are the predictions of a Regge–based GPD model [Gu04] and a model based on soft Valence quark structure. One of the most fundamental properties ofGaussianthe nucleonnucleonis the wstruc-avefunctions modifiedQ2Fbyp/Fa short–rangep = constaninteractiont [St03d]. ture of its valence quark distributions. Valence quarks are the irreducible kernel of each hadron, but 2 1 ! responsible for its charge, baryon numSCETber and other canmacroscopic be usedquantum tonum bers. In deep– inelastic scattering at average values of x, the valence quarks are “dressed”could bye quark-anmeasuredtiquarkup to 14 GeV2, probing the proton’s structure at distances as small as 0.1 fm. pairs produced by non-perturbativsysteematicaleffects at large distancely computscales ( 1Knofm),e wledgetheseas wellofasneutron by gluonform factors at high Q2 is equally important. With the 12 GeV Upgrade, bremsstrahlung at short distances.(A+...t higher) powx valueserthese correctionsqq¯ contributionsthe neutrondrop awmagneticay, and theform factor could be measured up to about 14 GeV2, and its electric form physics of the valence quarks is cleanly exposed [Is99]. factor up to 8 GeV2. While deep inelastic scattering and other experiments have provided a detailed map of the The 12 GeV form factor measurements will provide crucial constraints on the first moments of nucleon’s quark distributionsAsat waveerage lear( ned,0.3) and sosmall farv aluesQ^2of wax, theres nothas bignever enough,been an but with 12 the GPD’s and their t–dependence. Fig. 6 demonstrates the power of form factor measurements to experimental facility capableGeVof accurately upgrademeasuring thesethe cross correctionssections throughout bectheome“deep quitva- e interesting lence region” (x > 0.5) where the three basic valence quarks of the protondiscriminateand neutronbetdominateween GPD models. An equally important consideration is that form factors can be 2 the wavefunction. This represents a glaring gap in our knowledge of nucleonmeasuredstructure,up toesphighecially|t| (= Q for elastic scattering), while direct measurements of the GPD’s in 2 since there are qualitatively different predictions for the quark spin andexclusivflavor distributionse processesinaretherestricted to |t| ≤ 1 GeV . Thus, form factor measurements will continue to x 1 limit. The 12 GeV Upgrade will for the first time provide the necessaryprovidecomourbinationonly sourceof high of information about the quark distributions at small transverse distance beam intensity and reach in Q2 to allow us to map out the valence quarkscales.distributions at large x with high precision. These measurements will have a profound impact on our understanding of the structure of the proton and neutron. They will also provide crucial input for calculating cross sections for hard processes in high–energy hadron–hadron colliders such as the LHC, in searches for the Higgs boson or for physics beyond the Standard Model. GPD’s in deeply–virtual Compton scattering. GPD’s can be measured in certain exclusive processes in eN scattering, namely Deeply Virtual Compton Scattering (DVCS) and Deeply Virtual Meson Production (DVMP). At large Q2 and scattering energy, these processes are dom- Valence quark spin distributions at large x. The 12 GeV Upgradeinated willby theallowscatteringfor mea- from a single quark or antiquark, whose emission and absorption by the surements of inclusive spin structure functions at large x with unprecedennucleonted precision.is describedAs bany the GPD’s, see Fig. 7. The amplitude of the “hard” scattering process at n example, Fig. 12 shows the neutron polarization asymmetry, A1 , whichtheis determinedquark levelbycana ratiobe calculated in perturbative QCD (factorization). The measured cross sec- of spin-dependent to spin-averaged quark distributions. Most dynamicaltionsmoanddels predictspin asymmetriesthat in can then be used to extract the “soft” information about the nucleon the limit where a single valence up or down quark carries all of the nucleon’s momentum (x 1), n contained in the GPD’s.n it will also carry all of the spin polarization (i.e., A1 1 as x 1). Existing data on A1 end before reaching the region of valence quark dominance, and show no sign of making the predicted In the reaction eN eN at JLab energies, photons are produced not only via DVCS from the nucleon, but also (and even more copiously) via the electromagnetic Bethe-Heitler (BH) process. 17 The two processes interfere, and the BH term, which is completely determined by the well-known

12 which make them ill-defined, even in dimensional regularization. In previous computations, it has been argued that these pinch singularities should be dropped in evaluating box graphs at any order in v [9, 11, 12]. Pinch singularities are also a problem for the method of regions [13]. A direct application of the method of regions for d4k leads to ill-defined integrals, so it should only be applied to NRQCD after first doing the energy integrals. The zero-bin subtraction modifies the soft box graphs so that pinch singularities are absent, and the graphs are well defined.

2. The zero-bin subtraction automatically implements the previously studied pullup mechanism in NRQCD [14, 15], which was shown to be a necessary part of the deﬁni- tion of this type of theory with multiple overlapping low energy modes. Through the pullup, infrared (IR) divergences in soft diagrams are converted to ultraviolet (UV) divergences and contribute to anomalous dimensions.

3. There is a similar pullup mechanism at work in SCET for collinear diagrams. The anomalous dimensions of the SCET currents for endpoint B → Xsγ and B → Xu"ν¯ were computed in Ref. [1, 2] from the 1/$ and 1/$2 terms. Some of these terms in the collinear graphs are actually infrared divergences. The zero-bin subtraction converts these infrared divergences to ultraviolet divergences so that IR-logs in QCD can be resummed as UV-logs in the eﬀective theory. This formally justiﬁes the results used for anomalous dimensions in these computations, and in subsequent work for other processes with similar anomalous dimensions eg. [16, 17, 18, 19, 20, 21].

4. In high energy inclusive production such as γ∗ → qq¯g, there is a potential double counting at the corners of the Dalitz plot in SCET, which is resolved by properly 4 taking into account the zero-bin in both fully differential and partially integrated x) is suppressed by the Sudakov form factor, this Q2- observed consistency might be a sign of precocious scal- dependence reflects simply the kinematic broadening of ing as a consequence of delicate cancellations in the ratio. cross sections. the quark (and gluon) rapidity range with increasing nu- A more detailed discussion on this issue along with more cleon momentum. thorough phenomenological analyses will be given in a ∗ ∗ For an estimate, we use asymptotic wave functions [29], separate publication. 5. In high energy exclusive production, such as γ → πρ or γ → ππ, there are un- We thank V.M. Braun, H. Gao, G.P. Korchemsky, H.- physical singularities in convolution integrals of some hard kernels with the light-cone 3 = 120 x1x2x3 fN , 4 = 12 x1x2(fN + 1) , N. Li, and A.V. Radyushkin for useful discussions. This 4 = 12 x1x3(fN − 1) , (12) work was supported by the U. S. Department of Energy Endpoint Singularites 3 via grant DE-FG02-93ER-40762. −3 2 −2 2 wavefunctions φπ(x). For example with fN = 5.3 · 10 GeV and 1 = −2.7 · 10 GeV . where the argument on the left-hand side indicates the 2 6 p 2 With a choice of = (0.3 GeV) , Q F2 (Q ) is roughly 1 momentdxum being averaged. Averaging over k 2 does not 0.6 GeV6 for Q2 = (5 − 20) GeV2, about 1/3 of the φπ(x) yield independent information. Je erson Lab data at Q2 = 5 GeV2 [3]. Of course, to dx , = Using the tw=?ist-3 and -4 amplitudes, we can write the 2 2 get a more realistic PQCD prediction in this regime, one [1] H. Gao, Int. J. Mod. Phys. E12 (2003) 1. 0 x F2(Q0) foxrm factor in the following factorized form, [2] M.K. Jones et al., Phys. Rev. Lett. 84 (2000) 1398. must have the quark distribution amplitudes appropriate ! ! [3] O. Gayou et al., Phys. Rev. Lett. 88 (2002) 092301. 2 at this scale. However, from the comparison between the F2(Q ) = [dx][dy] x3 4(x1, x2, x3)T ({x}, {y}) ((72) ) 2 [4] V.M. Braun, A. Lenz, N. Mahnke, E. Stein, Phys. Rev. data and PQCD predictions for F1(Q ) [8], we believe D 65 (2002) 074011. At subleading order+ x1 4(onex2, x1, xoft3)T (en{x}, {y}) 3(y1, y2, y3) , that asymptotic PQCD is unlikely to be the dominant [5] G.A. Miller, Phys. Rev. C 66 (2002) 032201; G. A. Miller 2 2 2 which is divergent at x → 0 if φπ(x) vanishes linearly as x → 0. The same is true for contribution to F2(Q ) at Q = 3 − 5 GeV : one must and M. R. Frank, Phys. Rev. C 65 (2002) 065205. encounters endpointwhere {x} = ( xsingularities1, x2, x3). Let us see how to extract the take into account higher-order corrections and higher- [6] J.P. Ralston, P. Jain, hep-ph/0302043. exclusive light meson form factors at large Q2, as whealrdl paarst, pT ,rion Fcieg.s1s.es like B → π"ν¯ and twist e ects. [7] B.-Q. Ma, D. Qing, I. Schmidt, Phys. Rev. C 65 (2002) We use the nomenclature and strategy of Ref. [11] by 035205. FIG. 2: Perturbative diagrams contributing to the hard part calling the top quark line 1, the middle 2, etc., without [8] S.J. Brodsky, hep-ph/0208158. of F2 .0.3Mirror symmetric graphs have to added. B → ππ for Eπ " ΛQCD. The zero-bin subtraction icommmpilttiiengs tthehmatto athspeecsiefic kflaevrorn. eGlisvenmthuesstpinb- e !=400MeV [9] S.J. Brodsky, G.R. Farrar, Phys. Rev. Lett. 31 (1973)

Is this a breakdown of factorization? isospin structure of the initial and ﬁnal state nucleon 1 !=300MeV 1153; Phys. Rev. D 11 (1975) 1309.

/F !=200MeV treated as a distribution we call ø, and have a finite convolution with φπ(x): 2 [10] V.A. Matveev, R.M. Muradian, A.N. Tavkhelidze, Nuovo wave functions, we assume that the first and third quarks and T2 = T2 (1 it3), fiT3 ts!= T1 (1 3), an[2]d T3 = have spin up and the second one spin down. Call the ))F 0.2 [3] Cim. Lett. 7 (1973) 719. Belitsky, Ji, Yuan (‘03) T2 1 (1 3). The functions Gij are defined as ! [11] G.P. Lepage, S.J. Brodsky, Phys. Rev. Lett. 43 (1979) 1 1 1 amplitudes Ti when the e"lectromagnetic current acts on / 2 1 545; (E) 43 (1979) 1625; Phys. Rev. D 22 (1980) 2157. φπ(x) φπ(x) φπ(x)pa−rticlφe iπ. (C0le)ar−ly bxecφauπse(oxf )symmetry, T3 can be ob- (Q G11 = 2 2 ,

2 [12] A. Efremov, A. Radyushkin, Phys. Lett. B 94 (1980) 245. dx → dx = dx 2 ptained from T by a suitab2le exch

2 /Log ø 2 G12 = + 510; Phys. Rept. 112 (1984) 173. ! ! ! charge weighting, we find the flavor structure of the hard 2 x2x¯2y y¯2 x x x¯2y y¯2 ln 3 1 3 1 2 3 1 2 1 [14] I.G. Aznaurian, S.V. Esaibegian, K.Z. Atsagortsian, N.L. p 2 (Q part for the proton, 1 1 Ter-Isaakian, Phys. Lett. B 90 (1980) 151; (E) B 92 F ∼ Λ 0 + 2 2 2 − 2 , 1 2e e + e x¯ x y¯1y x2x x¯1y2y3y¯3 (1980) 371. 4 T p({x}, {y}) = u T + u d (T + T ) 0 1 1 23 3 3 34 5 6 3 1 3 2 3 1 [15] V.A. Avdeenko, S.E. Korenblit, V.L. Chernyak, Sov. J. ! " G = 2 , 2 from singularityeu ed 21 2 Q2 (GeV ) Nucl. Phys. 33 (1981) 252. x x3x¯3y y3y¯1 + (T1 + T3) + T2 . (8) 1 1 3 3 1 1 [16] C.E. Carlson, F. Gross, Phys. Rev. D 36 (1987) 2060. FIG. 3: GJL22ab=data p2lotte2d in+ter2m2s of the,leading(1P1Q)CD [17] N.G. Stefanis, Eur. Phys. J. direct C 1 (1999) 7. where we have omitted the argument of Ti on the right- x1x x¯2y y¯2 x x y1y3y¯1 scaling. The low, mid3dle, a3nd upp1er3data points correspond [18] A. Duncan, A.H. Mueller, Phys. Lett. B 90 (1979) 159; hand side, anneedd T has yrapidity1 and y3 inter,c h a n g e d,. dependentFor the distribution functions Resolution: i ζ to = 200, 300, 400, respectively. A.H. Mueller, Phys. Rept. 73 (1981) 689. neutron, u d. and G 21 = G21(1 3), G 22 = G22(1 3). 2 [19] A.V. Radyushkin, Acta Phys. Pol. B 15 (1984) 403. According to theφabove(cox,nventζion), we find the contri- To deterMminanohare the norma,l izI.Satio.n f(‘or06F2(Q) ), we need to [20] N. Isgur, C.H. Llewellyn-Smith, Nucl. Phys. B 317 (1989) bution to T1 from Fig.π1, knCowomthinegligbhatc-ckonteo dtishteribsuctaiolinngambpelhitauvdieosr o3f, t4heanrdatio 526. , w2hich can2only be obtained by solving QCD nonper- M 2C2 1 F24(Q )/F1(Q ) for which the Je erson Lab data has [21] H.-N. Li, G. Sterman, Nucl. Phys. B 381 (1992) 129. T fig.1({x}, {y}) = − B (4π )2 , turbatively. However, the scale evolution of these ampli- 1 Q6 s x¯2x y¯ y2 stimulated much discussion in the literature. PQCD [22] H.-N. Li, Phys. Rev. D 48 (1993) 4243. similar thing is known to happen1 3 for1 3 k tudependentdes selects at the asy mp.d.fptoticall’yslarg2e Q2 the leading [23] J. Bolz, R. Jakob, P. Kroll, M. Bergmann, N.G. Stefanis, 2 2 predicts the power-law scaling 1/Q . With the new fig.1 M CB 2 1 ⊥component with 2a fixed small-xi behavior. For example, Z. Phys. C 66 (1995) 267. T ({x}, {y}) = − (4π s) , (9) result for F2(Q ), we can determine its scaling up to fq(x,1 k , ζ) Q6for a rex¯ viex2y¯ wy2 see Collins (hep-ph/0304122) [24] C.R. Ji, A.F. Sill, Phys. Rev. D 34 (1986) 3350. 1 3 1 3 ltohgeaaristyhmmpitcotaicccfourrmacoyf. T3 hisexs1txr2oxn3g[1c1o],uwplhinergeacsotnhsattaonft in ⊥ 4 is x1x2. In Ref. [29] a set of phenomenological ampli- [25] S.J. Brodsky, S.D. Drell, Phys. Rev. D 22 (1980) 2236. where here and everywhere x¯ 1 − x. CB = 2/3 the ratio simply cancels. The wave function evolution tudes satisfying these asymptotic constraints have been [26] X. Ji, J.-P. Ma, F. Yuan, Phys. Rev. Lett. 90 (2003) is a color factor and the contribution from the twist-4 32 / (9 ) 2 2 8 / (3 ) 2 ypireoldpsosaedfaocntoar obfas is of con(fQorm) afolrexFp1a(nQsio)na.nd s (Q ) 241601. wave functions of the final-state nucleon has also been in- for F (Q2) from the leading non-vanishing contribution, [27] M. Burkardt, X. Ji, F. Yuan, Phys. Lett. B 545 (2002) cluded. The corresponding contribution to T can be ob- Wi2th the above wave functions, the integrals over mo- 3 where = 11 − 2n /3. Thus PQCD predicts that 345. tained by interchanging the labels, T fig.1 = T fig.1(1 3), mentum fractions xi anf d yi have logarithmic singulari- 3 1 2 2+8 / (9 ) 2 2 2 2 [28] X. Ji, J.-P. Ma, F. Yuan, Nucl. Phys. B 652 (2003) 383. fig.1 fig.1 (tQies,/ ilnndicating tQhat/ t he)(fFact(oQriz)a/tFion(Qbre)a)ksscdaolewsnaws haencon- T = T (1 3). This in fact is a general feature. 2 1 [29] V.M. Braun, R.J. Fries, N. Mahnke, E. Stein, Nucl. Phys. 3 1 one of the quar2ks in the wave function becomes soft To find a complete expression for the hard part, we stant at large Q , 8/(9 ) 1. Surprisingly, the Je erson B 589 (2000) 381. [19, 20]. It has been suggested that the higher-order must calculate perturbative diagrams displayed in Fig. Lab data plotted this way (ignoring the small 8/(9 )) [30] H.D. Politzer, Nucl. Phys. B 172 (1980) 349. PQCD resummat2ion, or the Sudakov form factor, sup- 2. A straightforward evaluation yields, exhibits little Q variation for a range of choices of [31] I. Musatov, A. Radyushkin, Phys. Rev. D 56 (1997) 2713. presses the contribution at small-x and provides an ef- [32] S. Descotes-Genon, C.T. Sachrajda, Nucl. Phys. B 625 2 2 as shown in Fig. 3. Since we do not exp2ect 2the asymp- M CB 2 fective cut-o for the integra2ls at x /Q , whe2re T ({x}, {y}) = (4π ) x (G − G ) , (10) totic predictions for F1,2(Q ) to work at these Q , the (2002) 239. 1 Q6 s 3 11 12 is a soft scale related to the size of the nucleon M 2C2 [21, 22, 23, 31, 32]. The outcome is that the xi integra- T ({x}, {y}) = B (4π )2 [−x G − x¯ G ] , 2 2 2 2 1 Q6 s 1 11 1 12 tions contribute an extra Q -dependent factor ln Q / , compared to F (Q2). Physically, the end-point diver- M 2C2 1 B 2 gencies indicate that quarks with di erent rapidity con- T2 ({x}, {y}) = 6 (4π s) Q tribute equally to the hard scattering. Since the con- × [x3(G22 − G 21 − G 22) − x¯3G21] , tribution from quarks with very large rapidity (small- Test SCET in B-Physics where Q is a bit bigger important Form Factors for measuring B ππ CP-violation p2 2 → ~ Q M' p 2 ~ !2

B M p2~ Q2

2 2 2 2 B p ~ Λ p2~ QΛ p ~ Λ M

2 2 p ~ ! p2~ Q! p 2 ~ !2 Single Jet Production

dΓ 2 = C(m , µ) dk+ImJ (k+, µ)S(2E m + k+, µ) dE b P γ − b γ " ! ! ! ! + 4 ik x J (k ) = Disc d x e− · 0 T χ¯ (x)n¯/ χ 0 Q ! | n,Q n| " ! jet of energy Q Test SCET in B-Physics where Q is a bit bigger important Form Factors for measuring B π"ν¯ B ππ CP-violation → p2 2 → ~ Q M' p 2 ~ !2

B M p2~ Q2

2 2 2 2 B p ~ Λ p2~ QΛ p ~ Λ M

2 2 p ~ ! p2~ Q! p 2 ~ !2 Single Jet Production

B Xsγ dΓ 2 = C(m , µ) dk+ImJ (k+, µ)S(2E m + k+, µ) → dE b P γ − b B Xu!ν¯ γ " → ! ! ! ! + 4 ik x J (k ) = Disc d x e− · 0 T χ¯ (x)n¯/ χ 0 Q ! | n,Q n| " ! jet of energy Q B¯0 D0π0 Factorization, Comparison to Data →

(Cleo, Belle, Babar) Mantry, Pirjol, I.S.

2.0 color allowed A(D*M) color suppressed D0$0 0 A(D M) # + # D # * 0.8 isospin triangle 1.5 0 D0!0 D " 0 0 0 = D # D K D "' * 0.6 = D* #

1.0 0.4 3 A 00 RI 0 - 0 - 2 A0_ D ! D % 0.2 + - + - + - D ! D % D $ 0 D $- " 0.5 ! 0 0 0.2 0.4 0.6 0.8 1 LO SCET prediction δ(Dπ) = 30.4 4.8◦ 0.0 ± δ(D∗π) = 31.0 5.0◦ ± DIS for large x

0.6 x = 1 20 0.8 0.5

GeV 2 15 > 0.6 W 0.4 ) 2

10 0.3 0.4 (GeV 2

Q 12 GeV 0.2 5 Upgrade 6 0.2 4 2 2 0.1 Q > 2 GeV JLab projected data 0 0 2 4 6 8 10 12 2 0 M! (GeV ) 0.2 0.4 0.6 0.8 1.0 x 2 Bj Figure 3: The kinematic coverage in momentum transfer, Q , as a function of the mass of (M ) and energy transferred to ( ) the target by the electron. This plot conveniently identifies the deep Figure 13: Projected measurement of the ratio of d– and u–quark momeninelastictumJlabscatteringdistributions, atregime 12 GeVas a triangle maybounded welonlthe proberight by thethisbeam region,energy. The fraction of 2 the proton’s momentum carried by the struck quark (given by the Bjorken variable, x) appears as d(x)/u(x), at large x, made possible2 by theQ12 (1GeV Upgrade.x) The shaded2 radialbandlines onrepresenthe plot.tsFwilromthethisl therefigure w ebecan infersurprises?the range of x and Q2 accessible to deep– uncertainty in existing measuremenmXts due=to nuclear F−ermi motion+ meffects.p inelastic eN scattering with the 12 GeV Upgrade of CEBAF and the large enhancement relative x to the original CEBAF energy of 4 GeV. a Jet! Collaboration, most of the experiments so far have focused2on measuring2QCDthedegrees-of-freedomtotal quarkbandecome visible. Kinematically this regime has been accessible before, for gluon contribution to the nucleonForspinxin inclusive(1),deep–inelasticmX scattering,Qexample, sinitiatingtatandarSLAC.nAnumerousessend DIStial feature of the Upgrade is that it provides not only the energy but ∼ O ∼ also the luminosity and polarization required to study low–rate processes such as deep–inelastic theoretical investigations of the role of gluon topology and the U2(1) axialscatteringanomaly2 at largeinx,hadronexclusiv2 e and semi-inclusive reactions, and measurements of polarization structure [Ba02, An95]. FollowingForon thesex results,1 inΛrecen/Qt y,earsΛthe foasymmetries.cusmhasXbeenAs anmoQexampleving,toofethendpnew reacoinh afftordedDISby the Upgrade, Fig. 3 shows the kinematic the investigation of specific aspects of the nucleon∼ spin,− such as the flavor#asymmetriescoverage for measuremen#of sea quarkts in the deep–inelastic region. The accelerator design will be comple- 2 2 2 mented by 2a set of detectors specifically designed to cover this kinematic region. Together, they distributions and quark transverseForspinx(transv1ersity) Λdistributions./Q , Λ will suppmortXa broad, notrange dof inclusiveepe, inelasticsemi-inclusive and exclusive measurements that will pro- ∼ − duce∼ a comprehensive picture of the quark and gluon structure of the nucleon. No other facility Properties such as these are of particular interest, as they are free ofpresencontributionstly available orinplannedvolvingoffers comparable capabilities for carrying out the envisaged research the axial anomaly, which affect the flavor–singlet quark spin distributions.program.They can therefore be related more directly to conventional models of nucleon structure. For example, the non-relativistic quark model, the meson cloud model, and the chiral quark–soliton model, 3makD qeuavreryk/gldiuoffnerenimatgipre-ng of the nucleon. It has long been known that the form factors dictions for the flavor asymmetry of the polarized sea quark distributions.ofTheelasticdiffeNerencescatteringbetwateenlow Q2 can be represented as the Fourier transform of the spatial transversely and longitudinally polarized quark distribution (transversity andelectrichelicitchargey) isanda currenmeasuret density in the Breit frame. Recently it was realized that this idea can be extended to much more general distributions of quarks and gluons in QCD. These are the of the relativistic nature of the motion of quarks in the nucleon – see Ref. [Ba01so-called] forgeneralizeda review.partonThedistributions (GPD’s), which unite the concept of the elastic form quark transversity distributions are also intimately related to the chiral propfactorertieswith thatof QCD,of the quarkbeingand antiquark distributions measured in inclusive eN scattering at 2 entirely due to chirally odd (helicity–flipping) effective interactions inducedlarge bQy [Co97the ,spJi97,ontaneousRa97]; see Refs. [Di03, Be05] for recent reviews. The GPD’s describe the amplitudes for a fast–moving nucleon to emit and absorb a quark (or antiquark). They depend on breaking of chiral symmetry in QCD. the longitudinal momentum fractions of the quark, as well as on the invariant momentum transfer

The mapping of the flavor dependence of polarized valence and sea quark distributions and 9 the determination of the quark transversity distributions require semi-inclusive measurements, in which the detected final–state hadron reveals information about the spin, flavor, and charge of the “struck” quark participating in the deep–inelastic process. The 12 GeV Upgrade will provide a unique opportunity to perform semi-inclusive measurements with high precision over a wide kinematic range, producing a detailed picture of the spin structure of the nucleon. Furthermore, semi-inclusive scattering allows us to probe the transverse momentum distribution of quarks in the nucleon via the azimuthal distribution of final–state hadrons. The study of these effects has made significant progress lately [Co93, Br02, Co02, Bu02a, Po03a] and is one of the frontiers of present research in QCD. Finally, the wide kinematic coverage of JLab at 12 GeV provides a

19 Recent work in SCET

Factorization theorem clariﬁed Manohar • (role of nonperturbative eﬀects) Chay and Kim Becher, Neubert, Peczak Chen, Idilbi, Ji Resum large logs, α ln(1 x), directly in momentum space. s − (No Landau pole problem.) The END