Collinear factorization for DIS at large Bjorken xb

Transcription

1 Collinear factorization for DIS at large Bjorken B Alberto Accardi Jlab & Hampton U. Jefferson Lab 6 March 008 Work in progress with: J.W. Qiu W. Melnitchouk, T. Keppel, E. Christy

2 Motivation and outline

3 Why large B and low Q? Large uncertainties in quark and gluon PDF at 0.5 u() at Q = 5:5 GeV d() at Q = 5:5 GeV Jlab Theory Cake Seminar, 6 Mar 008 3

4 Why large B and low Q? Large uncertainties in quark and gluon PDF at 0.5 Precise PDF at large are needed, e.g., at LHC, Tevatron ) New physics as ecess in large pt spectra large PDF ) DGLAP evolution feeds large, low Q into lower, large Q d/u ratio at = non perturbative structure of the nucleon Jlab Theory Cake Seminar, 6 Mar 008 4

5 Why large B and low Q? Large uncertainties in quark and gluon PDF at 0.5 Precise PDF at large are needed, e.g., at LHC, Tevatron ) New physics as ecess in large pt spectra large PDF ) DGLAP evolution feeds large, low Q into lower, large Q d/u ratio at = non perturbative structure of the nucleon JLAB has precision DIS data at large B, BUT low Q need of theoretical control over ) higher twist Q ) target mass corrections (TMC) B mn Q 3) jet mass corrections (JMC) mj Q Jlab Theory Cake Seminar, 6 Mar 008 this talk 5

6 OPE and Target Mass Corrections [Georgi, Politzer 976; see review of Schienbein et al. 007] 4 iq z d ze Ak = y¹ º hn jt [j (z)j (0)]jN i = k dy y F (y) 0 X k f ¹ :::¹k Ak hn jo¹ :::¹k (0)jN i symmetric, traceless X F (y)» e q(y) (at LO) = quark function y q q Mellin transform, sum, transform back: FGP (B ; Q )»= = B F (») 3 ½B + mn 6 Q 3B ½4B 0 0 d» F (» ) +» m4n 4 Q 4 ½5B» d» 0 d» 00 F (» 00 )»0 B B p = Nachtmann variable + ½ + + 4B mn =Q B Threshold problem: B implies th (B=) Inverse Mellin transform does not give back F(y)!! [Johnson, Tung 979] Unphysical region: F(y) ~ F(y) has support over 0<y< FGP(B) > 0 also for B>!! Jlab Theory Cake Seminar, 6 Mar 008 6

7 Collinear factorization - outline Target Mass Corrections O(B mn/q) momentum space, no need of Mellin transf. kinematics of handbag diagram no unphysical region at B (!!) any order in s at leading twist Jet Mass Corrections O(mj/Q) leading order in s, leading twist connection to lattice QCD?? Conclusions FT FT TMC naïve FT TMC TMC + JMC mn=939 MeV mn= B B Jlab Theory Cake Seminar, 6 Mar B 7

8 Target mass corrections

9 Kinematics with mn 0 W ¹º = 8¼ (p; q) = Collinear frames: [Aivazis et al 94] m ¹ p¹ = p+n¹ + N n p+ Q ¹ q ¹ =»p+n¹ + n»p+ k + kt ¹ ¹ ¹ + ¹ k = p n + n + k T p+ where: k+ = + p»= B p + + 4B mn =Q d4 z e iq z hpjj y¹ (z)j º (0)jpi Lorentz invariants: q B = p q q f = k q Q = q mn = p Light cone vectors: p p n =(= ; ~0? ; = ) p p n =(= ; ~0? ; = ) p a =(a0 a3 )= Bjorken limit: B recovers the massless (mn=0) kinematics Jlab Theory Cake Seminar, 6 Mar 008 9

10 Factorization theorem with mn 0 [see also Qiu's talk at CTEQ meeting 005] q» ~f = = ~ q k Epand around ~ ¹ = p+ n¹ ~ = 0 k k X d ¹º ~ ¹º WN (p; q) = Hf (k; q) 'f=n (; Q ) + O( =Q) f perturbative: doesn't know about the target's mass dynamical TMC only from nucleon w.f. Helicity structure functions FT, FL projected out of W : e.g., X d FT (B ; Q ) = hf T (~ f ; Q ) 'f =N (; Q ) + O( =Q ) f = / no kinematic prefactors [Aivazis, Olness, Tung 994] Jlab Theory Cake Seminar, 6 Mar 008 0

11 Kinematic constraints k = 0 ): General handbag diagram on shell gluons and light quarks ( e quark (nucleon) proton (nucleus) i.e., B ef»»=b Proof (can be generalized to heavy and off shell quarks and nuclei) 0 pj ³ = (e k + q) = Q =) ef ef s = (p + q) = (pj + py ) pj + py pj + mn 9 ³ > pj = Q = ef =) ef B ³ > s mn = Q ; B If baryon number flows in the upper blob (not the case for pqcd quarks) B ef + B mn =Q + mn =Q Jlab Theory Cake Seminar, 6 Mar 008

12 No unphysical region! TMC in collinear factorization: FT (B ; Q) = P f» B» ³» d hft ; Q 'f (; Q) FT (B ; Q) = 0 at B > Bjorken limit mn/q 0 recovers massless structure functions (mn=0) FT (B ; Q )! (0) FT (B ; Q ) P f B ³ d B hf T ; Q 'f (; Q ) Different from the naive collinear factorization TMC [Aivazis et al '94 (0) FTnv (B ; Q ) FT (»; Q ) = which does not vanish at B > P f» Kretzer,Reno '0] ³» d hf T ; Q 'f =N (; Q ) Jlab Theory Cake Seminar, 6 Mar 008

13 Target mass corrections F at NLO MRST00 MRST00 Fnv (B ) = + m 4B QN B (0) F (»)» Jlab Theory Cake Seminar, 6 Mar 008 3

14 Target mass corrections L/ T at NLO MRST00 MRST00 ¾L FL R= = ¾T F (0) nv F;L (B ) = F;L (») Jlab Theory Cake Seminar, 6 Mar 008 4

15 Polarized DIS [with Wally numerics soon to come] TMC for virtual photon asymmetries (leading twist): g(b ) g (B ) = X f (0) g;f '(») + HT g(b ) + g (B ) = 0 + HT TMC for g X (0) g(b ) = g;f '(») + HT + f where 'f () = 'f (; +) 'f (; )» ³» d (0) (0) B g;f 'f (») g;f ; Q 'f (; Q )» = mn 4B Q = ½B Jlab Theory Cake Seminar, 6 Mar 008 5

16 Jet mass corrections

17 Jet smearing at LO But... at leading order for FT, mf = 0» ³» hf T ; Q ) = ef ± FT TMC mn = 0 = P (0) FT (B ; Q ) = e ' (»; Q ) = F f f f T (»; Q ) at B < B Ansatz: jet with a non zero mass, smoohtly distributed in mj (k + q) = mj mj! ±» + Q jet mass distribution FT (B ; Q ) = note the limits = 0 dmj J (mj ) B B 0 Q» B» mj d eq ±» + 'f (; Q ) Q (0) dmj J (mj ) FT» mj + ;Q Q Jlab Theory Cake Seminar, 6 Mar 008 7

18 Jet smearing at LO FT FT TMC B TMC + JMC B Rigorously after some toil: J(mj) is the spectral function of a vacuum quark propagator dmj J (mj ) ¼±(l mj ) µ(l0 ) 0 4 iz l = d ze Tr h0jã(z)ã(0)j0i 4l Jlab Theory Cake Seminar, 6 Mar 008 8

19 Jet smearing at LO 3 Toy jet function log normal distribution <mj> = GeV = <mj> Jlab Theory Cake Seminar, 6 Mar 008 9

20 Jet smearing at LO 3 Toy jet function log normal distribution <mj> = GeV = <mj> Jlab Theory Cake Seminar, 6 Mar 008 0

21 Collinear factorization with a jet function [see also Collins, Rogers, Stasto, 007] Handbag diagram with a quark jet J^ T^ W ¹º (p; q) = eq 8¼ A hat denotes a Dirac matri: l =k+q T^ (k) = = ^ = J(l) = (color factors are included in ^ T) d4k º º ^ (k) J(l) ^ Tr T (¼)4 d4 z eiz k hpjã j (z)ã(0)jpi i d4 z eiz l h0jãj (z)ã(0)j0i Jlab Theory Cake Seminar, 6 Mar 008 i

22 Factorization procedure [Ellis, Furmanski, Petronzio, 983] Epand on a basis of Dirac matrices T^ (k) = (k)^ + (k)k= + 3 (k) (k)k= 5 contributes to higher twists = 0 (T odd) cancels for unpolarized targets ^ = j (k)^ J(l) + j (l)l= + j3 (l) 5 + j4 (l)l= 5 enters traces with odd no. of 's = 0 (T odd) cancels (quark spin is unobserved) Dominance of k+, l in Breit frame suggests to define h (k) = + Tr n =T^(k) = + d4 z eiz k hpjãj (z) + Ã(0)jpi 4k 4k h ^ j (l) = Tr n =J(l) = d4 z eiz l h0jã j (z) Ã(0)j0i 4l 4l Jlab Theory Cake Seminar, 6 Mar 008

23 Jet spectral representation = = j (l) = P 0 n (¼) ± 0 4 (4) (l Pn phi ) h i dmj J (mj )^ + J (mj )l= ¼±(l mj )µ(l0 ) dmj J (mj )¼±(l mj )µ(l0 ) with 0 dmj J (mj ) = I am assuming color neutralization J(mj ) through a (neglected) soft echange with the target jet q p J meson meson q m QCD vacuum is confining: no zero mass hadrons Jlab Theory Cake Seminar, 6 Mar 008 mj At least meson and parton 3

24 0 Jet function and lattice QCD dmj J (mj ) ¼±(l mj ) µ(l0 ) = 4l 4 d ze iz l Tr h0jã(z)ã(0)j0i Quark propagator in lattice QCD [e.g., Bowman et al. '05] (q ) 4 iz q d ze h0jã(z)ã(0)j0i = i q + M (q ) but: ) Landau gauge vs. light cone gauge ) Euclidean vs. Minkowski space Jlab Theory Cake Seminar, 6 Mar 008 4

25 Questions for you! What can I learn about the Jet Function from lattice QCD? from Dyson Schwinger equations? from e+e jets?...?? Should I ultimately regard it only as a phenomenological tool? fit it to DIS data, in the spirit of global QCD fits Can I compare the fitted J to lattice QCD computations?? (My formalism neglects a soft color echange) q p J meson meson q Jlab Theory Cake Seminar, 6 Mar 008 5

26 Conclusions Collinearly factorized DIS with Target and Jet Mass Corrections respects B, goes smoothly to 0 avoids threshold problem present in OPE formalism (Georgi Politzer) generalizable to other processes, and nuclear targets fully consistent with CTEQ / MRST global analysis correct way to etract PDF from large B structure fns! TMC derived at all orders numerical differences from OPE corrections 0 0% for F very large for FL JMC rigorously derived only at LO Clear physical picture: J(mJ) accessible in lattice QCD? only for unpolarized nucleons, as yet Jlab Theory Cake Seminar, 6 Mar 008 6

27 Open issues future work Open issues: Higher twist terms dynamical TMC JMC at NLO Fits to eperimental data on DIS [with W.Melnitchouk, T.Keppel, E.Christy] Large resummation,... What else? \ Global fits, CTEQ like [with W.Melnitchouk, T.Keppel,...] Etension to nuclear target ( Fermi motion ) [with J.W.Qiu & J.P.Vary] Neutron PDF in D(e,e'p)X PDF from neutrino DIS on nuclear targets B > events short range nucleon correlations superfast quarks in DIS, 6 quark bags,... Polarized structure functions [with W.Melnitchouk]... Jlab Theory Cake Seminar, 6 Mar 008 7

28 The end Jlab Theory Cake Seminar, 6 Mar 008 8

29 App. A F and GP Jlab Theory Cake Seminar, 6 Mar 008 9

30 Target mass corrections F at NLO MRST00 MRST00 Fnv (B ) = (0) F (») Jlab Theory Cake Seminar, 6 Mar

31 Georgi-Politzer Target Mass Corrections [Georgi, Politzer 976; see review of Schienbein et al. 007] In the OPE formalism h F (0) (»; Q ) m i B B FGP (B ; Q ) = + N ( ; Q ) B ½B» Q ½B i h (0) F (»; Q ) m B FGP (B ; Q ) = 3B +6 N ( ; Q ) B ½B» Q ½B h F (0) (»; Q ) i m B B L FLGP (B ; Q ) = + N ( ; Q ) B ½B» Q ½B where B»= ½B (B ; Q) = ½B = + 4B mn =Q» and, in my conventions, (0) F (v; Q ) mn B dv + (v») Q ½B v ½ FL (B ; Q ) = B F (B ; Q) F (B ; Q) B Jlab Theory Cake Seminar, 6 Mar 008 3

32 Georgi-Politzer Target Mass Corrections Why is the GP corrected FL so large?? FLGP (B ) (0) i B h FL (») mn B = + (B ) ½B» Q ½B (B ) (0) F mn B (v) = dv + Q ½B (v») v» Jlab Theory Cake Seminar, 6 Mar 008 3

33 Transverse structure function at LO in s with CTEQ5L parton distributions 0 FT 0 TMC std FT 0 TMC FT B 0.5 B TMC + JMC 0.5 B (the only cartoon in this talk) Jlab Theory Cake Seminar, 6 Mar

34 App. B collinear fact. and Jet function Jlab Theory Cake Seminar, 6 Mar

35 Collinear epansion - = ¹º H (k; l) ¼ kinematic constraints ) Epand H (k,l) around * leading twist contributes to Higher Twists [Qiu '90] NOTE: up to now no approimations especially, I did not approimate the final state kinematic Jlab Theory Cake Seminar, 6 Mar

36 Collinear epansion - ) Use spectral representation 3) Assume k, kt (/ )Q J J unapproimated! fat quark line: NOTE: Involves a shift in the final state momentum l evil!! see [CRS] but J(mJ) is unapproimated (improvement over mj=0 case) OK if d4l dominated by l such that j(l) has small slope. In terms of the spectral representation we need, Jlab Theory Cake Seminar, 6 Mar

37 Collinear epansion - 3 4) Ignore kinematic limits on k, kt : J where needed to define collinear PDF does not respect 4 momentum conservation evil!! e.g., 5) Set mj=0 ~~ inside H (k,l) [CRS] * Needed to: respect gauge invariance (otherwise q use Ward ids in proof of factorization not so evil: does not touch the final state kinematic Jlab Theory Cake Seminar, 6 Mar

38 Finally, as promised... Collinearly factorized DIS at LO with Target and Jet Mass Corrections respects B, goes smoothly to 0: J J FT J(mJ) m mj TMC + JMC 0 Jlab Theory Cake Seminar, 6 Mar B 38

39 Where can we trust the approimations? Neglect of integration limits on kt is OK if» m hkt i Q +» N k jma T 4» Q where h ³ Q i hkt i ¼ kt jintr. + s log kt jintr. solid line Replacement l¹! ^ l¹ is OK if Qma = Q & hmj i B Jlab Theory Cake Seminar, 6 Mar

40 Proof of collinear factorization - Generalized handbag diagram with a quark jet [Collins, Rogers, Stasto, 007] Factorization with fully unintegrated parton distributions (for an abelian theory of massive gluons QCD to come soon) [CRS] soft PCF target PCF Jlab Theory Cake Seminar, 6 Mar 008 jet PCF 40

41 Proof of collinear factorization - Start from J(kJ,ys m) = neglect soft jet target interactions, use P kt = k, kj=l the hard function H is the same as our ht,l,... integrate out kj, use spectral represantation for J(kJ) epand H, repeat approimations 3, 4 use ns A=0 gauge Jlab Theory Cake Seminar, 6 Mar 008 4