Universal Parton Distri. Fn s (non-pert.; QCD evolved)

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1 Charm Production at HERA and Theory of Heavy Quark Production Why is Heavy Quark Production a non-trivial problem in PQCD? it is a Multi-large-scale Problem Conventional approaches: one-scale problem approximate by a ) a Dichotomy: The simple zero-mass parton formalism The 3-flavor heavy-quark calculation An introduction to the Modern Approach (generalized MS formalism of Collins) Intuitively obvious a natural extension of the conventional MS m=0 approach Technically precise to all orders in ff s Now adopted by most (in various guises). Comparison with HERA data on Charm Production Inclusive F2 c (x; Q) Differential Distributions Conclusions... Amundson, Collins, Olness, Schmidt, Wang, Wu-Ki Tung

2 Why is Heavy Quark Production a Challenge? Conventional factorization (the master formula") holds only for one large scale problems. What about Heavy Quark Production?? A F A H A f A a ^ F a a H + O(ff n s log n (?? Q );:::) A f A a a Universal Parton Distri. Fn s (non-pert.; QCD evolved) H a F ^ a Partonic Hard (IRS) Xsec. ("LO", "NLO",...) The devil is in the details To give quantitative meaning to the master formula in PQCD, one must: ffl Specify precisely the Scheme used to organize the perturbation series how are the two factors defined. ffl Specify the accuracy of the calculation what is the magnitude of the residue term: e.g. O(ff n s log n ( Λ)) or Q O(ffn s log n ( m ))?; c Q O(m2 c ); O( Q2 ) :::? Q 2 m 2 c

3 Energy Scales in a High Energy Process Scales other then quark masses Small energy scales: m: mπ, m N,... Λ QCD,... Large energy scale: s: s, E, W,... Typical hard scale: Q: Q, M l1 l 2, P T, M j1 j 2,... (note: µ2 x s for lepto- & x 1 x 2 s for hadroprod.) Then, there is the heavy quark mass M H : If H is "heavy": m << M H For H to be produced: 2 M H < s However, relative magnitude of typical energy scale Q with respective to M H is crucial in determining the physics of the production mechanism!! Operationally, define c, b, t as heavy quarks: because M H >> Λ QCD

4 Heavy Quark Formalism-- 3-flavor Scheme Fixed-flavor-number (FFN) Scheme Σ a = g, u, d, s (light fl. only) a σ heavy quark: m H "large" No col. Subtr. f a number of quark flavor n fl =3 (for c) fixed, indep. of Q Lepto-production hadro-production "Heavy flavor creation" (HC) "Gluon-fusion process" Laenen, Smith, van Neerven...et.al NLO calculation: Nason, Dawson, Ellis Mangano, Ridolfi, Fixione. Advantage: Standard one-scale calculation. Limitations: no active H parton at any energy; contains powers of log(q 2 /m 2 H ) => Large theoretical uncertainty, hence suspect for Q >> m H.

5 3-flavor order αs 2 calculation (no charm parton at any energy scale) flavor O (α s 2 ) NLO α s 2 ln 2 ( ) F 2 c (x,q) NLO Q 2 ffl Implicitly assumes m 2 c Q 2 ; ffl Expansion is in powers of ff s ln(q 2 =m 2 c ) which is not small when Q 2 fl m 2 c the uncertainty band becomes wide in this region.

6 m= 0 Partonic Formalism of Heavy Quarks c, b, (t) Zero-mass Factorization Theorem Σ a = g, u, d, s, c, b, (t) (all active flavors) a σ^ All parton masses m a = 0 MSbar Subtr. f a active flavor : all quarks with m H < Q : n fl (Q) Usual parton distributions are generated in this scheme: EHLQ,..., MRS, CTEQ i.e. F 2 (x,q) = u + d + s + c + b NLO All popular applications are based on this scheme: Pythia, Herwig, Isajet ; EKS, JetRad, Advantage: Simple and intuitive Limitations: mass effects neglected (except for thresholds) Not appropriate for Q ~ m H

7 4-flavor order αs calculation (including the charm parton) 4-flavor O (α s ) NLO F 2 c (x,q) NLO Q 2 Implicitly assumes m 2 fi c Q 2 ; In the conventional approach, set m c! 0 in hard scattering calculation. ) Not a good approximation when m 2 c ο Q 2.

8 The Modern Approach Generalized MS Formalism of Collins (ACOT) Intuitively obvious: a composite scheme 4 flavor F 2 c (x,q) 3 flavor transition point Q 2 Technically precise: essential ingredients: ffl 3-flavor scheme near Q ο m c and up; ffl 4-flavor m c 6= 0 scheme at asymptotic Q fl m c and down; ffl a set of matching conditions which relate the two schemes at some matching scale μ m ; ffl a suitably chosen transition scale μ t at which one switches from one scheme to the other.

9 Matching of 3- and 4-flavor Schemes α s (µ) or f(x,µ) 4-flavor 3-flavor α 2 α s log(µ/m c ) + α 2 µ µ m1 =m c µ m2

10 General Formalism of Heavy Quark Partons c, b, (t) (Witten, Collins-Wilczek-Zee, Collins-Tung, ACOT, Collins97...) Generalized (m = / 0) Factorization Theorem Σ a = g, u, d, s, c, b, (t) (all active flavors) a σ^ quark mass m H = / 0 Mass Subtr. f a active flavor : all quarks with m H < Q : n fl (Q) Key features: f a (x,q) obey the usual (MSbar) evolution eqs. σ(q,m H ) is free from large log(q/m H ) after mass subtr. Proper limits: * for Q ~ m H : reduces to the 3flv heavy quark scheme; * for Q >> m H : reduces to the zero-ma ss parton model Advantage: intuitively "obvious"; theoretically more complete price to pay: more complicated to implement; mass subtr. needs to be calculated.

11 Some technical features of the General Scheme Renormalization Use the CWZ Renormalization Scheme (m H =0) / Rules for calculating Green s Functions: * H is "light" if : m H < µ ; H is "heavy" if : m H > µ ; Diagrams with heavy quark lines: Use BPHZ (zero momentum) subtraction (manifest decoupling of H below threshold) Diagrams without heavy quark lines: Use MS-bar subtraction (mass-indep. renorm. const. and anomalous dim.) Factorization All-order proof: Collins P.R. 97 (generalization of standard proof of factorization to the case of non-zero quark mass): σ ( Λ ) ( s m..) = f( x, µ ) σˆ ( sˆ, m, µ ) d( z, µ ) +, H H d j µ f i ( x, µ ) = P ( x) f j ( x, µ ) mass i dµ subtraction (mass-indep. splitting kernel) 2 Ο 2 Q

12 Phenomenology of Leptoproduction of Charm bñééêáãéåí~ä=oéëìäíëw= ebo^=ü~ë=ãé~ëìêéç=afp=åü~êã=éêççìåíáçåi=~åç= ÑçìåÇ=c Å O =L=c O= ==ú==orb==~í=ëã~ää=ñk= _çíü=ebo^=éñééêáãéåíë=ü~îé=ãé~ëìêéç=íüé= ÇáÑÑÉêÉåíá~ä=ÇáëíêáÄìíáçåë=çÑ=íÜÉ=éêçÇìÅÉÇ=ÅÜ~êã= áå=íüé=ñáå~ä=ëí~íé=eéåéêöói=ê~éáçáíói=é í I=ÁKF= müéåçãéåçäçöáå~ä=ëíìçáéëw= tü~í=çç=íüé=éñééêáãéåí~ä=êéëìäíë=íéää=ìë=~äçìí=íüé= ìåçéêäóáåö=mn`a=íüéçêó=eïüáåü=çñ=íüé= Å~äÅìä~íáçå~ä=ëÅÜÉãÉë=áë= êáöüíòf\= tü~í=~êé=íüé=ëáöåáñáå~åí=éêççìåíáçå=ãéåü~åáëãë\== EÜÉ~îóJÑä~îçê=ÅêÉ~íáçå=îëK=eÉ~îóJÑä~îçê=ÉñÅáí~íáçåF= fë=íüéêé=~åó=ëáöåáñáå~åí=åçåíêáäìíáçå=ñêçã=åçåj ééêíìêä~íáîé=e áåíêáåëáåòf=åü~êã=áå=íüé=åìåäéçå\= eçï=ççéë=íüé=~ççéíáçå=çñ=~=ãçêé=éêéåáëé= Ñçêãìä~íáçå=çÑ==n`a=íÜÉçêó=çÑ=ÅÜ~êã=éêçÇìÅíáçå= ~ÑÑÉÅí=íÜÉ=ÖäçÄ~ä=~å~äóëáë=çÑ=é~êíçå=ÇáëíêáÄìíáçå= ÑìåÅíáçåë=\= =

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14 flavor 1st order 3-flavor 2nd order x = Q^2 (GeV^2) flavor 1st order 3-flavor 2nd order x= Q^2 (GeV^2)

15 Complementarity between the 4-flavor Parton Picture and 3-flavor Heavy Quark scheme for Lepto-Production of H O(α s 2 ) 3-flv scheme O(α s ) 4-flv scheme O(α s 2 ) Full Theory F2 H (x,q) df2 H / dpt O(α s ) Parton Good Approx. Insufficient info. O(α s 2 ) FFN OK, but Large Uncertainties, lengthy calculations Good Approx.

16 Complementarity of the "Partonic" and "Heavy Quark" Aspects of c-quark HFE 0 Subtraction (Overlap) HFC 1 C C - + G G C C - µ/m --> 1 µ/m >> 1 G G Sub. term: f G x 1 f G H x HFE0 ; 1 fg H = αs P G->H ln (µ/m H )

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18 Conclusions ffl The natural theory of heavy quark production, valid at all energy scales, combines 3- flv and 4-flv schemes (m c 6= 0) in a seamless way. ffl Simplistic labels LO" & NLO" can be misleading: the power of ff s (# of loops) alone does not determine the accuracy in a multi-scale problem due to the presence of potentially large logarithms, e.g. ln(q 2 =m 2 c ). True LO & NLO results depend on the scheme and on the kinematic range of the process. ffl In practice, The order ff 2 s 3-flavor calculations (SB-Leiden) has been known to work very well over the available kinematic region. ffl The order ff s 4-flavor calculation is shown (in this talk) to also work very well over the full range, especially for F2 c (x; Q). ffl The order ff 2 s composite 3- and 4-flavor scheme represents the most complete theory for future experiments. It is also need to address questions such as intrinsic charm.

19 Renormalization and Factorization Schemes and Op. Procedure to calculate the Hard X-sects. (i) Within a given renormalization scheme (i.e. with specific ultra-violet counter-terms) in PQCD (including quark mass m c ), and to a given order in ff s, perform two independent calculations: ffl A set of partonic structure functions, e.g. represented by a set of Feynman diagrams, with on-shell parton targets:! a ( Q μ ;x;m c μ ; 1 ffl ;ff s (μ)) ffl A set of process-independent perturbative partonic distribution functions Λ f ~ b a (x; m c μ ; 1 ffl ;ff s (μ)) Additional counter-terms are required to regulate all ultra-violate singularities. These define the factorization scheme. Both will depend on the renormalization scale μ and contain collinear singularities (represented by 1 ) as well as potentially large logarithm ffl terms of the form (ff s ln( μ m c )) n (where in practice one chooses μ ο Q); Λ using either the (moment space) operator-product expansion or, equivalently, the (x-space) bi-local operator definition of the distribution functions.

20 (ii) Confirm that all collinear singularities, in the form of 1 as well as (ff ffl s ln( μ m c )) n terms, appearing in! a also appear in the universal functions f ~ b a, so that the result appears in form of a factorization theorem, P b! a ( Q μ ;x;m c μ ; 1 ffl ;ff s (μ)) = ~f a(x; b m c μ ; 1 ffl ;ff s (μ)) Ω b! b ( Q μ ;x;(m c μ ) R;ff s (μ)) with b! a being fully infra-red safe in the sense that it is free of 1 ffl as well as (ff s ln( μ m c )) n terms, and it is well-behaved as m c! 0; as indicated by the subscript R in the ( m c μ ) R dependence of b! a. (iii) Systematically invert this equation to solve for the set of finite hard cross-sections b! a ; P a! a( Q μ ;x;m c b! b ( Q μ ;x;m c μ ;ff s (μ)) R = μ ; 1 ffl ;ff s (μ)) Ω f ~ 1 (x; m c μ ; 1 ffl ;ff s (μ)) a b which are then used in the general factorization formula for physical structure functions in actual applications.