Heavy Quark Mass Effects and
Heavy Flavor Parton Distributions
DIS06
Tsukuba
Tung
Outline
• The importance of studying the Heavy Flavor Sector of
the Parton Structure of the Nucleon
• PQCD calculation including quark mass effects:
 General-Mass (GM) formalism (Collins);
 A coherent and efficient implementation of the GM
formalism, emphasizing the simplicity of the physical
effects of non-zero quark masses;
 Numerical results and comparisons, including FL.
• Applications:
(i) precision global analysis (cf. talk in SF group);
(ii) how well do current experimental data constrain the
charm distribution of the proton?
• Outlook
• Collaborators: Belyaev, Lai, Pumplin, Stump, Yuan
• Thanks to:
Collins, Kretzer, Olness, Schmidt
The General Mass (GM) PQCD Formalism
The
Parton
Picture:
wa
is based on:
Factorization Thm
(valid order-by-order, to all orders of PQCD)
• Conventional proofs: assume Zero Mass (ZM) partons;
• Collins (85-97): FacThm proof is independent of the
mass parameters in PQCD—General Mass (GM) case.
+ H.Order
Total Inclusive Structure Functions F tot — Forward Compton Amp.:
S final State
m(Q)
parton flavors
LO
F tot
NLO
NNLO
Only diagrams with
HQ lines are shown.
4-flavor
scheme
+…
mC
3-flavor
scheme
O(as0) terms with
light quarks only
O(as1) terms with
light quarks only
O(as2) terms with
light quarks only
+…
* Solid lines: heavy quark; * dashed line: light quark; * subtraction terms to remove overlap are not explicitly shown
m
New Implementation of the GM Formalism:
O(as0)
O(as1)
F tot
• Summing Initial S. partons
 Variable-flavor # schemes
(3,4,5: depends on Q)
• Summing Final S. partons
SACOTc
4-flavor
scheme
mC
3-flavor
scheme
 All flavors allowed by P.S.
• Kinematic Constraints:
SACOTc
O(as0) terms with
light quarks only
O(as1) terms with
light quarks only
 Phase space integration
limits;
 Rescaling—smooth and
physical threshold behavior
• CC (Barnett)
• NC (Acotc)
• Wilson Coefficients:
 Simplified ACOT (initial
state parton mass 0)—
more natural parton
kinematics and greatly
simplified W.C.
Some fine prints:
• Subtraction terms to remove overlap are not explicitly shown;
• Slightly different treatment of mass dependence in NC and CC cases.
When and where do mass effects matter?
• In the kinematic phase space:
 When c is different from x, and where f(x,Q) is
steep in x
NC: Kretzer, Schmidt,
wkt (cf. CC: Barnett)
 (slides)
• For Physics quantities that vanish in the zero-mass
limit, such as LO Flongitudinal.
 (slides)
• In real-life precision phenomenology:
 Certain HERA data sets—in the low Q2 region
(H1NCe+9697X and ZeusNCe+9697X)
(slides)
Comparison of
GM and ZM
Calculations:
where in the
(x,Q) plane do
the differences
matter?
F2(x,Q)
low Q2
mostly
GM
ZM
Comparison of
GM and ZM
Calculations:
where in the
(x,Q) plane do
the differences
matter?
FL(x,Q)
low Q2
mostly
GM
Ignore:
top mass
effect
ZM
The Longitudinal Structure Function
For the zero mass case, FL = 0 at LO.
 This is a good place to look for and test mass
effects (and NLO effect—e.g. measure the gluon).
Results:
• DFL(x,Q)/FL(x,Q) |M/=0 vs. M=0 (previous slide);
• FL(x,Q) vs. x for fixed Q;
• FL(x,Q) vs. Q for fixed Q;
• Comparison with data on reduced cross sections;
x-dependence of Flong at fixed Q
• Positive definite, as
physical observables
should be;
• Smooth dependence;
• Increase at small x
due to growth of
NLO contribution
from the gluon
distribution
Q-dependence of Flong for several x values
• Continuity across both the
charm and the bottom
quark thresholds (where
the flavor # increase by 1).
• This is guaranteed by the
ACOT formalism + the
correct treatment of
kinematics à la the Acotc
rescaling.
F2 —Heavy Flavor Threshold Behavior
* Smooth
behavior across
both charm and
bottom
thresholds
GM global analysis and HERA I Charm Production data
H1 NC
e+p
96-97
F2c
H1 NC
e+p
99-00
X
Zeus NC
e+p 9697 F2c
Zeus NC
e+p 9800 F2c
Remarks on the new implementation of the GM formalism
• Physical quantities, such as F2, FL, are positive definite and
smooth across heavy flavor thresholds.
• Simplicity of the general formalism + the new implementation
(including proper kinematics) underpin these results.
• This combination has proven to be very successful as the
basis of a comprehensive new Global Analysis, including all
relevant HERA I data. (Cf. talk in Session 1 of the SF group.)
• FL contributes to the xSec formula, especially at large y. The
success of the precision Global Analysis (including the large y
data) implies a good fit of theory to FL;
• Dedicated fits to FL can be done, but, in principle, does not
add anything, since the SFs are extracted from the xSec
data in the first place (often with additional assumptions or
approximations).
This will be looked into in much greater detail.
“Less is More?”
Application of the New Implementation of the GM
calculation,
(in addition to global analysis (cf. session 1 of SF group)
• First phenomenological study of the heavy flavor
parton distributions:
 Is there room for intrinsic charm in the nucleon?
 If yes, how much?
The Charm Content of the Nucleon
Why should we care about c(x,Q)?
 Intrinsic interest: the structure of the nucleon;
 Practical interests: collider phenomenology, especially
beyond the SM, e.g.
• Charged Higgs production, c + s-bar --> H+ ;
• Single top production in DIS (flavor-changing NC) …
• Conventional global analysis assume that heavy flavor
partons are exclusively generated “radiatively”, i.e. by
gluon splitting.
• This assumption/ansatz more or less agrees with
existing data on production of charm.
• “More or less” since: (i) experimentally, errors on data
are still large; and (ii) theoretically, the ansatz is
ambiguous: at what scale does the radiation start?
Is there a non-perturbative charm component in the
nucleon; and if so, how big can it be?
Theoretical preconceptions aside, let nature speaks for
herself:
Perform unbiased global analysis, allowing charm to have its
own degrees of freedom, in two scenarios:
 A sea-like component at some initial scale Q0;
 A light-cone model component (centered at moderate x)—
aka “intrinsic charm” (championed by you-know-who!).
(A hybrid model is also possible, but clearly there is not enough
experimental constraints yet to warrant a separate study.)
Method: (i) For various assumed input charm c(x,Q0), do
independent global fits, and compare the resulting
goodness-of-fit, c2global; (ii) Define the range of allowed
c(x,Q0) by the currently used Dc2global for defining PDF
uncertainties.
A little bit of detail (that is going to be asked anyway)
Since current experimental constraints are rather
loose, we must limit the new degrees of freedom:
• For the sea-like scenario, assume the shape of
c(x,Q0) is the same as s(x,Q0) and only vary the
normalization;
• For the light-cone model scenario, take the shape of
c(x,Q0) to be that of Brodsky etal, and only vary the
normalization.
Preliminary results on the nonperturbative charm content of the
nucleon …
Goodness-of-fit vs. input non-perturbative
Charm momentum fraction
D(charm mom. frac.)
Dc2
The appropriate value for Dc2 in the current global analysis
environment has not yet been investigated. Hence, the value for
the allowed charm mom. frac. should be taken as indicative only.
Parton Distributions in the presence of a nonzero component of charm
• Charm Distribution
 @ Q0, Q2 = 10 GeV, Q = 85 GeV & for Scenarios B—
light-cone like charm component.
• Gluon Distribution
 (same as above)
• Not shown due to lack of space:
 Strange Distribution;
 Ubar+dbar Distribution.
(these can be affected by the charm content in scenario A—the
sea-like input charm tied to the light flavors)
Charm and Gluon Distributions at Q = 1.3 GeV
Varying amounts of input lightcone charm components
(à la Brodsky etal.) : Momentum frac. at Q0 = 0 — 0.02.
Horizontal axis is scaled in x1/3—inbetween linear and log—
in order to exhibit the behavior at both large and small x.
Charm and Gluon Distributions at Q2 = 10 GeV2
Varying amounts of input lightcone charm components
(à la Brodsky etal.) : Momentum frac. at Q0 = 0 — 0.02.
* Two-component charm distr. is apparent! (The radiatively
generated component is represented by C6C0l (black) curve.
Charm and Gluon Distributions at Q2 = (85 GeV)2
Varying amounts of input lightcone charm components
(à la Brodsky etal.) : Momentum frac. at Q0 = 0 — 0.02.
* Very substantial amount of charm, over the radiatively generated
component (C6C0l), still persists at this very large scale  there can be
interesting phenomenological consequences even at LHC.
Outlook
• This is just the beginning. Looking forward to more
comprehensive and accurate data from HERA II
• With W/Z/g + tagged heavy flavor events at the hadron
colliders, we can get direct information on s/c/b quark
distributions;
• c-quark and b-quark are important phenomenologically
in the physics program at LHC for exploring beyond
the SM scenarios.
?
Outline
• The importance of studying the Heavy Flavor Sector
of the Parton Structure of the Nucleon
• PQCD calculation including quark mass effects:
 General-Mass (GM) formalism (Collins);
 A coherent and efficient implementation of the GM
formalism, emphasizing the simplicity of the physical
effects of non-zero quark masses;
 Numerical results and comparisons, including FL.
• Applications:
(i) precision global analysis (cf. talk in SF group);
(ii) how well do current experimental data constrain
the charm distribution of the proton?
It is interesting that a reason able limit
can be estimated already: < 0.015
Extras
Comments on NNLO
• In the perturbative approach, for the total inclusive
S.F.s and cross sections, once a comprehensive NLO
calculation is in place, it is straightforward to include
known NNLO corrections additively.
• However, one needs to realize that, unlike total
inclusive F2,L, quantities such as ”F2c” is not well
defined theoretically at NNLO and beyond. (It is not
infra-red safe!) It is rather misleading to talk about
a true “NNLO theory” of F2c (except within the 3-flv
scheme, which has only a limited range of
applicability).
• Extending global analysis to NNLO is certainly
desirable, but not necessarily urgent for current
applications (cf. excellent global fits), since
experimental errors for most measured quantities, as
well as other sources of uncertainties (such as
parametrization, power-law corrections …), largely
outweigh the NNLO corrections.