Sn

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Sense and nonsense in photon structure
Jirí Chýla
Institute of Physics, Prague

• The nature of PDF of the photon
• Why semantics matters
• Counting the powers of as
• Factorization scale and scheme (in)independence
• What is wrong with DIS factorization scheme?
• QCD analysis of photon structure function
• Numerical results (J. Hejbal)

J. Chýla JHEP04 (2000)007 J. Chýla hep-ph/05 J.
Hejbal talk at this Conference
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common sense
Sense and nonsense in photon structure
Jirí Chýla
Institute of Physics, Prague

• The nature of PDF of the photon
• Why semantics matters
• Counting the powers of as
• Factorization scale and scheme (in)independence
• What is wrong with DIS factorization scheme?
• QCD analysis of photon structure function
• Numerical results (J. Hejbal)

J. Chýla JHEP04 (2000)007 J. Chýla hep-ph/05 J.
Hejbal talk at this Conference
3
Basic facts
Colour coupling
is actually a function of renormalization scale µ
as well as renormalization scheme RS
At NLO renormalization scheme can be labeled by
?RS
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Parton distribution functions
quark singlet and nonsinglet densities
as well as gluon density depend on factorization
scale M and factorization scheme FS, i.e.
renormalization scale µ and factorization scale M
are independent parameters that should not be
identified!
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Evolution equations
inhomogeneous terms specific to photon
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Splitting functions
homogeneous
inhomogeneous
where
comes from pure QED!
free, defining the FS
unique,
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General solution of evolution equations
Point-like part (PL) particular solution of the
full inhomogeneous equation resulting from
resummation of the series of diagrams starting
with pure QED
Hadronic part (HAD) general solution of the
corresponding homoge- neous one
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All the difference between hadron and photon
structure functions comes from the pointlike part
of PDF of the photon. Only the pointlike
solutions of the evolution equations will be
therefore considered.
standard, but wrong interpretation
as switching off QCD by sending
we get
i.e. as expected the pure QED contribution!
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easy way to wrong interpretation of the correct
result start from pure QED (in units of a/2p)
integrating
integrating
define boundary condition
voilà
but it is clear that this is just mirage as q is
of pure QED nature and has nothing to do with QCD!
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Asymptotic pointlike solution
for asymptotic values of M
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Structure function
where the coefficient functions
QCD effects
pure QED
describe
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Why semantics matters
To avoid confusion, we should agree on the
meaning of the terms leading and next-to-leading
order. Recall their meaning for
QED part
where
contains QCD effects
For this quantity the QED contribution is
subtarcted and the terms leading and
next-to-leading orders are applied to QCD
contribution r(Q) only. This procedure that
should be adopted for other physical quantities
as well, including
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Counting powers of as
What does it mean that some quantity behaves as
?
The standard claim that PDF of the photon behave
as
is inconsistent with the fact that switching off
QCD we get, as we must, pure QED results for the
pointlike part of photon structure function.
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Factorization scale and scheme independence
recall the situation for proton structure
function
the scale and scheme independence implies
which in turm means that at NLO we get
where
is related to the nonuniversal NLO splitting
function P(1).
FS invariant
can be chosen freely to label different
factorization schemes.
or
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So either P(1) or C(1), but not both
independently, can be chosen at will to specify
the FS. For instance, F(Q) can be written as a
function of C(1) explicitly as
Mechanism guaranteeing FS invariance of F(Q)
used for technical reasons Fpq to all
orders EE in LO form to all orders
MSbar DIS ZERO
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Factorization scale and scheme invariance of
photon structure function
For the pointlike part of the nonsinglet quark
distribution function of the photon the situation
is more complicated as the expression involves
photonic coefficient function
Factorization scale invariance
the leads to the conclusion that the first
non-universal inhomogeneous splitting function
k(1) is, similarly as P(1), func-tion of C(1) (or
the other way around). So C(1) can again be used
to label the factorization scale ambiguity.
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The fact that photon structure function behaves
as . follows directly from factorization
scale independence
As this expression is independent of M, we can
take any M to evaluate it, for instance M0. For
MM0 the first term in vanishes and we get for
the r.h.s.
i.e. manifestly the expansion which starts with
O(a) pure QED contribution and includes standard
QCD cor-rections. This expansions vanishes when
QCD is switched off and there is no trace of the
supposed behaviour.
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What is wrong with DIS FS?
Recall Moch, Vermaseren, Vogt in Photon05
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In the standard approach the lowest order, purely
QED photonic coefficient function C(0), is
treated in the same way as genuine QCD NLO
coefficient function C(1) , i.e. is absorbed in
the definition of PDF of the photon in the
in the so called DIS? factorization scheme,
with the same boundary condition
But there is no mechanism guaranteeing the
invariance of photon structure function under
this change. Note that such mechanism must hold
also for pure QED contribution as it must operate
at any fixed order and cannot thus mix orders of
QED and QCD.
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However, getting rid of the troubling term
via this redefinition violates factorization
scheme invariance. The QED box diagram
regularized by quark mass mq gives
defines the QED part of the quark distribution
function of the photon
However, redefining the quark distribution
function via the substitution
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does not change the evolution equation in
f-scheme
Fundamental difference with respect to
redefini- tion procedure in QCD
Keeping the sum
f-independent implies
But we cannot impose on the same boundary
condition
in all f-schemes!!
since we would get manifestly f-dependent
expression
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QCD analysis of photon structure function
dropping charge factors we can separate
contributions of individual orders of QCD
in the following manner
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Comparison of the terms included in the standard
definition of the LO and NLO approximations with
those of mine
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Conclusions
The organization of finite order QCD
approximations of the photon structure function
which follows closely that of the ee-
annihilation to hadrons and separates the pure
QED contribution has been discussed. It differs
significantly from the standard one by the set
of terms included at each finite order. As shown
in the talk of J. Hejbal, this difference is
sizable and of clear phenomenological relevance.