1. Sense and nonsense in photon structure
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Sense and nonsense in photon structure
Jiří Chýla
Institute of Physics, Prague
The nature of PDF of the photon
Why semantics matters
Counting the powers of αs
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
Jiří Chýla
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2. Sense and nonsense in photon structure
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common
sense
Sense and nonsense in photon structure
Jiří Chýla
Institute of Physics, Prague
The nature of PDF of the photon
Why semantics matters
Counting the powers of αs
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
Jiří Chýla
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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 ΛRS
Jiří Chýla

4. 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!
Jiří Chýla

5. Evolution equations: inhomogeneous terms specific to photon 23.2.2019
Jiří Chýla

6. Splitting functions: homogeneous: inhomogeneous: where
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Splitting functions:
homogeneous:
inhomogeneous:
where
comes from pure QED!
free, defining
the FS
unique,
Jiří Chýla
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7. 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
Jiří Chýla

8. 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!
Jiří Chýla

9. easy way to wrong interpretation of the correct result:
start from pure QED (in units of α/2π)
replace
derivatives
using
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!
Jiří Chýla

10. Asymptotic pointlike solution
for asymptotic values of M
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11. Structure function: where the coefficient functions describe pure QED
QCD effects
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12. 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
Jiří Chýla

13. Counting powers of αs
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.
Jiří Chýla

14. 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
Jiří Chýla

15. is compensated by change of C(1) here
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
Choice of
C(1) here
is compensated by change of C(1) here
Mechanism guaranteeing FS invariance of F(Q):
MSbar:
DIS:
ZERO:
used for technical reasons
Fp=q to all orders
EE in LO form to all orders
Jiří Chýla

16. 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.
Jiří Chýla

17. i.e. manifestly the expansion which starts with O(α)
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 M=M0 the first term in vanishes and we get for the r.h.s.
i.e. manifestly the expansion which starts with O(α)
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.
Jiří Chýla

18. What is wrong with DIS FS?
Recall Moch, Vermaseren, Vogt in Photon05:
Jiří Chýla

19. with the same boundary condition
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 “DISγ”
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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20. The QED box diagram regularized by quark mass mq gives
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
Jiří Chýla

21. 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
Jiří Chýla

22. 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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23. Note, in particular, that the in standard approach the pure
Comparison of the terms included in the standard definition of the LO and NLO approximations with those of mine:
Note, in particular, that the in standard approach the pure
QED coefficient function enters first at the NLO
approximation
Jiří Chýla

24. Conclusions The organization of finite order QCD approximations of
the photon structure function which follows closely that
of the e+e- 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.
Jiří Chýla