Sense and nonsense in photon structure wwww 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 23.2.2019 Jiří Chýla eeee

Sense and nonsense in photon structure wwww 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 23.2.2019 Jiří Chýla eeee

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 23.2.2019 Jiří Chýla

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! 23.2.2019 Jiří Chýla

Evolution equations: inhomogeneous terms specific to photon 23.2.2019 Jiří Chýla

Splitting functions: homogeneous: inhomogeneous: where wwww Splitting functions: homogeneous: inhomogeneous: where comes from pure QED! free, defining the FS unique, 23.2.2019 Jiří Chýla eeee

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 23.2.2019 Jiří Chýla

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! 23.2.2019 Jiří Chýla

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! 23.2.2019 Jiří Chýla

Asymptotic pointlike solution for asymptotic values of M 23.2.2019 Jiří Chýla

Structure function: where the coefficient functions describe pure QED QCD effects 23.2.2019 Jiří Chýla

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 23.2.2019 Jiří Chýla

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. 23.2.2019 Jiří Chýla

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 23.2.2019 Jiří Chýla

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 23.2.2019 Jiří Chýla

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. 23.2.2019 Jiří Chýla

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. 23.2.2019 Jiří Chýla

What is wrong with DIS FS? Recall Moch, Vermaseren, Vogt in Photon05: 23.2.2019 Jiří Chýla

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. 23.2.2019 Jiří Chýla

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 23.2.2019 Jiří Chýla

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 23.2.2019 Jiří Chýla

QCD analysis of photon structure function dropping charge factors we can separate contributions of individual orders of QCD in the following manner 23.2.2019 Jiří Chýla

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 23.2.2019 Jiří Chýla

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. 23.2.2019 Jiří Chýla