Sense and nonsense in photon structure

1. 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

2. 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

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 byΛRS Jiří Chýla

4. Parton distribution functions: quark singlet and nonsinglet densities as well as gluon density depend onfactorization scale M and factorization scheme FS, i.e. renormalization scale μ and factorization scale M are independent parameters thatshould not be identified! Jiří Chýla

5. Evolution equations: inhomogeneousterms specific to photon Jiří Chýla

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

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 solutionof 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 thepure QED contribution! Jiří Chýla

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

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

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 QEDcontribution is subtarcted and the terms leading and next-to-leading orders are applied toQCD 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. Choice of C(1) here 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 Mechanism guaranteeing FS invariance of F(Q): used for technical reasons Fp=q to all orders EE in LO form to all orders MSbar: DIS: ZERO: 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 situationis more complicated as the expression involves photonic coefficient function Factorization scale invariance the leads to the conclusion thatthe first non-universal inhomogeneous splitting function k(1)is, similarly as P(1), func-tionof C(1) (or the other way around). So C(1)can again be used to label the factorization scale ambiguity. Jiří Chýla

17. 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 contributionand includes standard QCD cor-rections.Thisexpansions vanishes when QCD is switched off and there is no trace of the supposed behaviour. Jiří Chýla

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

19. 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. Jiří Chýla

20. However, getting rid of the troubling term via thisredefinition violates factorization scheme invariance. The QED box diagram regularizedby 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 Jiří Chýla

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