The Drell Levy Yan Relation: ep vs e + e Scattering to O(ff 2 s ) Johannes Blümlein DESY 1. The DLY Relation 2. Structure and Fragmentation Functions

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1 The DrellLevyYan Relation: ep vs e e Scattering to O(ff s ) Johannes Blümlein DESY. The Relation. Structure and Fragmentation Functions. Scheme Invariant Combinations 4. DrellYanLevy Relations for Evolution Kernels 5. Conclusions Based on : J. Blümlein, V. Ravindran, and W.L. van Neerven, hep-ph/48, Nucl. Phys. B in print J. Blümlein RDCOR, Carmel, C

2 . The Relation Hard Processes : Structure Functions: e p! e X Fragmentation Functions: e e! px d ff dxdq ο L μνw μν Drell, Levy and Yan (969) anticipated the crossing relation W (S) μν (q; p) = W μν (T ) (q; p) for the two reactions. ffl simple scalar-fermion ladder model with ffi( ) source J. Blümlein RDCOR, Carmel, C

3 What can we learn about the relation of these processes in QCD? ffl Factoriation of the Structure Functions ep and the Fragmentation Functions e e into: ffl nonperturbative input at Q ffl perturbative evolution kernel Q! Q The Crossing Relations can be studied in perturbative QCD only for the Evolution Kernels. Crossing for non-perturbative inputs?! LGT Scaling Variables: x B = Q p:q ;» x B» DIS x E = p:q Q ;» x E» e e annihilation Both domains are connected at x =, which is usually a singular point. What are the Conditions for Continuation from one Domain to the other for the Evolution Kernels? J. Blümlein RDCOR, Carmel, C

4 How is the Evolution of Structure Functions and Fragmentation Functions Related? To which Order does this Relation exist? History ffl Early Investigations Drell, Levy and Yan 969,97 Pestieau and Roy, 969 Gribov and Lipatov 97, 97 Fishbane and Sullivan 97 Suri 97; Gatto and Preparata 97, Dahmen and Steiner 97 Gatto, Menotti, Vendramin 97,97 Landshoff, Polkinghorne, Short ltarelli and Maiani 97/7 Bukhvostov, Lipatov and Popov 975 For a review see: P. Menotti, in: Proceedings of the Informal Meeting on Electromagnetic Interactions, Frascati, May, 97, Pisa Preprint SNS /7. ffl Later Developments Curci, Furmanski and Petronio 98 Floratos, Lacae and Kounnas 98 Stratmann and Vogelsang 997 Blümlein, Ravindran and van Neerven 998/ J. Blümlein RDCOR, Carmel, C 4

5 . Structure and Fragmentation Functions F i (x; Q ) = X k=q;g r Q F ik ff s (μ ); μ ; μ μ ;ffl Ω k ^f (x) ; r ^f is the bare parton density and Ω denotes the Mellinconvolution, with (f Ω g)() = Z Z d d f ( )g( )ffi( ) : ^F ik (; ff s (μ Q r ); X l=q;g μ ; μ μ ;ffl) = r ψc i;l ff s (μ Q r ); μ ; μ f f μ r l=q;g! Ω lk ff s (μ r ); μ f μ ; μ f μ ;ffl () : r lk denotes the transition functions and C i;l the coefficient functions. X! F i (x; Q ) = ψc i;l ff s (μ Q r ); μ ; μ f Ω f μ f l ff s (μ r ); μ f r μ ; μ f μ (x) ; r The structure functions contain the renormalied parton densities f l. f l ;ff s (μ r ); μ f μ ; μ f X! μ = ψlk ff s (μ r ); μ f r μ ; μ f μ ;ffl Ω ^f k () : r k=q;g The Transition Functions obey the RGE f fi(a s (μ s (μ f ) ff ffi lm P lm(a s (μ f );ffl) Ω mk a s (μ f ); μ f μ ; ;ffl () = where a s (μ f ) ff s(μ f ) 4ß ; = ffi( ) ; J. Blümlein RDCOR, Carmel, C 5

6 The strong coupling constant evolves as μ r d a s (μ r ) d μ r = fi a s (μ r ) fi a s (μ r ) ; The Coefficient Functions obey the RGE f fi(a s (μ s (μ f ) ff ffi lm P lm(a s (μ f );ffl) Ω C i;m a s (μ Q f ); μ ; () = : f Factoriation Scheme Transformations lk! X m=q;g Z lm Ω μ mk ; C i;l! X The splitting functions transform as: P lk = X fm;ng=q;g m=q;g Z lm Ω μ P mn Ω (Z ) nk fi(a s ) likewise the coefficient functions obey: ψ C i;q = ffi( ) a s C μ() i;q Z()! a s ψ μc i;m Ω Z ml X m=q;g Z lm Ω μc () i;q Z() (Z () )! Z () Ω Z gq () C μ() i;q Ω Z() C μ() i;g Ω Z() gq ; ψ C i;g = a μ s C () i;g Z()! a s! C μ() i;q Ω Z() C μ() i;g Ω Z() gg : ψ μc () i;g Z() Z () Ω Z gg () Z () d da s (Z ) mk J. Blümlein RDCOR, Carmel, C 6

7 The Mellin Integrals above can be turned into a simple algebraic structure by the transform with (f Ω g) N = Z f (N) = d N f () Z d N (f Ω g)() = f N g N : The combination of the above RGE's leads thus to the RGE for the structure and fragmentation functions: 4 f fi(a s f )) 5 N s (μ i f ) (x; Q ) = ; which are scheme invariant. Linearly Independent Pairs of Structure Functions can be expressed as combinations of the singlet quark and gluon densities f q;g and the respective coefficient functions. F N I (Q ) = f N q f N g ψ ψ a s (μ f ); μ f Q a s (μ f ); μ f Q!! C N I;q C N I;g ψ ψ a s (μ Q f ); μ f a s (μ Q f ); μ f This decomposition is scheme and scale (μ f ) F N F N B CN q CBq N C N g C N f N q f N g!! : ; I = ; B: J. Blümlein RDCOR, Carmel, C 7

8 . Scheme Invariant Combinations Evolution Equations of Structure or Fragmentation Functions do normally exhibit Factoriation and Renormaliation Scheme dependences. Instead of process-independent scheme-dependent Evolution equations for Partons one may think of Process-Dependent Scheme-Independent evolution equations for Observables. Evolution @ F N F N B = KN KB N KN B KN F N F N B ; evolution variable ) t = as (Q ) ln fi a s (Q ; physical evolution kernels K N IJ = 4 N I;m C N m;j (t) fi a s (Q ) fi(a s (Q )) CN I;m (t)fln mn (t) C N n;j (t) 5 with X KIJ N = a n s (Q ) K N (n) IJ n= Possible choices for F and F B are F =@t or F and F L. For these sets of physical observables we will examine the crossing-behaviour from S to T-Channel. The dependence on the renormaliation scheme is only removed if the perturbation series is summed to all orders. J. Blümlein RDCOR, Carmel, C 8

9 System : F (x; Q );@F =@t(x; Q ) Leading Order : K N() = K N() d K N() d = 4 = N() fl flgg N() fl N() fl N() gq K N() dd = fl N() fl N() gg Next-to-Leading Order : [Furmanski, Petronio 98] K N() = K N() d = K N() d = 4 4 N() fl gg fl N() fi fl N() flgg N() fl N() fl N() gg CN() N() ;q fl fi fi C N() ;g fl fl N() 4 (fl N() fl N() gq fl N() flgq N() fl N() flgg N() fi ) fl N() flgg N() fln() fl N() fl N() fl N() fl N() flgq N() fl N() gq 5 fi fl N() 5 () J. Blümlein RDCOR, Carmel, C 9

10 K N() dd = fl N() fi fl N() flgg N() N() fl fi 4 C N() ;g fl N() gg fl N() flgg N() fi 4fi C N() ;q fi fl N() 5 ( e F N L F N L =(a s(q )C N() )) System : F (x; Q );F L (x; Q ) Leading Order : [Catani 997] K N() = fl N() K N() L = fl N() K N() L = fl N() gq K N() LL = fl N() gg Next-to-Leading Order : CN() C N() fl CN() C N() fl N() C N() C N() fl N() K N() = fl N() fi fl N() fi C N() C N() C N() ;g fl N() CN() C N() C N() C N() fl N() flgg N() fl N() fi fl N() fi CN() C N() C N() ;g flgg N() [BRvN ] J. Blümlein RDCOR, Carmel, C

11 4 CN() C N() C N() ;g fl CN() C N() gq fi C C N() ;g CN() C N() ;q CN() C N() C N() ;g C N() C N() 5 fl N() K N() L = fl N() fi fl N() C N() ;q C N() ;g (fl N() C N() C N() C N() K N() L = fl N() gq fi fi fl N() CN() C N() CN() C N() 4 CN() C N() CN() C CN() C N() ;g CN() C N() C N() C N() fl N() fi fl N() fi flgg N() fi flgg N() fi flgg N() )fi C N() ;g fl N() fl N() fi fl N() fi CN() C N() C N() ;q C N() ;g C N() ;q 5 fl CN() C N() C N() ;g C N() CN() C N() C N() C N() C N() C N() C N() ;g C N() N() fl C N() CN() C N() 5 fl N() C N() C N() J. Blümlein RDCOR, Carmel, C

12 4 CN() C N() CN() C CN() C N() CN() C N() K N() LL = fl N() gg fi fi fl N() gg CN() C CN() C N() C N() ;g fl N() C N() ;g fl N() gq C N() ;g C N() C N() C N() ;g CN() C N() C N() C N() C N() ;q fl N() fi fl N() fi 4 CN() C N() 5 fl N() C N() C N() C N() ;g flgg N() CN() C N() C N() C N() C N() fi C N() 5 fl N() gg J. Blümlein RDCOR, Carmel, C

13 4. Relations for Evolution Kernels Original Crossing Relation: Wμν T (q; p) = W μν S (q; p) [Drell et al 969] Modified for particles with different spin s i in a simple ladder model F (S) i (x B ) = () (s s ) x E F (T ) i x E ; i = ; ;L: [Bukhvostov et al. 975] Similar relations are expected to hold for the QCD evolution kernels in LO. In the evolution kernels singular contributions like ln i ( ) ffi( ); = ffi( ) lni " ln i ( ) ( " ) (i ) ( ) arise, which have to be continued analytically. ; Continuation Rules: [ BRvN ] ln P ()! P(=) P ii! P ii P ;P gq! cross color pre factor Q =μf Q =μf spacelike! ln timelike iß : ffi( )! ffi( ) ln( )! ln( ) ln() iß ln(")! ln(") iß J. Blümlein RDCOR, Carmel, C

14 LO unpolaried and polaried Splitting Functions: P () () = P () () = 4C F» ( ) ffi( ) P () qg () = 8T RN f ( ) Λ P () qg () = 8T RN f ( ) Λ P () Gq () = 4C F P () Gq () = 4C F P () GG () = 8C P () GG () = 8C ( ) ( )»» ( ) ( ) ( ) fi ffi( ) fi ffi( ) Crossing Relations: μp () = P () μp gq () = N f T f P gq () C F Note the Color Factors! lready here μp () = C F P () N f T f μp gg () = P gg () ; ffi( )! ffi( ) : is required. J. Blümlein RDCOR, Carmel, C 4

15 NLO unpolaried and polaried Splitting Functions: μp ()S μp ()S μp ()S gq μp ()S gg P ()T = fi Z T () Z T () Ω μ P () gq Z T () gq Ω μ P () ; P ()T gq = fi Z T () Z T () Ω ( μ P () gg μ P () ) μ P () Ω (Z T () Z T () gg ) ; P ()T = fi Z T () gq Z T () gq Ω ( μ P () μ P () gg ) μ P () gq Ω (Z T () gg Z T () ) ; P ()T gg = fi Z T () gg Z T () gq Ω μ P () Z T () Ω μ P () gq ; where for unpolaried scattering Z T () ij = P () ji ln() a ji : a = a gg = ; a = ; a gq = ; and for polaried scattering a ij = : (c.f. also [Stratmann, Vogelsang 997]) J. Blümlein RDCOR, Carmel, C 5

16 NLO unpolaried and polaried Coefficient Functions:» ρ (T )() C ;q () C (S)() ;q ρ (T )() C ;g () C F C (S)() ;g N f T f ff ff (T )() = Z (T )() = Z : NLO unpolaried Longitudinal Coefficient Functions:» C (T )() ρ (T )() C () C F N f T f () C(S)() C (S)() ff = ; = : NNLO unpolaried Longitudinal Coefficient Functions: Coefficient fcts. see [Zijlstra, vn 99,994], [Rijken, vn 996,997]. C (T )()» ρ () C(S)() (T )() Z Ω C()S Lq ρ (T )() C () C F N f T f (T )() Z Ω C(S)() ff = ρ (T )() Z gq Ω C F N f T f C(S)() C (S)() ff = ρ (T )() Z gg Ω C F N f T f C()S ff ff ; : J. Blümlein RDCOR, Carmel, C 6

17 To derive these relations extensive use has to be made of convolution relations like ln( ) ln() Ω = 4 S; ( ) ln( ) ln( ) ln() ln() ln ( )ln() ln()ln( ) ln () ln( )ln () Li ( ) ln( ) Li ( ) ln()li ( ) Li ( ) 8 < ( Ω ) ln( ) = 4: S ; ( ) ln( ) 4 ln( ) 5 ln( ) ln() 4 ln() ln( )ln() 4 ln( )ln() ln( )ln () 4 ln() # " ln( ) Li ( ) 4 Li ( ) and relations between Nielsen integrals of various arguments (cf. [JB, Kurth 999]) Li S ; Li S ; = ln () Li ( ) ; = 6 ln () S ; ( ) ; = 6 ln () S ; ( ) Li ( ) ln()li ( ) ; = S ; () Li () ln()li () 6 ln () () : 9 = ; Transformations for other NNLO coefficient functions, see [BRvN ]. J. Blümlein RDCOR, Carmel, C 7

18 Transformation of the Physical Evolution Kernels Define ffik IJ := K T IJ KS IJ: The transforms for the F -F L system read: ffik N() = ffifl N() 4 fficn() μc N() μc N() μc N() ffiflgq N() μfl gq N() ffic N() ;g ffic N() ;q μn() μc N() μc N() μc N() μc N() μc N() μc N() μc N() ffic N() ;g μfl N() ffic N() ;g μfl gg N() ffic N() ;g ffic N() ;g μc N() μc N() ffic N() μc N() 5 μfl N() = ffifl N() (T )N() fi Z μfl N() gq (T )N() Z μfl N() (T )N() Z gq μc N() μc N() (ffiflgq N() (T )N() fi Z Z (T )N() μfl N() Z (T )N() μfl N() Z (T )N() gg μfl N() (T )N() Z μfl N() gg ) : J. Blümlein RDCOR, Carmel, C 8

19 ffik N() L = ffiflgq N() μfl N() (T )N() fi Z (T )N() (Z ffik N() LL = ffiflgg N() fi μfl gg N() Z μc N() μc N() ffik N() L = ffifl N() Z hffifl N() gq (T )N() μfl gg N() μc N() μc N() ffiflgg N() (T )N() μfl gq N() μc N() μc N() Z μn() μc N() (T )N() Z (T )N() Z gg ) ; (T )N() Z (T )N() fi Z Z (T )N() μc N() μc N() ffifl N() μfl N() gq μfl N() μn() μc N() (T )N() fi Z gq Z (T )N() hfi Z (T )N() μfl N() (μfl N() Z (T )N() Z Z (T )N() gg μfl gq N() Z (T )N() hfi Z (T )N() Z μfl N() gg ) (T )N() gq μfl N() (T )N() Z gg ffiflgq N() μfl N() (T )N() gq μfl N() (T )N() Z gq (T )N() μfl gq N() Z (T )N() μfl N() i μfl gg N() i μfl N() μfl N() gg (T )N() Z gq Z ; i μfl N() (T )N() gg μfl N() μc N() μc N() (T )N() hfi Z gg Z (T )N() μfl gq N() (T )N() Z gq i μfl N() The above operations are difficult to perform in x-space due to multiple direct & multiple inverse Mellin convolutions. J. Blümlein RDCOR, Carmel, C 9

20 The above substitutions yield: ffik N() = ffik N() L = ffik N() L = ffik N() LL = ) -Relation to O(ff s). The transforms for the F -@F =@t system read: ffik d = fi 4 ffic N() q fi μfl N() gq 4 (μfl N() Z 4 ffic N() g (T )N() Z ) μfl N() μfl gg N() 5 " μfl N() (T )N() 5 μfl N() μfl gq N() μfl N() gg fi # fi μfl N() 5 ffik dd = fi μfl N() gq 4fi "ffic N() q 4 ffic N() g Z Z (T )N() (T )N() # 5 " μfl N() μfl N() gg fi # J. Blümlein RDCOR, Carmel, C

21 The above substitutions yield: ffik d = ffik dd = ) -Relation to O(ff s). ) The Evolution of Observables using physical evolution kernels is related by an analytic continuation from the space-like to the time-like domain up to O(ff s). Gribov-Lipatov Relation (97) K(x E ;Q ) = K(x B ;Q ) This relation holds for the LO non-singlet contributions and some pieces in the NLO non-singlet contributions, but is generally violated beyond LO. J. Blümlein RDCOR, Carmel, C

22 5. Conclusions ffl The scale evolution of structure and fragmentation functions can be represented in terms of physical evolution kernels and observable non-perturbative input distributions. ffl The physical evolution kernels of either choice of observables are related for the evolution of structure and fragmentation functions by an analytic continuation ( relation) from» x < to < x < up to O(ff s). The GribovLipatov relation is violated beyond LO. ffl n extension of the present investigation to O(ff s) requires the knowledge of the hitherto unknown loop singlet anomalous dimensions. The relation for the evolution kernels is not necessarily expected to hold to arbitray high orders due to the emergence of new production thresholds for the s-channel process. ffl n interesting test of QCD can be carried out in comparing the scaling violations of structure and fragmentation functions using factoriation schemeindependent evolution equations. J. Blümlein RDCOR, Carmel, C