Phenomenology at low Q 2

Phenomenology at low Q 2 Lecture IV Barbara Badelek Warsaw University and Uppsala University HUGS @ JLAB June 26 B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 2 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 3 / 37

High energy DIS spin experiments Acceptance E42, E43, E54, E55, E55X at SLAC; electrons of < 5 GeV, targets: protons, deuterons, helium-3; 2 EMC, SMC, COMPASS at CERN; muons of 9 28 GeV, targets: protons, deuterons; 3 HERMES at DESY; electrons of 3 GeV, targets: protons, deuterons, (helium-3); 4 STAR, PHENIX at BNL; pp collider, s = 2 GeV; 5 Kinematic variables from incident and scattered leptons in, 2, 3; hadrons from target fragmentation often also measured and in case of 2, 3 identified if momenta larger than and 2.5 GeV respectively; 6 background due to µe scattering (at x =.545) in 2, 3; ) 2 (GeV 2 Q COMPASS 22-3 data 6 5 Q 2 (GeV 2 ) 2 SMC, low x trigger SMC, standard triggers - 4-2 -3-4 -6-5 -4-3 -2 - x 3 2 - -2-4 -3-2 - x B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 4 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 5 / 37

High energy DIS spin experiments...cont d Status of g measurements Spin-dependent cross sections are a small part of the DIS cross section = c.s. asymmetries = getting A then A then (using F 2 and R) g. Practical matters in: R. Windmolders, in Spin in physics, X Séminaire Rhodanien de Physique, eds Anselmino, Mila, Soffer, Frontier Group, 22. xg.6.4.2 proton SMC HERMES HERMES prel. HERMES prel. rev. E55 E43 xg.4 2 2 d.3.2. COMPASS 22-3, Q > GeV 2 2 COMPASS 22-3, Q < GeV ; this thesis 2 2 E43 Q > GeV 2 2 E55 Q > GeV 2 HERMES all Q 2 SMC all Q.4 deuteron -..2 -.2-5 -4-3 -2 - x Figure comes from M.Stolarski, PhD, Warsaw University, 26.2 neutron ( 3 He) All data at their quoted mean Q 2 ; errors are total; New precise results from COMPASS on g d, E42 -.2 E54 for x >few 6 ; JLab E99-7 prel. -.4 Lowest x for g p from SMC, x >.6;.... x Direct measurements on neutron for x >.2. Figure from Stösslein, Acta Phys. Pol. B33 (22) 283 No significant spin effects seen at lowest x! B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 6 / 37

High energy DIS spin experiments...cont d Status of g measurements...cont d 5% of momentum carried by gluons 2% of proton spin carried by quark spin Figure from R.Ent, DIS26 Scaling violation in g (x, Q 2 ) is weak. For g, Q 2 becomes > GeV 2 at x >.3 for SMC,.3 for HERMES and for COMPASS. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 7 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 8 / 37

Regge model predictions for g Remember: s W 2 = M 2 + Q 2 (/x ); thus low x behaviour of a structure function (F 2, g,...) reflects the high energy behaviour of the virtual Compton scattering cross section with the cms energy squared, s. This is the Regge limit of DIS. Regge gives for x (i.e. Q 2 W 2 ): g i (x, Q2 ) β(q 2 )x α i () () where i =singlet (s), nonsinglet (ns): g s = gp + gn, gns = gp gn. Possible trajectories: I = (g s; f trajectory) and I = (g ns; a trajectory). Expectations: α s,ns() < and α s() α ns(). Consequence: for Q 2, g (W 2 ) W 2α(). At large Q 2 : the DGLAP evolution and resummation of ln 2 (/x) generate more singular x dependence than that implied by eq.() for α s,ns() <. Other Regge isosinglet contributions to g at low x: a term ln x; a term 2 ln(/x) ; a perverse term /(xln 2 x) got invalidated. Perturbative QCD effects might modify the Regge expectations. In case of g it creates a more singular low x behaviour than the (nonperturbative) Regge expectations. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 9 / 37

Regge model predictions for g...cont d Testing Regge behaviour of g through its x dependence: choose high W 2 ; choose low x (i.e. Q 2 W 2 but not necessarily low Q 2 ); choose a bin of Q 2 (i.e. Q 2 =const); fit the x dependence of g. For the SMC: Testing not possible For COMPASS: Testing not possible either Observe: assuming g x to get x extrapolation of g to extract g moments is not correct! Evolve g to a common Q 2 before extrapolation! B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 / 37

Low x implications from the pqcd In the DGLAP the singular small x behaviour of the gluon and sea quark distributions (implied by the data) may originate from parametrization of the starting distributions at moderate Q 2 (equal to about 4 GeV 2 or so); evolution starting from non-singular valence-like parton distributions at a very low scale, µ.35 GeV 2. Glück, Reya, Vogt, Eur. Phys.J.C5 (998) 46 Then [ ] g (x, Q 2 ) exp A ξ(q 2 )ln(/x) (2) where ξ(q 2 ) = Q 2 µ 2 dq 2 α s (q 2 ) q 2 2π and A is different for the singlet and non-singlet case. (3) B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 2 / 37

Low x implications from the pqcd World data on g p and gd were NLO QCD analysed but at low x neither measurements nor reliable calculations exist. De Roeck et al., Eur. Phys. J. C6 (999)2 Bourrely, Soffer, Buccella, Eur. Phys. J. C23(22) 487. p g 2 5 x(ln(x)) 2 Extrapolation QCD-fit HERA SMC 5 Regge Extrapolation -5 - -5-2 -25 2.75.5.25.75.5.25 -.25 QCD-Extrapolation -.5-2 - -3-5 -4-3 -2 - X B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 3 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 4 / 37

Ln 2 (/x) corrections to g (x, Q 2 ) General Low x large parton densities = new dynamics? Small x behaviour of both g s and gns is controlled by terms corresponding to powers of α s ln 2 (/x) Bartels, Ermolaev, Ryskin, Z.Phys. C7 (996) 273; Z.Phys. C72 (996) 627. These terms generate the leading small x behaviour of g. They go beyond the standard QCD evolution of spin dependent parton densities which does not generate the double but only the single ln(/x) terms. They may be included in the QCD evolution; one of the methods: a formalism based on unintegrated parton distributions, f (x, k 2 ), where the conventional parton distributions p(x, Q 2 ) are p(x, Q 2 ) = Q 2 dk 2 k 2 f (x, k 2 ) (4) and k 2 is a transverse momentum squared of the partons. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 5 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 6 / 37

Ln 2 (/x) corrections to g ns are generated by ladder diagrams = or mathematically by an equation: f (x, k) = f () (x, k) + ᾱ s(k 2 ) x dz z k 2 /z k 2 dk 2 x k 2 f ( z, k 2 ) > q x p, k 2 x zp, k 2 > > q and g ns(x, Q2 ) = g () (x) + W 2 dk 2 k 2 k f (x = x( + k2 2 Q ), k 2 ) 2 > p > > > p where ᾱ s(k 2 ) = 2α s(k 2 )/3π and g () (x) is a nonperturbative part, corresponding to k 2 < k 2. Ln 2 (/x) terms originate from the z-dependent limit of the dk 2 /k 2 and x-dependent limit in W 2 (x). They create a leading small x behaviour of g ns if gns() and f () are non-singular at x. DGLAP evolution is incomplete at low x; only ln(/x) terms are present, originating from k 2 k 2 dk 2 /k 2. For fixed (i.e. non-running) ᾱ s(k 2 ) α s, small x behaviour is g ns (x, Q2 ) x λ where λ = 2 α s B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 7 / 37

Ln 2 (/x) corrections to g (x, Q 2 )...cont d Predictions for g ns...cont d A unified equation which incorporates the complete LO DGLAP at finite x and ln 2 (/x) effects at x was formulated. Potentially large lnq 2 and ln(/x) treated on equal footing. For the numerical results it was assumed that g ns() = 2g A ( x) 3 /3 where g A =.257 (axial vector coupling). At x, g ns() const, in agreement with the Regge expectation. The g ns() satisfies the Bjorken sum rule at LO: dxgns() (x) = g A /6. Parameter k 2= GeV2. To compare the g ns to the (SMC) data it was assumed: g ns gp gn = 2 [ g p gd /( ω D/3) ] ; ω D =.5 (D-state probability in the deuteron). g 3.5 3 2.5.3 < Q 2 /GeV 2 < 56.3 g 2.5.5 g ns vs x 6 continuous full calculations broken LO DGLAP 8 4 Q 2 = GeV 2 2 -.5 - -2 - -5-4 -3-2 - x x B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 8 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 9 / 37

Ln 2 (/x) corrections to g (x, Q 2 )...cont d Low Q 2 extrapolation of g ns Badelek, Kwiecinski, Phys. Lett. B48 (998) 229 For Q 2 (for fixed W 2 ), g should be a finite function of W 2, free from kinematical singularities or zeroes at Q 2 =. g ns from the above formalism and the above gns() fulfill this. If g ns() (x) has a singularity then it should be replaced by g ns() ( x) where x = x( + k 2/Q2 ). Remaining parts left unchanged. Then g ns can be extrapolated to the low Q2 for fixed 2Mν = Q 2 /x including Q 2 =. Observe! That is just the partonic contribution to the low Q 2 region; it may not be the only one there. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 2 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 2 / 37

Ln 2 (/x) corrections to g (x, Q 2 )...cont d Predictions for g Kwiecinski, Ziaja, Phys. Rev. D6 (999) 544 Results of full g (x, Q 2 ) calculations at Q 2 = GeV 2. At low x, the singlet part, g s dominates g ns. Apart of the standard ladder diagram, the following ones were taken into account for g ns(x, Q2 ): k 8 k 8 k k g p -5 - -5 a) b) c) = g p (x, Q2 ) vs x at Q 2 = GeV 2 red nonperturbative input, g () pink only LO DGLAP thick black full g blue LO DGLAP + ladder ln 2 (/x) -2-25 -3 e-5 e-4... x At low x, g x λ with λ.4 for g ns and λ.8 for g s. More singular than Regge expectations! B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 22 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 23 / 37

Ln 2 (/x) corrections to g (x, Q 2 )...cont d Low x contributions to g moments Fundamental tools: sum rules (Ellis Jaffe, Bjorken, DHG,...) which involve first moments of g, i.e. integrations over dx from to, e.g. Γ = g dx. Unmeasured regions: [,x min ], [x max,]. The [x max,] not critical but [,x min ] is very important. x min depends on ν max accessed in experiments at a given Q 2, e.g. SMC at 2 GeV and Q 2= GeV2 = x min.3; COMPASS at 6 GeV and Q 2= GeV2 = x min.2. Contribution to moments from the x <.3 has to be estimated phenomenologically. LO DGLAP + ln 2 (/x) resummation used to extrapolate polarised parton distributions and structure functions down to x 5 to calculate contributions to moments from 5 < x < 3. In 2 < Q 2 <5 GeV 2 interval, contributions to Γ p was 2% and 8% for Γn (however calculations of Γ n were below the data in the overlap region). Contributions with Q 2. Also estimated that 5 < x < 3 interval contributes % and 2% to the Bjorken and Ellis Jaffe s.r. Same formalism gave a contribution of.8 to the Bjorken integral from the unmeasured region, x <.3 at Q 2 = GeV 2 (LO DGLAP only gave.57 and assuming g = const resulted in.4). Kwiecinski, Ziaja, Phys. Rev.,6 (999)544 Extrapolation of the NLO DGLAP fits to the world data: in x <.3 is % of Γ p. NLO DGLAP for the SMC data at Q 2 = GeV 2 gave % contribution to the Bjorken integral. SMC, Phys. Rev., D58 (998) 22 (obs! assumptions!). B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 24 / 37

Nonperturbative effects in g Data on g (x, Q 2 ) extend to low Q 2. GeV 2. Nonperturbative mechanisms dominate the particle dynamics there; transition from soft to hard physics may be studied. Partonic contribution to g has to be suitably extrapolated to low Q 2 and complemented by a nonperturbative component. Low Q 2, spin-independent electroproduction well described by the GVMD = GVMD should be used to describe the g. Two attempts tried to extract from the data a contribution of nonperturbative effects at low x, low Q 2. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 25 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 26 / 37

Nonperturbative effects in g...cont d g at low Q 2, method I Badelek, Kiryluk, Kwiecinski, Phys. Rev. D6 (2) 49 The following representation of g was assumed: g (x, Q 2 ) = g VMD (x, Q 2 ) + g part (x, Q 2 ) (5) g part at low x is controlled by the ln 2 (/x) terms; it was parametrised as discussed in Kwiecinski, Ziaja, Phys. Rev. D6 (999) 544. g VMD (x, Q 2 ) was represented as: g VMD (x, Q 2 ) = Mν 4π V =ρ,ω,φ M 4 V σ V (W 2 ) γ 2 V (Q2 + M 2 V )2 (6) The unknown cross sections σ V (W 2 ) are combinations of the total cross sections for the scattering of polarised vector mesons and nucleons. At high W 2 : σ V = (σ /2 σ 3/2 )/2 Assume: Mν 4π M 4 V σ V γ 2 V (Q2 + M 2 V )2 = V =ρ,ω [ 4 ( ) C uval 9 (x) + 2 ū (x) + ( ) ] dval 9 (x) + 2 d Mρ 4 (x) (Q 2 + Mρ 2, (7) )2 Mν 4π Mφ 4 σ φp γφ 2 (Q2 + Mφ 2 = C 2 )2 9 s (x) Mφ 4 (Q 2 + Mφ 2, (8) )2 B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 27 / 37

Nonperturbative effects in g...cont d g at low Q 2, method I...cont d Each p j (x) x for x. Thus σ V /W 2 at large W 2, i.e. zero intercept of the appropriate Regge trajectories. Results for the spin asymmetry, A = g /F, for the proton, and for different C: C?? C <? A p Q 2 <. GeV 2 A p Q 2 < GeV 2.8.6.4 SMC data.2.5 E43 data, E beam = 6.2 GeV 2 C= C=-2.2 C=+4 C= C=-4..5 C=-4 -.2 Q 2 =.5.6.7.8.9. GeV 2 Q 2 =.2.6..2.3.6 GeV 2-4 -3 x.2.4.6.8. x B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 28 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 29 / 37

Nonperturbative effects in g...cont d g at low Q 2, method II Badelek, Kwiecinski, Ziaja, Eur. Phys. J. C26 (22) 45 The following representation of g was assumed, valid for fixed W 2 Q 2, i.e. small x = Q 2 /(Q 2 + W 2 M 2 ): g (x, Q 2 ) = g L (x, Q2 ) + g H (x, Q2 ) = Mν 4π V MV 4 σ V (W 2 ) γv 2 (Q2 + MV 2 + gas ( x, Q2 + Q 2 ). (9) )2 The first term sums up contributions from light vector mesons, M V < Q, Q 2 GeV2. The unknown σ V are expressed through the combinations of nonperturbative parton distributions, evaluated at fixed Q 2, similar to method I. The second term, g H(x, Q2 ), represents the contribution of heavy (M V > Q ) vector mesons to g (x, Q 2 ) can also be treated as an extrapolation of the QCD improved parton model structure function, g AS(x, Q2 ), to arbitrary values of Q 2 : g H(x, Q2 ) = g AS( x, Q2 + Q 2 ). The scaling variable x is replaced by x = (Q 2 + Q 2)/(Q2 + Q 2 + W 2 M 2 ). It follows that at large Q 2, g H(x, Q2 ) g AS(x, Q2 ). Thus: [ 4 g (x, Q 2 ) = C 9 ( u val (x) + 2 ū (x)) + ] 9 ( d val (x) + 2 d Mρ 4 (x)) (Q 2 + Mρ) 2 2 ] [ + C 9 (2 s (x)) Mφ 4 (Q 2 + Mφ 2 + gas ( x, Q2 + Q 2 )2 ). () B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 3 / 37

Nonperturbative effects in g...cont d g at low Q 2, method II...cont d Now fixing C in the photoproduction limit via the DHGHY sum rule. Digression: the DHGHY sum rule The γ p scattering amplitude fulfills the dispersion relation: S (ν, q 2 ) = 4 ν dν G (ν, q 2 ) q 2 /2M (ν ) 2 ν 2 () where G (ν, q 2 ) = M ν g (x, Q 2 ) (2) in the Q 2, ν limit. As a result of Low s theorem: S (, ) = κ 2 p(n), G in the Q 2 limit fulfills the DHGHY sum rule: dν ν G (ν, ) = 4 κ2 p(n). (3) End of digression - B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 3 / 37

Nonperturbative effects in g...cont d g at low Q 2, method II...cont d # Y & &! I!! ) & *( &P & I!& ) & *)H F?9 ' $ 9$% ) ""'! ) 6) & *'! V. Dharmawardane, DIS26 ; B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 32 / 37

Nonperturbative effects in g...cont d g at low Q 2, method II...cont d At ν, eq.() is: S (, q 2 dν ) = 4M Q 2 /2M ν 2 g (x(ν), Q 2 ). (4) Now we define the DHGHY moment, I(Q 2 ) as: I(Q 2 ) = S (, q 2 dν )/4 = M Q 2 /2M ν 2 g (x(ν), Q 2 ). (5) Before taking the Q 2 limit of (4), observe that it is valid only down to some threshold value of W, W th < 2 GeV (above resonances). Requirement W > W th gives the lower limit for integration over ν in (4), where ν t (Q 2 ) = (Wt 2 + Q 2 M 2 )/2M: I(Q 2 ) = I res(q 2 dν ( ) + M ν 2 g x(ν), Q 2). (6) ν t (Q 2 ) Here I res = contribution of resonances. The DHGHY sum rule now implies: dν I() = I res() + M ν 2 g (x(ν), ) = κ 2 p(n) /4. (7) ν t () B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 33 / 37

Nonperturbative effects in g...cont d g at low Q 2, method II...cont d Thus action plan for extracting C in eq.(): Taking: take g (x(ν), ), eq.(); C is the only free parameter, put it into eq.(7), get I res() from measurements, extract C from (7). I res() from photoproduction, W t =.8 GeV GDH, Nucl. Phys. 5 (22) 3, g AS prametrized by NLO GRSV2 Phys.Rev. D63 (2) 945 nonperturbative p () j (x) at Q 2 = Q 2 =.2 GeV2 from GRSV2 = C =.3 2 flat p () j (x) = N i ( x) η i = C =.24. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 34 / 37

Nonperturbative effects in g...cont d g at low Q 2, method II...cont d Byproducts: g from eq.() and the DHGHY moment, I(Q 2 ), eq.(5). Results for the proton: g (x,q 2 ) - - - Q 2 =. GeV 2 x =. Q 2 =. GeV 2 x =. Q 2 = GeV 2 x =. xg.9.8.7.6.5.4.3.2. -. -.2 TOT ASYM VMD (GRSV) SMC DATA... x data at Q 2 < GeV 2 - Q 2 = GeV 2-4 -3-2 - x =. - x Q 2 [GeV 2 ] broken lines g AS, dotted gl, continuous total g I(Q 2 ).6.4.2 -.2 -.4 -.6 -.8 TOT ASYM VMD (GRSV) E9-23 DATA -.2.4.6.8.2 Q 2 [GeV 2 ] points mark resonances at W < W t (Q 2 ) Figures from: Badelek, Kwiecinski, Ziaja, Eur. Phys. J. C26 (22)45. B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 35 / 37

Outline High energy DIS spin experiments Acceptance Status of g measurements 2 Regge model predictions for g 3 Low x implications from the pqcd 4 Ln 2 (/x) corrections to g (x, Q 2 ) General Predictions for g ns Low Q 2 extrapolation of g ns Predictions for g Low x contributions to g moments 5 Nonperturbative effects in g g at low Q 2, method I g at low Q 2, method II 6 Summary B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 36 / 37

Summary At high energies, low Q 2 region correlated with low x. Very important for understanding the nucleon structure is the transition from photoproduction to DIS; also for practical purposes. Several theoretical concepts relevant there. Both perturbative and nonperturbative contributions to the nucleon structure are present everywhere in Q 2 B. Badelek (Warsaw & Uppsala) Low Q 2, lecture IV HUGS, June 26 37 / 37