Dynamic versus Static Hadronic Structure Functions
DDynamic versus Static Hadronic Structure Functions (cid:63)
Stanley J. Brodsky
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA
Abstract “Static” structure functions are the probabilistic distributions computed from the square of thelight-front wavefunctions of the target hadron. In contrast, the “dynamic” structure functionsmeasured in deep inelastic lepton-hadron scattering include the effects of rescattering associ-ated with the Wilson line. Initial- and final-state rescattering, neglected in the parton model,can have a profound effect in QCD hard-scattering reactions, producing single-spin asymme-tries, diffractive deep inelastic scattering, diffractive hard hadronic reactions, the breakdownof the Lam-Tung relation in Drell-Yan reactions, nuclear shadowing, and non-universal nuclearantishadowing—novel leading-twist physics not incorporated in the light-front wavefunctions ofthe target computed in isolation. I also review how “direct” higher-twist processes – where aproton is produced in the hard subprocess itself – can explain the anomalous proton-to-pionratio seen in high centrality heavy ion collisions.
Key words:
Diffraction, QCD, Light-Front Wavefunctions, Hadronization, Multiple Scattering,Heavy-Ion Collisions
PACS:
1. Introduction
It is important to distinguish “static” structure functions which are computed directlyfrom the light-front wavefunctions of a target hadron from the nonuniversal “dynamic”empirical structure functions which take into account rescattering of the struck quarkin deep inelastic lepton scattering. [See fig. 1. ] The real wavefunctions underlying staticstructure functions cannot describe diffractive deep inelastic scattering nor single-spinasymmetries, since such phenomena involve the complex phase structure of the γ ∗ p am-plitude. One can augment the light-front wavefunctions with a gauge link corresponding (cid:63) This research was supported by the Department of Energy contract DE–AC02–76SF00515. SLAC-PUB-13507.
Email address: [email protected] (Stanley J. Brodsky).
Preprint submitted to Elsevier 19 October 2018 a r X i v : . [ h e p - ph ] J a n o an external field created by the virtual photon q ¯ q pair current [1,2], but such a gaugelink is process dependent [3], so the resulting augmented wavefunctions are not universal.[4,1,5].A remarkable feature of deep inelastic lepton-proton scattering at HERA is that ap-proximately 10% events are diffractive [6,7]: the target proton remains intact, and thereis a large rapidity gap between the proton and the other hadrons in the final state. Thepresence of a rapidity gap between the target and diffractive system requires that thetarget remnant emerges in a color-singlet state; this is made possible in any gauge by softrescattering. The multiple scattering of the struck parton via instantaneous interactionsin the target generates dominantly imaginary diffractive amplitudes, giving rise to aneffective “hard pomeron” exchange. The resulting diffractive contributions leave the tar-get intact and do not resolve its quark structure; thus there are contributions to the DISstructure functions which cannot be interpreted as parton probabilities [4]; the leading-twist contribution to DIS from rescattering of a quark in the target is thus a coherenteffect which is not included in the light-front wavefunctions computed in isolation.The shadowing of nuclear structure functions arises from destructive interference be-tween multi-nucleon amplitudes involving diffractive DIS and on-shell intermediate stateswith a complex phase. The physics of rescattering and nuclear shadowing is not includedin the nuclear light-front wavefunctions, and a probabilistic interpretation of the nuclearDIS cross section is precluded.Antishadowing of nuclear structure functions is also observed in deep inelastic lepton-nucleus scattering. Empirically, one finds R A ( x, Q ) ≡ (cid:0) F A ( x, Q ) / ( A/ F d ( x, Q ) (cid:1) > . < x < . i.e. , the measured nuclear structure function (referenced tothe deuteron) is larger than than the scattering on a set of A independent nucleons. IvanSchmidt, Jian-Jun Yang, and I [8] have extended the analysis of nuclear shadowing tothe shadowing and antishadowing of the electroweak structure functions. We note thatthere are leading-twist diffractive contributions γ ∗ N → ( q ¯ q ) N arising from Reggeon ex-changes in the t -channel [9]. For example, isospin–non-singlet C = + Reggeons contributeto the difference of proton and neutron structure functions, giving the characteristic Kuti-Weisskopf F p − F n ∼ x − α R (0) ∼ x . behavior at small x . The x dependence of thestructure functions reflects the Regge behavior ν α R (0) of the virtual Compton amplitudeat fixed Q and t = 0 . The phase of the diffractive amplitude is determined by ana-lyticity and crossing to be proportional to − i for α R = 0 . , which together withthe phase from the Glauber cut, leads to constructive interference of the diffractive andnondiffractive multi-step nuclear amplitudes. The nuclear structure function is predictedto be enhanced precisely in the domain 0 . < x < . θ W . We find that partof the anomalous NuTeV result [10] for θ W could be due to the non-universality of nu-clear antishadowing for charged and neutral currents. In fact, Schienbein et al. [11] haverecently given a comprehensive analysis of charged current deep inelastic neutrino-ironscattering, finding significant differences with the nuclear corrections for electron-ironscattering. 2 Square of Target LFWFs Modified by Rescattering: ISI & FSI • No Wilson Line Contains Wilson Line, Phases • Probability Distributions No Probabilistic Interpretation • Process-Independent Process-Dependent - From Collision • T-even Observables T-Odd (Sivers, Boer-Mulders, etc.) • No Shadowing, Anti-Shadowing Shadowing, Anti-Shadowing, Saturation • Sum Rules: Momentum and J z Sum Rules Not Proven • DGLAP Evolution; mod. at large x DGLAP Evolution • No Diffractive DIS Hard Pomeron and Odderon Diffractive DIS
Static Dynamic
General remarks about orbital angular mo-mentum Ψ n ( x i , ! k ⊥ i , λ i ) ! ni =1 ( x i ! R ⊥ + ! b ⊥ i ) = ! R ⊥ x i ! R ⊥ + ! b ⊥ i ! ni ! b ⊥ i = ! ⊥ ! ni x i = 1 S current quark jetfinal state interactionspectator systemprotone – ! * e – quark Fig. 1. Static versus dynamic structure functions
Diffractive multi-jet production in heavy nuclei provides a novel way to resolve theshape of light-front Fock state wavefunctions and test color transparency [12]. For ex-ample, consider the reaction [13,14]. πA → Jet + Jet + A (cid:48) at high energy where thenucleus A (cid:48) is left intact in its ground state. The transverse momenta of the jets balanceso that (cid:126)k ⊥ i + (cid:126)k ⊥ = (cid:126)q ⊥ < R − A . Because of color transparency, the valence wavefunc-tion of the pion with small impact separation will penetrate the nucleus with minimalinteractions, diffracting into jet pairs [13]. The x = x and x = 1 − x dependence of thedijet distributions thus reflects the shape of the pion valence light-cone wavefunction in x ; similarly, the (cid:126)k ⊥ − (cid:126)k ⊥ relative transverse momenta of the jets gives key informationon the second transverse momentum derivative of the underlying shape of the valencepion wavefunction [14]. The diffractive nuclear amplitude extrapolated to t = 0 will belinear in nuclear number A if color transparency is correct. The integrated diffractiverate will then scale as A /R A ∼ A / . This is in fact what has been observed by theE791 collaboration at FermiLab for 500 GeV incident pions on nuclear targets [15].
2. Single-Spin Asymmetries and Other Leading-Twist Rescattering Effects
Among the most interesting polarization effects are single-spin azimuthal asymmetriesin semi-inclusive deep inelastic scattering, representing the correlation of the spin of theproton target and the virtual photon to hadron production plane: (cid:126)S p · (cid:126)q × (cid:126)p H . Suchasymmetries are time-reversal odd, but they can arise in QCD through phase differences3n different spin amplitudes. In fact, final-state interactions from gluon exchange betweenthe outgoing quarks and the target spectator system lead to single-spin asymmetries(SSAs) in semi-inclusive deep inelastic lepton-proton scattering which are not power-lawsuppressed at large photon virtuality Q at fixed x bj [16]. In contrast to the SSAs arisingfrom transversity and the Collins fragmentation function, the fragmentation of the quarkinto hadrons is not necessary; one predicts a correlation with the production plane ofthe quark jet itself. Physically, the final-state interaction phase arises as the infrared-finite difference of QCD Coulomb phases for hadron wavefunctions with differing orbitalangular momentum. The same proton matrix element which determines the spin-orbitcorrelation (cid:126)S · (cid:126)L also produces the anomalous magnetic moment of the proton, the Pauliform factor, and the generalized parton distribution E which is measured in deeplyvirtual Compton scattering. Thus the contribution of each quark current to the SSA isproportional to the contribution κ q/p of that quark to the proton target’s anomalousmagnetic moment κ p = (cid:80) q e q κ q/p [16,17]. The SSA in the Drell-Yan process is thesame as that obtained in SIDIS, with the appropriate identification of variables, butwith the opposite sign. If both the quark and antiquark in the initial state of the Drell-Yan subprocess q ¯ q → µ + µ − interact with the spectators of the other incident hadron,one finds a breakdown of the Lam-Tung relation, which was formerly believed to bea general prediction of leading-twist QCD. These double initial-state interactions alsolead to a cos 2 φ planar correlation in unpolarized Drell-Yan reactions [18]. As noted byCollins and Qiu [19], the traditional factorization formalism of perturbative QCD forhigh transverse momentum hadron production fails in detail even at the LHC becauseof initial- and final-state rescattering. An important signal for factorization breakdownis a cos 2 φ planar correlation in dijet production.
3. Novel Intrinsic Heavy Quark Phenomena
The probability for Fock states of a light hadron such as the proton to have an extraheavy quark pair decreases as 1 /m Q in non-Abelian gauge theory [20,21]. The relevantmatrix element is the cube of the QCD field strength G µν , in contrast to QED where therelevant operator is F µν and the probability of intrinsic heavy leptons in an atomic stateis suppressed as 1 /m (cid:96) . The maximum probability occurs at x i = m i ⊥ / (cid:80) nj =1 m j ⊥ where m ⊥ i = (cid:112) k ⊥ i + m i . ; i.e. , when the constituents have minimal invariant mass and equalrapidity. Thus the heaviest constituents have the highest momentum fractions and thehighest x i . Intrinsic charm thus predicts that the charm structure function has supportat large x bj in excess of DGLAP extrapolations [22]; this is in agreement with the EMCmeasurements [23]. Intrinsic charm can also explain the J/ψ → ρπ puzzle [24]. It alsoaffects the extraction of suppressed CKM matrix elements in B decays [25]. The disso-ciation of the intrinsic charm | uudc ¯ c > Fock state of the proton can produce a leadingheavy quarkonium state at high x F = x c + x ¯ c in pN → J/ψXA (cid:48) since the c and ¯ c canreadily coalesce into the charmonium state. Since the constituents of a given intrinsicheavy-quark Fock state tend to have the same rapidity, coalescence of multiple partonsfrom the projectile Fock state into charmed hadrons and mesons is also favored. For ex-ample, one can produce a leading Λ c at high x F and low p T from the coalescence of the udc constituents of the projectile | uudc ¯ c > Fock state. In the case of a nuclear target, thecharmonium state will be produced at small transverse momentum and high x F with a4haracteristic A / nuclear dependence since the color-octet color-octet | ( uud ) C ( c ¯ c ) C > Fock state interacts on the front surface of the nuclear target [26]. This forward contri-bution is in addition to the A contribution derived from the usual perturbative QCDfusion contribution at small x F . Because of these two components, the cross section vio-lates perturbative QCD factorization for hard inclusive reactions [27]. This is consistentwith the observed two-component cross section for charmonium production observed bythe NA3 collaboration at CERN [28] and more recent experiments [29]. The diffractivedissociation of the intrinsic charm Fock state leads to leading charm hadron productionand fast charmonium production in agreement with measurements [30]. The hadropro-duction cross sections for double-charm Ξ + cc baryons at SELEX [31] and the productionof J/ψ pairs at NA3 are be consistent with the diffractive dissociation and coalescenceof double IC Fock states [32]. These observations provide compelling evidence for thediffractive dissociation of complex off-shell Fock states of the projectile and contradictthe traditional view that sea quarks and gluons are always produced perturbatively viaDGLAP evolution. It is also conceivable that the observations [33] of Λ b at high x F at theISR in high energy pp collisions could be due to the dissociation and coalescence of the“intrinsic bottom” | uudb ¯ b > Fock states of the proton. As emphasized by Lai, Tung, andPumplin [34], there are indications that the structure functions used to model charm andbottom quarks in the proton at large x bj have been underestimated, since they ignoreintrinsic heavy quark fluctuations of hadron wavefunctions.Goldhaber, Kopeliovich, Schmidt, Soffer, and I [26,35] have proposed a novel mech-anism for exclusive diffractive Higgs production pp → pHp and nondiffractive Higgsproduction in which the Higgs boson carries a significant fraction of the projectile protonmomentum. The production mechanism is based on the subprocess ( Q ¯ Q ) g → H wherethe Q ¯ Q in the | uudQ ¯ Q > intrinsic heavy quark Fock state has up to 80% of the projec-tile protons momentum. This mechanism provides a clear experimental signal for Higgsproduction at the LHC due to the small background in this kinematic region.
4. Color Transparency and the RHIC Baryon Anomaly
It is conventional to assume that leading-twist subprocesses dominate measurementsof high p T hadron production at RHIC energies. Indeed the measured cross section fordirect photon fragmentation Edσ/d p ( pp → γX ) = F ( x T , θ cm ) /p n e ffT is consistent with n eff ( pp → γX ) = 5 , as expected for the fixed- x T scaling of the gq → γq leading-twistsubprocess [36]. However, the measured fixed- x T scaling for proton production at RHICis anomalous: PHENIX reports n eff ( pp → pX ) (cid:39)
8. A review of this data is given byTannenbaum [37]. One can understand the anomalous scaling if a higher-twist subpro-cess [38] where the proton is made directly within the hard reaction, such as uu → p ¯ d and ( uud ) u → pu , dominates the reaction pp → pX at RHIC energies. The dominance ofdirect subprocesses is possible since the fragmentation of gluon or quark jets to baryonsrequires that the 2 to 2 subprocess occurs at much higher transverse momentum than the p T of observed proton because of the fast-falling (1 − z ) quark-to-proton fragmentationfunction. Thus the initial quark and gluon distributions have to be evaluated at higher x in leading twist fragmentation reactions compared to direct processes. Such “direct”reactions can readily explain the fast-falling power-law falloff observed at fixed x T andfixed- θ cm observed at the ISR, FermiLab and RHIC. Furthermore, the protons produced5irectly within the hard subprocess emerge as small-size color-transparent colored stateswhich are not absorbed in the nuclear target. In contrast, pions produced from jet frag-mentation have the normal cross section. This provides a plausible explanation of theRHIC data, [39] which shows a dramatic rise of the p/π ratio at high p T and a higher valuefor n eff at fixed x T when one compares peripheral with central (full overlap) heavy ioncollisions. The directly produced protons are not absorbed, but the pions are diminishedin the nuclear medium.References [1] A. V. Belitsky, X. Ji and F. Yuan, Nucl. Phys. B , 165 (2003) [arXiv:hep-ph/0208038].[2] J. C. Collins and A. Metz, Phys. Rev. Lett. , 252001 (2004) [arXiv:hep-ph/0408249].[3] J. C. Collins, Phys. Lett. B , 43 (2002) [arXiv:hep-ph/0204004].[4] S. J. Brodsky, P. Hoyer, N. Marchal, S. Peigne and F. Sannino, Phys. Rev. D , 114025 (2002)[arXiv:hep-ph/0104291].[5] J. C. Collins, Acta Phys. Polon. B , 3103 (2003) [arXiv:hep-ph/0304122].[6] C. Adloff et al. [H1 Collaboration], Z. Phys. C , 613 (1997) [arXiv:hep-ex/9708016].[7] J. Breitweg et al. [ZEUS Collaboration], Eur. Phys. J. C , 43 (1999) [arXiv:hep-ex/9807010].[8] S. J. Brodsky, I. Schmidt and J. J. Yang, Phys. Rev. D , 116003 (2004) [arXiv:hep-ph/0409279].[9] S. J. Brodsky and H. J. Lu, Phys. Rev. Lett. , 1342 (1990).[10] G. P. Zeller et al. Phys. Rev. Lett. , 091802 (2002) , 239902 (2003) [arXiv:hep-ex/0110059].[11] I. Schienbein, J. Y. Yu, C. Keppel, J. G. Morfin, F. I. Olness and J. F. Owens, arXiv:0806.0723.[12] S. J. Brodsky and A. H. Mueller, Phys. Lett. B , 685 (1988).[13] G. Bertsch, S. J. Brodsky, A. S. Goldhaber and J. F. Gunion, Phys. Rev. Lett. , 297 (1981).[14] L. Frankfurt, G. A. Miller and M. Strikman, Found. Phys. , 533 (2000) [arXiv:hep-ph/9907214].[15] E. M. Aitala et al. [E791 Collaboration], Phys. Rev. Lett. , 4773 (2001) [arXiv:hep-ex/0010044].[16] S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B , 99 (2002) [arXiv:hep-ph/0201296].[17] M. Burkardt, Nucl. Phys. Proc. Suppl. , 86 (2005) [arXiv:hep-ph/0408009].[18] D. Boer, S. J. Brodsky and D. S. Hwang, Phys. Rev. D , 054003 (2003) [arXiv:hep-ph/0211110].[19] J. Collins and J. W. Qiu, Phys. Rev. D , 114014 (2007) [arXiv:0705.2141 [hep-ph]].[20] M. Franz, M. V. Polyakov and K. Goeke, Phys. Rev. D , 074024 (2000) [arXiv:hep-ph/0002240].[21] S. J. Brodsky, J. C. Collins, S. D. Ellis, J. F. Gunion and A. H. Mueller,[22] S. J. Brodsky, P. Hoyer, C. Peterson and N. Sakai, Phys. Lett. B , 451 (1980).[23] B. W. Harris, J. Smith and R. Vogt, Nucl. Phys. B , 181 (1996) [arXiv:hep-ph/9508403].[24] S. J. Brodsky and M. Karliner, Phys. Rev. Lett. , 4682 (1997) [arXiv:hep-ph/9704379].[25] S. J. Brodsky and S. Gardner, Phys. Rev. D , 054016 (2002) [arXiv:hep-ph/0108121].[26] S. J. Brodsky, B. Kopeliovich, I. Schmidt and J. Soffer, Phys. Rev. D , 113005 (2006) [arXiv:hep-ph/0603238].[27] P. Hoyer, M. Vanttinen and U. Sukhatme, Phys. Lett. B , 217 (1990).[28] J. Badier et al. [NA3 Collaboration], Phys. Lett. B , 335 (1981).[29] M. J. Leitch et al. [FNAL E866/NuSea collaboration], Phys. Rev. Lett. , 3256 (2000) [arXiv:nucl-ex/9909007].[30] J. C. Anjos, J. Magnin and G. Herrera, Phys. Lett. B , 29 (2001) [arXiv:hep-ph/0109185].[31] A. Ocherashvili et al. [SELEX Collaboration], Phys. Lett. B , 18 (2005) [arXiv:hep-ex/0406033].[32] R. Vogt and S. J. Brodsky, Phys. Lett. B , 569 (1995) [arXiv:hep-ph/9503206].[33] G. Bari et al. , Nuovo Cim. A , 1787 (1991).[34] J. Pumplin, H. L. Lai and W. K. Tung, Phys. Rev. D , 054029 (2007) [arXiv:hep-ph/0701220].[35] S. J. Brodsky, A. S. Goldhaber, B. Z. Kopeliovich and I. Schmidt, Nucl. Phys. B , 334 (2009)[arXiv:0707.4658 [hep-ph]].[36] S. J. Brodsky, H. J. Pirner and J. Raufeisen, Phys. Lett. B , 58 (2006) [arXiv:hep-ph/0510315].[37] M. J. Tannenbaum, PoS C FRNC2006 , 001 (2006) [arXiv:nucl-ex/0611008].[38] S. J. Brodsky and A. Sickles, Phys. Lett. B , 111 (2008) [arXiv:0804.4608 [hep-ph]].
39] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. , 172301 (2003) [arXiv:nucl-ex/0305036]., 172301 (2003) [arXiv:nucl-ex/0305036].