Workshop on Intersections of Nuclear Physics with Neutrinos and Electrons JLab, May 4-5, 2006
Quark distributions at large x
Wally Melnitchouk Jefferson Lab 3.1 The New Standard PDF Sets
The standard set of parton distributions in the MS scheme, referred to as CTEQ6M, provides an excellent global fit to the data sets listed in Sec. 2.1. An overall view of these PDF’s is shown in Fig. 1, at two scales Q = 2 and 100 GeV. The overall χ2 for the CTEQ6M fit is 1954 for 1811 data points. The parameters for this fit and the individual χ2 values for the data sets are given in Appendix A. In the next two subsections, we discuss the comparison of this fit to the data sets, and then describe the new features of the parton distributions themselves. Quantitative comparison of data and fit is studied in more depth in Appendix B
Fig. 1 : Overview of the CTEQ6M parton distribution functions at Q = 2 and 100 GeV.
Lai et al., Eur. Phys. J. C12 (2000) 375
Significant advances in determination of quark and gluon 3.1.1 Comparison with Data distributions at small x in recent years The fact that correlated systematic errors are now fully included in the fitting procedure allows a more detailed study of the quality of fits than was possible in the past. We can take the correlated systematic errors into account explicitly when comparing data and theory, by using the procedure discussed in Sec. B.2 of Appendix B. In particular, based on the formula for the extended χ2 function expressed in the simple form Eq. (11), we obtain a precise graphical representation of the quality of the fit by superimposing the theory curves on the shifted data points D containing { i} the fitted systematic errors. The remaining errors are purely uncorrelated, hence are properly represented by error bars. We use this method to present the results of our fits when!ever possible. Figure 2 shows the comparison of the CTEQ6M fit to the latest data of the H1 experiment [14]. The extensive data set is divided into two plots: (a) for x < 0.01, and (b) for x > 0.01. In order to keep the various x bins separated, the values of F2 on the plot have been offset vertically 2 for the kth bin according to the formula: ordinate = F2(x, Q ) + 0.15 k. The excellent fit seen in the figure is supported by a χ2 value of 228 for 230 data points. Similarly, Fig. 3 shows the comparison to the latest data from ZEUS [15]. One again sees very good overall agreement.
8 0.6 2 2 Q = 25 GeV uv d v 0.4 !u d! s
x f (x,Q) g/15 0.2
0 0 0.2 0.4 0.6 0.8 1 x sea quarks & gluons valence quarks q¯ = u¯, d¯, s¯... u g q = u, d, s... p d Valence quarks Nucleon structure at intermediate & large x dominated by valence quarks Most direct connection between quark distributions and models of the nucleon is through valence quarks
0.6 uv d v 0.4 !u d! s
x f (x,Q) g/15 0.2
0 0 0.2 0.4 0.6 0.8 1 x Valence quarks At large x, valence u and d distributions extracted from p and n structure functions 4 1 F p ≈ u + d 2 9 v 9 v 4 1 F n ≈ d + u 2 9 v 9 v u quark distribution well determined from p d quark distribution requires n structure function 4 − n p d ≈ F2 /F2 n p u 4F2 /F2 − 1 Higher twists
(a) (b) (c) Valence quarks RatioHigher of d to twists u quark distributions particularly sensitive to quark dynamics in nucleon
SU(6) spin-flavour symmetry τ = 2 τ > 2 (a) (b) (c) proton wave function single quark qq and qg ↑ 1 ↑ √2 ↓ scattering correlations p = d (uu)1 d (uu)1 −3 − 3 Higher twists √2 ↑ 1 ↓ 1 ↑ + u (ud)1 u (ud)1 + u (ud)0 τ = 2 6 τ >− 23 √2 (τ = 2) (b) (c) (a) diquark spin single quark interactingqq and qg corquarkrelationsspectator scattering diquark τ = 2 τ > 2
single quark qq and qg scattering correlations Valence quarks Ratio of d to u quark distributions particularly sensitive to quark dynamics in nucleon
SU(6) spin-flavour symmetry proton wave function
↑ 1 ↑ √2 ↓ p = d (uu)1 d (uu)1 −3 − 3
√2 ↑ 1 ↓ 1 ↑ + u (ud)1 u (ud)1 + u (ud)0 6 − 3 √2
u(x) = 2 d(x) for all x n F2 2 p = F2 3 Valence quarks
scalar diquark dominance
M∆ > MN =⇒ (qq)1 has larger energy than (qq)0
. =⇒ scalar diquark dominant in x → 1 limit .
. =⇒ since only u quarks couple to scalar diquarks . d → 0 u
n F2 1 p → F2 4
Feynman 1972, Close 1973, Close/Thomas 1988 Valence quarks
hard gluon exchange at large x, helicity of struck quark = helicity of hadron
q↑ ! q↓ . =⇒ . . =⇒ helicity-zero diquark dominant in x → 1 limit . d 1 → u 5 F n 3 2 → F p 2 7 Farrar, Jackson 1975 Valence quarks
BUT no free neutron targets! (neutron half-life ~ 12 mins)
use deuteron as ‘‘effective neutron target’’
d p n However: deuteron is a nucleus, and F2 != F2 + F2
nuclear effects (nuclear binding, Fermi motion, shadowing) obscure neutron structure information
“nuclear EMC effect” Nuclear ‘‘EMC effect’’
anti-shadowing FA 2 pions? d F2 Fermi motion shadowing
multiple ! Ca, SLAC scattering " Ca, NMC # Fe, SLAC $ Fe, BCDMS
x
A d 40 56 “EMC effect” Fig. 3.1. The structure function ratio F2 /F2 for Ca and Fe. The data are taken from NMC [71], SLACwhat[72], an aboutd BCDMS d[73 /]. N? binding, N off-shell
A d Figure 3.1 presents a compilation of data for the structure function ratio F2 /F2 over A the range 0 x 1. Here F2 is the structure function per nucleon of a nucleus with ≤ ≤ d mass number A, and F2 refers to deuterium. In the absence of nuclear effects the ratios A d d F2 /F2 are thus normalized to one. Neglecting small nuclear effects in the deuteron, F2 can N approximately stand for the isospin averaged nucleon structure function, F2 . However, the more detailed analysis must include two-nucleon effects in the deuteron. Several distinct regions with characteristic nuclear effects can be identified: at x < 0.1 one observes a A d systematic reduction of F2 /F2 , the so-called nuclear shadowing. A small enhancement is seen at 0.1 < x < 0.2. The dip at 0.3 < x < 0.8 is often referred to as the traditional “EMC effect”. For x > 0.8 the observed enhancement of the nuclear structure function is associated with nuclear Fermi motion. Finally, note again that nuclear structure functions can extend beyond x = 1, the kinematic limit for scattering from free nucleons.
Shadowing region • Measurements of E665 [76,77,78] at Fermilab and NMC [71,75,79,80,81,82] at CERN provide detailed and systematic information about the x- and A-dependence of the A d structure function ratios F2 /F2 . Nuclear targets ranging from He to Pb have been used. A sample of data for several nuclei is shown in Fig.3.2. While most experiments −4 Xe d cover the region x > 10 , the E665 collaboration provides data for F2 /F2 [76] down to x 2 10−5. Given the kinematic constraints in fixed target experiments, the small " ·
30 EMC effect in deuteron
896 G. PILLER, W. MELNITCHOUK, AND A. W. THOMAS 54
2 2 2 ∗ A␣͑k,q͒ϭi͑q ⑀k␣ϩ͑k Ϫm ͒⑀q␣ γ Nuclear “impulse approximation’’
Ϫ2͑k⑀kq␣Ϫk⑀kq␣͒͒, ͑8c͒
2 incoherent scattering A␣͑k,q͒ϭϪim͑q g␣gϩ2q␣͑kgϪkg͒͒. ͑8d͒ N from individual nucleons Here k is the interacting quark four-momentum, and m is its in deuteron ␣  mass. We use the notation ⑀kqϵ⑀␣k q . ͑The com- plete forward scattering amplitude would also contain a crossed photon process which we do not consider here, since d in the subsequent model calculations we focus on valence quark distributions.͒ The function (k,p) represents the soft FIG. 1. DIS from a polarized nucleus in the impulse approxima- quark-nucleon interaction. SinceHone is calculating the tion. The nucleus, virtual nucleon, and photon momenta are denoted imaginary part of the forward scattering amplitude, the inte- by P, p, and q, respectively,d and S stands for the nuclear spin N δ(off)F d x gration over the quark momentum k is constrained by ␦ vector. The upper blob Frepresents2 (x) the= truncateddy antisymmetricfN/d(y) F2 (x/y) + 2 ( ) ˆ ! functions which put both the scattered quark and the nonin- nucleon tensor G , while the lower one corresponds to the polar- ˆ teracting spectator system on-mass-shell: ized nucleon-nucleus amplitude A.
4 2 2 2 2 d k 2␦͓͑kϩq͒ Ϫm ͔2␦͓͑pϪk͒ ϪmS͔ H␣ϭ͑p␣kϪpk␣͒p k f1ϩ͑p␣Ϫp␣͒ dk˜ϵ , nucleon momentum distribution off-shell correction ͑2͒4 ͑k2Ϫm2͒2 ϫ͑p f2ϩk f3͒ϩ͑k␣Ϫk␣͒͑p f4ϩk f5͒ ͑9͒ ϩ␣f6ϩ⑀␣p k ␥5͑p” f7ϩk”f8͒ where m2ϭ(pϪk)2 is the invariant mass squared of the S ϩ⑀ ␥ ␥͑pf ϩkf ͒, ͑13b͒ spectator system. ␣ 5 9 10 Taking the trace over the quark spin indices we find where the functions g and f are scalar functions of 1•••9 1•••10 ␣ ␣ Tr͓ r͔ϭA␣H ϩA␣H , ͑10͒ p and k. H Performing the integration over k in Eq. ͑7͒ and using where H␣ and H␣ are vector and tensor coefficients, respec- Eqs. ͑13͒, we obtain expressions for the truncated structure tively. The general structure of H␣ and H␣ can be deduced functions G(i) in terms of the nonperturbative coefficient from the transformation properties of the truncated nucleon functions f i and gi . The explicit forms of these are given in ˆ Appendix I. From Eq. ͑4͒ we then obtain the leading twist tensor G and the tensors A␣ and A␣ . Namely, from ␣ ˜ contributions to the truncated nucleon tensor Gˆ .Itisim- A*␣(k,q)ϭA␣(k,q) and A (k,˜q)ϭϪA␣(k,q), we have portant to note that at leading twist the non-gauge-invariant ˆ contributions to G vanish, so that the expansion in Eq. ͑4͒ ␣ ˜ † is the most general one which is consistent with the gauge H ͑p,k͒ϭϪ H␣͑˜p,k͒ , ͑11a͒ P P invariance of the hadronic tensor. H␣͑p,k͒ϭ͑ H ͑˜p,˜k͒ †͒*, ͑11b͒ T ␣ T III. NUCLEAR STRUCTURE FUNCTIONS ␣ ϭ ␣† H ͑p,k͒ ␥0H ͑p,k͒␥0 . ͑11c͒ Our discussion of polarized deep-inelastic scattering from nuclei is restricted to the nuclear impulse approximation, il- ␣ ˜ Similarly, since A*␣(k,q)ϭA␣(k,q) and A (k,˜q ) lustrated in Fig. 1. Nuclear effects which go beyond the im- ϭA␣(k,q), one finds pulse approximation include final state interactions between the nuclear debris of the struck nucleon ͓17͔, corrections due ␣ ˜ † H ͑p,k͒ϭ H␣͑˜p,k͒ , ͑12a͒ to meson exchange currents ͓18–20͔ and nuclear shadowing P P ͑see ͓21–24͔ and references therein͒. Since we are interested in the medium- and large-x regions, coherent multiple scat- H␣͑p,k͒ϭϪ͑ H ͑˜p,˜k͒ †͒*, ͑12b͒ T ␣ T tering effects, which lead to nuclear shadowing for xՇ0.1, will not be relevant. In addition, it has been argued ͓6͔ that ␣ ␣† H ͑p,k͒ϭ␥0H ͑p,k͒␥0 . ͑12c͒ meson exchange currents are less important in polarized deep-inelastic scattering than in the unpolarized case since With these constraints, the tensors H␣ and H␣ can be pro- their main contribution comes from pions. jected onto Dirac and Lorentz bases as follows: Within the impulse approximation, deep-inelastic scatter- ing from a polarized nucleus with spin 1/2 or 1 is then de- H␣ϭp␣␥5͑p” g1ϩk”g2͒ϩk␣␥5͑p” g3ϩk”g4͒ scribed as a two-step process, in terms of the virtual photon- nucleon interaction, parametrized by the truncated ϩi␥5p k ͑p␣g5ϩk␣g6͒ϩ␥␣␥5g7 ˆ antisymmetric nucleon tensor G(p,q), and the polarized ˆ ϩi␥5␣͑p g8ϩk g9͒, ͑13a͒ nucleon-nucleus scattering amplitude A(p,P,S). The anti- EMC effect in deuteron
Nucleon momentum distribution in deuteron relativistic dNN vertex function 2 pmax 1 2 Ep 2 2 fN/d(y) = Md y dp Ψd(p! ) ! 0 4 −∞ p " " " "
momentum fraction of deuteron carried by nucleon EMC effect in deuteron
Nucleon momentum distribution in deuteron relativistic dNN vertex function 2 pmax 1 2 Ep 2 2 fN/d(y) = Md y dp Ψd(p! ) ! 0 4 −∞ p " " " " EMC effect in deuteron
Nucleon momentum distribution in deuteron relativistic dNN vertex function 2 pmax 1 2 Ep 2 2 fN/d(y) = Md y dp Ψd(p! ) ! 0 4 −∞ p " " " "
Wave function dependence only at large |y-1/2|
sensitive to large p components of wave function not very well known EMC effect in deuteron
Nucleon momentum distribution in deuteron relativistic dNN vertex function 2 pmax 1 2 Ep 2 2 fN/d(y) = Md y dp Ψd(p! ) ! 0 4 −∞ p " " " "
Nucleon off-shell correction δ(off)F d δ(Ψ)F d negative energy components 2 2 of d wave function 2 δ(p )F d off-shell N structure function 2 Off-shell correction
total
2 δ(p )F d 2
δ(Ψ)F d 2
≤ 1 − 2 % effect
WM, Schreiber, Thomas, Phys. Lett. B 335 (1994) 11 EMC FeffIGUectRES in deuteron
full model light-cone
Fermi motion only
with binding + off-shell
D N FIG. 1. F2 /F2 ratLargerio as a func tEMCion of x f oeffr theectoff-sh e(smallerll model of R d/Nefs. [4 ,5ratio)] (solid) and the on-shell model of Ref. [6] (dotted)F. n 2 underN estimated at large x In Refs. [4,5] the structure function F2 was modeled in terms of relativistic quark– nucleon vertex functions, which were parametrized by comparing with available data for the parton distribution functions. The off-shell extrapolation of the γ∗N interaction was modeled assuming no additional dynamical p2 dependence in the quark–nucleon vertices. (off) D This enabled an estimate of the correction δ F2 to be made, which was found to be quite small, of the order 1 2% for x < 0.9. The result of the fully off-shell calculation from Ref. ∼ − [4] is shown in Fig.1 (solid curve),∼where the ratio of the total deuteron to nucleon structure D N functions (F2 /F2 ) is plotted. Shown also is the result of an on-mass-shell calculation from Ref. [6] (dotted curve), which has been used in many previous analyses of the deuteron data [7,8]. The most striking difference between the curves is the fact that the on-shell ratio has a very much smaller trough at x 0.3, and rises faster above unity (at x 0.5) than the off-shell curve, which has a deeper≈trough, at x 0.6 0.7, and rises above ≈unity somewhat ≈ − later (at x 0.8). The beh≈avior of the off-shell curve in Fig.1 is qualitatively similar to that found by Uchiyama and Saito [9], Kaptari and Umnikov [10], and Braun and Tokarev [11], who also used off-mass-shell kinematics, but did not include the (small) non-convolution correction (off) D term δ F2 . The on-shell calculation [6], on the other hand, was performed in the infinite momentum frame where the nucleons are on their mass shells and the physical structure functions can be used in Eq.(1). One problem with this approach is that the deuteron wave functions in the infinite momentum frame are not explicitly known. In practice one usually makes use of the ordinary non-relativistic S- and D-state deuteron wave functions
3 Unsmearing
n Note: calculated d/N ratio depends on input F2 extracted n depends on input n ... cyclic argument
Solution: iteration procedure
δ(off)F d F d → F d − δ(off)F d 0. subtract 2 from d data: 2 2 2
p p p 1. smear F 2 with f N / d : fN/d ⊗ F2 ≡ Sp F2
n d p 2. extract neutron via F2 = Sn(F2 − F2 /Sp)
starting with e.g. Sn = Sp
n 3. smear F 2 with to get new Sn
4. repeat 2-3 until convergence DEEP INELASTIC SCATTERINGUnsmearingFROM Aϭ3 NUCLEI... PHYSICAL REVIEW C 68, 035201 ͑2003͒
N 2 2 structure function F2 (xЈ,p ,Q ) can be evaluated in terms of a relativistic quark spectral function N as
xЈ2 dk2 FN͑xЈ,p2,Q2͒ϭ ͑k2͑p͒,p2 ͒, 2 ͚ ͵ 2 3 N X 1ϪxЈ X kmin 4͑2͒ ͑45͒
2 where N depends on the virtualities of the struck quark, k , 2 2 2 and spectator system, pX , and the limit kminϭkmin(xЈ,p ,pX) Afnan, Bissey, Gomez, Liuti, WM, Thomasfollows et al., from the positivity constraint on the struck quark’s FIG. 8. Neutron to proton structure function ratioPhys.extracted Rev. C68 (2003)from 035201 2 2 3 3 transverse momentum kЌу0. The dependence of kmin on p He H n p 2 the Fg2ood/F 2conratiovergencevia the iterationafter seprocedure.veral iterationsThe input is F2/F2 (*M ) generates an off-shell correction which grows with A ϭ1, and the rationafter ϳ3 iterations is indistinguishable from the 2 resulting F 2 independent of starting assumptions due to the A dependence of the virtuality p of the bound exact result. nucleon. This serves to enhance the EMC effect at large x in depends only on smearing function fN/d comparison with naive binding model calculations which do this section, we examine the accuracy of this assumption. not take into account nucleon off-shell effects ͓45͔. Assum- N The off-shell dependence of F2 is, as a matter of principle, ing that the spectator quarks can be treated as a single system 2 not measurable, since one can always redefine the nuclear with a variable mass mX , the off-shell structure function in spectral function to absorb any p2 dependence in the bound Eq. ͑45͒ can be related to the on-shell function by a nucleon structure function. However, off-shell effects can be p2-dependent rescaling of the argument xЈ, namely ͓35͔, identified once a particular form of the interaction of a nucleon with the surrounding nuclear medium is specified. N N 2 F2 ͑xЈ͉͒p2*M 2 F2 ͑xЈ͑p ͒ϾxЈ͉͒p2ϭM 2. ͑46͒ The discussion of off-shell modification of the nucleon struc- → ture function in the nuclear medium is therefore understood to be within the framework of the nuclear spectral functions It is this ͑further͒ rescaling in x that is responsible for the defined in Sec. III. larger effect at large x. The effect of the off-shell correction on the ratio , illus- In convolution models, off-shell corrections can arise both R kinematically, through the transverse motion of the nucleon trated in Fig. 9, is a small (Շ1%) increase in the ratio at x in the nucleus, and dynamically, from modifications of the ϳ0.6. Off-shell effects of this magnitude can be expected in bound nucleon’s internal structure. Kinematical off-shell ef- models of the EMC effect where the overall modification of fects are essentially model independent, as discussed in Ref. the nuclear structure function arises from a combination of ͓35͔, while dynamical off-shell effects do depend on descrip- conventional nuclear physics phenomena associated with tions of the intrinsic deformation of the bound nucleon struc- nuclear binding, and a small medium dependence of the ture and are therefore model dependent. The latter have been nucleon’s intrinsic structure ͓1,33,46,78͔. modeled, for instance, in a covariant spectator model ͓33͔,in Other models of the EMC effect, such as the color screen- which the DIS from a bound nucleon is described in terms of ing model for suppression of pointlike configurations ͑PLC͒ relativistic vertex functions which parametrize the nucleon- in bound nucleons ͓79͔, attribute most or all of the EMC quark-spectator ‘‘diquark’’ interaction. The dependence of effect to a medium modification of the internal structure of the vertex functions on the quark momentum and the diquark the bound nucleon, and consequently predict larger devia- energy is constrained by fitting to the on-shell nucleon ͑pro- tions of from unity ͓77͔. However, recent 4He(eជ,eЈpជ ) R ton͒ structure function data, while the additional dependence polarization transfer experiments ͓80͔ indicate that the mag- on the virtuality of the off-shell nucleon can be constrained nitude of the off-shell deformation is indeed rather small. by comparing the calculated nuclear structure function with The measured ratio of transverse to longitudinal polarization A of the ejected protons in these experiments can be related to the inclusive F2 data. Taking the nucleon’s off-shellness into account, the bound the medium modification of the electric to magnetic elastic nucleon structure function in Eq. ͑8͒ can be generalized to form factor ratio. Using model-independent relations derived ͓33,35,46͔ from quark-hadron duality, the medium modifications in the form factors were related to a modification at large x of the
A 2 2 2 2 N 2 2 deep inelastic structure function of the bound nucleon in Ref. ϭ 4 F2 ͑x,Q ͒ ͵ dy͵ dp ͑y,p ,Q ͒F2 ͑xЈ,p ,Q ͒, ͓81͔.In He, for instance, the effect in the PLC suppression ͑44͒ model was found ͓81͔ to be an order of magnitude larger than that allowed by the data ͓80͔, and with a different sign for xտ0.65. The results therefore place rather strong con- where xЈϭx/y and the function (y,p2,Q2) depends on the straints on the size of the medium modification of the struc- nuclear wave functions. In the absence of p2 dependence in ture of the nucleon, suggesting little room for large off-shell N 2 F2 , the light-cone momentum distribution f (y,Q ) in Eq. corrections, and support a conventional nuclear physics de- ͑8͒ would correspond to the p2 integral of (y,p2,Q2). In scription of the 3He/3H system as a reliable starting point for the approach of Ref. ͓35͔, the medium modified nucleon nuclear structure function calculations.
035201-11 A similar result is also obtained in the treatment of Brodsky et al. [21] (based on counting-rules), where the large-x behavior of the parton distribution for a quark polar- ized parallel (∆Sz = 1) or antiparallel (∆Sz = 0) to the proton helicity is given by: q↑↓(x) = (1 x)2n−1+∆Sz , where n is the minimum number of non-interacting quarks − (equal to 2 for the valence quark distributions). In the x 1 limit one therefore predicts: → n F2 3 d 1 p , [Sz = 0 dominance]. (11) F2 → 7 u → 5 Note that the d/u ratio does not vanish in this model. Clearly, if one is to understand the dy- namics of the nucleon’s quark distributions at large x, it is imperative that the consequences of these models be tested experimentally. The reanalyzed SLAC [7,22] data points themselves are plotted in Fig.3, at an average value of Q2 12 GeV2. The very small error bars are testimony to the quality of the SLAC p and D data.≈The data represented by the open circles have been extracted with the on-shell deuteron model of Ref. [6], while the filled circles were obtained using the off-shell model of n p Refs. [4,5]. Most importantly, the F2 /F2 points obtained with the off-shell method appear to approach a value broadly consistent with the Farrar-Jackson [20] and Brodsky et al. [21] prediction of 3/7, whereas the data previously analyzed in terms of the on-shell formalism produced a ratio that tended to the lowEffer vectalue oonf 1/ 4n/p. ratio
with binding SU(6) & off-shell helicity retention scalar Fermi motion only diquarks
WM, Thomas Phys. Lett. B 377 (1996) 11 n p FIG. 3. Deconvoluted F2 /F2 ratio extracted from the SLAC p and D data [7,22], at an average value of Q2 12 GeV2, assuming no off-shell effects (open circles), and including off-shell effects ≈ n (full circles). without EMC effect in d, F 2 underestimated at large x n p The d/u ratio, shown in Fig.4, is obtained by inverting F2 /F2 in the valence quark dominated region. The points extracted using the off-shell formalism (solid circles) are
7 again significantly above those obtained previously with the aid of the on-shell prescription. In particular, they indicate that the d/u ratio may actually approach a finite value in the x 1 limit, contrary to the expectation of the model of Refs. [17,18], in which d/u tends to→zero. Although it is a priori not clear at which scale the model predictions [17,18,20,21] should be valid, for the values of Q2 corresponding to the analyzed data the effects of Q2 evolution are minimal. Effect on n/p ratio
FIG. 4. Extracνtepd d/u ratio, using the off-shell deuteron calcuνlatpion (full circles) and using on-shell kinemFatic2s (ope=n circ2lesx). A(lsdo sh+own uf¯or)comparisonxisFthe3ratio =extraxcted(frdom−neutur¯in)o measurements by the CDHS collaboration [23]. ν¯p ν¯p F n x NaturallyFit 2would=be p2rexferab(leuto+extrda¯c)t 2 at larxgeF3withou=t haxving(tuo d−eal wd¯it)h uncertainties in the nuclear effects. In principle this could be achieved by using neutrino and antineutrino beams to measure the u and d distributions in the proton separately, and n reconstructing F2 from these. Unfortunately, as seen in Fig.4, the neutrino data from the CDHS collaboration [23] do not extend out to very large x (x < 0.6), and at present cannot discriminate between the different methods of analyzing the e∼lectron–deuteron data. The results of our off-shell model are qualitatively similar [22] to those obtained using the nuclear density method suggested by Frankfurt and Strikman [24]. There the EMC effect in deuterium was assumed to scale with that in heavier nuclei according to the ratio of the D N respective nuclear densities, so that the ratio F2 /F2 in the trough region was depleted by n p about 4%, similar to that in Fig.1 (solid curve). This would give an F2 /F2 ratio broadly consistent with 3/7. We should also point out similar consequences for the spin-dependent neutron structure n function g1 , where the models of Refs. [17,18] and Refs. [20,21] also give different predictions for gn/gp as x 1, namely 1/4 and 3/7, respectively. Quite interestingly, while the ratio of 1 1 →
8 Diquarks as Inspiration and as Objects
Frank Wilczek∗ mass difference. The Λ is isosinglet, so it features [ud]; while Σ, being isotriplet, features (ud). The Σ is indeed heavier, by aSbeoputtem8b0eMr 1e7V,.2O00f4course, this comparison of diquarks is not ideal, since the spectator s quark also has significant spin-dependent interactions. A cleaner comparison involves thearXiv:hep-ph/0409168charm analogues, whe rv2e Σ 17 ΛSep= 2004215 MeV. (Actually this c − c comparison is not so clean either, as we’lAl bdsitsrcaucsts later. One sign of uncleanliness is that 3 + 1 + b p t h e w t m Q I 1 1 t g A
there either Σ (2520) or Σ (2455) mighn t be used for comparison; here I’ve taken the h h Attraction between quarks o is a fundamental aspect of QCD. It is plausible that n
c c e a a e . h n C
2 2 o g n e a 1 e i l ∗ t i s r f p e e r t e D S c i t s t g e l r s a o h o
several of the most profound aspects of loe w-energy QCD dynamics are connected to t h D , n y a y l s h weighted average.) t e n i I f p m d s a h c i c e e D t e e e : m c i s u t i n e h e d e p a s a r e i o q s v r n i r r u t m d t i o l e t g m e o d u i q o c e t s diquark correlations, including: paucit ty of exotics (which is the foundation of the quark l A j h t u r n i w q y r y d i w w o c u b e a d n h i a a r c t e e t , u
One of the oldest observations in deep inelastic scatteriu ng is that the ratio of neutron l n s r e u e s e i y t o e h e c l r l t k t s a i r d t v a r l a s t o r r e a - n n a i i s y model and of traditional nuclear physics), similarity of ms esons and baryons, color su- l o a n e u c h q n f n c r y c o . h 1 a r u f c h r r i o t a d c t , o d u e f s q t m t a o , r l k o t l l l r k T to proton structure functions approaches in the limit x 1 t o e a i h i s o u i l o w p r h o o b a 3 ¯ a s
4 v t v f perconductivity at high densif ty, hyperfine splittings, ∆I = 1/2 rule, and some striking i r s t h e r e l D s i n f r u c r e t s i . k h y , e t c l e a e d t → u t s t a i fl e t l o m e y a s i r t m s n i , t c b r t c i a o m e q u u h a t h t a i t i q
features of structure and fragmentatioo n functions. After a brief overview of these issues, e n a m t n o o m t u p m u c x s h r e d e t a h l t M r t n s a s t r u n w r o t y i e s e a u t i p q i e e o a t s r i h s r a s t I o e m c i
F (x) 1 , k c , a s u - o i i o
I discuss how diquarks can be studied in isolation, both phenomenologp ically and numer- a r p e b t w 2 n t n l h g s i s s c c i h y r a o n n n r n p r i a s a t o h o . t i v o r c e
lim a (1.1) r a d s c i a r r l k p s c a m n a n e f b k d l o q o l l y d i t I T p o i u
ically, and present n approximate mass differences for diquarks with different quantum f n c o a x - s e u
r x 1 a e s r n e u g Fe (x) → 4 d t a t c n d h e t i a a t i n i
2 c u b s n h e m h 3 n i n a → i e u r g r a s f n p c n K n o e d g e a k t k d t o a i m a s l
numbers. The mass-loaded generaliz⊗ ation of the Chew-Frautschi formula provides an a g t o t e e i e s p e o s n s r i b t y q 3 ¯ : e a n r n s t e r a g b w t e n , i o 3 r u u a s s n a u i e S a r p g y c In i s a c p e i h u i r t d m n t c a n a , e b o essential tool. t h p c c l a e a e y t t i r a u h b p d
In terms of the twist-two operator matrix elemQ ents used in the formal analysis of deep l e a a t m F c h l a t a i p k n e i b s c o i d r i a t s e t n t y n z n r e o o f e t i i s s n C p h e e r w s a u t 3 s a r n m h n n i n s y l b m i A n fi t e n a d o n o g c i e e f i t ⊗ D i o f t n b e o d t f o s i o w r inelastic scattering, this ta ranslates into the statement l i u i b k r b b h ff fl . t ) n f s e l n a f e i n a n 3 ¯ l e u , l a s o r e e e u o m g t o e W W j i d s 1 c a l t r l z t n w r o s t w t u x x w a p → t r e w c e i v e i h f t d i i o t m n s n e l a - e t 1 n r o o h i s i e i o e e
1 Diquarks as Inspiratiot n a t a t e l i c t c o g l o t . n 7 i e n e i o u r h c 1 m t s n m e a l t 1 c , n n s n , i a e m z n s e s l E . s l n t . , q o
i ¯ r o r c e e t 2 h g t
Folklore e that experiment gives 1/4 limiting ratio... g g b ( a i f C u w S p dγ n d p 1 ←→ 2 ←→ a A o r n k c
e µ µ µ l w t a , o w 0 h s e y i c o i s a y i i l v t h ∗ k r , - g f e a n 0 p h t o o t h n a t r e e s d lim e 0 (1.2)
# | ∇ e · · · ∇ | & g Q h ∆ t r e o 4 e c k d i n l y n w e . - c e ! e q d
1.1 Diquarks in Microscopic QCD c r f e q i s c C x I m p e n h r d t r - q M u
h → a u T m h e r F p h q i D u t p u¯γ n u p 1 ←→ 2 ←→ = a n i m g µ µ µ p →∞ a o a . r a a e b o u s e w w a e r y a f g r n r r S l c g r s a r e d k t 1 s f g o u k e i o i
# | ∇ · · · ∇ | & o h t k h e Q t r t / e y Q s e c t s m n i a s t r h k i e w f 2 f t n e s
In electrodynamics the basic interaction bed tween like-charged particles is repulsive. In o O C s h i s m s C c h - o n e c a w f c n i a r h e D o a n v d i a i a m t a D p u p d n i r n s b 6 i u e n a o n t e . n
where spin averaging of forward matrix elements, symmetrizatil on over the µs, and removal o a e i t r s h n . e l t f l d a w
QCD, however, the primary interaction between two quarks can bc e attractive. At the t o v i j o w d o n I y i r i , d s q t h n t o b e g t i r r e d y s b t a a e e u s m i e g n i a e a r i c n c w a r i n t c t e ff l a l s ; a r t i f a l u h r
of traces is implicit, and a commono tensorial form is factored out, together with similar equa- p d e 3 a t most heuristic level, this comes about as follows. Each quark is in o the 3 representation, so e e u e r i a c y o l w i l o e m p a n k r s l s f a t l t o s f y a r c e n l r e i t o n a a e t o i t o n n o w a a p i r r a c m h h n s u
p ¯ f s n t n t a t t # r n tions where operators wits h strange quarks, gluons, etc. appear in the numerator. Equation
that the two-quark color state 3 3 can be either the symmetric 6, or the antisymmetric 3. o e s t e i t t r t e n o q e c . ¯ d - q u i s i h r c r h y e d t v e t b q f s e u e o a s s l o e
⊗u i c e i y n $ l d t p v e l d a a q c i e t n u o r m q s n b e e t Antisymmetry, of course, is not possible with just 1 color! Twn o widely separated quarks each g r e u u
(1.2) states that in the valence regime x 1, where the struck pi arton carries all the longitu- = % s r m d u a i d k . e t s t t l e u l a m o h u a u a p a s i s e e r a n n b r i t u a r m m o → t p 0 r s e k v n A k o g o r i t - - , n
generate the color flux associated with tt he fundamental representation; if they are brought e o d e s o e dinal momentum of the proton, tu hat struck parton must be a u quark. It implies, by isospin r t o a . n r n u e i g fi f r c i , a t g t h v e
¯ a h c q I h s t 3 l e ¯ together in the 3, they will gel nerate the flux associated with a single anti-fundamental, h l d n y o o q e ¯ s t , symmetry, the corresponding relation fo. r the neutron, namely that in the valence regime withwinhiachniesujtursotnhtahlfeapsamrtuocnh.mTuhsutsbbeyabdriqnguianrgk.thTe hqeunartkhsetoragetitoheorfwneeulotwroenr tthoepgrloutoonnfimeldatrix ¯ 1 2 energy: there is attraction in the 3 channel. We might expect this attraction to be r(ou3g)hly
arXiv:hep-ph/0409168 v2 17 Sep 2004 1 elements will be governed by the ratio of the squares of quark charges, namely −2 = . half as powerful as the quark-antiquark 3 3¯ 1. Since quark-antiquark attraction(dr)2ives 4 ⊗ → 3 Anyt(hiesoesninergglyet)incothnetaamttirnaactivoen cfhroamnneolthbeerloswouzrecreos, wtrilglgceorinntgricbountdeeenqsautaiollny tq¯oqnu=me0raotfoqrq¯and # $ % denopmaiirnsaatnodr,cthhirearlesbyymimncetrreyasbinregaktihnigs, raantiaot.traEcqtiuonateiovnen(h1a.2lf) aiss,pforwoemrfutlhweopulodinatppoefavriteow of symmbeeptroyt,enatipalelycuqluiaitreriemlaptoirotna:ntitforeuqnudierersstaanndienmg elorwge-enntercgoynsQpCirDacdyynbaemtwicese.n isosinglet and isotripl∗eStoliocipteedracotnotrrsib.utiIotn itso,thfreoIman Kaoggaennmereamlorpiahl yvosliucmale, pedo.inMt. Sohfifvmiaenw. , most remarkable: it is one of the most direct manifestations of the fractional charge on quarks; and it is a sort of hadron = quark identity, closely related to1 the quark-hadron continuity conjectured to arise in high density QCD. It is an interesting challenge to derive (1.2) from microscopic QCD, and to estimate the rate of approach to 0. A more adventurous application is to fragmentation. One might guess that the formation of baryons in fragmentation of an energetic quark or gluon jet could proceed stepwise, through the formation of diquarks which then fuse with quarks. To the extent this is a tunneling-type process, analogous to pair creation in an electric field, induced by the decay of color flux tubes, one might expect that the good diquark would be significantly more
3 uncertainty due to nuclear effects in neutron (full range of nuclear models)
d distribution poorly known beyond x ~ 0.5 “Cleaner” methods of determining d/u
e∓ p → ν(ν¯)X need high luminosity
ν(ν¯) p → l∓ X low statistics ± p p(p¯) → W X need large lepton rapidity
!eL(!eR) p → e X low count rate ± e p → e π X need z ~ 1, factorization 3 3 e He( H) → e X tritium target “Spectator Tagging”
“Cleaner” methods ofW determining2 = M 2 + 2M" # Qd/u2 e d → e p X r ! pn = (M D "E S ," p S ); #n = 2"#S target d
!
recoil pr 2 r ES # p S $ qˆ W 2 = p + q = p µ p + 2 (M " E )# " pr $ qr "Q 2 pS = (E S , p S ); "S = ( n ) n nµ ( D s n ) M D /2 % M * 2 +2M# (2"& )"Q 2 slow backward p S Q 2 Q 2 neutron nearly on-shellx *= "JLab Hall B experiment (‘‘BoNuS’’) µ ! 2 p q 2Mcompleted# (2$%S ) run Dec. 2005 minimize! r escattering n µ
! Issues at large x Target mass corrections 2 2 finite M / Q effects (but leading twist!) Georgi, Politzer, PRD14 (1976) 1829 Kretzer, Reno, PRD69 (2004) 034002 Higher twists 2 dynamical quark-gluon correlations, 1 / Q suppressed
Quark-hadron duality low-W resonances conspire to produce scaling function WM, Ent, Keppel, Phys. Rept. 406 (2005) 127 Large-x resummation extend validity of pQCD by resumming large-x logs arising from soft & collinear gluons Sterman, NPB281 (1987) 310; Catani, Trentadue, NPB327 (1989) 323 Corcella, Magnea, hep-ph/050742; Vogelsang, AIP Conf. Proc. 747 (2005) 9 Issues at large x Target mass corrections
Georgi-Politzer (GP) prescription
2 2 3 1 4 4 1 1 GP 2 x M x ! ! M x ! !! !! F2 (x, Q ) = 3 F (ξ) + 6 2 4 dξ F (ξ ) + 12 4 5 dξ dξ F (ξ ) r Q r !ξ Q r !ξ !ξ! 2x ξ = 1 + r = 1 + 4 2 2 2 “quark distribution function” r ! x M /Q F2(y) F (y) = y2
... and similar for other structure functions numerically...
Christy et al. (2005)
TMC no TMC
2 2 TMCs significant at large x / Q , especially for FL Threshold problem if F ( y ) ∼ ( 1 − y ) β at large y then since ξ0 ≡ ξ(x = 1) < 1
F (ξ0) > 0
TMC 2 Fi (x = 1, Q ) > 0 is this physical?
problem with GP formulation? Possible solutions
Johnson/Tung - modified threshold factor
Nachtmann moment 1 n+1 2 n 2 ξ 3 + 3(n + 1)r + n(n + 2)r 2 µ2 (Q ) = dx 3 F2(x, Q ) !0 x " (n + 2)(n + 3) #
n fixed, Q2 → ∞
n 2 2 2 −λn µ2 (Q ) → (ln Q /Λ ) An 1 n An = dξ ξ F (ξ) !0
2 “regularized” amplitudes n → ∞, Q fixed (threshold-independent) n 2 n 2 n 2 µ2 (Q ) → ξ0 (Q ) µ!2 (Q ) Possible solutions
Johnson/Tung - modified threshold factor
Nachtmann moment 1 n+1 2 n 2 ξ 3 + 3(n + 1)r + n(n + 2)r 2 µ2 (Q ) = dx 3 F2(x, Q ) !0 x " (n + 2)(n + 3) #
n 2 n 2 2 2 −λn ansatz µ2 (Q ) = ξ0 (Q ) (ln Q /Λ ) An
consistent with asymptotic pQCD behavior
not unique!
Bitar, Johnson, Tung PLB 83B (1979) 114 Possible solutions
Johnson/Tung - modified threshold factor
1 n n−2 moreover, if identify A n with M2 = dx x F2(x) !0
n 2 n 2 n 2 µ2 (Q ) = ξ0 (Q ) M2 (Q ) 2 n 2 n 2 nM n M2 (Q ) = µ2 (Q ) + M2 + · · · Q2 cf. exact expression 2 n n n(n − 1) M n M (Q2) = µ (Q2) + M +2 + · · · 2 2 n + 2 Q2 2
inconsistency at low Q 2 ? TMC that the target mass corrected inelastic structure functions F2 remain finite as x 1 even if the LT terms vanish in this limit. Clearly, the region x close to 1 is beyond the ap→pli- that the target mass corrected inelasticPsossibletructure func tisolutionsons F TMC remain finite as x 1 cability of Eqs.(23). However, in the applications to nuclear stru2 cture functions at large x it even if the LT terms vanish in this limit. Clearly, the region x close to 1 is beyond the ap→pli- is important to meet the threshold conditPioossiblen. One possib lsolutionse way to deal with this problem is that the target mass corrected cianbeillaitsytiocfsEtrqus.c(t2u3r)e. fHuonwcetivoenr,s inF TthMeCaprepmlicaaintiofinnsitoe nauscxlear st1ructure functions at large x it to expand Eqs.(23) in power serie2s in Q−2 and keep a finite number of terms. In parti2cular, even if the LT terms vanish in thisisimlimpoirtt.aCntlKetaorulagin/Plmy,eetht ethreegtihorneettisxhoclldoP sc-eossibleo ntexpandodi1tioisnb. eOy nosolutionsendp oexprtshseibalep→pwlessionsai-y to deal wi tinh th i1s p/rQobl e m is keeping thPe LossibleT and the 1/ Qsolutions2 term we have cability of Eqs.(23). However, intothexepaapnpdliEcaqtsi.o(2n3s)tionnpuocwleearrssetrrieusctinurQe −fu2 nacntdioknesepatalafirngietexniutmber of terms. In parti2cular, Kulagin/P2etti - expand expressions in 1/Q is important to meet the threshokledepcionngdtihtieonLT.TMOaCnde pthoes2s1ib/Qle wtaeLyrTmtowdee2haalvweith this problem is 2 KFulagin/PT (x, Qetti) = - FexpandT (x, Q )expr+ essions in 1/Q to expand Eqs.(23) in power series in Q−2 and keep a finite number of terms. In parti2cular, Kulagin/Petti - expandTMC expr2 xessions3MLT 2 2 1 d zin 1/Q ∂ keeping the LT and the 1/Q2 term we have FT (x, Q ) = FT (x, Q2 ) + F LT(z, Q2) F LT(x, Q2) , (24a) 2 1 2 2 T xQ3M 2 x zdz − ∂∂x ! 2" F LT(z, Q2) F LT(x, Q2)# , (24a) TMC 2 LT 2 2 4x2M 2 2 2 T FT (x, Q ) = FT (x, Q ) + TMC 2 Q x z LT 2 − ∂x F2 (x, Q ) = 1 ! " F2 (x, Q ) + # x3M 2 1 dz ∂ − 4xQ22M 2 2 F TFMLCT((xz,,QQ22))= ! 1F LT(x, Q#2) F,LT(x, Q2) +(24a) Q2 z22 2 −x∂3xM−2T Q2 1 dz 2 ∂ ! x ! 6 ##F LT(z, Q2) F LT(x, Q2) , (24b) 2 " 2 3 2 2 1 2 2 2 4x M xQM x zdz − ∂∂x TMC 2 LT 2 ! 6" F LT(z, Q2) F LT(x, Q2)# , (24b) F2 (x, Q ) = 1 2 F2 (x, Q ) + 2 2 2 2 2 2 − Q TMC 2 Q 2x Mx z LT 2 − ∂x ! xF#3 (x, Q ) = 1 ! " xF3 (x, Q ) + # 3 2 1 − 2xQ22M 2 x M dzTMLCT 22 !∂ LT #2 LT 2 6 xF3F ((xz,,QQ))= 1F (x, Q1 ) xF, 3 (x, Q )(2+4b) 2 2 2 x3M−22 Q2 dz ∂ Kulagin, Petti Q x z − ∂!x 2 # zF LT(z, Q2) Kulagin,xF L TPetti,(x, NPQ2A765) .(2006) 126(24c) ! " 3 2 2 1 #2 3 3 hep-ph/0412425 2 2 xQM x zdz − ∂∂x Kulagin, Petti TMC 2 2x M LT 2 ! 2" zF LT(z, Q2) xF LT(x, Q2)# . (24c) xF3 (x, Q ) = 1 xF3 (x, Q ) + 2 2 3 3 hep-ph/0412425 − Q2 Q x z − ∂x I!n this approx#imation the structure fu!nct"ions have a correct threshould behav#ior and vanish 3 2 1 ixnIntMhtehilsimapitproofxdxizmathas1iLo,Tnptr hocorveis2dterrduectctthu∂a rthrte tfuhLesholdneTcLtiTontse2rh mabehavseaandcviorotrhreecirt dtKulagin,hereivsahtoiuv lePettids bveahnaisvhioirnatnhdisvalinmisith. 2 →zF (z, Q ) xF (x, Q ) . (24c) T2he target m2ass 3corrections should3 also be applied to thehep-ph/0412425HT terms in the higher order iQn the limixt ozf x has1, pr ocorvide−drect∂thxat tthrhe LTesholdterms an dbehatheir dviorerivatives vanish in this limit. termTs hi!en tta"hregettwmisat→sesxhascpoarnres cicorotinon(s2r0sect)h.ouFl odthrratlshoiesholdsbreeaa#spopnliew dbehaetdoothneoviortHcTontesridmesr i1n/Qth4e theirgmhesrionrdthere In this approximation the struTcttMuerCme fsfuoinrnmcttuiholaen,stwwhhiasivtceheaxaprcaeonrssrmieocantllt(ih2n0re)t.shheFocouorldntshbidiesehrraeevdaisokorinnaewnmdeavdtaiocnanilsohrtancogne.sidWere a1/lsQo 4notetrem, tshiant tthhee in the limit of x 1, providedtaTtrhgMaetCt tmfhoaersmsLTucloatr,erewrcmhtisochnansadfroetrhsameniarloldffei-nrsihvtehaletlitvcaoernsgsveidat,enrii.esedh. kiwninhteehmnisaptl2iicm=alitMr.a2n,gseh. oWuledablseotrneoattee,dthaastptahret 2 → more limited range of applicability# (not too low Q ) ? The target masshascorre ccortionosrftsectathrhogeuetln dumthrcaallessasoresholdcboeeffrreaeccpttpsiolainen sddbehafotworialtlnhbeovioreffHd-Tsishcetulelsrstmeadrsgient,Stihe.ece.t.hwiIgVhheAner6po2.r=deMr 2, should be treated as part 2 terms in the twist expansion (2o0f)t.heFonructlheaisr reeffaescotmorsn awned dweoi lllimitednboet dciosncusisds erangedr 1in/QSe4c ttof.erImV asApplicabilityi6n. #the (not too low Q ) ? TMC formula, which are small in the considered kinematical range. We also note, that the B. Structure function phenomenology target mass corrections for an off-shell target, i.e. when p2 = M 2, should be treated as part 2 B. Structure function phenomenology of the nuclear effectmors and wei lllimitedbe discuss eranged in Sec tof. IV aApplicability6. # (not too low Q ) ? The twist expansion and PDFs as universal, process-independent characteristics of the targeTthaeretwaitsttheexpbaanssiisoonf aenxdtePnsDivFesQasCDunpivheernsaolm, epnroocloegssy-ionfdehpigehn-deennetrgcyhaprraocctersissetsi.csInofpthhee- B. Structure function phneotanmrogemnetoelanorgoeilcoaagtlytshteudbiaessi,s tohfeexPtDenFssivaerQe CexDtrpahcteendomfreonmoloQgCy Dof ghliogbha-lenfietrsg.y Aproncuemssbees.r Ionf psuhceh- annaolmyseensolaorgeicaavlasitlaubdliees[,3t9h,e4P0,D4F1s].arIeneoxutrasctueddiefrsomof QnuCcDleagrlodbaatlafidtse.scrAibneduminbeSreocft.suVchF toanVaIlyDseswearuesaevathileabrleesu[3lt9s, 4b0y, A41le].khIin o[3u9r] s2tuwdhieos pofronvuidcleesarthdeatsaetdoesfctrhibeednuicnleSoenctP. DVFFs The twist expansion and PDFs as universal, process-independent chara2cteristics of the target are at the basis of extenobstiotvaeVinQIeDdCDwwietphhuescneooetmffiheecnieroenlstouglatynsdobfyshpiAlgitlhet-iknehngienrfug[3ny9cp]tirooncwsehscsoaelspc.urolIavntidepdehset-tohethseetNoNfLtOheanpupcrloexoinmPatDioFns. nomenological studies, the PDFFuobrstthaaerinetmedeoxrwteri,attchhteecdoHefffiTroctmiernmQt sCanaDdndgspltohlbiteatiPlnDfigtFsfu.unncActieonrntuasmincbateilecrsuolhafatvesudecahtlosothbeeeNnNeLvaOluaaptepdroixnim[3a9t]i.on. FuItrtshheotmulodreb,ethaelsoHTremtearmrksedantdhatthethPeDtFwiustnceexrptaainnstiioens haanvde paelsrotubrebeantievvealQuCatDedaipnp[a3r9e]n. tly analyses are available [39, 40, 41]. In our studies 2of nuclear data described in Sect. V F breaIkts sdhoowunldabte2loawlsoQre.mFaurrktehdertmhaotret,hethtewcisotnseexrpvaantsioionnoafnedlepcterrotmurabgantievteicQcCurDrenapt praerqeunitrleys to VI D we use the results by Alekhin [39] who p2 rovides the set of2 the nu2cleon PDFs thberesatkrsucdtouwren fautnlcotwionQ F. 2Ftuorthvaernmisohrea,sthQe cfoonrseQrvation0o.f eTlehcetrdomataagnseeetimc ctuorrienndticraetqeuitrhese obtained with coefficient and splitting functions calculated to the NNL2 O app2ro→ximation. prtehseensctreuoctfuaretrfaunnscittiioonn Fre2gitoonvbanetiswheeans pQertfuorrbaQtive an0d. nTohne-pderattuarbseaetmivetoreginimdiecsataet tQhe2 Furthetmore, the HT terms and the PDF uncertainties have also been evaluated i→n [39]. 2 abporeusten1cGe eoVf 2a. tInranousirtisotnudrieegsioonf nbuetcwleeaerneffpercttusribnatihvee satnrudctnuorne-pfuerntcutriobnatsivsoemreegdimateas paotinQts It should be also remarked tahbaotuth1eGtweVis2t. eIxnpoaunrsisotnudaiensdopf enrutculrebarateiffvecQtsCinDthapepsatrruecnttulyre functions some data points breaks down at low Q2. Furthermore, the conservation of electromagnetic current requires 2 In our analys2is we use P2 DFs obtained from new fits optimized in the low Q2 region and including additional the structure function F2 to va2nish as Q for Q 0. The data seem to indicate the 2 presence of a transition region dbaIenttawowuerietanhnarpleysespriestcuwtretbuoase[t3iP9v]D→e. FTasnhoidbsteanxinoternad-cfptrioeomrntunoferwbPaDfittFsivsoepatlrsimoegitziaemkdeeisnsitnahteot laoQwcc2oQunrtegthioennauncdleianrclcuodrirnegctaidodnistitoonaDl about 1 GeV2. In our studies ofdnadtuaactaldeewascirtrhiebffreedescpitnescthinteopt[r3he9es]e.snTttrhpuiascpteeuxrtrr(eSaceftcuitoninocntoifVoPnGDs)F.ssRoamelssueolttdsaakfrteoasmipnttohoienantcsceowunfitsthweillnubceleraerpocortrerdecetilosenws hteoreD. data described in the present paper (Section V G). Results from the new fits will be reported elsewhere.
2 2 10 In our analysis we use PDFs obtained from new fits optimized in the low Q region an1d0including additional data with respect to [39]. This extraction of PDFs also takes into account the nuclear corrections to D data described in the present paper (Section V G). Results from the new fits will be reported elsewhere.
10 Alternative solution work with ξ 0 dependent PDFs
n-th moment A n of distribution function
ξmax n An = dξ ξ F (ξ) !0
what is ξ m ax ?
GP use ξ m ax = 1 , ξ 0 < ξ < 1 unphysical
strictly, should use ξ m ax = ξ 0
Steffens, WM nucl-th/060314, PRC (2006) Alternative solution
what is effect on phenomenology? try several “toy distributions” standard TMC (“sTMC”) −1/2 3 q(ξ) = N ξ (1 − ξ) , ξmax = 1 modified TMC (“mTMC”) −1/2 3 q(ξ) = N ξ (1 − ξ) Θ(ξ − ξ0), ξmax = ξ0 threshold dependent (“TD”)
TD −1/2 3 q (ξ) = N ξ (ξ0 − ξ) , ξmax = ξ0 TMCs in F2 8
2 0.3 Q = 1 GeV2 scaling sTMC mTMC 0.2 TD
(x) q
2 ! F TD 0.1
0 0 0.2 0.4 0.6 0.8 1 x correct threshold behavior for “TD” correction
2 0.3 Q = 5 GeV2 scaling sTMC mTMC 0.2 TD
(x) q
2 ! F TD 0.1
0 0 0.2 0.4 0.6 0.8 1 x
2 2 2 FIG. 3: The x dependence of the F2 structure function at Q = 1 GeV (upper) and 5 GeV (lower). The effects of TMCs on the (input) scaling distribution (dotted curve) are illustrated for the sTMC (dashed) and mTMC (double-dot–dashed) TD prescriptions, and compared with the effects on the (input) TD-distribution ξq (ξ) (dot-dashed) using the TD approach (prescription C, solid). the sTMC and mTMC prescriptions, the corrected structure function is significantly larger in magnitude than for the TD prescription at intermediate and large x. For the sTMC case in particular, it is also seen to approach a nonzero value in the x → 1 limit. This result suggests that the evaluation of the twist-two part of the longitudinal structure function at low Q2 may also need to be reassessed in phenomenological analyses, especially at intermediate and large x. 8
2 0.3 Q = 1 GeV2 scaling sTMC mTMC 0.2 TD
(x) q
2 ! F TD 0.1
0 0 0.2 0.4 0.6 0.8 1 TMCsx in F2
2 0.3 Q = 5 GeV2 scaling sTMC mTMC 0.2 TD
(x) q
2 ! F TD 0.1
0 0 0.2 0.4 0.6 0.8 1 x
2 2 2 FIG. 3: The x dependence of the F2 structure function at Q = 1 GeV (upper2) and 5 GeV (lower). The effects of TMCs on the (input) scaling distribution (doeffttedectcurv esmall) are illu statrat ehigherd for the s TQMC (dashed) and mTMC (double-dot–dashed) TD prescriptions, and compared with the effects on the (input) TD-distribution ξq (ξ) (dot-dashed) using the TD approach (prescription C, solid). the sTMC and mTMC prescriptions, the corrected structure function is significantly larger in magnitude than for the TD prescription at intermediate and large x. For the sTMC case in particular, it is also seen to approach a nonzero value in the x → 1 limit. This result suggests that the evaluation of the twist-two part of the longitudinal structure function at low Q2 may also need to be reassessed in phenomenological analyses, especially at intermediate and large x. TMCs in FL 10
2 0.06 Q = 1 GeV2 sTMC mTMC
0.04 TD (x) L F
0.02
0 0 0.2 0.4 0.6 0.8 1 x
2 2 FIG. 5: Longitudinal structure function FL at Q = 1 GeV for the sTMC (dashed), mTMC (double-dot–dashed) and TD- distribution (solid) prescorcriptiornects. Not ethrthatesholdthe scaling lbehaongitudinvioral dist rfiborutio n“TD”is zero. correction
reduced2 TMC effect cf. sTMC and mTMC structure functions at finite Q , and produces vanishing structure functions as x → 1. This is true for both the F2 and FL structure functions. n The Nachtmann moments µ2 of the F2 structure function, calculated with the threshold dependent distributions TD TD 2 2 q , agree with the moments An of q to within 1% for the n = 2, 4 and 6 moments for Q as low as 1 GeV and even lower. In contrast, the deviation for the standard or modified TMC procedure (sTMC or mTMC prescriptions) is more than an order of magnitude larger at the same Q2 values, and grows rapidly with increasing n. Furthermore, 2 2 6 n for Q > M one can show analytically that, at least to O(1/Q ), the moments µ2 and An are identical. Similarly, n for the longitudinal structure function FL, the Nachtmann moments µL with the threshold dependent distribution are considerably smaller (i.e. closer to the asymptotic value of zero) than the moments in the sTMC or mTMC prescriptions. A consequence of our formulation is that the moments of the threshold dependent distributions will in general be M 2/Q2 dependent. This dependence is not associated with either perturbative QCD effects or higher twists, but comes entirely from the leading twist, target mass effects. Our analysis suggests that it may be necessary to reassess the interpretation of a parton distribution in the presence of the finite M 2/Q2, or ξ, corrections, as well as the implementation of the qTD distributions in the Q2 evolution equations. We will address these problems in future work [15]. At the same time, our numerical results give impetus to investigating the impact of TMCs on phenomenological fits to structure functions at low Q2 [16] and the extraction of twist-two parton distributions.
Acknowledgments
We would like to thank J. Blu¨mlein, M. E. Christy, C. E. Keppel, and A. W. Thomas for helpful discussions. This work was supported by the U.S. Department of Energy contract DE-AC05-84ER40150, under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility (Jefferson Lab). FMS is also supported by FAPESP (03/10754-0) and CNPq (308932/2003-0).
[1] W. Melnitchouk, R. Ent and C. Keppel, Phys. Rept. 406, 127 (2005). 5 from 0 to 1 (specifically, in the integrals for An, H(ξ) and G(ξ)). Here the normalization N ensures that the distribution integrates to unity. We denote this prescription the “standard TMC” (sTMC). 1 (B) Integrate a modified distribution which vanishes for ξ > ξ0, as implied by Eq. (7) :
−1/2 3 q(ξ) = N ξ (1 − ξ) Θ(ξ − ξ0) . (19)
We denote this prescription the “modified TMC” (mTMC). (C) Use a “threshold dependent” (TD) quark distribution which vanishes in the physical limit:
TD −1/2 3 q (ξ) = N ξ (ξ0 − ξ) . (20) Nachtmann F 2 moments
1.1 n=2
1
n sTMC / A n 2 0.9 mTMC µ TD
0.8
0 1 2 3 4 Q2 (GeV 2 )
FIG. 1: Ratio of the n = 2 Nachtmann moment of the F2 structure function and the n = 2 moment of the quark distribution, 2 moment of structure function agrees with as a function of Q . The curves correspond to prescriptions A [“sTMC”] (dotted), B [“mTMC”] (dashe2d) and C [“TD”] (solid). moment of PDF to 1% down 2to2 very low Q Note that because of the upper limit in Eq. (7), An itself will be M /Q dependent for prescriptions B and C. The n results for the ratio µ2 /An of the n = 2 moments are displayed in Fig. 1 for the three cases, with prescriptions A, B and C corresponding to the dotted, dashed and solid curves, respectively. Comparing the sTMC and mTMC results, 2 one can see a reduced Q dependence when the integrals are restricted to ξ < ξ0. However, a much more dramatic change occurs when the quark distribution is constrained to vanish at ξ0. This renders the Nachtmann moment almost equal to the moment of the quark distribution for virtually all Q2 considered. Certainly for Q2 > 1 GeV2 there is no visible deviation of the ratio from unity. Even for very small Q2, Q2 ∼ 0.3 GeV2, the ratio differs from unity by only ∼ 0.7% (of course the OPE itself may not be valid at such low values of Q2). Similarly, the ratios for the n = 4 and n = 6 moments are shown in Fig. 2. The deviation of the ratio from unity 2 < 2 for the sTMC approach is between 10% − 20% for Q ∼ 1 GeV , while that for the modified TMC with prescription B is of the order of 5% over the same Q2 region. On the other hand, for the threshold dependent prescription C, the deviation from unity remains around 1% even at these low Q2 values. A consequence of prescription C is that the moments of the parton distribution are Q2 dependent. This seems to be an inevitable consequence if the Nachtmann moments of the structure function are to be equal to the moments of the parton distribution for all Q2. Note that this Q2 dependence is not of higher twist or perturbative QCD origin, but arises solely from kinematics. Nevertheless, this avoids the more serious problems which arise within the sTMC
1 We believe this was also the implication of De Ru´jula et al. [11] 6
1.1 n=4 1
n 0.9 sTMC / A n 2 0.8 mTMC µ TD Nachtmann F 2 moments0.7 0.6
0 1 2 3 4 6 Q2 (GeV 2 )
1.2 1.1 n=4 n=6 1 1 n 0.9 n 0.8 sTMC sTMC / A / A n n 2 0.8 mTMC 2 mTMC
µ 0.6 µ
0.7 TD TD 0.4 0.6 0.2 0 1 2 3 4 0 1 2 3 4 2 2 Q (GeV ) Q2 (GeV 2 )
FIG. 2: Ratios of the n = 4 (upper graph) and n = 6 (lower graph) Nachtmann moment of the F2 structure function and the 2 corresponding moments of the quark distribution, as a function of Q . The curves are as in Fig. 1. 1.2 2 n=6 approach (prescription A), where the Nachtmann moments below Q2 ∼ 1 GeV2 start to deviate significantly from the higher momentsmom eshonts of thewqua rkmdistuchributions. Iwn adeakdition, inerthe sTMQC formulation one is faced with the so-called “threshold 1 2 problem”. Namely, if the moments An of the quark distributions are Q independent, then one should have: dependence than sTMC & mTMC1 prescriptions1 n 2 n 2 dξ ξ F (ξ, Q1) = dξ ξ F (ξ, Q2) (21) n 0.8 sTMC !0 !0
/ A 2 2 2
n for any two momentum scales Q1 and Q2. Since F (ξ, Q ) must vanish in the kinematically forbidden region ξ > ξ0, 2 0.6 mTMC µ TD 0.4
0.2 0 1 2 3 4 Q2 (GeV 2 )
FIG. 2: Ratios of the n = 4 (upper graph) and n = 6 (lower graph) Nachtmann moment of the F2 structure function and the 2 corresponding moments of the quark distribution, as a function of Q . The curves are as in Fig. 1. approach (prescription A), where the Nachtmann moments below Q2 ∼ 1 GeV2 start to deviate significantly from the moments of the quark distributions. In addition, in the sTMC formulation one is faced with the so-called “threshold 2 problem”. Namely, if the moments An of the quark distributions are Q independent, then one should have:
1 1 n 2 n 2 dξ ξ F (ξ, Q1) = dξ ξ F (ξ, Q2) (21) !0 !0
2 2 2 for any two momentum scales Q1 and Q2. Since F (ξ, Q ) must vanish in the kinematically forbidden region ξ > ξ0, 6
1.1 n=4 1
n 0.9 sTMC / A n 2 0.8 mTMC µ TD Nachtmann F 2 moments0.7 0.6
0 1 2 3 4 6 Q2 (GeV 2 )
1.2 1.1 n=4 n=6 1 1 n 0.9 n 0.8 sTMC sTMC / A / A n n 2 0.8 mTMC 2 mTMC
µ 0.6 µ
0.7 TD TD 0.4 0.6 0.2 0 1 2 3 4 0 1 2 3 4 2 2 Q (GeV ) Q2 (GeV 2 )
FIG. 2: Ratios of the n = 4 (upper graph) and n = 6 (lower graph) Nachtmann moment of the F2 structure function and the 2 corresponding moments of the quark distribution, as a function of Q . The curves are as in Fig. 1. 1.2 n 2 n 2 → ∞ µ2 (finite Qn=6) approaµch2(pr(esQcription A), where the)Nachtmann moments below Q2 ∼ 1 GeV2 start to deviate significantly from the =moments of the quark distributions. In addition, in the sTMC formulation one is faced with the so-called “threshold 1 2 2 2 An(finite Q probleAm”.nNa(mQely, if th→e mom∞ents A)n of the quark distributions are Q independent, then one should have: 1 1 n 2 n 2 dξ ξ F (ξ, Q1) = dξ ξ F (ξ, Q2) (21) n 0.8 sTMC !0 !0
/ A 2 2 2
n for any two momentum scales Q1 and Q2. Since F (ξ, Q ) must vanish in the kinematically forbidden region ξ > ξ0, 2 0.6 extract mTMCPDFs from structure function data µ TD 2 0.4 at lower Q
0.2 0 1 2 3 4 Q2 (GeV 2 )
FIG. 2: Ratios of the n = 4 (upper graph) and n = 6 (lower graph) Nachtmann moment of the F2 structure function and the 2 corresponding moments of the quark distribution, as a function of Q . The curves are as in Fig. 1. approach (prescription A), where the Nachtmann moments below Q2 ∼ 1 GeV2 start to deviate significantly from the moments of the quark distributions. In addition, in the sTMC formulation one is faced with the so-called “threshold 2 problem”. Namely, if the moments An of the quark distributions are Q independent, then one should have:
1 1 n 2 n 2 dξ ξ F (ξ, Q1) = dξ ξ F (ξ, Q2) (21) !0 !0
2 2 2 for any two momentum scales Q1 and Q2. Since F (ξ, Q ) must vanish in the kinematically forbidden region ξ > ξ0, 9
0.05 n=2 0.04
0.03
n sTMC L µ 0.02 mTMC TD Nachtmann F L 0.01moments
0 0 1 2 3 4 9 Q2 (GeV 2 ) n L
0.05 µ 0.001 n=2 n=4 0.04 0.0008 0.03
n sTMC 0.0006 sTMC L µ 0.02 mTMC mTMC 0.0004 TD TD 0.01 0.0002
0 0 0 1 2 3 4 0 1 2 3 4 2 2 Q (GeV ) Q2 (GeV 2 )
2 FIG. 4: Nachtmann moments of the longitudinal structure function for n = 2 (upper) and n = 4 (lower) as a function of Q . n L µ 2 IV. CONCLUSION 0.001 weaker Q n=4dependence for TD prescription In this work we have revisited the long-standing problem of target mass corrections to nucleon structure functions. 0.0008 The standard procedure for implementing target mass effects suffers from the well known threshold problem, in which the corrected, leading twist structure function does not vanish at x = 1. We have proposed a solution to this problem by introducing a finite-Q2, “threshold dependent” parton distribution function that explicitly depends on the 2 0.0006 sTMC kinematical threshold ξ0, which is smooth in the entire physical region, and approaches the ordinary, Q -independent parton distribution in the limit Q2 → ∞. Our prescription avoids any discontinuities in the parton distributions and mTMC 0.0004 TD 0.0002
0 0 1 2 3 4 Q2 (GeV 2 )
2 FIG. 4: Nachtmann moments of the longitudinal structure function for n = 2 (upper) and n = 4 (lower) as a function of Q .
IV. CONCLUSION
In this work we have revisited the long-standing problem of target mass corrections to nucleon structure functions. The standard procedure for implementing target mass effects suffers from the well known threshold problem, in which the corrected, leading twist structure function does not vanish at x = 1. We have proposed a solution to this problem by introducing a finite-Q2, “threshold dependent” parton distribution function that explicitly depends on the 2 kinematical threshold ξ0, which is smooth in the entire physical region, and approaches the ordinary, Q -independent parton distribution in the limit Q2 → ∞. Our prescription avoids any discontinuities in the parton distributions and Summary d quark distribution poorly known at large x
(anti)neutrino data can help determine d/u ratio at large x complement e scattering data (e.g. BONUS) alternative formulation of TMC in GP approach without threshold problem
much faster approach to scaling for ξ 0 dependent PDF