I review the phenomenology of unpolarized cross sections and azimuthal
asymmetries in semi-inclusive deeply inelastic scattering (SIDIS). The general
theoretical framework is presented and the validity of the Gaussian model is
discussed. A brief account of the existing analyses is provided.Comment: 12 pages, 5 figures, invited talk at "Transversity 2011", Veli
  Losinj, Croati

Barone, Vincenzo

English

arXiv

I review the phenomenology of unpolarized cross sections and azimuthal asymmetries in semi-inclusive deeply inelastic scattering (SIDIS). The general theoretical framework is presented and the validity of the Gaussian model is discussed. A brief account of the existing analyses is provided

Barone, V.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2012-11173-1Colloquia: Transversity 2011IL NUOVO CIMENTO Vol. 35 C, N. 2 Marzo-Aprile 2012Phenomenology of SIDIS unpolarized cross sectionsand azimuthal asymmetriesV. BaroneDi.S.T.A., Universita` del Piemonte Orientale “A. Avogadro”INFN, Gruppo Collegato di Alessandria - 15121 Alessandria, Italyricevuto il 20 Ottobre 2011; approvato il 25 Novembre 2011pubblicato online l’1 Marzo 2012Summary. — I review the phenomenology of unpolarized cross sections and az-imuthal asymmetries in semi-inclusive deeply inelastic scattering (SIDIS). The gen-eral theoretical framework is presented and the validity of the Gaussian model isdiscussed. A brief account of the existing analyses is provided.PACS 13.85.Ni – Inclusive production.PACS 12.38.-t – Quantum chromodynamics.1. – IntroductionIt has been known since the early years of quantum chromodynamics that azimuthalasymmetries in unpolarized processes are generated by gluon radiation and splitting. Theobservation of these asymmetries was in fact proposed as a test of perturbative QCD [1].The first experimental results date back to the Eighties and showed that unpolarizedsemi-inclusive deeply inelastic scattering (SIDIS) indeed displays non vanishing azimuthalmodulations of the type cosφh and cos 2φh [2, 3].While the pQCD processes account for these asimmetries at large transverse momenta,in the low-Ph⊥ region it is the intrinsic transverse motion of quarks that plays a key roˆle(for a recent review see [4]). There are two non-perturbative sources of azimuthal asym-metries in unpolarized processes: 1) the Cahn effect [5,6], a purely kinematic correctiondue to the quark transverse momentum: 2) the Boer-Mulders effect [7], arising from thecorrelation between the transverse momentum and the transverse spin of quarks insidean unpolarized nucleon.2. – Theoretical frameworkWe will consider unpolarized SIDIS, l() + N(P ) → l(′) + h(Ph) + X(PX), in thecurrent fragmentation region. At low Ph⊥ the cross section of this process factorizesinto a transverse-momentum–dependent quark distribution (TMD) and a fragmentationfunction [8].Let us consider the transverse kinematics in a reference frame where the virtualphoton and the nucleon are collinear (we call it γ∗N collinear frame, see fig. 1). Thec© Societa` Italiana di Fisica 121122 V. BARONEFig. 1. – Transverse kinematics in the photon-nucleon collinear frame.transverse momenta are denoted as follows: k⊥, transverse momentum of the initialquark; p⊥, transverse momentum of the hadron w.r.t. to the fragmenting quark; Ph⊥,transverse momentum of the hadron w.r.t. to the γ∗N axis. Neglecting 1/Q2 corrections,these momenta are related by(1) p⊥  Ph⊥ − zh k⊥.The SIDIS structure functions are actually expressed in terms of the quark transversemomentum kT defined in the frame where the produced hadron h and the nucleon arecollinear (hN collinear frame). However, if one neglects 1/Q2 corrections, the transversemomenta in the γ∗N and hN collinear frame coincide: k⊥  kT .The general expression of the cross section for unpolarized SIDIS is [9, 10]dσdxB dy dzh dφh dP 2h⊥=2π α2emxByQ2{(1− y + 12y2)FUU,T + (1− y)FUU,L(2)+(2− y)√1− y cosφhF cosφhUU + (1− y) cos 2φh F cos 2φhUU}.The azimuthal asymmetries are defined as(3) Acosφh = 2〈cosφh〉 = 2∫dφh cosφh dσ∫dφh dσ,and similarly for Acos 2φh .In the extended parton model, the structure functions appearing in eq. (2) are given byconvolutions of transverse-momentum–dependent quark distributions and fragmentationfunctions. Up to order 1/Q one hasFUU,T = C [f1D1] ,(4)F cosφhUU,Cahn =2MQC[− hˆ · kTMf1D1],(5)F cosφhUU,BM =2MQC[− (hˆ · k′T )k2TMhM2h⊥1 H⊥1],(6)F cos 2φhUU,BM = C[−2(hˆ · kT )(hˆ · k′T )− kT · k′TMMhh⊥1 H⊥1],(7)PHENOMENOLOGY OF SIDIS UNPOLARIZED CROSS SECTIONS AND AZIMUTHAL ETC. 123where C[w f D] is a transverse-momentum convolution.Equation (5) represents the Cahn contribution to the cosφh structure function. Itoriginates from the elementary lepton-quark cross section in the presence of transversemomenta, which reads(8) dσˆ ∼ 1y2[(1− 4k⊥Q√1− y cosϕ)+(1− 4k⊥Qcosϕ√1− y)]+O(k2⊥Q2).Equations (6) and (7) are the Boer-Mulders (BM) contributions to the azimuthalmodulations of unpolarized SIDIS. They couple the Boer-Mulders function h⊥1 to theCollins function H⊥1 [11], which describes the fragmentation of transversely polarizedquarks into an unpolarized hadron. Notice that the BM effect contributes to cosφh atorder 1/Q (i.e., at twist three) and to cos 2φh at leading twist.At twist three one should also take into account quark-gluon interactions, which giverise to the so-called “tilde” distributions. Thus F cosφhUU gets the additional term [10]F cosφhUU =2MQC[− hˆ · k′TMh(xh˜H⊥1 +MhMf1D˜⊥z)(9)− hˆ · kTM(xf˜⊥D1 +MhMh⊥1H˜z)].We recall however that TMD factorization is not proven beyond leading twist, so eq. (9)must be taken with a grain of salt.The Cahn effect produces at order 1/Q2 (twist four) a further contribution to thecos 2φh structure function,(10) F cos 2φhUU,Cahn =M2Q2C[2(hˆ · kT )2 − k2TM2f1 D1].This expression incorporates only part of the 1/Q2 kinematic corrections. Besides them,there are also dynamical 1/Q2 corrections arising from twist-four TMD’s and fragmenta-tion functions. Therefore, F cos 2φhUU,Cahn is just an approximate estimate of the full O(1/Q2)structure function.In summary, ignoring unknown higher-twist terms, the two azimuthal asymmetries ofunpolarized SIDIS are symbolically given by(11) 〈cosφh〉 = 1QCahn +1QBM, 〈cos 2φh〉 = BM+ 1Q2Cahn.3. – The Gaussian approachIn most phenomenological analyses, the TMD’s and fragmentation functions areparametrized by Gaussians:(12) f1(x,k2⊥) = A f1(x) e−k2⊥/k2⊥ , D1(z,p2⊥) = BD1(z) e−p2⊥/p2⊥ .124 V. BARONEHere x is the fraction of the nucleon light-cone momentum carried by the quark, and z isthe fraction of the light-cone momentum of the struck quark carried by the final hadron.These two variables coincide with xB and zh, respectively, modulo 1/Q2 corrections.We stress that there is a strong assumption—with no fundamental basis—behind theGaussian model, namely that distributions factorize in x and k⊥.In principle, one has to distinguish the Gaussian widths k2⊥ and p2⊥ from the averagesquared momenta, defined as(13) 〈k2⊥〉 ≡∫d2k⊥ k2⊥ f1(x,k2⊥), 〈p2⊥〉 ≡∫d2p⊥ p2⊥D1(z,p2⊥).Average values coincide with the Gaussian widths, 〈k2⊥〉 = k2⊥, 〈p2⊥〉 = p2⊥, only if weintegrate over k⊥ and p⊥ between 0 and ∞.Bounds on x ≡ k−/P− and k⊥ can be obtained in the quark parton model by insertinga set of physical intermediate states in the quark-nucleon amplitudes [12]. Since themomentum of these states is(14) Pμn =(xM2 − k2⊥2xP−, (1− x)P−,−k⊥),the condition M2n ≥ 0 yields(15) k2⊥ ≤ x(1− x)M2.This is the bound on the truly intrinsic quark transverse momentum [13] (for recentdiscussions see [14, 15]). Numerically the upper limit (15) is k2⊥ < 0.25GeV2. Clearly,the average squared momentum 〈k2⊥〉must be smaller than this value. On the other hand,various phenomenological analyses give much larger figures, 〈k2⊥〉 ∼ 0.25–0.40GeV2.Is there a contradiction between these results? The answer is that the bound (15)actually refers to a “static” nucleon (Q2 = 0), whereas the value of 〈k2⊥〉 extractedfrom experiments effectively accounts for the “non-intrinsic” transverse momentum, ra-diatively generated at a given Q2. As a consequence, 〈k2⊥〉 is not a fixed, universal,quantity.Consider now the unpolarized and angle-independent structure function:(16) FUU =∑ae2a∫d2k⊥∫d2p⊥ δ2(p⊥ + zhk⊥ − Ph⊥) fa1 (xB ,k2⊥)Da1(zh,p2⊥).Using Gaussian parametrizations and integrating between 0 and ∞, one gets(17) FUU =∑ae2a fa1 (xB)Da1(zh)e−P2h⊥/P2h⊥π P 2h⊥with(18) P 2h⊥ = p2⊥ + z2h k2⊥.This is a popular relation appearing in many analyses (where it is often quoted as arelation between average momenta). It must used with some caution. In fact: i) it isPHENOMENOLOGY OF SIDIS UNPOLARIZED CROSS SECTIONS AND AZIMUTHAL ETC. 1250.010.110 0.2 0.4 0.6    R (P    h⊥)P  2h⊥  (GeV2)Gauss modelFig. 2. – The ratio R(Ph⊥) = dσUU (Ph⊥)/dσUU (0): Gaussian model vs. CLAS data (from [16]).invalidated by a possible cutoff on k2⊥; ii) it has (complicated) 1/Q2 corrections. iii) dueto experimental cuts, P 2h⊥ differs from the measured 〈P 2h⊥〉.4. – Transverse-momentum dependence of cross sectionsThe Gaussian Ansatz for the unpolarized SIDIS cross section,(19)dσUU (Ph⊥)dz dP 2h⊥=dσUU (0)dz dP 2h⊥e−P2h⊥/P2h⊥ ,has been tested by Schweitzer, Teckentrup and Metz [16] in the CLAS kinematics (xB =0.24, zh = 0.34, Q2 = 2.4GeV2) [17]. As shown in fig. 2, the description of data is verygood, with a Gaussian width P 2h⊥ = 0.17GeV2.The results of a Gaussian analysis of the P 2h⊥ distributions measured by COMPASSare displayed in fig. 3 (left) [18]. The fitted P 2h⊥ is found to have a mild dependence onxB and to increase with Q2. A check of the relation (18) has been also performed, withthe result shown in fig. 3 (right). One sees a clear departure from a linear dependence2 (GeV/c)2Tp0 0.2 0.4 0.6 0.8 12 Tdzdpμ/dN+dh-110110PreliminaryCOMPASS 2004 LiD (part)+h0.20<z<0.250.25<z<0.300.30<z<0.350.35<z<0.400.40<z<0.500.50<z<0.600.60<z<0.700.70<z<0.802 = 7.57 (GeV/c)〉2Q〈 = 0.0932〉Bjx〈2z0 0.1 0.2 0.3 0.4 0.5 0.6 0.72 (GeV/c)〉2 Tp〈0.10.20.30.40.50.60.7> = 0.25  2> = 0.28, <p2Ref: <k 0.01 ±> = 0.43  2 0.05, <p±> = 0.53  2: <k+h 0.01 ±> = 0.47  2 0.05, <p±> = 0.36  2: <k-h = 7.36〉2Q〈 = 0.0550〉Bjx〈COMPASS 2004 LiD (part) PreliminaryFig. 3. – Left: P 2h⊥ distributions fitted by Gaussians (COMPASS data) [18]. Right: Fitted P 2h⊥(called 〈p2T 〉 in the figure) vs. z2h (COMPASS data) [18].126 V. BARONE00.20.40.60 100 200 300  〈P 2h⊥ 〉 (GeV2 )s (GeV2 )JLabHERMESCOMPASS)2 (GeV2W0 50 100 150 200 2502 (GeV/c)〉2 Tp〈0.20.250.30.350.40.450.50.550.60.65 〉2Q〈 4.074.474.574.622<6.0 (GeV/c)23.5<Q0.5<z<0.6Charged hadronsCOMPASS 2004 LiD (part) PreliminaryFig. 4. – Left: Dependence of 〈P 2h⊥〉 on the lepton-nucleon c.m. energy squared of various ex-periments (from [16]). Right: Dependence of 〈P 2h⊥〉 on the photon-nucleon c.m. energy squaredW 2 (COMPASS data, from [18]).of P 2h⊥ on z2h. The dashed and dot-dashed curves in the figure are fits of the typeP 2h⊥ = z0.5h (1− zh)1.5 p2⊥ + z2h k2⊥.Finally, it has been noticed in [16] that the average transverse momentum of hadronsextracted in different experiments (JLab, HERMES, COMPASS) has an approximatelylinear rise in s (fig. 4, left). This energy dependence is confirmed by a COMPASSanalysis [18], which shows that 〈P 2h⊥〉 increases with W 2 (fig. 4, right), which may be anindication of a possible broadening of 〈k2⊥〉 and 〈p2⊥〉.5. – Azimuthal asymmetriesThe first analysis of the cosφh asymmetry in SIDIS in terms of the Cahn effect waspresented in 2005 by Anselmino et al. [19]. They fitted the EMC data [2] with a Gaussianmodel, obtaining the values k2⊥ = 0.25GeV2, p2⊥ = 0.20GeV2, but did not consider theBoer-Mulders term, which is of the same order as the Cahn contribution.The cos 2φh asymmetry was extensively studied in [20, 21]. Both the leading-twistBoer-Mulders contribution and the twist-4 Cahn term were included in the analysis. Signsand magnitudes of the Boer-Mulders function, extracted from a fit to HERMES [22] andCOMPASS [23] preliminary data, were found to be in agreement with the theoreticalexpectations (based on the impact-parameter approach, lattice studies and large Ncarguments): h⊥u1 ∼ 2f⊥u1T , h⊥d1 ∼ −f⊥d1T .In particular, it was pointed out that, since the Cahn effect is the same for π+ andπ−, and the favorite and unfavorite Collins functions are related by H⊥fav1 ≈ −H⊥unf1 , asshown by the fits to the Collins effect in SIDIS, a signature of the Boer-Mulders effect is〈cos 2φh〉π− > 〈cos 2φh〉π+ . Another interesting output of the analysis in [20, 21] is thatthe Cahn contribution turns out to be relatively large in spite of being O(1/Q2). Anexample of the fits presented in [21] is shown in fig. 5.The HERMES and COMPASS Collaborations have recently presented new prelimi-nary data on azimuthal asymmetries in unpolarized SIDIS [24, 25]. While the two mea-surements are in fairly good agreement for 〈cosφh〉, the COMPASS result for the cos 2φhasymmetry is systematically larger than the corresponding HERMES result.PHENOMENOLOGY OF SIDIS UNPOLARIZED CROSS SECTIONS AND AZIMUTHAL ETC. 127-0.15-0.1-0.05 0 0.05 0.1 0.1510-2 10-1  Acos 2φxCOMPASS Deuteronπ−CahnBoer-MuldersCahn+BM-0.15-0.1-0.05 0 0.05 0.1 0.15  Acos 2φ π+CahnBoer-MuldersCahn+BM π+CahnBoer-MuldersCahn+BM 0.3  0.4  0.5  0.6  0.7zπ−CahnBoer-MuldersCahn+BM π+CahnBoer-MuldersCahn+BM 0.1  0.3  0.5  0.7  0.9P  (GeV)Tπ−CahnBoer-MuldersCahn+BMFig. 5. – The fit of [21] to the COMPASS preliminary data on 〈cos 2φh〉 with a deuteron target.The CLAS Collaboration at JLab has released final results on the azimuthal modula-tions [17]. A large discrepancy is found between the data on 〈cosφh〉 and the Cahn effectpredictions (extrapolated from other analyses). The 〈cos 2φh〉 signal is non-zero only inthe small-zh region, where target fragmentation effects are important and may affect theinterpretation of data.From the theoretical viewpoint, the situation is quite unclear, as only partial analysesare presently available. If we simply extrapolate them, we find that the Cahn contri-bution to 〈cosφh〉 is huge and largely overshoots the data, and that the Boer-Mulderscontribution to 〈cos 2φh〉 is small and apparently goes in the wrong direction, in the sensethat it has an opposite sign in 〈cosφh〉 and 〈cos 2φh〉, and therefore should produce a〈cosφh〉π− smaller than 〈cosφh〉π+ , whereas the data show a similar π+-π− pattern forboth asymmetries.We have already said that the intrinsic transverse momentum of quarks is kinemati-cally bounded. We expect that the radiatively generated transverse momentum shouldbe limited as well. The effect of an upper bound on k⊥ has been recently explored byBoglione, Melis and Prokudin [26]. They derived a cutoff of the form k2⊥ ≤ η(xB)Q2in the γ∗N center-of-mass frame. The consequence of setting this bound is that theCahn contribution gets largely suppressed, while the BM contribution remains almostunaffected. This correction goes in the right direction, but further work is needed toestablish a frame-independent condition on k⊥.6. – ConclusionsThere is nothing magic about the Gaussian approach. It is just a parametrization,which happens to work rather well for cross sections at low Ph⊥. Since this model lacksa solid basis, one should not be surprised if some simple regularities based on it turn outto fail.The parameters k2⊥ and p2⊥ in the Gaussian distributions are likely to be Q2 (andW 2) dependent. This means that they cannot be fixed once forever and used anywhere.One should do as for normal PDF’s: take a TMD at small Q2 from some model, evolveit and fit the data.Although we have clear signals of relatively large azimuthal asymmetries, the situationis still unsettled. A combined state-of-the-art fit of 〈cosφh〉 and 〈cos 2φh〉 would be highlydesirable in order to understand the origin of these observables. We should be prepared tothe possibility that the scheme (11) might not work. We recall that the corrections to thisscheme are of three types: 1) “genuine” (i.e. “tilde”) twist-3 contributions (originatingfrom quark-gluon correlations); 2) further 1/Q2 kinematic terms; 3) dynamical twist-4contributions.128 V. BARONEA lesson we have learned is that higher twists are relevant in the presently exploredregion of SIDIS. Clearly, it would be important to disentangle them from leading twistcontributions. In order to do so, a wider Q2 range is needed—an important task forfuture experiments. In the meanwhile, the 1/Q2 contributions should be fully workedout, computing both the kinematic corrections and the target mass effects.∗ ∗ ∗I would like to thankAnna Martin and Franco Bradamante for their kind invita-tion to Transversity 2011, and Elena Boglione, Stefano Melis, Alexei Prokudin,Giulio Sbrizzai for useful discussions. This work is partially supported by the ItalianMinistry of Education, University and Research (MIUR) in the framework of a ResearchProject of National Interest (PRIN 2008).REFERENCES[1] Georgi H. and Politzer H. D., Phys. Rev. Lett., 40 (1978) 3.[2] Aubert J. J. et al. (EMC Collaboration), Phys. Lett. B, 130 (1983) 118.[3] Arneodo M. et al. (EMC Collaboration), Z. Phys. C, 34 (1987) 277.[4] Barone V., Bradamante F. and Martin A., Progr. Part. Nucl. Phys., 65 (2010) 267.[5] Cahn R. N., Phys. Lett. B, 78 (1978) 269.[6] Cahn R. N., Phys. Rev. D, 40 (1989) 3107.[7] Boer D. and Mulders P. J., Phys. Rev. D, 57 (1998) 5780.[8] Ji X.-D., Ma J. 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Phenomenology of SIDIS unpolarized cross sections and azimuthal asymmetries

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