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Transverse spin structure of hadrons from lattice QCD
Progress in Particle and Nuclear Physics 61 (2008) 73–80 www.elsevier.com/locate/ppnp
Review
Transverse spin structure of hadrons from lattice QCD D. Br¨ommel a , M. Diehl b , M. G¨ockeler c , Ph. H¨agler d,∗,1 , R. Horsley e , Y. Nakamura f , D. Pleiter f , P.E.L. Rakow g , A. Sch¨afer c , G. Schierholz b,f , H. St¨uben h , J.M. Zanotti e (QCDSF/UKQCD Collaborations) a School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK b Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany c Institut f¨ur Theoretische Physik, Universit¨at Regensburg, 93040 Regensburg, Germany d Institut f¨ur Theoretische Physik T39, Physik-Department der TU M¨unchen, James-Franck-Str., 85747 Garching, Germany e School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK f John von Neumann-Institut f¨ur Computing NIC / DESY, 15738 Zeuthen, Germany g Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK h Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin, 14195 Berlin, Germany
Abstract This paper presents recent lattice QCD calculations of transverse spin densities of quarks in hadrons.2 Based on our simulation results for the tensor generalized form factors, we find substantial correlations between spin and coordinate degrees of freedom in the nucleon and the pion. They lead to strongly distorted transverse spin densities of quarks in the nucleon and a surprisingly non-trivial transverse spin structure of the pion. Following recent arguments by Burkardt [M. Burkardt, Phys. Rev. D 72 (2005) 094020], our results imply that the Boer–Mulders function h ⊥ 1 , describing correlations of the transverse spin and intrinsic transverse momentum of quarks, is large and negative for up-quarks in the proton and the π + . This supports the recent hypothesis that all Boer–Mulders functions are alike [arXiv:0705.1573], and also provides additional motivation for future studies of azimuthal asymmetries in π p Drell-Yan production at, e.g., COMPASS. c 2008 Elsevier B.V. All rights reserved.
Keywords: Lattice QCD; GPDs; Transverse spin
1. Introduction In recent years, transverse spin phenomena have provided new insights into QCD factorization [3] and the origin of, for example, single spin asymmetries measured in semi-inclusive deep inelastic scattering experiments [4]. A ∗ Corresponding author. Tel.: +49 89 289 14403.
E-mail address: [email protected] (Ph. H¨agler). 1 Speaker. 2 This contribution is based on Refs. [M. G¨ockeler, et al., Phys. Rev. Lett. 98 (2007) 222001; D. Br¨ommel, et al., [QCDSF Collaboration]. arXiv:hep-lat/0708.2249]. c 2008 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2007.12.035
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Fig. 1. Illustration of the quark distribution in a proton in impact parameter space.
central object of interest is the quark transversity distribution δq(x) = h 1 (x), which describes the probability of finding a transversely polarized quark with longitudinal momentum fraction x in a transversely polarized nucleon [5]. Substantial efforts have been made to understand the so-called transverse momentum dependent PDFs (tmdPDFs) ⊥ (x, k 2 ) [6], which measures the correlation of the intrinsic quark transverse like, for example, the Sivers function f 1T ⊥ 2 momentum k⊥ and the transverse nucleon spin S⊥ , as well as the Boer–Mulders function h ⊥ 1 (x, k⊥ ) [7], describing the correlation of k⊥ and the transverse quark spin s⊥ . A complementary approach to the transverse spin structure of hadrons is based on generalized parton distributions (GPDs) and the associated impact parameter densities of quarks in hadrons [8]. Whereas it is unclear to date whether tmdPDFs can be directly accessed in lattice QCD, moments of GPDs can and have already been studied for several years in lattice simulations. In the following sections, we give a brief introduction to the transverse spin densities and their relation to GPDs. We then present our lattice calculations of the relevant tensor generalized form factors and discuss the results for the densities of transversely polarized quarks in the nucleon [1] and the pion [2]. 2. Transverse spin structure of the nucleon We begin by introducing the density ρ(x, b⊥ , s⊥ , S⊥ ) [8], which represents the probability of finding a quark with longitudinal momentum fraction x and transverse spin s⊥ at distance b⊥ from the center-of-momentum of the nucleon with transverse spin S⊥ . The probability interpretation in impact parameter space has been developed in [9] and is illustrated in Fig. 1. Lattice calculations give access to x-moments of transverse spin densities [8] Z 1 1 1 2 2 2 i i e S⊥ A T n0 (b⊥ )− 1 A (b ) dx x n−1 ρ(x, b⊥ , s⊥ , S⊥ ) = An0 (b⊥ ) + s⊥ ρ n (b⊥ , s⊥ , S⊥ ) = b T n0 ⊥ 2 4m 2 ⊥ −1 ) j b ji i 0 2 j 1 e00 2 i i j i 0 2 ij S⊥ Bn0 (b⊥ ) + s⊥ + ⊥ B T n0 (b⊥ ) + s⊥ (2b⊥ b⊥ − b⊥ δ )S⊥ 2 A (b2 ) , (1) m m T n0 ⊥ 2∂ f. where m is the nucleon mass, and where the derivatives are defined by f 0 ≡ ∂b2 f and 1b⊥ f ≡ 4∂b2 b⊥ 2 b ⊥ ⊥ ⊥ The right-hand side of Eq. (1) can be interpreted as a “multipole-expansion”, consisting of two orbitally symmetric j i and b j ji S i , and a quadrupole term monopole terms (in the first line), two dipole structures proportional to b⊥ ji s⊥ ⊥ ⊥ i (2bi b j −b2 δ i j )S j . The (derivatives of the) three nucleon generalized form factors (GFFs) B (b ), proportional to s⊥ n0 ⊥ ⊥ ⊥ ⊥ ⊥ eT n0 (b⊥ ) thus determine how strongly the orbital symmetry in the transverse plane is distorted by the B T n0 (b⊥ ) and A 2 ), A 2 dipole and the quadrupole terms. The b⊥ -dependent GFFs An0 (b⊥ T n0 (b⊥ ), . . . in Eq. (4) are related to GFFs in momentum space An0 (t), A T n0 (t), . . . by a Fourier transformation Z 2 d 1⊥ −ib⊥ ·1⊥ 2 f (b⊥ )≡ e f (t = −12⊥ ), (2) (2π )2 where 1⊥ is the transverse momentum transfer to the nucleon. For the lowest moment, n = 1, one finds A10 (t) = F1 (t), B10 (t) = F2 (t) and A T 10 (t) = gT (t) where F1 , F2 and gT are the Dirac, Pauli, and tensor nucleon form factors, respectively. The GFFs discussed here are also directly related to x-moments of the corresponding vector and
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Fig. 2. Results for the generalized form factors B T (n=1,2)0 (t).
tensor GPDs (for a review see [10]). They parametrize off-forward nucleon matrix elements of certain local quark operators, for example the tensor operators µνµ1 ···µn−1
OT ↔
→
↔ µ1
= AS q σ µν γ5 i D
↔ µn−1
···iD
q
(3)
←
with D = 12 ( D − D ), and where AS denotes symmetrization in ν, . . . , µn−1 followed by anti-symmetrization in µ, ν and subtraction of traces. A concrete example of the corresponding parametrization for n = 1 is given by [12,11,8]
0 0 µν t e P Λ OT |PΛi = u(P 0 , Λ0 ) σ µν γ5 A T 10 (t) − A (t) T 10 2m 2 µναβ 1α γβ 1[µ σ ν]α γ5 1α e A (t) u(P, Λ), + B T 10 (t) − T 10 2m 2m 2 where 1 = P 0 − P and t = 12 . Parameterizations for n ≥ 1 in terms of tensor GFFs and their relation to GPDs are given in [11]. Since it is very challenging to access tensor GPDs in experiment [13], lattice QCD calculations of the eT n0 (t) are particularly interesting. GFFs A T n0 (t), B n0 (t) and A 2.1. Simulation results Our numerical lattice results are based on configurations generated with n f = 2 dynamical non-perturbatively O(a) improved Wilson fermions and Wilson gluons. Simulations have been performed on lattices of V ×T = 163 ×32 and 243 × 48 at four different couplings β = 5.20, 5.25, 5.29, 5.40 with up to five different κ = κsea values per β. The lattice scale a in physical units has been set using a Sommer scale of r0 = 0.467 fm [14,15]. The range of available pion masses extends down to 400 MeV, with lattice spacings below 0.1 fm and spatial volumes as large as (2.1 fm)3 . We note that the computationally demanding disconnected contributions, which in principle are needed for isosinglet quantities, are not included in this analysis. We expect, however, that they are small for the tensor GFFs [16]. Our results have been transformed to the MS scheme at a scale of 4 GeV2 using non-perturbative renormalization [17] of the corresponding operators. The calculation of GFFs in lattice QCD follows standard methods based on two- and three-point functions, and is described in, e.g., Refs. [18–21]. u Fig. 2 shows results for the GFFs B T (n=1,2)0 (t) as a function of the momentum transfer squared t, for a pion mass of m π ≈ 600 MeV, a lattice spacing of a ≈ 0.08 fm and a volume of V ≈ (2 fm)3 . In order to obtain a functional parametrization of the lattice results, we fit all GFFs using a p-pole ansatz F(t) = F0 /(1 − t/( p m 2p )) p with the three parameters F0 , m p and p for each GFF, where F0 = F(t = 0) provides us with the forward value of the GFF. Unfortunately, the statistics are not sufficient to determine all three parameters from a single fit to the lattice data. For a given generalized form factor, we therefore fix the power p first, guided by fits to selected datasets, and subsequently determine the forward value F0 and the p-pole mass m p by a full fit to the lattice data. The corresponding fits of B T (n=1,2)0 (t) are shown in Fig. 2 as shaded error bands, where we have set p = 2.5. We have checked that the final p-pole parameterizations only show a mild dependence on the value of p chosen prior to the fit. Our analysis indicates that the discretization errors are smaller than the statistical errors, and we will neglect any dependence of
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Fig. 3. Pion mass dependence of the generalized form factors B T (n=1,2)0 (t = 0) for up-quarks.
the GFFs on the lattice spacing in the following. Guided by our investigations of finite size effects in the case of the nucleon mass and the axial vector form factor g A [14,22], we conclude that the volume effects for the lattices and nucleon observables studied here are small and may be neglected. In Fig. 3, we show as an example the pion mass u dependence of the GFFs B T (n=1,2)0 (t = 0). Since the pion masses are still rather large for most of our data points, we cannot expect results from one-loop chiral perturbation theory [23] to be applicable to the available data. We therefore extrapolate the forward moments and the p-pole masses using an ansatz linear in m 2π to get a first estimate of the GFFs phys at the physical point. The corresponding fits are shown as shaded error bands in Fig. 3. At m π = 140 MeV, we find u d u d B T 10 (t = 0) = 2.93(13), B T 10 (t = 0) = 1.90(9) and B T 20 (t = 0) = 0.420(31), B T 20 (t = 0) = 0.260(23). As will be discussed in some more detail below, these comparatively large values already indicate a significant impact of this tensor GFF on the transverse spin structure of the nucleon. Noticing that the (tensor) GFFs B T n0 are analogous to the R (vector) GFFs Bn0 , we define an anomalous tensor magneticR moment [24], κT ≡ dx E T (x, ξ, t = 0) = B T 10 (t = 0), similar to the standard anomalous magnetic moment κ = dx E(x, ξ, t = 0) = B10 (t = 0) = F2 (t = 0). Based on up our results, we find large positive values for the anomalous tensor magnetic moment for both flavors, κT,latt ≈ 3.0 and down ≈ 1.9, in contrast to the u- and d-quark contributions to the anomalous magnetic moment, which are both large κT,latt up down ≈ −2.03. For the numerical evaluation of the density ρ n (b , s , S ) but of opposite sign, κexp ≈ 1.67 and κexp ⊥ ⊥ ⊥ in Eq. (1), we Fourier-transform the p-pole parametrization to impact parameter (b⊥ ) space. The parameterizations of the impact parameter dependent GFFs then depend only on the power p, the p-pole masses m p and the forward values F0 , which have been obtained from fits to the lattice data. In Fig. 4, we show the lowest moment n = 1 of the density for u- and d-quarks in the nucleon. We note that the (n = 1)-moment of the density can be written as the difference of the corresponding moments for quarks and antiquarks, ρ n=1 = ρqn=1 − ρqn=1 . The result for ρ n=1 (b⊥ , s⊥ , S⊥ ) is therefore not necessarily positive, although we expect contributions from antiquarks to be small in general. Due to the rather large anomalous magnetic moments κ u,d , we find strong distortions for unpolarized quarks in transversely polarized nucleons (the left part of the figure). This has already been noticed in [25] and can serve as a dynamical explanation of the experimentally observed Sivers-effect. Remarkably, we find even stronger distortions for transversely polarized quarks s⊥ = (sx , 0) in an unpolarized nucleon, as can be seen on the right-hand side of Fig. 4. In strong contrast to the distortions visible on the left-hand side of Fig. 4, we find that the densities for transversely polarized u- and d-quarks are both deformed in the positive b y direction due to the large positive u
d
values for the tensor GFFs B T 10 (t = 0) and B T 10 (t = 0) discussed above. For a first interpretation of these results, let us assume that transverse quark spin and orbital angular momentum are aligned for u- and d-quarks. Following this assumption, we expect that the densities for polarized u- and d-quarks in an unpolarized (polarization-averaged) nucleon are shifted in the same direction, for example upwards for quarks with spin in x-direction in a nucleon moving towards the observer in z-direction, as can be seen on the right-hand side of Fig. 4. Further, taking into account that the u-quark (d-quark) spin is predominantly oriented parallel (antiparallel) to the nucleon spin, we predict that densities of unpolarized quarks in a polarized nucleon are shifted in opposite directions for u- and d-quarks, for example that the orbital motion of d-quarks around the (−x)-direction leads to a larger d-quark density in the lower half plane, as can indeed be seen on the left-hand side of Fig. 4. Burkardt [24] has argued that the deformed densities on the
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Fig. 4. Lowest moment (n = 1) of the densities of unpolarized quarks in a transversely polarized nucleon (left) and transversely polarized quarks in an unpolarized nucleon (right) for up (upper plots) and down (lower plots) quarks. The quark spins (inner arrows) and nucleon spins (outer arrows) are oriented in the transverse plane as indicated.
right-hand side of Fig. 4 are related to a non-vanishing Boer–Mulders function h ⊥ 1 [7], which describes the correlation of the intrinsic quark transverse momentum and the transverse quark spin s⊥ . According to this conjecture we have in particular, κT ∼ −h ⊥ 1 , so that our lattice results indicate that the Boer–Mulders function for both u- and d-quarks in the nucleon is large and negative. Since h ⊥ 1 is time reversal odd, it enters with a different overall sign in SIDIS and Drell-Yan production [26]. The results just quoted refer to the functions relevant for SIDIS. 3. Spin structure of the pion The probability density ρ(x, b⊥ , s⊥ ) for polarized quarks in the pion can be directly obtained from the nucleon density in Eq. (1) by setting S⊥ = 0. The result is considerably simpler than for the nucleon but still contains a dipole i i j b j , term proportional to s⊥ ⊥ " # Z 1 i i j b j s 1 ∂ 2 2 ⊥ BTπn0 (b⊥ ) . ρ n (b⊥ , s⊥ ) = dx x n−1 ρ(x, b⊥ , s⊥ ) = Aπn0 (b⊥ (4) )− ⊥ 2 2 mπ ∂b⊥ −1 Here, the b⊥ dependent vector and tensor GFFs of the pion, Aπn0 and BTπn0 , are moments of the corresponding GPDs 2 ) and E π (x, ξ = 0, b2 ), respectively. A non-vanishing B π H π (x, ξ = 0, b⊥ ⊥ T T n0 in Eq. (4) would lead to a dipole-like distortion of the quark density in the transverse plane, and thereby imply a non-trivial transverse spin structure of the pion. Thus, a computation of BTπn0 from first principles in lattice QCD provides a crucial insight into the pion structure. We note that the GFF Aπ10 (t) for up-quarks in the π + is identical to the electromagnetic pion form factor Fπ (t), which we investigated in detail in [27]. The momentum-space GFFs BTπn0 (t) are related to the impact parameter dependent GFFs in Eq. (4) by a Fourier transformation as in Eq. (2). They parameterize pion matrix elements of the tensor operators in Eq. (3), µνµ1 ···µn−1
hπ + (P 0 )|OT
|π + (P)i = AS
n−1 µναβ 1α P¯β X 1µ1 · · · 1µi P¯ µi+1 · · · P¯ µn−1 BTπni (t) 2m π i=0
(5)
even
where P¯ = 21 (P 0 + P). Similarly, the GFFs Aπni (t) parameterize pion matrix elements of local vector quark π,u + operators [28]. In the following, we consider for definiteness the GFFs Aπ,u n0 (t) and BT n0 (t) for up-quarks in a π .
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Fig. 5. Results for the generalized form factors BTπ,u n0 (t) as functions of the invariant momentum transfer t.
The corresponding down-quarks GFFs and the GFFs for π − or π 0 can be easily determined by charge conjugation and isospin symmetry [29]. 3.1. Simulation results In Fig. 5, we show as an example the t dependence of the GFFs BTπ,u n0 (t) for n = 1, 2 at β = 5.29 and a pion mass of m π ≈ 600 MeV. A parametrization of the t dependence of the lattice results is obtained from a fit based on the p-pole ansatz discussed in Section 2.1. To avoid irregular densities at b⊥ = 0, we restrict the power p to p > 3/2 for BTπ,u n0 (t) according to [8]. Our analysis shows that in general, smaller values of p are preferred in the fits. We π,u therefore set in the following p = 1.6 for BTπ,u n0 (t) and choose a standard monopole-ansatz ( p = 1) for An0 (t). In a lattice calculation, we clearly cannot resolve structures at arbitrarily small distances. Hence, these powers should be considered as effective values, suitable for the range of momenta accessible in our simulation. Taking p = 2, which corresponds to the behavior expected for −t → ∞, changes the results for BTπ,u (n=1,2)0 by less than the statistical error. Concerning the pion mass dependence of our results it is important to note that, due to the prefactor m −1 π in the parameterization (5), the GFFs BTπn0 (t) must vanish like m π for m π → 0 [29]. We therefore consider in the following the ratio BTπn0 /m π , which tends to a constant in the chiral limit. A study of the lattice spacing dependence of BTπ,u 10 (t = 0)/m π shows that discretization errors are smaller than the statistical errors, and we therefore neglect any dependence of the GFFs on a in the following analysis. Fig. 6 shows the pion mass dependence of the forward 2 values BTπ,u n0 (t = 0)/m π , together with a linear extrapolation in m π , indicated by the shaded error band. For the same reasons as discussed in Section 2.1, we do not attempt to fit the available lattice data to the one-loop result of −1 ChPT [29,30]. The linear chiral extrapolation to m π = 140 MeV gives BTπ,u 10 (t = 0)/m π = 1.64 ± 0.11(stat) GeV π,u −1 and BT 20 (t = 0)/m π = 0.307 ± 0.032(stat) GeV , for p = 1.6. Within the present statistics, we can only give an π estimate of possible finite size effects, which are of order 20% for BTπ,u (1,2)0 (t = 0)/m π . For the GFFs An0 (t) with n n = 1, 2, we refer to our results in [28,27]. Finally, we evaluate the density ρ (b⊥ , s⊥ ) in Eq. (4) using the Fourier transformed p-pole parameterizations of the momentum-space GFFs. The lowest moment of the density for up-quarks in a π + is shown in Fig. 7. We find a substantial deformation on the right-hand side of Fig. 7 for transversely polarized quarks, compared to the unpolarized case on the left. We conclude that the pion has indeed a pronounced, non-trivial transverse spin structure. Using Eq. (4), we can easily calculate an average transverse shift R 2 y d b⊥ b ρ n (b⊥ , s⊥ ) 1 BTπn0 (t = 0) y (6) hb⊥ in = R 2 ⊥ n = 2m π Aπn0 (t = 0) d b⊥ ρ (b⊥ , s⊥ ) y
in the y direction for transverse quark spin s⊥ = (1, 0) in the x direction. We obtain hb⊥ i1 = 0.162(11) fm and y hb⊥ i2 = 0.117(13) fm. The corresponding average transverse shift of transversely polarized quarks in an unpolarized y nucleon is given by hb⊥ in = B T n0 (t = 0)/ (2m N An0 (t = 0)), where the An0 (t = 0) are the moments of the y unpolarized quark distribution. With the results of [1] reported in Section 2.1, we find hb⊥ i1 = 0.154(6) fm and y hb⊥ i2 = 0.101(8) fm for up-quarks in the proton. Comparing the average transverse shifts for the pion and the nucleon, we find the remarkable result that the distortion in the distribution of a transversely polarized up-quark is
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Fig. 6. Pion mass dependence of BTπ,u n0 (t = 0)/m π . The stars represent the extrapolated values at the physical point.
Fig. 7. The lowest moment of the impact parameter densities of unpolarized (left) and transversely polarized (right) up-quarks in a π + . The quark spin (inner arrow) is oriented in the transverse plane as indicated.
within errors of the same strength in a π + and in the proton. In the same way as for the nucleon, we can relate the pion ⊥,π GPD E Tπ , or equivalently the GFFs BTπ,u [24,31], which describes the correlation n0 , to the Boer–Mulders function h 1 between the transverse spin and the intrinsic transverse momentum of quarks in the pion [7]. Our results for BTπn0 then imply that the Boer–Mulders function for up-quarks in a π + is large and negative. Furthermore, a comparison of our results for the pion and the nucleon provides strong support for the arguments in [32], which suggest that all Boer–Mulders functions for valence u- and d-quarks are alike, that is that they are all negative and of similar size relative to the unpolarized distributions for all hadrons. 4. Conclusions We have presented lattice QCD calculations of the tensor generalized form factors which give access to impact parameter densities of transversely polarized quarks in the nucleon and the pion. Our results provide evidence for strong correlations of transverse coordinate and spin degrees of freedom, giving rise to significantly distorted densities for transversely polarized quarks, and in particular a surprisingly non-trivial pion spin structure. Following the arguments in [24,31], our results imply that the transverse momentum dependent Boer–Mulders function is not only large and negative for both up and down quarks in the proton, but also large and negative for up quarks in the π + . Since the effects we observe for the pion and for the nucleon are the same within errors, we find clear support for the hypothesis brought forward in [32] that all Boer–Mulders functions are alike. Since large Boer–Mulders functions give rise to potentially large azimuthal asymmetries, our results provide additional motivation for future studies of such asymmetries in semi-inclusive deep inelastic scattering at, for example, JLab [33] and π p Drell-Yan production at, for example, COMPASS. Acknowledgments The numerical calculations have been performed on the Hitachi SR8000 at LRZ (Munich), the apeNEXT at NIC/DESY (Zeuthen) and the BlueGene/L at NIC/FZJ (J¨ulich), EPCC (Edinburgh) and KEK (by the Kanazawa
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group as part of the DIK research programme). This work was supported by DFG (Forschergruppe Gitter-HadronenPh¨anomenologie and Emmy-Noether programme), HGF (contract No. VH-NG-004) and EU I3HP (contract No. RII3CT-2004-506078). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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Review
Transverse spin structure of hadrons from lattice QCD D. Br¨ommel a , M. Diehl b , M. G¨ockeler c , Ph. H¨agler d,∗,1 , R. Horsley e , Y. Nakamura f , D. Pleiter f , P.E.L. Rakow g , A. Sch¨afer c , G. Schierholz b,f , H. St¨uben h , J.M. Zanotti e (QCDSF/UKQCD Collaborations) a School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK b Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany c Institut f¨ur Theoretische Physik, Universit¨at Regensburg, 93040 Regensburg, Germany d Institut f¨ur Theoretische Physik T39, Physik-Department der TU M¨unchen, James-Franck-Str., 85747 Garching, Germany e School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK f John von Neumann-Institut f¨ur Computing NIC / DESY, 15738 Zeuthen, Germany g Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK h Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin, 14195 Berlin, Germany
Abstract This paper presents recent lattice QCD calculations of transverse spin densities of quarks in hadrons.2 Based on our simulation results for the tensor generalized form factors, we find substantial correlations between spin and coordinate degrees of freedom in the nucleon and the pion. They lead to strongly distorted transverse spin densities of quarks in the nucleon and a surprisingly non-trivial transverse spin structure of the pion. Following recent arguments by Burkardt [M. Burkardt, Phys. Rev. D 72 (2005) 094020], our results imply that the Boer–Mulders function h ⊥ 1 , describing correlations of the transverse spin and intrinsic transverse momentum of quarks, is large and negative for up-quarks in the proton and the π + . This supports the recent hypothesis that all Boer–Mulders functions are alike [arXiv:0705.1573], and also provides additional motivation for future studies of azimuthal asymmetries in π p Drell-Yan production at, e.g., COMPASS. c 2008 Elsevier B.V. All rights reserved.
Keywords: Lattice QCD; GPDs; Transverse spin
1. Introduction In recent years, transverse spin phenomena have provided new insights into QCD factorization [3] and the origin of, for example, single spin asymmetries measured in semi-inclusive deep inelastic scattering experiments [4]. A ∗ Corresponding author. Tel.: +49 89 289 14403.
E-mail address: [email protected] (Ph. H¨agler). 1 Speaker. 2 This contribution is based on Refs. [M. G¨ockeler, et al., Phys. Rev. Lett. 98 (2007) 222001; D. Br¨ommel, et al., [QCDSF Collaboration]. arXiv:hep-lat/0708.2249]. c 2008 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2007.12.035
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Fig. 1. Illustration of the quark distribution in a proton in impact parameter space.
central object of interest is the quark transversity distribution δq(x) = h 1 (x), which describes the probability of finding a transversely polarized quark with longitudinal momentum fraction x in a transversely polarized nucleon [5]. Substantial efforts have been made to understand the so-called transverse momentum dependent PDFs (tmdPDFs) ⊥ (x, k 2 ) [6], which measures the correlation of the intrinsic quark transverse like, for example, the Sivers function f 1T ⊥ 2 momentum k⊥ and the transverse nucleon spin S⊥ , as well as the Boer–Mulders function h ⊥ 1 (x, k⊥ ) [7], describing the correlation of k⊥ and the transverse quark spin s⊥ . A complementary approach to the transverse spin structure of hadrons is based on generalized parton distributions (GPDs) and the associated impact parameter densities of quarks in hadrons [8]. Whereas it is unclear to date whether tmdPDFs can be directly accessed in lattice QCD, moments of GPDs can and have already been studied for several years in lattice simulations. In the following sections, we give a brief introduction to the transverse spin densities and their relation to GPDs. We then present our lattice calculations of the relevant tensor generalized form factors and discuss the results for the densities of transversely polarized quarks in the nucleon [1] and the pion [2]. 2. Transverse spin structure of the nucleon We begin by introducing the density ρ(x, b⊥ , s⊥ , S⊥ ) [8], which represents the probability of finding a quark with longitudinal momentum fraction x and transverse spin s⊥ at distance b⊥ from the center-of-momentum of the nucleon with transverse spin S⊥ . The probability interpretation in impact parameter space has been developed in [9] and is illustrated in Fig. 1. Lattice calculations give access to x-moments of transverse spin densities [8] Z 1 1 1 2 2 2 i i e S⊥ A T n0 (b⊥ )− 1 A (b ) dx x n−1 ρ(x, b⊥ , s⊥ , S⊥ ) = An0 (b⊥ ) + s⊥ ρ n (b⊥ , s⊥ , S⊥ ) = b T n0 ⊥ 2 4m 2 ⊥ −1 ) j b ji i 0 2 j 1 e00 2 i i j i 0 2 ij S⊥ Bn0 (b⊥ ) + s⊥ + ⊥ B T n0 (b⊥ ) + s⊥ (2b⊥ b⊥ − b⊥ δ )S⊥ 2 A (b2 ) , (1) m m T n0 ⊥ 2∂ f. where m is the nucleon mass, and where the derivatives are defined by f 0 ≡ ∂b2 f and 1b⊥ f ≡ 4∂b2 b⊥ 2 b ⊥ ⊥ ⊥ The right-hand side of Eq. (1) can be interpreted as a “multipole-expansion”, consisting of two orbitally symmetric j i and b j ji S i , and a quadrupole term monopole terms (in the first line), two dipole structures proportional to b⊥ ji s⊥ ⊥ ⊥ i (2bi b j −b2 δ i j )S j . The (derivatives of the) three nucleon generalized form factors (GFFs) B (b ), proportional to s⊥ n0 ⊥ ⊥ ⊥ ⊥ ⊥ eT n0 (b⊥ ) thus determine how strongly the orbital symmetry in the transverse plane is distorted by the B T n0 (b⊥ ) and A 2 ), A 2 dipole and the quadrupole terms. The b⊥ -dependent GFFs An0 (b⊥ T n0 (b⊥ ), . . . in Eq. (4) are related to GFFs in momentum space An0 (t), A T n0 (t), . . . by a Fourier transformation Z 2 d 1⊥ −ib⊥ ·1⊥ 2 f (b⊥ )≡ e f (t = −12⊥ ), (2) (2π )2 where 1⊥ is the transverse momentum transfer to the nucleon. For the lowest moment, n = 1, one finds A10 (t) = F1 (t), B10 (t) = F2 (t) and A T 10 (t) = gT (t) where F1 , F2 and gT are the Dirac, Pauli, and tensor nucleon form factors, respectively. The GFFs discussed here are also directly related to x-moments of the corresponding vector and
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Fig. 2. Results for the generalized form factors B T (n=1,2)0 (t).
tensor GPDs (for a review see [10]). They parametrize off-forward nucleon matrix elements of certain local quark operators, for example the tensor operators µνµ1 ···µn−1
OT ↔
→
↔ µ1
= AS q σ µν γ5 i D
↔ µn−1
···iD
q
(3)
←
with D = 12 ( D − D ), and where AS denotes symmetrization in ν, . . . , µn−1 followed by anti-symmetrization in µ, ν and subtraction of traces. A concrete example of the corresponding parametrization for n = 1 is given by [12,11,8]
0 0 µν t e P Λ OT |PΛi = u(P 0 , Λ0 ) σ µν γ5 A T 10 (t) − A (t) T 10 2m 2 µναβ 1α γβ 1[µ σ ν]α γ5 1α e A (t) u(P, Λ), + B T 10 (t) − T 10 2m 2m 2 where 1 = P 0 − P and t = 12 . Parameterizations for n ≥ 1 in terms of tensor GFFs and their relation to GPDs are given in [11]. Since it is very challenging to access tensor GPDs in experiment [13], lattice QCD calculations of the eT n0 (t) are particularly interesting. GFFs A T n0 (t), B n0 (t) and A 2.1. Simulation results Our numerical lattice results are based on configurations generated with n f = 2 dynamical non-perturbatively O(a) improved Wilson fermions and Wilson gluons. Simulations have been performed on lattices of V ×T = 163 ×32 and 243 × 48 at four different couplings β = 5.20, 5.25, 5.29, 5.40 with up to five different κ = κsea values per β. The lattice scale a in physical units has been set using a Sommer scale of r0 = 0.467 fm [14,15]. The range of available pion masses extends down to 400 MeV, with lattice spacings below 0.1 fm and spatial volumes as large as (2.1 fm)3 . We note that the computationally demanding disconnected contributions, which in principle are needed for isosinglet quantities, are not included in this analysis. We expect, however, that they are small for the tensor GFFs [16]. Our results have been transformed to the MS scheme at a scale of 4 GeV2 using non-perturbative renormalization [17] of the corresponding operators. The calculation of GFFs in lattice QCD follows standard methods based on two- and three-point functions, and is described in, e.g., Refs. [18–21]. u Fig. 2 shows results for the GFFs B T (n=1,2)0 (t) as a function of the momentum transfer squared t, for a pion mass of m π ≈ 600 MeV, a lattice spacing of a ≈ 0.08 fm and a volume of V ≈ (2 fm)3 . In order to obtain a functional parametrization of the lattice results, we fit all GFFs using a p-pole ansatz F(t) = F0 /(1 − t/( p m 2p )) p with the three parameters F0 , m p and p for each GFF, where F0 = F(t = 0) provides us with the forward value of the GFF. Unfortunately, the statistics are not sufficient to determine all three parameters from a single fit to the lattice data. For a given generalized form factor, we therefore fix the power p first, guided by fits to selected datasets, and subsequently determine the forward value F0 and the p-pole mass m p by a full fit to the lattice data. The corresponding fits of B T (n=1,2)0 (t) are shown in Fig. 2 as shaded error bands, where we have set p = 2.5. We have checked that the final p-pole parameterizations only show a mild dependence on the value of p chosen prior to the fit. Our analysis indicates that the discretization errors are smaller than the statistical errors, and we will neglect any dependence of
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Fig. 3. Pion mass dependence of the generalized form factors B T (n=1,2)0 (t = 0) for up-quarks.
the GFFs on the lattice spacing in the following. Guided by our investigations of finite size effects in the case of the nucleon mass and the axial vector form factor g A [14,22], we conclude that the volume effects for the lattices and nucleon observables studied here are small and may be neglected. In Fig. 3, we show as an example the pion mass u dependence of the GFFs B T (n=1,2)0 (t = 0). Since the pion masses are still rather large for most of our data points, we cannot expect results from one-loop chiral perturbation theory [23] to be applicable to the available data. We therefore extrapolate the forward moments and the p-pole masses using an ansatz linear in m 2π to get a first estimate of the GFFs phys at the physical point. The corresponding fits are shown as shaded error bands in Fig. 3. At m π = 140 MeV, we find u d u d B T 10 (t = 0) = 2.93(13), B T 10 (t = 0) = 1.90(9) and B T 20 (t = 0) = 0.420(31), B T 20 (t = 0) = 0.260(23). As will be discussed in some more detail below, these comparatively large values already indicate a significant impact of this tensor GFF on the transverse spin structure of the nucleon. Noticing that the (tensor) GFFs B T n0 are analogous to the R (vector) GFFs Bn0 , we define an anomalous tensor magneticR moment [24], κT ≡ dx E T (x, ξ, t = 0) = B T 10 (t = 0), similar to the standard anomalous magnetic moment κ = dx E(x, ξ, t = 0) = B10 (t = 0) = F2 (t = 0). Based on up our results, we find large positive values for the anomalous tensor magnetic moment for both flavors, κT,latt ≈ 3.0 and down ≈ 1.9, in contrast to the u- and d-quark contributions to the anomalous magnetic moment, which are both large κT,latt up down ≈ −2.03. For the numerical evaluation of the density ρ n (b , s , S ) but of opposite sign, κexp ≈ 1.67 and κexp ⊥ ⊥ ⊥ in Eq. (1), we Fourier-transform the p-pole parametrization to impact parameter (b⊥ ) space. The parameterizations of the impact parameter dependent GFFs then depend only on the power p, the p-pole masses m p and the forward values F0 , which have been obtained from fits to the lattice data. In Fig. 4, we show the lowest moment n = 1 of the density for u- and d-quarks in the nucleon. We note that the (n = 1)-moment of the density can be written as the difference of the corresponding moments for quarks and antiquarks, ρ n=1 = ρqn=1 − ρqn=1 . The result for ρ n=1 (b⊥ , s⊥ , S⊥ ) is therefore not necessarily positive, although we expect contributions from antiquarks to be small in general. Due to the rather large anomalous magnetic moments κ u,d , we find strong distortions for unpolarized quarks in transversely polarized nucleons (the left part of the figure). This has already been noticed in [25] and can serve as a dynamical explanation of the experimentally observed Sivers-effect. Remarkably, we find even stronger distortions for transversely polarized quarks s⊥ = (sx , 0) in an unpolarized nucleon, as can be seen on the right-hand side of Fig. 4. In strong contrast to the distortions visible on the left-hand side of Fig. 4, we find that the densities for transversely polarized u- and d-quarks are both deformed in the positive b y direction due to the large positive u
d
values for the tensor GFFs B T 10 (t = 0) and B T 10 (t = 0) discussed above. For a first interpretation of these results, let us assume that transverse quark spin and orbital angular momentum are aligned for u- and d-quarks. Following this assumption, we expect that the densities for polarized u- and d-quarks in an unpolarized (polarization-averaged) nucleon are shifted in the same direction, for example upwards for quarks with spin in x-direction in a nucleon moving towards the observer in z-direction, as can be seen on the right-hand side of Fig. 4. Further, taking into account that the u-quark (d-quark) spin is predominantly oriented parallel (antiparallel) to the nucleon spin, we predict that densities of unpolarized quarks in a polarized nucleon are shifted in opposite directions for u- and d-quarks, for example that the orbital motion of d-quarks around the (−x)-direction leads to a larger d-quark density in the lower half plane, as can indeed be seen on the left-hand side of Fig. 4. Burkardt [24] has argued that the deformed densities on the
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Fig. 4. Lowest moment (n = 1) of the densities of unpolarized quarks in a transversely polarized nucleon (left) and transversely polarized quarks in an unpolarized nucleon (right) for up (upper plots) and down (lower plots) quarks. The quark spins (inner arrows) and nucleon spins (outer arrows) are oriented in the transverse plane as indicated.
right-hand side of Fig. 4 are related to a non-vanishing Boer–Mulders function h ⊥ 1 [7], which describes the correlation of the intrinsic quark transverse momentum and the transverse quark spin s⊥ . According to this conjecture we have in particular, κT ∼ −h ⊥ 1 , so that our lattice results indicate that the Boer–Mulders function for both u- and d-quarks in the nucleon is large and negative. Since h ⊥ 1 is time reversal odd, it enters with a different overall sign in SIDIS and Drell-Yan production [26]. The results just quoted refer to the functions relevant for SIDIS. 3. Spin structure of the pion The probability density ρ(x, b⊥ , s⊥ ) for polarized quarks in the pion can be directly obtained from the nucleon density in Eq. (1) by setting S⊥ = 0. The result is considerably simpler than for the nucleon but still contains a dipole i i j b j , term proportional to s⊥ ⊥ " # Z 1 i i j b j s 1 ∂ 2 2 ⊥ BTπn0 (b⊥ ) . ρ n (b⊥ , s⊥ ) = dx x n−1 ρ(x, b⊥ , s⊥ ) = Aπn0 (b⊥ (4) )− ⊥ 2 2 mπ ∂b⊥ −1 Here, the b⊥ dependent vector and tensor GFFs of the pion, Aπn0 and BTπn0 , are moments of the corresponding GPDs 2 ) and E π (x, ξ = 0, b2 ), respectively. A non-vanishing B π H π (x, ξ = 0, b⊥ ⊥ T T n0 in Eq. (4) would lead to a dipole-like distortion of the quark density in the transverse plane, and thereby imply a non-trivial transverse spin structure of the pion. Thus, a computation of BTπn0 from first principles in lattice QCD provides a crucial insight into the pion structure. We note that the GFF Aπ10 (t) for up-quarks in the π + is identical to the electromagnetic pion form factor Fπ (t), which we investigated in detail in [27]. The momentum-space GFFs BTπn0 (t) are related to the impact parameter dependent GFFs in Eq. (4) by a Fourier transformation as in Eq. (2). They parameterize pion matrix elements of the tensor operators in Eq. (3), µνµ1 ···µn−1
hπ + (P 0 )|OT
|π + (P)i = AS
n−1 µναβ 1α P¯β X 1µ1 · · · 1µi P¯ µi+1 · · · P¯ µn−1 BTπni (t) 2m π i=0
(5)
even
where P¯ = 21 (P 0 + P). Similarly, the GFFs Aπni (t) parameterize pion matrix elements of local vector quark π,u + operators [28]. In the following, we consider for definiteness the GFFs Aπ,u n0 (t) and BT n0 (t) for up-quarks in a π .
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Fig. 5. Results for the generalized form factors BTπ,u n0 (t) as functions of the invariant momentum transfer t.
The corresponding down-quarks GFFs and the GFFs for π − or π 0 can be easily determined by charge conjugation and isospin symmetry [29]. 3.1. Simulation results In Fig. 5, we show as an example the t dependence of the GFFs BTπ,u n0 (t) for n = 1, 2 at β = 5.29 and a pion mass of m π ≈ 600 MeV. A parametrization of the t dependence of the lattice results is obtained from a fit based on the p-pole ansatz discussed in Section 2.1. To avoid irregular densities at b⊥ = 0, we restrict the power p to p > 3/2 for BTπ,u n0 (t) according to [8]. Our analysis shows that in general, smaller values of p are preferred in the fits. We π,u therefore set in the following p = 1.6 for BTπ,u n0 (t) and choose a standard monopole-ansatz ( p = 1) for An0 (t). In a lattice calculation, we clearly cannot resolve structures at arbitrarily small distances. Hence, these powers should be considered as effective values, suitable for the range of momenta accessible in our simulation. Taking p = 2, which corresponds to the behavior expected for −t → ∞, changes the results for BTπ,u (n=1,2)0 by less than the statistical error. Concerning the pion mass dependence of our results it is important to note that, due to the prefactor m −1 π in the parameterization (5), the GFFs BTπn0 (t) must vanish like m π for m π → 0 [29]. We therefore consider in the following the ratio BTπn0 /m π , which tends to a constant in the chiral limit. A study of the lattice spacing dependence of BTπ,u 10 (t = 0)/m π shows that discretization errors are smaller than the statistical errors, and we therefore neglect any dependence of the GFFs on a in the following analysis. Fig. 6 shows the pion mass dependence of the forward 2 values BTπ,u n0 (t = 0)/m π , together with a linear extrapolation in m π , indicated by the shaded error band. For the same reasons as discussed in Section 2.1, we do not attempt to fit the available lattice data to the one-loop result of −1 ChPT [29,30]. The linear chiral extrapolation to m π = 140 MeV gives BTπ,u 10 (t = 0)/m π = 1.64 ± 0.11(stat) GeV π,u −1 and BT 20 (t = 0)/m π = 0.307 ± 0.032(stat) GeV , for p = 1.6. Within the present statistics, we can only give an π estimate of possible finite size effects, which are of order 20% for BTπ,u (1,2)0 (t = 0)/m π . For the GFFs An0 (t) with n n = 1, 2, we refer to our results in [28,27]. Finally, we evaluate the density ρ (b⊥ , s⊥ ) in Eq. (4) using the Fourier transformed p-pole parameterizations of the momentum-space GFFs. The lowest moment of the density for up-quarks in a π + is shown in Fig. 7. We find a substantial deformation on the right-hand side of Fig. 7 for transversely polarized quarks, compared to the unpolarized case on the left. We conclude that the pion has indeed a pronounced, non-trivial transverse spin structure. Using Eq. (4), we can easily calculate an average transverse shift R 2 y d b⊥ b ρ n (b⊥ , s⊥ ) 1 BTπn0 (t = 0) y (6) hb⊥ in = R 2 ⊥ n = 2m π Aπn0 (t = 0) d b⊥ ρ (b⊥ , s⊥ ) y
in the y direction for transverse quark spin s⊥ = (1, 0) in the x direction. We obtain hb⊥ i1 = 0.162(11) fm and y hb⊥ i2 = 0.117(13) fm. The corresponding average transverse shift of transversely polarized quarks in an unpolarized y nucleon is given by hb⊥ in = B T n0 (t = 0)/ (2m N An0 (t = 0)), where the An0 (t = 0) are the moments of the y unpolarized quark distribution. With the results of [1] reported in Section 2.1, we find hb⊥ i1 = 0.154(6) fm and y hb⊥ i2 = 0.101(8) fm for up-quarks in the proton. Comparing the average transverse shifts for the pion and the nucleon, we find the remarkable result that the distortion in the distribution of a transversely polarized up-quark is
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Fig. 6. Pion mass dependence of BTπ,u n0 (t = 0)/m π . The stars represent the extrapolated values at the physical point.
Fig. 7. The lowest moment of the impact parameter densities of unpolarized (left) and transversely polarized (right) up-quarks in a π + . The quark spin (inner arrow) is oriented in the transverse plane as indicated.
within errors of the same strength in a π + and in the proton. In the same way as for the nucleon, we can relate the pion ⊥,π GPD E Tπ , or equivalently the GFFs BTπ,u [24,31], which describes the correlation n0 , to the Boer–Mulders function h 1 between the transverse spin and the intrinsic transverse momentum of quarks in the pion [7]. Our results for BTπn0 then imply that the Boer–Mulders function for up-quarks in a π + is large and negative. Furthermore, a comparison of our results for the pion and the nucleon provides strong support for the arguments in [32], which suggest that all Boer–Mulders functions for valence u- and d-quarks are alike, that is that they are all negative and of similar size relative to the unpolarized distributions for all hadrons. 4. Conclusions We have presented lattice QCD calculations of the tensor generalized form factors which give access to impact parameter densities of transversely polarized quarks in the nucleon and the pion. Our results provide evidence for strong correlations of transverse coordinate and spin degrees of freedom, giving rise to significantly distorted densities for transversely polarized quarks, and in particular a surprisingly non-trivial pion spin structure. Following the arguments in [24,31], our results imply that the transverse momentum dependent Boer–Mulders function is not only large and negative for both up and down quarks in the proton, but also large and negative for up quarks in the π + . Since the effects we observe for the pion and for the nucleon are the same within errors, we find clear support for the hypothesis brought forward in [32] that all Boer–Mulders functions are alike. Since large Boer–Mulders functions give rise to potentially large azimuthal asymmetries, our results provide additional motivation for future studies of such asymmetries in semi-inclusive deep inelastic scattering at, for example, JLab [33] and π p Drell-Yan production at, for example, COMPASS. Acknowledgments The numerical calculations have been performed on the Hitachi SR8000 at LRZ (Munich), the apeNEXT at NIC/DESY (Zeuthen) and the BlueGene/L at NIC/FZJ (J¨ulich), EPCC (Edinburgh) and KEK (by the Kanazawa
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group as part of the DIK research programme). This work was supported by DFG (Forschergruppe Gitter-HadronenPh¨anomenologie and Emmy-Noether programme), HGF (contract No. VH-NG-004) and EU I3HP (contract No. RII3CT-2004-506078). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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