First simultaneous extraction of spin-dependent parton distributions and fragmentation functions from a global QCD analysis

TL;DR

This paper presents the first comprehensive global QCD analysis that simultaneously extracts spin-dependent parton distributions and fragmentation functions from diverse scattering data, resolving longstanding puzzles about strange quark polarization.

Contribution

It introduces a novel simultaneous extraction method using Monte Carlo techniques, avoiding SU(3) symmetry assumptions, and provides new insights into strange quark polarization and axial charges.

Findings

Strange polarization is consistent with zero.

The analysis resolves the strange quark polarization puzzle.

Results support a small SU(2) flavor asymmetry in the polarized sea.

Abstract

We perform the first global QCD analysis of polarized inclusive and semi-inclusive deep-inelastic scattering and single-inclusive $e^+e^-$ annihilation data, fitting simultaneously the parton distribution and fragmentation functions using the iterative Monte Carlo method. Without imposing SU(3) symmetry relations, we find the strange polarization to be very small, consistent with zero for both inclusive and semi-inclusive data, which provides a resolution to the strange quark polarization puzzle. The combined analysis also allows the direct extraction from data of the isovector and octet axial charges, and is consistent with a small SU(2) flavor asymmetry of the polarized sea.

Figures

Tables

Table 1. Table 1: Summary of χ 2 superscript 𝜒 2 \chi^{2} values and number of data points N dat subscript 𝑁 dat N_{\rm dat} for the various processes used in this analysis.

process	target	$N_{\mathrm{dat}}$	${\chi }^2$
DIS	$p,d,{}^3$He	854	854.8
SIA (${\pi }^{\pm }$)		459	600.1
SIA ($K^{\pm }$)		391	397.0
SIDIS (${\pi }^{\pm }$)
HERMES HERMES05	$d$	18	28.1
HERMES HERMES05	$p$	18	14.2
COMPASS COMPASS09	$d$	20	8.0
COMPASS COMPASS10	$p$	24	18.2
SIDIS ($K^{\pm }$)
HERMES HERMES05	$d$	27	18.3
COMPASS COMPASS09	$d$	20	18.7
COMPASS COMPASS10	$p$	24	12.3
Total:		1855	1969.7

Equations

$$A_1^h\left(x,z,Q^2\right)=\frac{g_1^h(x,z,Q^2)}{F_1(x,z,Q^2)},$$

$$g_1^h(x,z,Q^2)$$

$$\mathrm{T}(x;\mathbf{a})=\frac{M{x}^a(1-x{)}^b(1+c\sqrt{x})}{B(n+a,1+b)+cB(n+\frac{1}{2}+a,1+b)},$$

Full text

The decomposition of the proton’s spin into its quark and gluon helicity and orbital angular momentum contributions has been one of the defining problems that has engaged the hadron physics community for the better part of three decades Aidala12 . Initial explanations of the small fraction of the proton spin found to be carried by quarks focused on a large gluonic contribution generated through the axial anomaly Altarelli88 , or a large negative polarization of the strange quark sea. Subsequent experiments failed to find compelling evidence to support either of these scenarios, although recent results from RHIC have provided the first clear indications for a nonzero gluon polarization, $\Delta g$ DSSV14 . Complementing this has been a growing effort to determine the quark and gluon orbital angular momentum components of the proton spin, through measurements of generalized parton distributions in exclusive processes Diehl03 . Critical to all these endeavors is the necessity to reliably extract from the experimental data the fundamental parton distribution functions (PDFs) that characterize the partons’ spin and momentum distributions through global QCD analysis.

Typically, global QCD analyses JAM15 ; LSS15 ; NNPDF14 ; BB10 ; AAC09 ; DSSV09 ; KTA17 of inclusive deep-inelastic scattering (DIS) and other polarized data extract spin-dependent PDFs using constraints from weak baryon decays under the assumption of SU(3) flavor symmetry. This puts significant restriction on the first moment of the polarized strange PDF, with $\Delta s^{+}\equiv\Delta s+\Delta\bar{s}\approx-0.1$. Further assumptions about the behavior of the PDFs at large parton momentum fractions $x$ induces a shape for $\Delta s^{+}(x)$ with magnitude peaking at $x\sim 0.1$. With the inclusion of semi-inclusive DIS (SIDIS) data, a strikingly different shape for the strange polarization emerges DSSV09 ; LSS10 , changing sign to become positive at $x\sim 0.1$. This was found, however, to be strongly dependent on the assumed $s\to K$ fragmentation function (FF), which enters in the calculation of the SIDIS cross section LSS10 ; LSS11 . Ideally, an unambiguous determination of the strange quark polarization requires a simultaneous QCD analysis of both the PDFs and FFs.

In this paper we report on the first such analysis, using data from inclusive and semi-inclusive DIS and single-inclusive $e^{+}e^{-}$ annihilation (SIA) to simultaneously constrain the spin-dependent PDFs and $\pi^{\pm}$ and $K^{\pm}$ FFs. To avoid biasing the extraction of $\Delta s^{+}$ by assumptions about SU(3) symmetry, we allow for the combined data sets to determine the octet axial charge directly. This is not feasible in a DIS-only analysis, but becomes viable with the flavor separation capability of SIDIS data. We perform the analysis within the iterative Monte Carlo (IMC) approach JAM15 ; JAM16 , which avoids potential bias in single-fit analyses introduced by fixing parameters not well constrained by data, and allows a statistically rigorous determination of PDF and FF uncertainties by an efficient exploration of the parameter space.

In this first combined study of PDFs and FFs, which is performed within collinear factorization at next-to-leading order (NLO) in the $\overline{\rm MS}$ scheme, and referred to as “JAM17”, we simplify the analysis by placing cuts on the DIS and SIDIS kinematics to avoid higher twist contributions, with the four-momentum transfer squared $Q^{2}>1$ GeV2 and hadronic final state mass squared $W^{2}>10$ GeV2. The higher twists were extracted in a previous IMC analysis JAM15 , with a lower cut $W>2$ GeV, but did not significantly affect the determination of the leading twist PDFs.

The detailed expressions for DIS and SIA observables can be found Refs. JAM15 and JAM16 , respectively. For the SIDIS data, the observables measured are the longitudinal double spin asymmetries $A_{1}^{h}$ for the production of a hadron $h$,

[TABLE]

where the semi-inclusive spin-dependent $g_{1}^{h}$ and spin-averaged $F_{1}^{h}$ structure functions depend on both $x$ and the fraction $z=p\cdot p_{h}/p\cdot q$ of the virtual photon’s momentum ($q$) carried by the hadron ($p_{h}$), with $p$ the target momentum.

The polarized $g_{1}^{h}$ function in Eq. (1) is defined in terms of the spin-dependent PDFs $\Delta q$ and FFs $D_{q}^{h}$,

[TABLE]

where the ${\cal O}(\alpha_{s})$ corrections are given in Ref. Stratmann01 . The unpolarized structure function $F_{1}^{h}$ is defined analogously, with the spin-dependent PDFs replaced by their spin-averaged counterparts.

Following Refs. JAM15 ; JAM16 , we parameterize both the polarized PDFs and FFs at the input scale $Q_{0}^{2}=1$ GeV2 using template functions of the form

[TABLE]

where ${\bm{a}}=\{M,a,b,c\}$ are the fitting parameters, and $B$ is the Euler beta function. For the polarized PDFs we set $n=1$ so that $M$ corresponds to the first moment. This template is used for all the fitted polarized PDFs, which we choose to be $\Delta q^{+}$, $\Delta\bar{q}$ and $\Delta g$, for flavors $q=u,d$ and $s$. The FFs are also given by Eq. (3) (with $x$ replaced by $z$), setting $c=0$ and $n=2$, so that $M$ corresponds to the average momentum fraction carried by the produced hadron. For the FFs $D_{u^{+}}^{\pi^{+}}\equiv D_{u}^{\pi^{+}}+D_{\bar{u}}^{\pi^{+}}=D_{d^{+}}^{\pi^{+}}$, $D_{u^{+}}^{K^{+}}$ and $D_{s^{+}}^{K^{+}}$, which contain both favored and unfavored distributions, we assign two template functions, while for the remaining unfavored FFs, $D_{\bar{u}}^{\pi^{+}}=D_{d}^{\pi^{+}}$, $D_{s}^{\pi^{+}}=(1/2)D_{s^{+}}^{\pi^{+}}$, $D_{\bar{u}}^{K^{+}}=(1/2)D_{d^{+}}^{K^{+}}$ and $D_{s}^{K^{+}}$, along with the heavy quarks and gluons, a single template function is used. Following Ref. JAM16 , we use the zero mass variable flavor scheme and parametrize the heavy quark FFs discontinuously at their mass thresholds.

The resulting $\chi^{2}$ values for each process fitted in our analysis are presented in Table 1. For inclusive DIS we use the data sets from Refs. EMC89 ; SMC98 ; SMC99 ; COMPASSdis10 ; COMPASSdis07 ; SLAC-E130 ; SLAC-E142 ; SLAC-E143 ; SLAC-E154 ; SLAC-E155p ; SLAC-E155d ; SLAC-E155_A2pd ; SLAC-E155x ; HERMES97 ; HERMES07 ; HERMES12 , and for SIA from Refs. TASSO80 ; TASSO83 ; TASSO89 ; ARGUS89 ; TPC84 ; TPC86 ; TPC88 ; HRS87 ; SLD04 ; OPAL94 ; OPAL00 ; ALEPH95 ; DELPHI95 ; DELPHI98 ; TOPAZ95 ; Belle13 ; Leitgab13 ; BaBar13 . The SIDIS data sets are from HERMES HERMES05 for $\pi^{\pm}$ and $K^{\pm}$ production from the deuteron, and $\pi^{\pm}$ production from the proton, and from COMPASS with $\pi^{\pm}$ and $K^{\pm}$ production from deuterium COMPASS09 and hydrogen COMPASS10 targets. Overall, the $\chi^{2}$ per datum for all the SIDIS $\pi^{\pm}$ data is 68.5/80, and 49.3/71 for the $K^{\pm}$ data, while the $\chi^{2}$ per datum for the combined inclusive DIS, SIDIS and SIA data is $1969.7/1855\approx 1.06$.

The polarized quark and antiquark PDFs from the combined fit are illustrated in Fig. 1, together with their 1$\sigma$ uncertainties. (The polarized gluon PDF is essentially unchanged from the earlier JAM15 analysis JAM15 .) For the denominator of the asymmetries $A_{1}^{h}$, we use spin-averaged PDFs from the CJ12 NLO global fit CJ12 . Using the MMHT14 MMHT14 PDFs instead gives a difference of $\approx 2\%-5\%$, which is insignificant on the scale of the experimental uncertainties of the asymmetries. The $\Delta u^{+}$ and $\Delta d^{+}$ PDFs, which are determined largely by the inclusive DIS data, are similar to those in the JAM15 analysis JAM15 , giving only marginally harder distributions at large $x$ values. The difference in the magnitudes of $\Delta d^{+}$ at $x\sim 0.2$ arises from anticorrelation with $\Delta s^{+}$; since the latter is less negative, it requires some compensation to describe the DIS observables.

Unlike inclusive DIS, the SIDIS observables can in principle discriminate between different quark and antiquark flavors, and in Fig. 1 we also show the light sea quark polarizations for the isoscalar and isovector combinations, $\Delta\bar{u}\pm\Delta\bar{d}$. Our results suggest a slightly positive isovector sea polarization in the range $x\approx 0.01-0.1$, with the isoscalar combination more consistent with zero. This is similar to the expectations in some nonperturbative models Schreiber91 ; Diakonov96 that predict larger isovector than isoscalar sea polarization, as well as in recent lattice simulations HueyWen ; Fernanda . The signal is relatively weak, however, and can be attributed to several $\pi^{\pm}$ and $K^{\pm}$ SIDIS data sets that marginally favor a nonzero sea polarization.

An example of this is illustrated in Fig. 2 for the COMPASS $\pi^{-}$ asymmetry COMPASS10 , which, because of the valence ($\bar{u}d$) structure of the $\pi^{-}$, is the most sensitive observable to $\bar{u}$ polarization. Comparing the fitted proton $A_{1p}^{\pi^{-}}$ asymmetry with that obtained by setting $\Delta\bar{u}=\Delta\bar{d}=0$, the difference is rather small, but noticeable for $x\lesssim 0.1$, where the asymmetry with the unpolarized sea lies at the edge of the 1$\sigma$ envelope of the full result. Similar effects are found for other SIDIS asymmetries that depend explicitly on $\Delta\bar{u}$ or $\Delta\bar{d}$. The results are also qualitatively similar to those found in the DSSV09 global analysis DSSV09 , although the magnitude of the sea quark asymmetries here is somewhat smaller.

For the strange quark polarization, the results in Fig. 1 suggest that $\Delta s^{+}$ is small at all $x$, albeit within relatively large uncertainties. While consistent with zero within 1$\sigma$, there does appear some indication of a positive $\Delta s^{+}$ at $x\approx 0.1$. This can be attributed directly to the HERMES deuteron $K^{-}$ production data HERMES05 , illustrated in Fig. 2. Since $\Delta s$ is weighted by the (large) favored $D_{s}^{K^{-}}$ FF, the $A_{1d}^{K^{-}}$ asymmetry is most sensitive to strange quark polarization. In contrast, $K^{+}$ production, which is sensitive to $\Delta\bar{s}$, is dominated by the much larger $\Delta u$ PDF weighted by the favored $D_{u}^{K^{+}}$.

Compared with the full result, the asymmetry computed with a negative $\Delta s^{+}$, as in the JAM15 analysis of inclusive DIS JAM15 , gives a significantly worse fit to the HERMES $A_{1d}^{K^{-}}$ data, with $\chi^{2}$ increasing from 5.7 to 18.5 for 9 data points. A similar effect is seen for the COMPASS $K^{-}$ data on protons (deuterons), which prefer a non-negative strangeness, with $\chi^{2}$ increasing from 4.8 to 9.0 (12.0 to 18.5) for 12 (10) data points.

In addition to $\Delta s^{+}$, we also explored the sensitivity to a nonzero strange–antistrange asymmetry, $\Delta s^{-}$. While most global PDF analyses assume $\Delta s=\Delta\bar{s}$, a nonzero asymmetry is expected from chiral symmetry breaking in QCD Signal87 ; Malheiro97 ; Thomas00 . In principle, the availability of precise $K^{\pm}$ SIDIS data could discriminate between $s$ and $\bar{s}$ polarization; however, as Fig. 1 illustrates, the current experimental errors render extraction of a nonzero $\Delta s^{-}$ signal impractical.

Of course, preference for a positive or negative strangeness depends rather strongly on the FFs used in the evaluation of the asymmetry LSS10 ; LSS11 ; LSS15 . The solution to this problem is to simultaneously determine both PDFs and FFs, as we seek to do here. The results for the FFs extracted from the combined fit are displayed in Fig. 3 for the most relevant quark flavors fragmenting to $\pi^{+}$ and $K^{+}$, at a scale $Q^{2}=5$ GeV2 appropriate for the SIDIS data.

For the pion, the $D_{u^{+}}^{\pi^{+}}$ FF is relatively well constrained by the SIA data, compared with the unfavored $D_{\bar{u}}^{\pi^{+}}$. The $\pi^{+}$ FFs are also similar to those from the previous JAM analysis of SIA data JAM16 , as well as from other parametrizations DSS ; HKNS . For kaons, the uncertainties for the favored $D_{u^{+}}^{K^{+}}$ and unfavored $D_{s}^{K^{+}}$ are generally larger because of the lower precision of the $K$ data.

One of the most important features in Fig. 3 is the difference between the $D_{s^{+}}^{K^{+}}$ FF for the various parametrizations, which has profound impact on $\Delta s$ extraction. Here the JAM17 result is more comparable with the DSS fit DSS , while the magnitude of the HKNS HKNS result is somewhat smaller. The $D_{s^{+}}^{K^{+}}$ FF is also qualitatively similar to the recent NJL–Jet model calculation Hrayr11 , specifically in the large-$z$ region where $D_{s^{+}}^{K^{+}}>D_{u^{+}}^{K^{+}}$. The $D_{s^{+}}^{K^{+}}$ obtained from the SIA-only analysis JAM16 is very similar to that in Fig. 3, with the SIDIS data pulling the JAM17 result slightly larger at low $z$. Recall that the smaller $s^{+}\to K^{+}$ fragmentation in the HKNS fit is what allowed a more negative $\Delta s^{+}$ at $x\sim 0.1$ in the combined DIS and SIDIS analysis of Ref. LSS11 , similar to the shape of $\Delta s^{+}$ in DIS-only analyses such as JAM15 in Fig. 1. The shapes of the heavy quark and gluon FFs are essentially unchanged from Ref. JAM16 .

If it is SIDIS data that restrict the $\Delta s^{+}(x)$ PDF to be small and positive at intermediate $x$ values, a natural question to ask is what drives $\Delta s^{+}$ to be large and negative in DIS-only analyses? The answer appears to be the imposition of the SU(3) constraint on the octet axial charge, which is related to the lowest moment of the SU(3) nonsinglet combination, $a_{8}\equiv\Delta u^{+}+\Delta d^{+}-2\Delta s^{+}$. We have verified that in the absence of this constraint, it is indeed possible to fit the DIS data sets with zero $\Delta s^{+}$ at the input scale, with identical $\chi^{2}$ values as in a fit with the strange distribution free to vary. This confirms that the sensitivity to $\Delta s^{+}$ from DIS data alone, in combination with $Q^{2}$ evolution, is negligible. In contrast, the SU(3) assumption tends to pull the strange PDF to be negative across all $x$ in order to generate a negative moment, $\Delta s^{+}(Q_{0}^{2})\approx-0.1$.

To understand the origin of the large negative peak in $\Delta s^{+}$ at $x\approx 0.1$ in DIS-only analyses, we examine the behavior of $\Delta s^{+}$ in the absence of low-$x$ DIS data ($x\lesssim 0.02$), where sea quarks are expected to play a greater role. Starting from a shape of $\Delta s^{+}(x)$ at the input scale that is negative at small $x$ and positive at large $x$, such as in the DSSV09 fit DSSV09 , we find that the strange PDF remains qualitatively unchanged when fitting to the reduced DIS data set. Upon closer examination, around 5 data points at the lowest $x$ bins from the COMPASS deuterium data are found to favor small negative values for $\Delta s^{+}$, which then drives the strange PDF in the intermediate-$x$ region to be more negative in order to satisfy the SU(3) constraint. Finally, the characteristic negative peak at $x\sim 0.1$ observed in most global analyses is generated by fixing the parameter $b\approx 6-10$, as for typical sea quark PDFs. Such a peak is artificial since there is no direct sensitivity to $\Delta s^{+}(x)$ in current inclusive DIS data.

The “strange quark polarization puzzle” LSS11 ; LSS15 can therefore be understood by simply relaxing the SU(3) constraint, which then produces a strange distribution with shape and magnitude that agree well with DIS and SIDIS asymmetries. While both the (mostly positive) JAM17 and (mostly negative) JAM15 strange PDFs give nearly identical $\chi^{2}$ for DIS data, the latter will be strongly disfavored by the SIDIS asymmetries. In fact, the positive shape of $\Delta s^{+}$ at $x\sim 0.1$ is obtained even when samples consistent with SU(3) symmetry are selected, as Fig. 1 illustrates. Such samples prefer more negative PDFs at lower $x$ values, $x\lesssim 10^{-2}$, where the shape is not well constrained, and restrict the first moment to $\Delta s^{+}\sim-0.1$. In contrast, our new results gives a smaller averaged value, $\Delta s^{+}(Q_{0}^{2})=-0.03(10)$, but now of course with larger uncertainty. Interestingly, the central value agrees with the recent lattice QCD determination of strangeness polarization, $\Delta s^{+}_{\rm latt}=-0.02(1)$ at $Q^{2}\approx 7$ GeV2 Bali12 .

Our result for the strange moment translates to a central value of the octet axial charge $a_{8}=0.46(21)$ that is $\approx 20\%$ smaller than the traditional SU(3) value $0.586(31)$, as suggested in earlier theoretical studies Bass10 . Even though the uncertainty is somewhat large, the peaking of the $a_{8}$ distribution around $\sim 0.5$ is entirely data driven, as Fig. 4 illustrates with the comparison of the flat prior distributions sampled in the range $[-0.2,1.2]$. Future higher precision SIDIS kaon data would be needed to reduce the uncertainty on both the polarized strangeness and test the degree of SU(3) breaking in the octet axial charge.

Another consequence of the more positive value of $\Delta s^{+}$ (smaller $a_{8}$) is an $\approx 25\%$ larger total spin carried by quarks and antiquarks in the nucleon Bass10 , $\Delta\Sigma(Q_{0}^{2})=0.36(9)$. Within the larger uncertainties resulting from the relaxing of the SU(3) constraint in our simultaneous analysis, this is compatible with the singlet charge of 0.28(4) obtained in the JAM15 fit JAM15 . In fact, the simultaneous fit can also be used to determine the triplet axial charge $a_{3}\equiv\Delta u^{+}-\Delta d^{+}$ preferred by the data, without assuming SU(2) symmetry. We find that $a_{3}=1.24(4)$, in good agreement with the standard value $g_{A}=1.269(3)$ from neutron weak decay. This is a remarkable empirical confirmation of the equality between $a_{3}$ and $g_{A}$, and of QCD itself, to almost 2%!

Finally, as suggested in Fig. 1, the antiquark component of the isovector axial charge prefers slightly positive values, $\Delta\bar{u}-\Delta\bar{d}=0.05(8)$, but is consistent with zero within the uncertainty. The recent polarized $pp$ scattering data from PHENIX PHENIX-W on asymmetries from $W^{\pm}+Z$ decays and from STAR STAR-W on $W^{\pm}$ asymmetries also indicate a slightly larger $\Delta\bar{u}$ in the $x\sim 0.16$ range.

In the future, the IMC analysis will be extended to include NNLO Anderle:2015lqa ; Bertone:2017tyb and small-$x$ Kovchegov:2017jxc ; Kovchegov:2017lsr corrections, as well as unpolarized SIDIS data for better determination of the unfavored FFs DSSpi15 ; DSSk17 . Since the unpolarized strange quark PDF is currently not well determined, useful constraints on $\Delta s$ from these data will necessitate a simultaneous analysis of spin-averaged and spin-dependent PDFs, in addition to FFs. This remains an important challenge for future global QCD analyses.

We are grateful to A. Accardi, S. D. Bass and A. W. Thomas for helpful discussions. This work was supported by the US Department of Energy (DOE) contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab. N.S. was supported by the DOE contract DE-FG-04ER41309.