The spin of the proton in chiral effective field theory
Proton spin is investigated in chiral effective field theory through an examination of the singlet axial charge, , and the two non-singlet axial charges, and . Finite-range regularization is considered as it provides an effective model for estimating the role of disconnected sea-quark loop contributions to baryon observables. Baryon octet and decuplet intermediate states are included to enrich the spin and flavour structure of the nucleon, redistributing spin under the constraints of chiral symmetry. In this context, the proton spin puzzle is well understood with the calculation describing all three of the axial charges reasonably well. The strange quark contribution to the proton spin is negative with magnitude 0.01. With appropriate evolution, we find the singlet axial charge at the experimental scale to be , consistent with the range of current experimental values.
In 1988 the European Muon Collaboration (EMC) published their polarized deep inelastic measurement of the proton’s spin dependent structure function . Their result suggested that the quark spins summed over the up, down and strange quark flavors contribute only a small fraction of the proton’s spin EMC (). The EMC data shocked the particle physics community, because it was thought to be contradictory to the apparently successful, naive quark model descriptions of proton structure where the constituent quarks carry the total proton spin. It inspired a vigorous global program of experimental and theoretical developments to understand the internal spin structure of the proton extending for nearly three decades. For reviews of the spin structure of the proton, see for example Refs. Aidala:2012mv (); Myhrer:2009uq (); Jaffe:2001dm (); Bass:2004xa (); Burkardt:2008jw (); Ellis:1995de (); Shore:1998dn (); Bass:1992bk (); Filippone:2001ux ().
The experimental efforts at CERN Ashman:1987hv (); Adeva:1998vw (); Ageev:2005gh (); Alexakhin:2006oza (), DESY Airapetian:2007mh (), JLab Dharmawardane:2006zd (), RHIC Abelev:2006uq (); Adare:2007dg () and SLAC Hughes:1990sf () have been impressive. A summary of the status and recent experimental results on the spin structure of the nucleon can be found in Ref. Aidala:2012mv (). Unlike the early EMC result which suggested that the quark spin contribution, , might be consistent with zero ( EMC ()), today the experimental measurements indicate the nucleon’s flavor-singlet axial charge measured in polarized deep inelastic scattering is at GeV. This tends to about one-third of the total spin as Airapetian:2007mh (); Alexakhin:2006oza (); Alekseev:2010hc ().
The matrix elements of the non-singlet axial current and the singlet axial current are defined as follows
where are generators of the flavor group and is a vector in flavor space. The singlet axial current is not conserved due to the Adler-Bell-Jackiw anomaly. As a result, the flavor-singlet matrix element can receive an additional contribution from gluon polarization Carlitz:1988ab (); Altarelli:1988nr (); Bass:1991yx (); Efremov:1984ip (). This led to the early idea that the measured singlet component receives an important contribution from the gluon polarization , i.e.
The polarized gluon distribution function was estimated to be less than 0.3 at a scale of 1 GeV in the MIT bag model Chen:2006ng (). From the extensive experimental studies one finds that the absolute value is of the order 0.2 - 0.3 for = 3 GeV Adolph:2012ca (); Nocera:2014gqa (). This amount of gluon polarization, by itself, is far too small to resolve the problem of the small value of through the axial anomaly.
Another explanation for the small value of draws on the strange quark contribution to the proton spin. The non-singlet axial charge extracted from hyperon beta-decays under the assumption of flavour symmetry is Close:1993mv (). If the strange quark contribution to the proton spin were around , the proton spin, , expressed as , would be close to the experimental data. However, the uncertainty of could be as large as Jaffe:1989jz (); Ratcliffe:2004jt (). A recent re-evaluation of the nucleon’s axial-charges in the Cloudy Bag model, taking into account the effect of the one-gluon-exchange hyperfine interaction and the meson cloud, led to the value Bass:2009ed (). In this case, was found to be of order 0.01 in magnitude (and negative), with the small value of a consequence of SU(3) breaking.
Soon after the release of the EMC data it was realized that the effect of the pion cloud of the nucleon, associated with chiral symmetry breaking, would be to lower the quark spin content of the nucleon Schreiber:1988uw (). This is because pion emission tends to flip the nucleon spin and hence the spins of the quarks in it, while the quarks in the pion necessarily carry orbital angular momentum but no spin. This effect was calculated in the cloudy bag model and the effect of the pion cloud together with the relativistic motion of the light quarks in the bag Chodos:1974je () reduced to around 0.5. An alternative approach to the problem recognised that, given the standard spin dependent one-gluon-exchange correction to the energy of the nucleon, there must be a corresponding exchange current correction to the proton spin Myhrer:1988ap (). This too reduces the proton spin by around 0.15 below the naive bag model result of 0.65. It is only recently that studies of the nucleon mass splitting in lattice QCD Young:2002cj () provided the justification for combining the pion cloud and one-gluon exchange effects Myhrer:2009uq (). This led to a theoretical result in the range to , which is compatible with the aforementioned experimental value and, after QCD evolution, with the results of lattice QCD for the angular momentum carried by the quarks in the proton Thomas:2008ga ().
The scale dependence of presents another consideration in understanding the fraction of the proton spin carried by quarks Jaffe:1987sx (). Consideration of the genral features of QCD evolution long ago led to the conclusion that the natural scale at which to match a quark model to QCD is quite low, so that most of the momentum of the proton is carried by valence quarks and one can think of the gluons as having been integrated out of the theory. In Jaffe’s scenario, the small value of the experimental proton spin is due to differences in the energy scale of the experimental result and the quark model results. Since the anomalous dimension of the singlet axial current is nontrivial, its matrix element is scale dependent. With the evolution, it is possible that the large proton spin at low will be reduced through evolution to the large of the experimental result. As mentioned in Ref. Jaffe:1987sx (), it is difficult to get a reliable evolution at low- because perturbative QCD is not applicable. More specifically, one cannot determine which evolution line presented in Ref. Jaffe:1987sx () is correct. One needs a direct calculation of at the low energy scale.
In this paper, we will investigate the proton spin carried by the quarks in the framework of effective field theory, assuming that at the corresponding low scale the gluons have been integrated out, with the only residue being a spin-dependent effective interaction between quarks. , and the non-singlet axial charges and will be calculated simultaneously in the chiral effective field theory. In this approach the proton structure is enhanced through the dressing of the proton by octet-meson and both octet and decuplet baryon intermediate states. These processes enrich the spin and flavour structure of the nucleon, redistributing spin under the constraints of chiral symmetry. As we will see, this formalism is able to describe all three axial charges in a reasonable manner.
We consider heavy baryon chiral perturbation theory and include octet and decuplet intermediate-state baryons. The lowest-order chiral Lagrangian used in the calculation of the nucleon spin distribution function is expressed as
where is the covariant spin operator defined as
Here, is the nucleon four velocity. In the rest frame, we have ). , and are the standard -flavour coupling constants.
According to the Lagrangian, the one-loop Feynman diagrams, which contribute to the quark spin fraction of the proton, are plotted in Fig. 1. Working with the chiral coefficients of full QCD Leinweber:2002qb (); Wang:2007iw (), the contribution of the doubly-represented -quark sector of the proton to the proton spin, described by diagram (a) of Fig. 1, is expressed as
where the first through fourth terms in the bracket are the contributions from the , , the transition and the intermediate states, respectively. The -quark contribution with a intermediate state vanishes. The coefficients, , of the integrals, , are expressed as
These coefficients reflect the -flavour symmetry considered in obtaining the meson-baryon couplings (proportional to and ), the angular-momentum composition of the intermediate meson-baryon intermediate states, and the -spin-flavour wave function of the intermediate state baryon, considered in assigning a quark-sector spin contribution. The latter is discussed in further detail below.
With the above coefficients, one can write the quark-sector contribution to the proton spin of Fig. 1a as
Similarly, the strange quark contribution to the proton spin from diagram (a) of Fig. 1 is written as
In the above equations, the low energy coefficients () describe the tree-level quark contribution to the baryon spin. For example, for the intermediate proton and neutron, their spins are expressed as
In the naive quark model, the value of is 1. However, it is smaller than 1 due to the relativistic and confinement effects Chodos:1974je ().
Diagram (b) of Fig. 1 illustrates decuplet baryon intermediate states. The -sector contribution to the proton spin from this diagram is
where the coefficients and are
The and quark-sector contributions are
In deriving these equations, the tree-level quark contributions to the spin of decuplet baryons are used. For example
These contributions will also be reduced upon taking relativistic and confinement effects into account.
Diagrams (c) and (d) of Fig. 1 provide contributions from intermediate states involving an octet-decuplet transition. The quark-sector contribution to the proton spin from these diagrams is expressed as
The and quark-sector contributions are
The integrals in the above equations, , and are defined in Ref. Wang:2007iw ().
Including the tree-level contribution, the total -, - and -quark sector contributions to the spin of the proton are
Here is the wave-function renormalization constant calculated from the standard diagrams corresponding to those of Fig. 1. Values are listed in Table I.
In the numerical calculations, the -flavour couplings are , . The decuplet coupling Jenkins:1992pi (). The regulator in the integrals is chosen to be of a dipole form
The final quark spin contributions are related to the low-energy coefficients , and . These are the tree-level values of the quark spin and are unity in the naive constituent-quark model. Relativistic and confinement effects associated with light quarks suppress this value. We begin by assuming and treat as a parameter constrained by the axial charge . With GeV, the central value of that we find is , less than one as expected. Since the strange quark is expected to be less relativistic, this value may be an overestimate of the spin suppression in that case. However, the strange quark contribution to the proton spin is small and so the approximation is adequate for this purpose.
With , the , and quark contributions to the proton spin are
The axial charge and . Before considering the necessary evolution to the value GeV relevant to the experimental data, it is interesting to consider other improvements to our use of -spin-flavour wave functions in attributing quark spin to intermediate meson-baryon states.
Although it lies outside the framework of chiral effective field theory, the effect of one-gluon-exchange (OGE) is particularly important for spin dependent quantities. Hogaason and Myhrer Hogaasen () showed that the incorporation of the exchange current correction arising from the effective one-gluon-exchange (OGE) force shifts the tree-level non-singlet charge, , from to , where is about 0.05. Thus, if one were to include the OGE correction, would be somewhat larger at if one chose it to reproduce the axial charge . For the charges, and , the OGE correction shifts their tree-level values from to Myhrer:2009uq (). In this case, and . Correspondingly, the quark contributions to the proton spin are
The results show that the strange quark contribution to the proton spin is very small relative to the and contributions. The axial charge, , is intermediate between the value extracted under the assumption of SU(3) symmetry from hyperon -decay, Close:1993mv (), and that obtained in the cloudy bag model, Bass:2009ed ().
To provide an estimate of the uncertainty in these results, we vary the regulator parameter, , governing the size of meson cloud contributions to proton structure. Considering GeV, the uncertainties in the quark contributions to proton spin are
The axial charges with the corresponding error bars are
The non-singlet axial current is conserved in the limit of massless quarks and the anomalous dimension for the non-singlet axial current vanishes. Therefore, the non-singlet matrix elements and are scale independent. However, the anomalous dimension of the singlet axial current is nontrivial, and is a scale dependent quantity. Consistent with the idea that at a sufficiently low scale the valence quarks dominate and the gluons have been effectively integrated out of the theory, we set at that scale. Then to compare the result for calculated within chiral effective field theory to experiment, is obtained through NNLO QCD evolution to GeV.
The evolution equation has the form Gensini:1995ev ()
where . After integrating in from a normalization scale of to , one obtains Gensini:1995ev ()
with the NNLO calculation of the anomalous dimension, , taken from Ref. Larin:1993tq ().
In Fig. 2 we illustrate the evolution of commencing with our result of Eq. (34) attributed to the scale GeV as in Ref. Jaffe:1987sx (). Initially, decreases rapidly with increasing raising concerns about the application of an NNLO calculation for . However, in the context of the model uncertainty presented in Fig. 2 the present evolution will suffice.
At GeV, our calculation of the proton spin can be compared with experiment. Our model provides
which agrees with the experimental measurement of at GeV.
In summary, we have examined the proton spin fractions carried by quarks using a model in which the meson-cloud dressings of the proton are characterized by chiral effective field theory, regularized through a regulator characterizing the nontrivial size of the source of the meson cloud. Finite-range regularization provides an effective model for estimating the role of disconnected sea-quark loop contributions to baryon observables Leinweber:2004tc (); Wang:1900ta (); Wang:2012hj (); Hall:2013dva (); Wang:2013cfp (); Wang:2015sdp (). Both baryon octet and decuplet intermediate states are included to enrich the spin and flavour structure of the nucleon, redistributing spin under the constraints of chiral symmetry. Drawing on extensive experience Leinweber:2004tc (); Wang:1900ta (); Wang:2013cfp (); Wang:2012hj (); Young:2002ib (); Leinweber:2003dg (); Wang:2007iw (); Wang:2010hp (); Allton:2005fb (); Armour:2008ke (); Hall:2013oga (), the preferred regulator parameter is GeV. To gain insight into the role of the meson cloud and uncertainties associated in determining the size of the meson-cloud contributions, we have varied from 0.6 GeV to 1 GeV.
The coefficient , which takes relativistic and confinement effects into account is constrained by the experimental axial charge . The one-gluon-exchange correction to the axial charges is also taken into consideration. Because each quark-sector contribution is calculated separately, the non-singlet charges, and , and the singlet charge are obtained simultaneously. The results are summarized in Table I.
Our model provides significant insight into the the proton spin puzzle. The main conclusions are:
At low energy scales the total quark spin contribution to the proton spin, , is only of order one half in the valence region.
As indicated in Table 1, all three of the quark spin contributions , and decrease in value as one increases the size of the meson-cloud contribution by increasing . As a result the net spin carried by the quarks, , diminishes with increasing meson-cloud contributions. This is in accord with the increased role of orbital angular momentum Thomas:2008ga () between the odd-parity mesons and the even-parity baryons of the proton’s meson cloud considered herein.
The parameter reflecting the role of relativistic and confinement effects and constrained by is around 0.82, smaller than 1 as expected but larger than the typical “ultra-relativistic” value of 0.65. Again, increasing the size of the meson-cloud contributions diminishes this value. For example, at GeV, .
The non-singlet charge lies between the value extracted from the hyperon -decays under the assumption of SU(3) symmetry, , and the value obtained in the cloudy bag model Bass:2009ed (). Because the experimental value of extracted from DIS data depends on this quantity, further work to pin down the extent of SU(3) breaking would be valuable.
The strange quark contribution to the proton spin is negative and its absolute value is of the order 0.01. Larger values admit stronger hyperon contributions which act to increase this magnitude.
The experimental value of the at 3 GeV is reproduced through a combination of the chiral correction and evolution of from a scale of 0.5 GeV Jaffe:1987sx (). We find (3 GeV) is which agrees with the experimental measurement of .
Future work should explore the role of higher order terms in the evolution of and explore non-perturbative treatments that can provide further insight into the connection between models of hadron structure and modern experimental results.
This work is supported in part by DFG and NSFC (CRC 110), by the National Natural Science Foundation of China (Grant No. 11475186) and by the Australian Research Council through grants DP140103067, DP150103164 (DBL), FL0992247 and DP151103101 (AWT) and through the ARC Centre of Excellence for Particle Physics at the Terascale.
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