SStrangeness in the Meson Cloud Model
A. I. Signal
Institute of Fundamental Sciences PN461Massey UniversityPalmerston North 4442New Zealand
Abstract.
I review progress in calculating strange quark and antiquark distributions of the nucleon using themeson cloud model. This progress parallels that of the meson cloud model, which is now a usefultheoretical basis for understanding symmetry breaking in nucleon parton distribution functions. Iexamine the breaking of symmetries involving strange quarks and antiquarks, including quark -antiquark symmetry in the sea, SU(3) flavour symmetry and SU(6) spin-flavour symmetry.
Keywords:
Strange and antistrange quark distributions, meson cloud model, nucleon
PACS:
BEGINNINGS - QUARK-ANTIQUARK ASYMMETRY
Tony Thomas and I met when I came to Adelaide as a new PhD student in early 1985.We agreed that I would work in the area of deep inelastic scattering, and in the first yearTony gave me a number of projects to work on. One of these was to try to extend hiswork from 1983 on the role of the non-perturbative pion cloud of the nucleon in DIS [1]to include kaons. We soon realized that the strangeness carrying components of the cloudwould have different characteristics to the non-strange components. This is because allthe ¯ s antiquarks in the cloud come from the kaon, whereas all the s quarks come from thehyperons. So immediately we saw the possibility that quark and antiquark could havedifferent momentum distributions in the cloud. This was one of the first calculations totake into account the contributions to nucleon quark distribution functions coming frombaryons in the cloud via the Sullivan process [2], see fig. 1.We were able to show that the contributions to the quark and antiquark distributionsare given by convolutions between the distribution functions of quarks or antiquarks in FIGURE 1.
Non-perturbative contributions to the strange sea of the nucleon. (a) The incoming photonis absorbed by a virtual kaon. (b) The in coming photon is absorbed by a virtual hyperon a r X i v : . [ nu c l - t h ] A p r he hyperon or kaon with the momentum distribution, or fluctuation function, of thesehadrons in the cloud: x δ ¯ s ( x ) = (cid:90) x dy f K ( y ) (cid:18) xy (cid:19) ¯ s K (cid:18) xy (cid:19) , (1) x δ s ( x ) = (cid:90) x dy f H ( y ) (cid:18) xy (cid:19) s H (cid:18) xy (cid:19) . (2)Using covariant perturbation theory we found [3] that the meson cloud contributionto the antistrange distribution is softer than the contribution to the strange distribution.This arises mainly because we used a ¯ s K distribution in the kaon that was fairly soft, andthe fluctuation function f K ( y ) = f H ( − y ) (3)is also softer for the kaon than the hyperons.As this was the first attempt to calculate strange contributions from the meson cloud,there were a number of shortcomings with this paper. The first was the use of thecovariant formulation of perturbation theory in the calculation. Unfortunately, using thisformulation required us to make ansatze for the structure functions of the struck, off-shell, hadrons. We chose these to be the same as on-shell structure functions, which isnot correct [4]. A better formulation to use for the meson cloud model is time orderedperturbation theory in the infinite momentum frame, as shown by Wally Melnitchoukand Tony Thomas in an important paper for the development of the model [5]. Asimilar approach was also used by Zoller [6]. Using the time ordered approach has theadvantages that the struck hadrons remain on-mass-shell, so avoiding any ambiguitiesand allowing us to use experimental input to the structure functions. Also the momentumdistributions in the cloud can be shown to satisfy the relation (3) exactly, rather thanthis being imposed by fiat. In the infinite momentum frame, diagrams where the struckhadron is moving backwards in time are suppressed by powers of the longitudinalmomentum, and do not contribute as the limit p L → ∞ is taken.We also had to make educated guesses for the strange and antistrange distributions inhyperons and kaons respectively. For the kaon we used an experimental determinationof the pion structure function [7] which is fairly soft, whereas for the hyperons we useda simple valence distribution of the form s H ( x ) = N s x − / ( − x ) . There was also no Q dependence of our input or output distributions.The question of a possible quark - antiquark asymmetry in the strange sea receivednew interest in the early 2000’s as a result of the interesting experimental result fromthe NuTeV collaboration [8]. NuTeV measured NC to CC ratios in deep-inelastic ν ( ¯ ν ) - nucleon scattering. This enabled them to determine the effective couplings to left andright-handed quarks ( g L and g R ) and, via the Paschos - Wolfenstein (PW) ratio, R PW = σ ν NC − σ ¯ ν NC σ ν CC − σ ¯ ν CC = g L − g R = − sin θ W , (4)the value of the weak mixing anglesin θ W = . ± . ( stat ) ± . ( syst ) , (5)hich is 2% smaller than the world average value, or a 3 σ discrepancy. However, thePW ratio receives corrections from both charge symmetry breaking in the nucleon partondistributions (which Tim Londergan and Tony Thomas have investigated in detail [9]),and quark - antiquark symmetry breaking in the sea: R PW = − sin θ W + b + b (cid:104) x ( u V + d V ) (cid:105) / (cid:20) −(cid:104) x ( s − ¯ s ) (cid:105) + ( (cid:104) x δ u V (cid:105) − (cid:104) x δ d V ) (cid:105) (cid:21) (6)where δ u V = u pV − d nV ; δ d V = d pV − u nV (7)are the charge symmetry breaking valence distributions and b = ∆ u = g L u − g R u ; b = ∆ d = g L d − g R d . (8)At the NuTeV scale ( Q =
16 GeV ) the coefficient in front of the square bracketsof eqn. (6) is about 1.3, so a symmetry breaking term inside the square brackets of − . θ W . We note that the CTEQ group has analyzed the uncertainties aroundthe experimental results for strange and anti-strange distributions in some detail [10].They place bounds on the second moment of the quark - antiquark asymmetry − . < (cid:104) x ( s − ¯ s ) (cid:105) < . . (9)This provided impetus to revisit our calculation of the asymmetry. Now we do thecalculation using time ordered perturbation theory, with on-shell structure functions.For the strange distribution in the hyperons we now use a bag model calculation [11],evolved using next-to-leading order QCD evolution to Q =
16 GeV . The valence ¯ s ( x ) distribution in the kaon is taken from the parameterization of the Dortmund group [12].We also note that the form factors cutting off the NKH vertex are fairly soft ( Λ c ∼ K ∗ meson Fock states. This can have a significant effect on the calculations, as the couplingconstants for K ∗ NH are fairly large [13]. Also the fluctuation functions for N → K ∗ H peak close to y = .
5, meaning that the final convolutions to obtain the contributions to s and ¯ s reflect the underlying hardness or softness of the valence quark distribution in thehadron. However, we realize that we are pushing the bounds of the cloud model here, asit is not clear that K ∗ H final states would have a clear rapidity gap.We find that the fluctuation functions for kaons are softer than for hyperons, whereasthe s quark distributions in Λ and Σ hyperons are now softer than the ¯ s distributionin the K and K ∗ . This means that once the quark distributions are convoluted with thefluctuation functions, there is only a small s − ¯ s difference, see fig. 2. The second momentof the asymmetry has a magnitude around 10 − , and positive (negative) sign without(with) K ∗ states included. As this is significantly smaller than the size of effect neededto move the NuTeV result into agreement with the world data, we conclude that thestrange sea asymmetry is probably not responsible for the NuTeV anomaly. IGURE 2.
The strange sea asymmetry calculated in the meson cloud model. The solid and dashedcurves are the results without and with the K ∗ contributions respectively. FIGURE 3.
Comparison of MCM calculations for x ( ∆ s + ∆ ¯ s ) with the HERMES data at Q = . . POLARISED STRANGE SEA
The calculational techniques outlined in the above section can be generalized to thepolarized quark distributions ∆ s ( x ) and ∆ ¯ s ( x ) without too much difficulty. Polarizedquark distributions have been of interest for over 20 years, since the EMC collaborationmeasured a very small fraction of the nucleon spin being carried by quarks [14]. Thisis usually interpreted in the context of SU(3) flavour and implies that the strange sea isstrongly polarized opposite to the proton ∆ S (cid:39) − .
15 [15]. It has been pointed out thata natural consequence of the meson cloud model is that the cloud is capable of carryinga significant proportion of the proton’s angular momentum [16].The HERMES collaboration have carried out an extensive programme of flavour anal-ysis of their polarized DIS data [17, 18], which shows that the polarized sea quark distri-butions are fairly small. Our calculations in the MCM, which include the contributionsfrom K ∗ states, are consistent with HERMES data, see fig. 3. x FIGURE 4.
The sum of the strange and antistrange quark distributions from the MCM calculations (thethick solid curve), the HERMES measurements (the data points) and the global fit results from CTEQ6.6M(the thick dashed curve), MSTW2008 (the dash curve) and CTEQ6.5 (the shaded area), and the next-to-leading order analysis of NuTeV dimuon data (the thin solid curve).
SU(3) FLAVOUR SYMMETRY BREAKING
The unpolarized strange sea is less well constrained by experimental data than thelight ( ¯ u , ¯ d ) sea. For instance the CTEQ6.5 pdf set [10] has a very large variance in theparameters describing the s and ¯ s distributions ( ±
50% in some instances). The recentHERMES data [18] on the strange sea highlights this problem, as it does not agree wellwith the NuTeV determination [19] - though we note that the HERMES analysis oftheir data is only to leading order in QCD, whereas the NuTeV analysis goes to next-to-leading order.In the MCM, we can estimate the strange sea via the SU(3) flavour breaking asym-metry ∆ ( x ) = ¯ u ( x ) + ¯ d ( x ) − s ( x ) − ¯ s ( x ) (10)which has leading contributions in the cloud coming from the differences between e. g. | N π (cid:105) − | Λ K (cid:105) Fock states. On the other hand, there are no leading contributions to ∆ ( x ) in perturbative QCD (and next-to-leading contributions can also be expected to besmall). Having calculated ∆ ( x ) in the MCM, we can subtract the light sea distributions,which are experimentally well constrained, and estimate the total strange sea. Our resultsare shown in fig. 4, and are generally consistent with the HERMES data. We haveagain included K ∗ states in the calculation of ∆ ( x ) , but they do not dominate the finalresults, and removing them has about a 10% effect on our total s ( x ) + ¯ s ( x ) . We notethat our calculation becomes negative at x ≈ .
25, which is unphysical. This could bedue to either the MCM calculation overestimating ∆ ( x ) or the CTEQ6.6 pdf set [20]underestimating the light sea [ ¯ u ( x ) + ¯ d ( x )] or both.In conclusion, the meson cloud model remains an excellent non-perturbative labo-ratory for exploring and understanding symmetry breaking among the nucleon partondistribution functions. There are still important questions around the polarized and un-polarized strange sea distributions, andthe model can help to provide solutions to these. CKNOWLEDGMENTS
I am happy to acknowledge the contributions to my understanding of the meson cloudmodel that have come from many colleagues and friends. Firstly to Tony Thomas,who introduced me to this problem, guided me through my PhD studies, and has beenincredibly generous with his time and support over 25 years. Also I have learnt a greatdeal from other Adelaide students especially Andreas Schreiber, Wally Melnitchouk andFernanda Steffans. My colleagues at Massey University, Fu-Guang Cao and FrancoisBissey, have provided many insights, and I am grateful to them for many years ofenjoyable collaboration.
REFERENCES
1. A. W. Thomas,
Phys. Lett. B , , 97 (1983); M. Ericson and A.W. Thomas, Phys. Lett. B , , 122(1983).2. J. D. Sullivan, Phys. Rev. D , , 1732 (1972).3. A. I. Signal and A. W. Thomas, Phys. Lett. B , , 205 (1987).4. W. Melnitchouk, A. W. Schreiber and A. W. Thomas, Phys. Rev. D , , 1183 (1994).5. W. Melnitchouk and A. W. Thomas, Phys. Rev. D , , 3794 (1993).6. V. Zoller, Z. Phys. C , , 443 (1992).7. J. Badier et al., Z. Phys. C , , 291(1983).8. G. P. Zeller et al. (NuTeV collaboration), Phys. Rev. Lett. , , 091802 (2002).9. J. T. Londergan and A. W. Thomas, Phys. Rev. D , , 111901(R) (2003).10. H. L. Lai, et. al. (CTEQ collaboration), J. High Energy Phys. , , 089 (2007).11. C. Boros and A. W. Thomas, Phys. Rev. D , , 074017 (1999); F. G. Cao and A. I. Signal, Phys. Lett.B , , 138 (2000); F. G. Cao and A. I. Signal, Phys. Lett. B , , 229 (2003).12. M. Glück, E. Reya and I. Schienbein, Eur. Phys. J. C , , 313 (1999).13. H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A , , 631 (1996).14. J. Ashman et al. (EMC collaboration), Phys. Lett. B , , 364 (1988).15. S. J. Brodsky, J. Ellis and M. Karliner, Phys. Lett. B , , 309 (1988); S. D. Bass, The Spin Structureof the Proton , World Scientific, Singapore, 2007.16. J. Speth and A. W. Thomas,
Adv. Nucl. Phys. , , 83 (1997); F. Bissey, F. G. Cao and A. I. Signal, Phys. Rev. D , , 094008 (2006).17. A. Airapetian et al. (HERMES collaboration), Phys. Rev. Lett. , , 012005 (2004).18. A. Airapetian et al. (HERMES collaboration), Phys. Lett. B , , 446 (2008).19. D. Mason et al. (NuTeV collaboration), Phys. Rev. Lett. , , 192001 (2007).20. P. M. Nadolsky, et. al. (CTEQ collaboration), Phys. Rev. D ,78