Chiral expansion of nucleon PDF at x∼ m π / M N
aa r X i v : . [ h e p - ph ] N ov LU TP 13-39Nov. 2013
Chiral expansion of nucleon PDF at x ∼ m π /M N A.Moiseeva
Institut f¨ur Theoretische Physik II, RuhrUniversity Bochum, 44780 Bochum, Germany ∗ A.A.Vladimirov
Department of Astronomy and Theoretical Physics, Lund University,S¨olvegatan 14A, S 223 62 Lund, Sweden † Based on the chiral perturbation theory, we investigate the low-energy dynamics of nucleonparton distributions. We show that in different regions of the momentum fraction x thechiral expansion is significantly different. For nucleon parton distributions these regions arecharacterized by x ∼ x ∼ m π /M N and x ∼ ( m π /M N ) . We derive extended counting rulesfor each region and obtain model-independent results for the nucleon parton distributionsdown to x & m π /M N ≈ − . I. INTRODUCTION
The investigation of the pion contribution to nucleon parton distribution functions (PDFs)started already in the early 70’s [1]. Roughly speaking, one can single out two utmost approaches.The first approach is based on the convolution model and on the interpretation of the pion clouddistribution as the amplitude of the Sullivan process, for details see [2]. The second one is basedon the straightforward application of low-energy effective theories, such as chiral perturbationtheory (ChPT), to PDFs, see e.g. [3]. Both of these approaches have their own advantages anddisadvantages: the convolution model provides a simple and demonstrative interpretation, whereasthe effective theory approach is based on a systematical expansion. In spite of superficial similarityof the approaches some of the results for the meson cloud contributions are in contradiction, for arecent comparison see [4].The application of ChPT to pions shows itself to be very efficient and describes from the model-independent field theory-based point of view some well-known effects, such as, the increasingof the pion size in the chiral limit [5]. On the other hand the application of the low-energy ∗ Electronic address: [email protected] † Electronic address: [email protected]
FIG. 1: Diagrammatical illustration of a hard scattering process. The coefficient function C is computablewithin pertubative QCD, whereas the coefficient function W is computable within ChPT. The horizontallines represent the Mellin convolution and the black blob is the parton distribution in the chiral limit. effective theory to the non-local operators, appearing in the definition of parton distributions, isnot straightforward. This is caused by the necessity to resum higher order terms in the chiralexpansion [6]. Recently such an analysis has been performed for the nucleon case [7]. It has beenshown that in the nucleon case the situation is more distinct, i.e., there are several regions in x with significantly different structure of the chiral expansion, which has been earlier recognized inmodel considerations, see e.g.[8]. In the following we shortly explain the uses of the meson-nucleonChPT framework for the evaluation of nucleon parton distributions. The details and results of ourcalculation are/will be given in [7, 9, 10]. II. EFFECTIVE OPERATOR FOR NUCLEON PARTON DISTRIBUTIONS
Generalized parton distributions (GPDs) are very intricate and rich objects. To point outthe framework, we consider for simplicity only non-skewed GPDs (∆ + = 0) with nonzero ∆ -dependance, which reduce at ∆ = 0 to nucleon PDFs, where we restrict us to the isovectorcombination. Expressions for ∆ + = 0, as well as, for other flavor combinations can be found in[9]. These parton densities are defined via the nucleon matrix elements of the light-cone quarkoperators Z dλ π e − ixλP + h p | O ( λ ) | p ′ i (cid:12)(cid:12)(cid:12)(cid:12) ∆ + =0 = 1 P + ¯ u (cid:18) γ + q u − d ( x, ∆ ) + iσ + ν ∆ ν M E u − d ( x, ∆ ) (cid:19) u, (1)where O ( λ ) is the unpolarized (or vector) quark operator of twist-two, M is the nucleon mass, and∆ = p ′ − p .To calculate the matrix element (1) in an effective field theory, we have to find an effectiveoperator in terms of hadronic degrees of freedom that possesses the symmetry properties as theQCD operator O ( λ ). This procedure can be understood as some kind of low-energy factorization,which separates the very low-energy dynamics of large distance interactions of a hadron with itsmeson cloud (which is governed by the spontaneous chiral symmetry braking) from the unknowndynamics of the hadron core. Visual representation of such a double factorization is presentedin fig.1. The unknown part ˚ q ( x ), representing the hadron core dynamics, enters via a generatingfunction in the effective operator. Since the construction of the effective operator is only restrictedby the quantum numbers of the QCD operator, the amount of possible effective operators is infinite.However, their number can be restricted by choosing a proper counting hierarchy. Note that theeffective operator has indefinite twist since the effective degrees of freedom are of indefinite twistby themselves.In order to calculate the chiral corrections to parton distributions, one should first find thecounting rules for all dimensional quantities. In the meson-baryon ChPT one has the followingcounting rules ∂ µ π ∼ m, ∂ µ N ∼ M, (2)where the pion mass m ≪ M . The pion mass is the small parameter of the chiral expansion m πF π = a χ ≪
1, whereas, the nucleon mass violates the low-energy expansion: M πF π ∼
1. There areseveral methods to bypass the violation of the chiral expansion by nucleon mass, such as, the heavybaryon theory [11], the extended on-mass-shell (EOMS) scheme [12], and several others. However,for the consideration of a nonlocal operator one should go beyond the standard power counting rulesof meson-baryon ChPT. The reason is that the nonlocal operator has its own intrinsic dimensionalscale: the light-cone separation λ . The chiral counting for λ should be defined additionally.At all that, the situation is significantly different for pion and nucleon parton distributions. Theorigin of the difference is the counting rules for the derivatives of pion and nucleon fields (2). Letus demonstrate this fact explicitly. First of all, in order to apply the counting rules, we have toexpand the matrix element of the light-cone operator in the set of local operators, O ( λ ) = O (0) + λO (1) + λ O (2) + ... , (3)where O ( n ) ∼ ∂ n + O (0).Let us suppose now that the operator (3) contains only pion fields and that only a pion is presentin the in/out-state. Then the chiral expansion for every individual local operator O ( n ) starts from a nχ : h π | O π ( λ ) | π i = h q (0 , + a χ q (0 , + ... i + a λ h a χ q (1 , + a χ q (1 , + ... i + a λ h a χ q (2 , + ... i + ... , (4)where a λ = (4 πF π ) λ . One can see that if a λ ∼ a χ or a λ ∼ O ( a χ ). Such a picture corresponds to rather small light-cone separation, λ ∼ (4 πF π ) − . In the regime a λ ∼ a − χ , which implies λ ∼ m − , the series reorganizes: h π | O π ( λ ) | π i = h q (0 , + a χ a λ q (1 , + ( a χ a λ ) q (2 , + ... i + a χ h q (0 , + a χ a λ q (1 , + ... i + ... , (5)where all terms in the brackets are of the same order. For large λ , the higher order contributions ofthe expansion should be taken into account. However, in the definition of the parton distribution(1) all possible λ ’s contribute, except for very large λ at which the Fourier exponent starts tooscillate. The effective region of integration is 0 < λ . ( xp + ) − . Therefore, in order to obtain thecorrect chiral expansion in, say x ∼
1, one should take a λ ∼ a − χ at least (since p + ∼ m ). Thedetailed discussion can be found in [13]. For lower x the higher order terms should be taken intoaccount [6].For the nucleon operator, or for the pion operator in the nucleon brackets, the structure of thechiral expansion is different. The point is that there is a possibility to get the nucleon mass scalevia derivatives acting on the nucleon field, and therefore h N | O N ( λ ) | N i = h q (0 , + a χ q (0 , + ... i + a λ h q (1 , + a χ q (1 , + ... i + a λ h q (2 , + ... i + ... . (6)One can see that at a λ ∼ λ ∼ M − ) the expansion (6) contains infinitely many terms of O ( a χ ).Integration over the region 0 < λ . M − corresponds to x ∼
1. This regime has been consideredin [3, 16, 17] and in many other articles. At such small λ the operator is almost local, and theresult of calculation leads to the generally incorrect expression q ( x, ∆ ) = q ( x ) F (∆ ), where F isthe corresponding form factor. The first significant reorganization of the series (6) takes place at a λ ∼ a − χ . This allows us to obtain corrections to parton distributions down to x ∼ mM = α . Thenext significant reorganization takes place at a λ ∼ a − χ , which corresponds to x ∼ α . More detailsof the chiral expansion analysis for nucleon parton distributions are given in [7].It is very inconvenient to deal with the parameter λ in a straightforward manner, moreover, dueto the λ -integration we loose the guidance of the counting rules. In order to bypass these difficulties,we suggest to transfer the counting rules of λ to the light-cone vector n µ , which accompanies theparameter λ in the definition of the parton distribution. Also we suggest to work directly withnonlocal operator (i.e., without expansion in λ ). Note that one can rescale n µ without damagingthe operator properties. Thus, we assume that λ ∼
1, whereas n µ changes its counting dependingon x . In the x -region, interesting for us, the counting rule for n µ reads n µ ∼ m for x > m M = α . (7)The assumption λ ∼ x ∼ α for the vector and axial-vector cases in both theisovector and isoscalar sectors, which arise altogether from four different generating functions.Only two of them appear at the tree level and in the chiral limit they have the meaning of thecorresponding parton distributions. The remaining two functions appear at one loop level and theyhave no simple interpretation. The isovector vector operator is a typical representative, O a ( λ ) = Z − dβ ¯ N (cid:18) − βλ (cid:19) γ + h ˚ q ( β ) t a + + q ( β ) γ t a − + ˚ q ( β ) − q ( β )4 ˜ t a + (8)+ ∆˚ q ( β ) − q ( β )4 γ ˜ t a − i N (cid:18) βλ (cid:19) , where ˚ q ( β ) and ∆˚ q ( β ) are the isovector combinations of PDFs in the chiral limit, q ( β ) and q ( β )are additional generating functions. The pion field combinations are t a ± = 12 (cid:18) u † (cid:18) − βλ (cid:19) τ a u (cid:18) βλ (cid:19) ± u (cid:18) − βλ (cid:19) τ a u (cid:18) βλ (cid:19)(cid:19) , ˜ t a ± = 12 (cid:18) u † (cid:18) − βλ (cid:19) (cid:20) τ a , U (cid:18) βλ (cid:19)(cid:21) u † (cid:18) βλ (cid:19) ± u (cid:18) − βλ (cid:19) (cid:20) τ a , U † (cid:18) βλ (cid:19)(cid:21) u (cid:18) βλ (cid:19) + ... (cid:19) , where u = U = exp( iπ a τ a / F π ), and the dots denote the terms with commutators at the point( − βλ/ O aπ ( λ ) = − iF π Z − dβ ˚ Q ( β )Tr (cid:20) τ a (cid:18) U † (cid:18) − βλ (cid:19) ↔ ∂ + U (cid:18) βλ (cid:19) + U (cid:18) − βλ (cid:19) ↔ ∂ + U † (cid:18) βλ (cid:19)(cid:19)(cid:21) , (9)where ˚ Q ( x ) is the isovector pion PDF in the chiral limit. The complete expressions for the operators,and they normalizations can be found in [9, 10]. III. CHIRAL STRUCTURE OF THE NUCLEON
One of the main consequence of the different counting rules, applied to different regions in x ,is that the resulting expression for the leading chiral correction is non-linear in a χ rather thanlinear. The situation should be understood in the following way. On one hand, there is an “exact”(containing all orders of the perturbative expansion) expression q ( x ) = P ∞ n =0 a nχ q ( n ) ( x ). On theother hand, there is a truncated expression which contains, say, the leading term and the next-to-leading term: q t ( x ) = q (0) ( x )+ a χ q (1) ( x ). The difference between these two functions, q ( x ) − q t ( x ), is FIG. 2: Diagrams relevant for the leading chiral correction to the unpolarized (vector) nucleon partondistribution. The crossed circle denotes the pure pion operator (9), the crossed box denotes the pion-nucleonoperator (8). not necessarily O ( a χ ), but depending on x it can be O ( a χ ) or O ( a χ ). Resumming particular higherorder contributions, we obtain the expression which differs from the unknown “true” expression by O ( a χ ) contributions.Using the operators (8-9) and the counting rules (7), one can evaluate the leading nonanalyticalcontribution to the nucleon parton distributions. The analytical part cannot be evaluated thateasy because it contains a large set of new generating functions. The diagram representation of theleading contribution is shown in fig. 2. The nucleon parton distribution is conveniently presentedin the following form q ( x, ∆ ) = ˚ q ( x ) + M (4 πF π ) Z − dβ | β | θ (cid:18) < xβ < (cid:19) h ˚ q (cid:18) xβ (cid:19) C ( β, ∆ ) (10)+ ∆˚ q (cid:18) xβ (cid:19) − ˚ q (cid:16) xβ (cid:17) ∆ C ( β, ∆ ) + ˚ Q (cid:18) xβ (cid:19) C π ( β, ∆ ) i , where C ( x, ∆ ) = − (cid:18) g a (cid:19) δ (¯ x ) α ln α + g a Z ¯ x dη xα − ∆ M (¯ x − η (1 + x ))¯ x + α x − ∆ M η (¯ x − η ) , (11)∆ C ( x, ∆ ) = g a δ (¯ x ) α ln α + 2 g a ¯ x ln (cid:18) α x ¯ x (cid:19) , (12) C π ( x, ∆ ) = (1 − g a ) δ ( x ) Z dη (cid:18) α − η ¯ η ∆ M (cid:19) ln (cid:18) α − η ¯ η ∆ M (cid:19) (13) − g a x ln (cid:18) α ¯ xx (cid:19) + 4 g a Z ¯ x dη xα − xη ∆ M x + ¯ xα − ∆ M x (¯ x − η ) , with the axial-vector coupling constant g a ≈ . α = mM ≈ .
15, and ¯ x = 1 − x . The evaluationhas been done within the EOMS renormalization scheme. The structure of our result typicallyappear in a dispersive framework, see e.g. [14, 15], but never occurred in straightforward ChPTcalculations [3, 16].Indeed, one can see that in the region x ∼ α . The first terms of the expansion coincide with the results obtained by means of Mellin FIG. 3: Left panel: relative chiral corrections to the PDF (thick curve), without pion operator (dashedcurve), and only pion operator (thin curve). Right panel: The transverse size of the nucleon (thick curve)and only pion operator contribution (thin curve). moments, see e.g. [17]. The region x ∼ α is not described by the first terms of the α -expansion,although it is still of order O ( α ). In the region x ∼ α one cannot expand the functions C, ∆ C and C π in α , since they are altogether of order α . These results are in agreement with [4] and canbe compared in parts with dispersion analysis in [15].Having at hand the model-independent result for the low-energy parameter behavior of thenucleon parton distributions one can consider various interesting aspects parton dynamics, suchas, the role of the pion cloud in the nucleon, the size of the nucleon, the large-distance behaviorof the nucleon, and many others. A brief inspection of expressions (10) shows the significantdominance of the pion operator in the region x < α . This is also visualized in the left panel offig. 3, where we show the relative chiral corrections for a standard PDF parametrization (taken asPDF in the chiral limit ). One can see that the main contribution comes from the pion operatorand that it this unevenly grows in the region α . x . α . The behavior demonstrated in fig. 3 isuniversal, and holds for all quark and also antiquark PDFs. Recently, experimental data have beenconfronted with the description of the pion cloud model by means of the Sullivan process [18]. Themain disagreement of data and the model prediction takes place in the region x ∼ α . Possibly, thisdisagreement can be reduced in our approach.A more clear and model-independent result follows from the ∆ -dependance. The point is thatthe parton distribution in the chiral limit can be expressed via the standard PDF by solving theexpression (10) at ∆ = 0. Moreover, the transverse size of the nucleon, defined as h b ( x ) i = A more realistic treatment requires to take a PDF that is calculated in some model with massless quarks, e.g., inthe light-front formulation of QCD. d ln q ( x, ∆ ) /d ∆ at ∆ = 0, is insensible to the difference between q ( x ) and ˚ q ( x ) and also doesnot contain the unknown functions ˚ q ( x ) and ˚ q ( x ). Therefore, one can very precisely and in amodel-independent way obtain an expression for the transverse size of the nucleon for the region x & α . In the right panel of fig. 3 we show our result for the transverse size vs. x . Technicaldetails of our approach and qualitative estimates will be presented in [10]. Acknowledgements
A.M. is supported in part by DFG (SFB/TR 16, “Subnuclear Structure ofMatter”) and by the European Community-Research Infrastructure Integrating Activity “Study of StronglyInteracting Matter” (acronym HadronPhysics3, Grant Agreement n. 283286) under the Seventh FrameworkProgramme of EU. A.V. thanks the organize committee of Light-Cone 2013 for support and also A.V.is supported in part by the European Community-Research Infrastructure Integrating Activity Study ofStrongly Interacting Matter (HadronPhysics3, Grant Agreement No. 28 3286) and the Swedish ResearchCouncil grants 621-2011-5080 and 621-2010-3326.[1] S. D. Drell, D. J. Levy and T. -M. Yan, “A Theory of Deep Inelastic Lepton Nucleon Scattering andLepton Pair Annihilation Processes. 2. Deep Inelastic electron Scattering,” Phys. Rev. D (1970) 1035.[2] J. Speth and A. W. Thomas, “Mesonic contributions to the spin and flavor structure of the nucleon,”Adv. Nucl. Phys. (1997) 83.[3] D. Arndt and M. J. Savage, “Chiral corrections to matrix elements of twist-2 operators,” Nucl. Phys.A (2002) 429 [nucl-th/0105045].[4] M. Burkardt, K. S. Hendricks, C. -R. Ji, W. Melnitchouk and A. W. Thomas, “Pion momentumdistributions in the nucleon in chiral effective theory,’ Phys. Rev. D (2013) 056009 [arXiv:1211.5853[hep-ph]].[5] I. A. Perevalova, M. V. Polyakov, A. N. Vall and A. A. Vladimirov, arXiv:1105.4990 [hep-ph].[6] N. Kivel and M. V. Polyakov, “Breakdown of chiral expansion for parton distributions,” Phys. Lett. B (2008) 64 [arXiv:0707.2208 [hep-ph]].[7] A. M. Moiseeva and A. A. Vladimirov, “On chiral corrections to nucleon GPD,” Eur. Phys. J. A (2013) 23 [arXiv:1208.1714 [hep-ph]].[8] M. Strikman and C. Weiss, “Chiral dynamics and partonic structure at large transverse distances,”Phys. Rev. D (2009) 114029 [arXiv:0906.3267 [hep-ph]].[9] A.Moiseeva, “Nucleon parton distributions in Chiral Perturbation Theory”, PhD thesis, Bochum (2013).[10] A. M. Moiseeva, M. V. Polyakov, A. A. Vladimirov, in preparation.[11] E. E. Jenkins and A. V. Manohar, “Baryon chiral perturbation theory using a heavy fermion La-grangian,” Phys. Lett. B (1991) 558.[12] T. Fuchs, J. Gegelia, G. Japaridze and S. Scherer, “Renormalization of relativistic baryon chiral per- turbation theory and power counting,” Phys. Rev. D (2003) 056005 [hep-ph/0302117].[13] N. Kivel and M. V. Polyakov, “One loop chiral corrections to hard exclusive processes: 1. Pion case,”hep-ph/0203264.[14] M. Alberg and G. A. Miller, “Taming the Pion Cloud of the Nucleon,” Phys. Rev. Lett. (2012)172001 [arXiv:1201.4184 [nucl-th]].[15] C. Granados and C. Weiss, “Chiral dynamics and peripheral transverse densities,” arXiv:1308.1634[hep-ph].[16] J. -W. Chen and X. -d. Ji, “Is the Sullivan process compatible with QCD chiral dynamics?,” Phys.Lett. B (2001) 107 [hep-ph/0105197].[17] M. Diehl, A. Manashov and A. Schafer, Eur. Phys. J. A31