Quark fragmentation in the θ -vacuum
aa r X i v : . [ h e p - ph ] J un Quark fragmentation in the θ -vacuum Zhong-Bo Kang and Dmitri E. Kharzeev RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: October 27, 2018)The vacuum of Quantum Chromodynamics is a superposition of degenerate states with differenttopological numbers that are connected by tunneling (the θ -vacuum). The tunneling events aredue to topologically non-trivial configurations of gauge fields (e.g. the instantons) that induce local P -odd domains in Minkowski space-time. We study the quark fragmentation in this topologicallynon-trivial QCD background. We find that even though QCD globally conserves P and CP symme-tries, two new kinds of P -odd fragmentation functions emerge. They generate interesting dihadroncorrelations: one is the azimuthal angle correlation ∼ cos( φ + φ ) usually referred to as the Collinseffect, and the other is the P -odd correlation ∼ sin( φ + φ ) that vanishes in the cross sectionsummed over many events, but survives on the event-by-event basis. Using the chiral quark modelwe estimate the magnitude of these new fragmentation functions. We study their experimentalmanifestations in dihadron production in e + e − collisions, and comment on the applicability of ourapproach in deep-inelastic scattering, proton-proton and heavy ion collisions.
1. Introduction.
Quantum Chromodynamics (QCD) isat present firmly established as the theory of the stronginteractions. Equations of motion in QCD possess topo-logically non-trivial solutions [1] signaling the presence ofdegenerate ground states differing by the value of topo-logical charge [2]. The physical vacuum state of the the-ory is a superposition of these degenerate states, so-called θ -vacuum [3]. To reflect this vacuum structure one mayequivalently introduce a θ -term in the QCD Lagrangian.Unless θ is identically equal to zero, this term explicitlybreaks P and CP symmetries of QCD. However stringentlimits on the value of θ < × − deduced from theexperimental bounds on the electric dipole moment ofthe neutron [4] indicate the absence of global P and CP violation in QCD.Nevertheless it has been proposed that the local P - and CP -odd effects due to the topological fluctuations char-acterized by an effective θ = θ ( ~x, t ) varying in space andtime could be directly observed through multi-particlecorrelations [5]. In heavy ion collisions, the existence ofmagnetic field (and/or the angular momentum) in thepresence of topological fluctuations can induce the sep-aration of electric charge with respect to the reactionplane, so-called Chiral Magnetic Effect [6–9]. There is arecent experimental evidence for this effect from STARCollaboration at RHIC [10]. The interpretation of STARresult in terms of the local parity violation is under in-tense scrutiny at present, see e.g. [11–13].In this paper, we study the role of QCD topology inhard processes using the formalism based on factorizationtheorems [14]. From the QCD factorization point of view,the cross section in high energy collision can be factor-ized into a convolution of perturbatively calculable par-tonic cross section and the non-perturbative but univer-sal parton distribution and fragmentation functions. Inthe conventional formalism, these distribution and frag-mentation functions are required to be P -even because of the parity-conserving nature of the strong interaction.However, in the presence of local (in space and time) P -odd domains P -odd fragmentation functions can emerge[15]; note that only the cross section of the entire processhas to be P -even, not the fragmentation function.In this letter, we derive the most general form of thequark fragmentation function for a quark fragmentinginto a pseudoscalar meson which is consistent with theLorentz invariance. Abandoning the parity constraint,we obtain two P -odd fragmentation functions besides thewell-known P -even spin-averaged fragmentation function[16] and Collins function [17]. We obtain the exact op-erator definitions and estimate the size of these new P -odd fragmentation functions using the chiral quark model[18]. As a first step, we present their observable effect inthe back-to-back dihadron production in e + e − collisions.We encourage the experimentalists to carry out the re-lated analyses at RHIC and elsewhere.
2. Quark fragmentation functions in locally P -oddbackground. The quark fragmentation functions are de-fined through the following matrix [19]:∆ ( z, p ⊥ ) = 1 z Z dy − d y ⊥ (2 π ) e ik · y h |L y ψ ( y ) | P X ih P X | ¯ ψ (0) L † | i| y + =0 , (1)where p is the momentum of the final state hadron witha transverse momentum p ⊥ relative to the fragment-ing quark k . We choose the hadron moving along +ˆ z direction, and define the light-cone momentum p ± =( p ± p z ) / √
2. For convenience, we define two light-like vectors: ¯ n µ = δ µ + and n µ = δ µ − . The momen-tum fraction z = p + /k + , and ~k ⊥ = − ~p ⊥ /z . L y = P exp (cid:0) ig R ∞ dλn · A ( y + λn ) (cid:1) is the gauge link neededto make ∆ ( z, p ⊥ ) gauge invariant.Since QCD is a theory conserving C , P , and T globally,one usually expands the above matrix using the followingconstraints [20]:Hermiticity : ∆ † ( p, k ) = γ ∆( p, k ) γ (2)Parity : ∆( p, k ) = γ ∆(¯ p, ¯ k ) γ (3)Time − reversal : ∆ ∗ ( p, k ) = V T ∆(¯ p, ¯ k ) V − T (4)where V T = iγ γ and ¯ p µ = p µ = ( p , − ~p ). Using thebasis of gamma matrices Γ = { , γ µ , γ µ γ , σ µν , iγ } , andthe available momenta p and k , one can expand ∆( p, k )in the most general form:∆( p, k ) = (cid:20) M A A /p + A /k + A σ µν k µ p ν M (cid:21) + (cid:20) A /pγ + A /kγ + M A iγ + A σ µν iγ k µ p ν M (cid:21) , (5)where M is the hadron mass used to make all A i ’s havethe same dimension. Since the time-reversal changes out-state to in-state, it does not really give any constrainton the coefficients A i [20]. One the other hand, if oneapplies the Hermiticity constraint, all of the A i ’s haveto be real. If one further applies Parity constraint, onefinds A = A = A = A = 0. However, as we statedin the Introduction, we are interested in the situation inwhich a local P -odd domain develops in space-time, andthe quark fragmentation happens inside such a P -odddomain (or in other words, the quark scatters off the non-trivial gauge field configuration prior to transforming intoa pseudoscalar meson). In this case, the P -odd modesin the quark fragmentation could be populated and onehas to release the parity constraint in Eq. (3). Withoutparity constraint, we thus need to keep all 8 terms A through A in Eq. (5). Applying the twist-expansionby parametrizing the momenta as p µ ≈ p + ¯ n µ and k µ ≈ ( p + ¯ n µ − p µ ⊥ ) /z and keeping the leading terms we obtain∆( z, p ⊥ ) = 12 h D ( z, p ⊥ ) / ¯ n + H ⊥ ( z, p ⊥ ) σ µν p ⊥ µ ¯ n ν M i + 12 (cid:20) e D ( z, p ⊥ ) / ¯ nγ + e H ⊥ ( z, p ⊥ ) σ µν iγ p ⊥ µ ¯ n ν M i . (6)where D ( z, p ⊥ ) and H ⊥ ( z, p ⊥ ) are the usual P -evenfragmentation functions: D ( z, p ⊥ ) is the transverse mo-mentum dependent spin-averaged fragmentation function[16], and H ⊥ ( z, p ⊥ ) is the Collins function describinga transversely polarized quark fragmenting into an un-polarized hadron [17]. Now besides the two conven-tional P -even fragmentation functions, we also obtaintwo new P -odd fragmentation functions: e D ( z, p ⊥ ) and e H ⊥ ( z, p ⊥ ). As we will show below, e H ⊥ ( z, p ⊥ ) generatesa new kind of azimuthal correlation. Its role is simi-lar to H ⊥ ( z, p ⊥ ): H ⊥ ( z, p ⊥ ) represents an asymmetric distribution ∝ (ˆ p × p ⊥ ) · ~s q , while e H ⊥ ( z, p ⊥ ) representsan asymmetric distribution ∝ p ⊥ · ~s q for a transverselypolarized quark with spin vector ~s q to fragment into apseudoscalar meson. The newly derived P -odd fragmen-tation functions will lead to interesting P -odd effects inexperiment as we will show in the next section.In order to study the experimental effects generatedby these P -odd fragmentation functions, we need to es-timate their magnitude. For this purpose we use theeffective chiral quark model developed by Manohar andGeorgi [18], which is an effective theory of QCD at lowenergy scale. This model has also been adopted for anestimate of the Collins functions in [21, 22]. The effec-tive Lagrangian describing the interaction between thequarks and the pion in the leading order is given by L qq Π = − g A f π ¯ ψ q γ µ γ ~τ · ∂ µ ~πψ q (7)where f π ≈
93 MeV is the pseudoscalar decay constant. kp FIG. 1: Lowest-order Feynman diagram for a quark with mo-mentum k fragmenting into a π meson with momentum p . At tree level, the fragmentation of a quark is modeledthrough the process q ∗ → πq , see Fig. 1. One can obtainthe unpolarized quark fragmentation function D ( z, p ⊥ )from the definition in Eq. (6) which has been done in [21].The Collins function can be calculated similarly, thoughone needs to go beyond the tree diagram and considerthe π -loop to obtain the final result, see Ref. [21].Since the chiral quark model Lagarangian in Eq. (7)conserves parity, it does not generate P -odd fragmenta-tion functions e D ( z, p ⊥ ) and e H ⊥ ( z, p ⊥ ). As we statedin the Introduction, QCD contains topological gaugefield configurations, and their effect can be mimicked byan effective space-time dependent θ field [5, 6]. Onecan thus add to the Lagrangian of QCD the term( g / π ) θ ( x, t ) F µνa ˜ F aµν ; performing an axial U (1) rota-tion this term can be transformed into N f ∂ µ θ ¯ ψ q γ µ γ ψ q [9]. Let us define an effective ¯ θ µ = ∂ µ θ/ N f , whose zero(time) component is the chiral chemical potential µ in-troduced in [9]. The existence of this new term will yielda modified quark propagator i e S ( p, ¯ θ ) = i/ ( /p − m + / ¯ θγ )given by i ˜ S ( p, ¯ θ ) = i (cid:2) P R S ( p + ¯ θ ) + P L S ( p − ¯ θ ) (cid:3) × (cid:2) mγ (cid:0) S ( p + ¯ θ ) − S ( p − ¯ θ ) (cid:1)(cid:3) × " m ¯ θ (cid:0) ( p + ¯ θ ) − m (cid:1) (cid:0) ( p − ¯ θ ) − m (cid:1) − (8)where P L , R are the left (right) projection operator P L , R = (1 ± γ ) /
2, and iS ( p ) = i ( /p + m ) / ( p − m )is the conventional quark propagator.With the modified quark propagator in the P -odd”bubble”, both the P -odd fragmentation functions can begenerated at the tree level as shown in Fig. 1. We furtherfind that the new fragmentation functions are suppressedby a factor ¯ θ , /p + when ¯ θ is along time or z -direction.Since p + is a large component in our twist-expansion, wethus neglect the effect from the 0 , θ tobe self-consistent. On the other hand, if the ¯ θ is alongthe transverse direction that is perpendicular to p ⊥ , wefind that the P -odd fragmentation functions vanish. Wethus only consider the situation when ¯ θ is along the p ⊥ direction: ¯ θ µ = ¯ θ ⊥ ˆ p µ ⊥ , in which case we find e D ( z, p ⊥ ) = g A f π π z θ ⊥ p ⊥ p ⊥ + z m q + (1 − z ) m π (cid:20) − z − − z ) z m q m π (cid:0) p ⊥ + z m q + (1 − z ) m π (cid:1) , (9) e H ⊥ ( z, p ⊥ ) = g A f π m q m π π ¯ θ ⊥ p ⊥ (cid:0) p ⊥ + z m q + (1 − z ) m π (cid:1) × (cid:20) (cid:0) p ⊥ + z m q (cid:1) ( z −
2) + (1 − z ) m π × (cid:2) (3 z − m π − p ⊥ − z m q ) (cid:3) (cid:21) , (10)where we have neglected terms ∼ O (¯ θ ). From aboveequations, we see that both of the P -odd fragmentationfunctions are proportional to ¯ θ ⊥ /p ⊥ . Since p ⊥ is a smallcomponent, the effect is not suppressed. We will nowestimate the size of the observable effect generated by the P -odd fragmentation functions within the same model.
3. Observable effect of parity-odd fragmentation func-tions.
Let us now discuss the experimental consequencesof the P -odd fragmentation functions. As a first step, westudy a relatively simple process, the back-to-back di-hadron production in e + e − collisions e + e − → h h + X .The method we presented here can be generalized tostudy P -odd effects in heavy ion collisions.At leading order in QCD coupling, the two hadrons h and h in e + e − collisions are the fragmentation productsof a quark and an antiquark originating from e + e − → q ¯ q annihilation. Following Ref. [23], we choose a referenceframe such that the e + e − → q ¯ q annihilation occurs in the x - z plane, with the back-to-back quark and antiquarkmoving along the z -axis. The final hadrons h and h carry light-cone momentum fractions z and z and haveintrinsic transverse momenta p ⊥ and p ⊥ with respectto the directions of the fragmenting quarks. Using thefragmentation parameterization in Eq. (6), one can derive the differential cross section as dσd PS = σ X q e q (cid:26) (1 + cos θ ) × h D q ( z ) D ¯ q ( z ) − e D q ( z ) e D ¯ q ( z ) i + sin θ cos( φ + φ ) × h H ⊥ q ( z ) H ⊥ ¯ q ( z ) + e H ⊥ q ( z ) e H ⊥ ¯ q ( z ) i + sin θ sin( φ + φ ) × h H ⊥ q ( z ) e H ⊥ ¯ q ( z ) − e H ⊥ q ( z ) H ⊥ ¯ q ( z ) i (cid:27) , (11)where the phase space d PS = dz dz d cos θd ( φ + φ ), σ = N c α em / Q , and θ is the angle between the ini-tial beam direction and the z -axis, not to be confusedwith the θ ( x ) field. In Eq. (11), we have integrated overthe moduli of the intrinsic momenta, p ⊥ and p ⊥ , andover the azimuthal angle φ . The p ⊥ -integrated functions D q ( z ) and H ⊥ q ( z ) are defined as D q ( z ) = Z d p ⊥ D q ( z, p ⊥ ) , (12) H ⊥ q ( z ) = Z d p ⊥ | ~p ⊥ | M H ⊥ q ( z, p ⊥ ) . (13)The definition of e D q ( z ) (or e H ⊥ ¯ q ( z )) is similar to D q ( z )(or H ⊥ q ( z )).The cos( φ + φ ) correlation is usually referred to asthe Collins effect, analyzed recently by BELLE Collab-oration [24]. However, we find that the product of two P -odd fragmentation functions e H ⊥ q ( z ) leads to the sameazimuthal correlation. This complicates the extractionof the Collins function, but may in effect provide an al-ternative view of the origin of the Collins effect and putsan experimental constraint of the P -odd fragmentationfunction e H ⊥ q ( z ). It is interesting that a new azimuthalcorrelation also emerges: the sin( φ + φ ) term, whichis explicitly P -odd. Notice that for the sin( φ + φ )contribution, the first term corresponds to the situationwhen the antiquark fragments inside the P -odd bubble,see Fig. 2(a), whereas the second term corresponds tothe situation when the quark fragments inside the P -oddbubble, see Fig. 2(b). They have the opposite sign, andthus when averaged over the large number of events, theeffect will vanish. Thus a P -odd effect happens only onthe event-by-event basis [8]. FIG. 2: Illustration of the antiquark (a) and the quark (b)fragmenting in the P -odd bubble. To estimate the effect, let us assume that the anti-quark fragments inside the P -odd bubble; the relativemagnitude of the correlation will depend on the fol-lowing factor I (¯ θ, z , z ), besides the kinematic factorsin θ/ (1 + cos θ ), I (¯ θ, z , z ) = H ⊥ q ( z ) e H ⊥ ¯ q ( z ) D q ( z ) D ¯ q ( z ) − e D q ( z ) e D ¯ q ( z ) . (14)Certainly I (¯ θ, z , z ) depends on the size of ¯ θ ⊥ . To esti-mate ¯ θ ⊥ , we resort to the instanton vacuum model (fora review, see [25]). According to [25], the two most im-portant parameters are the mean size of the instanton ρ ∼ / R between instan-tons which satisfies ρ/R ∼ /
3. We thus estimate ¯ θ ⊥ asfollows:¯ θ ⊥ ∼ N f ∂ ⊥ θ ( ~x, t ) ∼ N f · ρ · ρ R ∼
10 MeV , (15)where we have used N f = 3, and the factor ρ /R repre-sents the probability for a quark scatters off an instanton;note that in Minkowski space-time the instanton event iselongated along the light cone [15]. With ¯ θ ⊥ = 10 MeV,and other standard parameters of the chiral quark model[18], and using the calculation of the fragmentation func-tions taken from Ref. [21], we find I (¯ θ, z , z ) ∼ .
5% fora typical z = z = 0 . π + and π − . We urge the experi-mentalists at BELLE, RHIC and elsewhere to carry outan analysis to constrain the P -odd fragmentation func-tions. Because of the universality of the fragmentationfunctions, we expect that the formalism developed herecould be generalized to other processes.
4. Conclusion.
In this letter we have studied the quarkfragmentation in the topologically non-trivial QCD back-ground. We have found two new fragmentation func-tions besides the well-known spin-averaged fragmenta-tion function and the Collins function. Both of the newfragmentation functions are P -odd. We have related themagnitude of these functions to the typical size of thetopological fluctuations (described by the effective θ ( x )field). We have studied the observable effects of the P -odd fragmentation functions in back-to-back dihadronproduction in e + e − collisions, and have found that a newazimuthal correlation ∝ sin( φ + φ ) appears. Since thenew azimuthal correlation is explicitly P -odd, it can beobserved only on an event-by-event basis. Our resultsalso offer a new interpretation of the Collins correlation.We encourage the experimentalists to carry out an anal-ysis to constrain the P -odd fragmentation functions, andanticipate new applications, e.g. in heavy ion collisions. We thank J. Liao, R. Millo, M. Grosse Perdekamp,J. Qiu, E. Shuryak, S. Taneja, A. Vossen and F. Yuan forhelpful discussions and useful comments. We are grate-ful to RIKEN, Brookhaven National Laboratory, and theU.S. Department of Energy (Contract No. DE-AC02-98CH10886) for supporting this work. [1] A. A. Belavin, A. M. Polyakov, A. S. Schwartz andYu. S. Tyupkin, Phys. Lett. B (1975) 85.[2] S. S. Chern and J. Simons, Annals Math. (1974) 48.[3] G. ’t Hooft, Phys. Rev. Lett. , 8 (1976); Phys. Rev.D , 3432 (1976) [Erratum-ibid. D , 2199 (1978)];R. Jackiw and C. Rebbi, Phys. Rev. Lett. , 172 (1976);C. G. Callan, R. F. Dashen and D. J. Gross, Phys. Lett.B , 334 (1976).[4] C. A. Baker et al. , Phys. Rev. Lett. , 131801 (2006).[5] D. Kharzeev, R. D. Pisarski and M. H. G. Tytgat,Phys. Rev. Lett. , 512 (1998); arXiv:hep-ph/9808366;arXiv:hep-ph/0012012.[6] D. Kharzeev, Phys. Lett. B , 260 (2006); AnnalsPhys. , 205 (2010).[7] D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A , 67(2007).[8] D. E. Kharzeev, L. D. McLerran and H. J. Warringa,Nucl. Phys. A , 227 (2008).[9] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys.Rev. D , 074033 (2008).[10] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett. , 251601 (2009); arXiv:0909.1717 [nucl-ex].[11] A. Bzdak, V. Koch and J. Liao, Phys. Rev. C , 031901(2010); arXiv:1005.5380 [nucl-th].[12] F. Wang, arXiv:0911.1482 [nucl-ex].[13] M. Asakawa, A. Majumder and B. Muller,arXiv:1003.2436 [hep-ph].[14] J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser.Direct. High Energy Phys. , 1 (1988).[15] A. Efremov and D. Kharzeev, Phys. Lett. B , 311(1996).[16] J. C. Collins and D. E. Soper, Nucl. Phys. B , 445(1982).[17] J. C. Collins, Nucl. Phys. B , 161 (1993).[18] A. Manohar and H. Georgi, Nucl. Phys. B , 189(1984).[19] See e.g. A. Bacchetta et al. , JHEP , 093 (2007).[20] See e.g. D. Boer, P. J. Mulders and F. Pijlman, Nucl.Phys. B , 201 (2003).[21] A. Bacchetta et al. , Phys. Rev. D , 094021 (2002).[22] A. Bacchetta, A. Metz and J. J. Yang, Phys. Lett. B ,225 (2003); D. Amrath et al. , Phys. Rev. D , 114018(2005).[23] M. Anselmino et al. , Phys. Rev. D , 054032 (2007).[24] R. Seidl et al. [Belle Collaboration], Phys. Rev. Lett. ,232002 (2006); Phys. Rev. D , 032011 (2008).[25] T. Schafer and E. V. Shuryak, Rev. Mod. Phys.70