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Chiral-Odd and Spin-Dependent Quark Fragmentation Functions and their Applications


CHIRAL-ODD AND SPIN-DEPENDENT* QUARK FRAGMENTATION FUNCTIONS AND THEIR APPLICATIONS

Xiangdong Ji Center for Theoretical Physics Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts 02139

arXiv:hep-ph/9307235v1 8 Jul 1993

ABSTRACT We de?ne a number of quark fragmentation functions for spin-0, -1/2 and -1 hadrons, and classify them according to their twist, spin and chirality. As an example of their applications, we use them to analyze semi-inclusive deep-inelastic scattering on a transversely polarized nucleon.

Submitted to: Phys. Rev. D

CTP#2219

hep-ph/9307235

June 1993

* This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069.

I. INTRODUCTION In high-energy processes, the structure of hadrons is described by parton distributions, or in broader sense, parton correlations. In previous work1?5 , we have introduced and exploited a number of low-twist parton distributions, with some producing novel spin-dependent and chiral-?ip e?ects in hard scattering processes. These processes in turn allow us to gain access to these distributions experimentally and thereby help us to learn the non-perturbative QCD physics of hadrons. Among the distributions that we have discussed, the quark transversity distribution in the nucleon, which is de?ned by the following light-cone correlation,1,6 h1 (x) = 1 2 dλ iλx ? e P S⊥ |ψ(0) n γ5 S ⊥ ψ(λn)|P S⊥ , / / 2π (1)

is particularly interesting: it is one of the three distributions which characterize the state of quarks in the nucleon in the leading-order high-energy processes; its unweighted sum rule measures the tensor charge of the nucleon, which is identical to the axial charge in nonrelativistic quark models; it is a chiral-odd distribution (containing both left and right handed quark ?elds) so it does not appear in many well-known inclusive hard processes such as deep-inelastic scattering. Because of this last feature, we ?nd it cannot be easily measured experimentally. To see what criteria an underlying physical process has to meet in order to measure the transversity distribution, we consider the so-called “cut diagrams” for the cross section of the process, which are obtained by gluing together the Feynman diagrams for the amplitude and its complex conjugate. In a cut diagram, a quark ?owing out of a hadron will come back to it after a series of scattering. For h1 (x) to appear, the chirality of the quark must be ?ipped when it returns. This occurs if the quark goes through some soft processes during scattering, as shown in Fig. 1a. The only exception, a hard process which ?ips chirality, is a mass insertion, shown in Fig. 1b. For the light (u or d) quarks, the mass insertion is suppressed by m/ΛQCD and is ignorable. [Mass insertions might be signi?cant for heavy quarks but they are not the subject of this paper.] Chirality can be ?ipped in a parton distribution as in the Drell-Yan process shown in Fig. 1c, where the quark line goes through the interior of another hadron, or in a quark fragmentation process in hadron production shown in Fig. 1d, where the quark line goes through a fragmentation vertex). To measure the transversity distribution utilizing the second mechanism, we must clarify the structure of fragmentation vertices. The semi-inclusive hadron production from a quark fragmentation is described by fragmentation functions. As is shown in ref. 7, parton fragmentation functions in QCD are de?ned as matrix elements of quark and gluon ?eld operators at light-cone separations. Thus, their twist, spin, and chirality structures shall be as rich as parton distribution functions. In –1–

particular, there shall be a correspondent fragmentation function for each parton distribution function de?ned in ref. 2. As we shall show blow, there also exit additional fragmentation functions due to hadron ?nal state interactions. Despite their similarity, fragmentation functions are more di?cult to calculate than distribution functions. However, our purpose here is to de?ne them and to study the circumstances under which they contribute to scattering processes. This paper is organized as follows. In section II, we introduce complete twist-two and three and a part of twist-four quark fragmentation functions for production of spin-0, -1/2, and -1 hadrons. In section III, we study measurement of the nucleon’s transversity distribution in deep-inelastic scattering using the chiral-odd quark fragmentation functions de?ned in section II. We speci?cally consider three hadron-production processes: single-pion production, spin1/2 baryon production, and vector meson and two-pion production. The ?rst process uses polarized beam and target, and the double spin asymmetry vanishes in high-energy limit. The second process uses unpolarized lepton beam, however, requires measurement of the spin-polarization of the produced baryons. The third process is a single-spin process, which utilizes a fragmentation function arising from hadron ?nal state interactions. We conclude the paper in section IV. II. QUARK FRAGMENTATION FUNCTIONS Quark fragmentation functions were introduced by Feynman to describe hadron production from the underlying hard parton processes.8 In QCD, it is possible to obtain analytical formulas for these functions in terms of the matrix elements of the quark and gluon ?elds as for quark distributions in a hadron.7 In addition to the well-known spin-independent, chiral-even ? fragmentation function D(z) (we shall call it f1 (z)) widely discussed in literature, one can introduce various chiral-odd and spin-dependent fragmentation functions, which are capable of producing novel e?ects in lepton-hadron and hadron-hadron scattering.9 In this section, we de?ne fragmentation functions involving quark bilinears for production of spin-0, -1/2, and -1 hadrons. The discussion here can be easily generalized to gluons and more complicated fragmentation processes. 1. Fragmentation Functions For Spin-0 Meson Let us consider pion production, or equivalently, production of any hadron whose spin is not observed. In this case, generalizing the procedure in refs. 2 and 7, we can de?ne three fragmentation functions with quark ?elds alone, z dλ ?iλ/z ? ? ? e 0|γ ?ψ(0)|π(P )X π(P )X|ψ(λn)|0 = 4[f1 (z)p? + f4 (z)M 2 n? ], 2π –2– (2)

dλ ?iλ/z ? e 0|ψ(0)|π(P )X π(P )X|ψ(λn)|0 = 4M e1 (z), ? (3) 2π where P is the four-momentum of the pion and p and n are two light-like vectors such that p2 = n2 = 0, p? = n+ = 0, p· n = 1, and P = p+nm2 /2. All Dirac indices on quark ?elds are π implicitly contracted. [Our notation for fragmentation functions is analogous to the notation for distribution functions developed in refs. 2 and 5, the caret denotes fragmentation.] The mass M is a generic QCD mass scale, and we avoid use of the produced hadron mass because of the singular behavior introduced in the chiral limit (the left hand side of (3) does not vanish as mπ → 0). The summation over X is implicit and covers all possible states which can be populated by the quark fragmentation. The state |π(P )X is an incoming scattering state between π and X. The renormalization point (?2 ) dependence is suppressed in (2) and (3). QCD radiative corrections induce log?2 dependence in the fragmentation functions, which is compensated by the logQ2 /?2 -dependence of their coe?cients in expressions for observed cross sections. The resulting logQ2 dependence, or the Alteralli-Parisi evolution,10 is an important aspect of fragmentation processes which we put aside while we classify their spin and chirality properties. Here we work in n · A = 0 gauge, otherwise gauge links have to be added to ensure ? the color gauge invariance. From a simple dimensional analysis, we see that f1 (z), e1 (z), and ? ? (z) are twist-two, -three, and -four, respectively; and from their γ-matrix structure, f1 (z) ? f4 ? and f4 (z) are chiral-even and e1 (z) is chiral-odd. Hermiticity guarantees these fragmentation ? functions are real. The chiral-odd fragmentation function e1 (z) involves both “good” and “bad” components ? 1 ? ? ? of quark ?elds on the light cone (ψψ = ψ+ ψ? + ψ? ψ+ , where ψ± = P± ψ with P± = 2 γ ? γ ± ). Using the QCD equation of motion (neglecting the masses for light quarks), → 1 d / / (4) i ψ? (λn) = ? n i D ⊥ (λn)ψ+ (λn), dλ 2 α → →α → →α →α → ? where D ⊥ = D ? D · npα + D · pnα and i D (λn) = i ? ? gAα (λn). we rewrite e1 (z) in (3) as, → z2 dλ ?iλ/z ? e1 (z) = ? ? e 0| n i D⊥ (0)ψ+ (0)|π(P )X π(P )X|ψ+(λn)|0 / / 8M 2π (5) ← ? + 0|ψ+ (0)|π(P )X π(P )X|ψ+(λn) n i D⊥ (λn)|0 . / / z
α ←α ← where i D (λn) = i ? + gAα (λn). Thus the twist-three fragmentation explicitly involves three parton ?elds: two quark and one gluon. The appearance of eq. (5) motivates us to introduce a fragmentation density matrix, →α dλ d? ?iλ/z ?i?(1/z1 ?1/z) ? ?α e e 0|i D⊥ (?n)ψρ (0)|π(P )X π(P )X|ψσ(λn)|0 Mρσ (z, z1 ) = 2π 2π (6) ←α dλ d? iλ/z i?(1/z1 ?1/z) ?σ(0)i D (?n)|0 . + e e 0|ψρ (λn)|π(P )X π(P )X|ψ ⊥ 2π 2π

–3–

where α is restricted to transverse dimensions. It has the following expansion in the Dirac spin space, ? E(z, z1 ) ? + ... (7) M α (z, z1 ) = M γ α / p z ? where E(z, z1 ) is a real, chiral-odd fragmentation function involving two light-cone fractions ? and the ellipsis denotes higher-twist contributions. The function E(z, z1 ) can be isolated from ? ? ? M α through a projection: E(z, z1 ) = z/(8M )Tr n γα M α (z, z1 ). From eqs. (5)-(7), it is easy / to prove, e1 (z) = ?z ? 1 ? E(z, z1 )d . z1 (8)

? Therefore, e1 (z) is just a special moment of E(z, z1 ). As a consequence, a measurement of ? e1 (z) at one momentum scale is not su?cient to determine its value at other scales, because ? ? an Alteralli-Parisi type of evolution equation exists only for E(z, z1 ), not for a subset of its moments.11 This property of e1 (z) contrasts that of twist-two fragmentation functions, such ? ? as f1 (z). 2. Fragmentation Functions Arising From Hadron Final-State Interactions The quark fragmentations introduced above have a one-to-one correspondence with the quark distributions introduced for a spin-0 meson. In practice, one can de?ne one additional fragmentation function for the pion, z dλ ?iλ/z ? e 0|σ ?ν iγ5 ψ(0)|π(P )X π(P )X|ψ(λn)|0 = 4M ??ναβ pα nβ e? (z) ?1 2π (9)

If there were no ?nal state interactions between π and X, the state |π(P )X transforms as a free state under time-reversal symmetry and e? (z) vanishes identically. Thus the magnitude ?1 of e? (z) depends crucially on the e?ects of hadron ?nal state interactions. ?1 To illustrate that such fragmentation functions do exit, we consider production of an electron-positron pair from a virtual photon of mass 4m2 < q 2 < 16m2 in Quantum Electroe e dynamics. The production cross section is proportional to the vacuum tensor, W ?ν = 1 2π d4 xeiqx 0|J ? (x)|e+ (P )e? e+ (P )e? |J ν (0)|0 (10)

And this, according to Lorentz invariance, has the following decomposition in terms of Lorentz scalers, q? qν (11) W ?ν = (?g ?ν + 2 )W1 + ... + i(P ? q ν ? P ν q ? )W6 q If neglecting the ?nal state interactions between the electron and positron, one can prove immediately W6 = 0 due to time-reversal invariance. –4–

However, if taking into account one-photon exchange, one ?nds, W6 = C 1 ? 4m2 /q 2 θ(q 2 ? 4m2 ) e e (12)

where C is an unimportant numerical constant. The θ function indicates the ?nal state interaction vanishes if q 2 < 4m2 , in particular, if q 2 < 0, W ?ν is proportional to the photone electron scattering cross section, to which we know W6 does not contribute. The fragmentation function e? (z) is chiral-odd and twist-three. It is intimately related ?1 to e1 (z) introduced in the previous subsection. It is simple to show that it contributes to W6 ? type of terms in semi-inclusive production of hadrons in e+ e? annihilation. 3. Fragmentation Functions For Spin-1/2 Baryon Now we turn to consider the quark fragmentation for a spin-1/2 baryon. Eight more fragmentation functions can be introduced through bilinear quark ?elds besides these in eqs. (2), (3) and (9). They all depend on the polarization of the baryon: four of them are related to the longitudinal polarization and the other four to the transverse polarization, z dλ ?iλ/z ? e 0|γ ?γ5 ψ(0)|B(P S)X B(P S)X|ψ(λn)|0 2π
? ? M gT (z)S⊥ ?

(13)

= 4 g1 (z)p (S|| · n) + ? z

+ M g3 (z)(S|| · n)n ?

2

?

dλ ?iλ/z ? e 0|σ ?ν iγ5 ψ(0)|B(P S)X B(P S)X|ψ(λn)|0 2π ? ? = 4 h1 (z)(S ? pν ? S ν p? ) + hL (z)M (p? nν ? pν n? )(S|| · n)
⊥ ⊥

(14)

? ν ? + h3 (z)M 2 (S⊥ nν ? S⊥ n? ) + ... ,

z =4 z

dλ ?iλ/z ? e 0|γ ? ψ(0)|B(P S)X B(P S)X|ψ(λn)|0 2π
? gT (z)M T⊥ ??

(15)

+ ...

dλ ?iλ/z ? e 0|γ5 ψ(0)|B(P S)X B(P S)X|ψ(λn)|0 2π ? = 4M h ? (z)(S|| · n)
L

(16)

where dots denote terms already appeared in (2), (3) and (9), B(P S) represents the spin-1/2 ? baryon with the four-momentum P and polarization S (we write S ? = S ·np? +S ·pn? +MB S⊥ ? with the baryon mass MB ), and T ? = ??ναβ S⊥ν pα nβ is a transverse vector orthogonal to S⊥ . ? ? Again, through dimensional analysis, g1 (z) and h1 (z) are twist-two; gT (z), hL (z), gT (z) and ? ? ?? ? ? (z) are twist-three; and g3 (z) and h3 (z) are twist-four. The fragmentation functions g ? (z) ? hL ? ?T ?? and hL (z) vanish identically if without ?nal-state interactions. –5–

? As was the case for e1 (z), at the level of twist-three, gT (z) and hL (z) are not the most ? ? general fragmentation functions. Using (4), we derive, gT (z) = ? ? → dλ ?iλ/z z2 e 0|i D⊥ (0) · S⊥ n γ5 ψ+ (0)|B(P S⊥)X / 8M 2π ? × B(P S⊥ )X|ψ+(λn)|0 dλ ?iλ/z → z2 e 0| D⊥ (0) · T⊥ n ψ+ (0)|B(P S⊥ )X / + 8M 2π ? × B(P S⊥ )X|ψ+(λn)|0 + C.C.

(17)

→ dλ ?iλ/z z2 ? hL (z) = ? e 0|i D⊥ (0) n γ5 ψ+ (0)|B(P S|| )X / / 8M 2π ? × B(P S|| )X|ψ+ (λn)|0 + C.C.

(18)

where C.C. stands for complex conjugate. The generalization of eqs. (17) and (18) to two? light-cone-fraction distributions can be made by de?ning M α (z, z1 ) for the baryon just as for the pion in eq. (6). In addition, we need to de?ne a new fragmentation density matrix, →α dλ d? ?iλ/z ?i?(1/z1 ?1/z) e e 0|i D⊥ (?n)ψρ (0)|B(P S)X 2π 2π ? × B(P S)X|ψσ (λn)|0 ←α dλ d? iλ/z i?(1/z1 ?1/z) ? e e 0|ψρ (λn)|B(P S)X B(P S)X|ψσ(0)i D⊥ (?n)|0 . + 2π 2π (19) α ? which is the same as M except the minus sign for the ?rst term. Making expansion in the ?α Nρσ (z, z1 ) = ? spin space, we have,
α ? M α (z, z1 ) = M γ α / E(z, z1 )/z + iM T⊥ / G1 (z, z1 )/z + ... p ? p ? α ? ? N α (z, z1 ) = M S⊥ γ5 / G2 (z, z1 )/z + M γ α / γ5 H(z, z1 )/z + ... p ? p

(20)

The fragmentation functions can be projected from the density matrices: G1 = iz/(4M )Tr n / γ5 T⊥α M α , G2 = ?z/(4M )Tr n γ5 S⊥α N α , and H = ?z/(8M )Trγ⊥α n γ5 N α . It is easy to / / prove that gT (z) = ? ? z 2 d( 1 ? ? ) G1 (z, z1 ) + G2 (z, z1 ) , z1 d( 1 ? )H(z, z1 ). z1 (21) (22)

? hL (z) = ?z

These relations are useful for proving electromagnetic gauge invariance of scattering amplitudes, as an example shows in section III. –6–

4. Fragmentation Functions For Spin-1 Meson Finally, we consider quark fragmentation functions for vector meson production. To facilitate counting, let us de?ne the quark-meson forward scattering amplitudes, AhH;h′ H ′ , where h(h′ ) and H(H ′ ) are quark and meson helicities, respectively. The combination
1 1 A 2 1; 2 1 + A 1 ?1; 1 ?1 + A 1 0; 1 0 is independent of the meson polarization, from which we de2 2 2 2 ? ? ? ?ne four fragmentation functions f1 , e1 and e? , and f4 , depending on what components of ?1

quark ?elds form the amplitude: good-good (++), good-bad (+?), or bad-bad (??). Of course, they are what we have just de?ned in (2), (3) and (9). Similarly, the combination
1 A 1 1; 2 1 + A 1 ?1; 1 ?1 ? 2A 1 0; 1 0 depends on the LL type of tensor polarization of the meson 2 2 2 2 2 (see below for de?nition) and the corresponding four fragmentation functions are ?1 , ?2 and b b

?? , and ?3 ; the combination A 1 1 ? A 1 1 depends on the TT type of vector polarizab2 b 2 1; 2 1 2 ?1; 2 ?1 ?2 tion and the associated fragmentation functions are g1 , h2 and h? , and g3 ; the combination ? ? ?
1 A 1 0;? 2 1 ? A? 1 1; 1 0 is related to the LT type of vector polarization and the associated frag2 2 2 ? ? ? mentation functions are de?ned as h1 , g2 and ??? , and h3 ; and ?nally, the combination g2 1 1 1 A 2 0;? 2 1 + A? 2 1; 1 0 is related to the LT type of tensor polarization and the associate frag2 ? 1 ?2 ?3 mentation functions are de?ned as h? , g? , ??2 , and h? . The spin and twist structure of g

these twenty fragmentation functions are shown in Table 1, and the ones with bar on their subscripts arising from hadron ?nal state interactions.

Table I.

Quark fragmentation functions for vector mesons.

Note: the functions with bar vanish if there are no ?nal state interactions. Twist-2 ++
1 1 1 1 1 A 2 1→ 2 1 + A 2 ?1→ 2 ?1 + A 2 0→ 1 0 2

Twist-3 +?(S) e1 ? ?1 b ? h2 g2 ? g? ?2 +?(A) e? ?1 ?? b2 ?2 h? ??? g2 ??2 g

Twist-4 ?? ? f4 ?3 b g3 ? ? h3 ?3 h?

meson polarization S TLL VT T VLT TLT

? f1 ?1 b g1 ? ? h1 ?1 h?

1 1 1 1 A 2 1→ 2 1 + A 2 ?1→ 1 ?1 ? 2A 2 0→ 1 0 2 2

1 1 1 A 2 1→ 1 1 ? A 2 ?1→ 2 ?1 2

1 1 A 2 0→? 2 1 ? A? 1 1→ 1 0 2 2

1 1 A 2 0→? 2 1 + A? 1 1→ 1 0 2 2

–7–

Now we relate these fragmentation functions to the matrix elements of the bilinear quark operators. Since the meson polarization vector ?? appears in bilinear form in all the matrix elements, we introduce a rank-two tensor T ?ν = ?? ??ν . Its trace T ? = S, antisymmetric ? [?ν] ? ?ν ν ?? {?ν} ? ?ν part T = ? ? ? ? ? , and traceless-symmetric part T = ? ? + ?ν ??? ? (? · ?? )g ?ν /2 represent the scalar, vector, and tensor polarization of the meson. Together with p? and n? , they can be used to build various Lorentz structures to expand the quark matrix elements. The coe?cients of the expansion, depending on the polarization and dimension of the associated structures, can be uniquely identi?ed with the fragmentation functions in Table 1. To illustrate this, take the scalar polarization S, from which one can form one scalar S, two vectors Sp? and Sn? , and one tensor ??ναβ pα nβ S, and the coe?cients of these structure ? shall be e1 , f1 and f4 , and e? , respectively. For the case of tensor polarization, consider ? ? ?1 the projection of T {?ν} in longitudinal directions, T{αβ} pα nβ (= T )(n? pν + nν p? ), which characterizes the LL type of tensor polarization. With this one can construct one scalar T , two vectors T p? and T nν , and one tensor ??ναβ pα nβ T and their coe?cients are ?2 , ?1 and ?3 , b b b and ?? , respectively. Proceeding in this way, de?ne A = i?αβγδ ?α ?? pγ nδ to characterize the b2 β ? TT type of vector polarization, S⊥ = i??αβγ pα nβ T[γδ] nδ the LT type of vector polarization, ? and T⊥ = ??αβγ pα nβ T{γδ} nδ the LT type of tensor polarization, and construct all possible structures with them. The complete expansion of quark matrix elements reads, z dλ ?iλ/z ? e 0|γ ? ψ(0)|V (P ?)X V (P ?)X|ψ(λn)|0 2π [?ν] ? ? = 4 f1 (z)Sp? + f4 (z)M 2 Sn? + ??? (z)M iT g2 nν
⊥ {?ν}

(23)

+ ??2 (z)M T⊥ g z

nν + ?1 (z)T p? + ?4 (z)M 2 T n? , b b (24) (25)

dλ ?iλ/z ? e 0|ψ(0)|V (P ?)X V (P ?)X|ψ(λn)|0 = 4M S?1 (z) + T ?2 (z) , e b 2π dλ ?iλ/z ? z e 0|iγ5 ψ(0)|V (P ?)X V (P ?)X|ψ(λn)|0 = 4M Ah? (z), 2 2π dλ ?iλ/z ? e 0|γ ? γ5 ψ(0)|V (P ?)X V (P ?)X|ψ(λn)|0 z 2π = 4 g1 (z)Ap + ?
? ? M g2 (z)S⊥ ?

(26)

+

? M g? (z)T⊥ ?2

+ M g3 (z)An ?

2

?

,

z

dλ ?iλ/z ? e 0|σ ?ν iγ5 ψ(0)|V (P ?)X V (P ?)X|ψ(λn)|0 2π ? ? ν ν ? ?1 ? = 4 h1 (z)(S⊥ pν ? S⊥ p? ) + h? (z)(T⊥ pν ? T⊥ p? ) + h2 (z)M A(p? nν ? pν n? )

(27)

? ? ν ν ? ?3 + h3 (z)M 2 (S⊥ nν ? S⊥ n? ) + h? (z)M 2 (T⊥ nν ? T⊥ n? )

+ e? (z)M ??ναβ pα nβ S + b? (z)M ??ναβ pα nβ T . 1 2 Thus, all the twenty fragmentation functions in Table 1 have expressions in QCD. –8–

III. MEASURING THE TRANSVERSITY DISTRIBUTION FROM DEEP-INELASTIC SCATTERING As an example of applying the fragmentation functions de?ned in the last section, we consider measuring the nucleon’s transversity distribution h1 (x) through deep-inelastic scattering. Because h1 (x) is chiral-odd, it does not appear in inclusive deep-inelastic cross section if the current quark masses are neglected. However, h1 (x) does appear in semi-inclusive hadron production if one takes into account the e?ects of the chiral-odd fragmentation of the struck quark. For pseudo-scalar meson production, the leading chiral-odd quark fragmen? tation function is e1 (z), for spin-1/2 baryon production, it is h1 (z), and for vector meson ? ?1 production, it is h? (z) or its generalization to a fragmentation function for two pions. In the following, we discuss these cases separately. 1. Single-Pion Production We consider deep-inelastic scattering with longitudinal polarized lepton on transversely polarized nucleon target, focusing on pion production in the current fragmentation region. Since there is no chiral-odd twist-two fragmentation function for the pion to couple with the h1 (x) distribution in the nucleon, nor is there chiral-even twist-two transverse-spin-dependent ? distribution in the nucleon to couple with the fragmentation function f1 (z) for the pion, the spin-dependent cross section vanishes at the leading order in Q. At the twist-three level (the order of 1/Q), h1 (x) contributes through coupling with the chiral-odd fragmentation function e1 (z), and so does the chiral-even transverse-spin distribution gT (x) through the fragmen? ? tation function f1 (z). Both contributions exist in Fig. 2a. At the same order, we have to consider also Figs. 2b and 2c, in which one radiative gluon takes part in quark fragmentation, and Figs. 2d and 2e, in which one gluon from the nucleon participates in hard scattering. These processes, representing coherent parton scattering, introduce dependences on the twolight-cone-fraction parton distributions, G1,2 (x, x1 ), which are the parents of gT (x),2 and ? fragmentation function, E(z, z1 ), which is the parent of e1 (z). However, as we shall show be? low, they can be eliminated by using QCD equations of motion, and the ?nal result contains only e1 (z) and gT (x). ? Let us ?rst consider the contribution from the diagram in Fig. 2a. Using the de?nition of the nucleon tensor, W?ν = we obtain from this diagram,
a W?ν =

1 4π

eiq·ξ d4 ξ P S|J? (ξ)Jν (0)|P S ,

(28)

1 4π

d4 k (2π)4

d 4 Pπ 2 ? 2πδ(Pπ ? m2 ) Tr MN (P, S⊥ , k)γ? Mπ (k + q, Pπ )γν π (2π)4 –9–

(29)

where mπ is the pion mass, MN (P, S⊥ , k)αβ = ? d4 ξeiξ·k P S⊥ |ψβ (0)ψα (ξ)|P S⊥ (30)

is quark’s spin-density matrix for the nucleon and ? Mπ (k, Pπ )αβ =
X

? d4 ξe?iξ·k 0|ψα (0)|π(Pπ )X π(Pπ )X|ψβ (ξ)|0

(31)

is the quark fragmentation density matrix for the pion. Here q is the four-momentum of the virtual photon, and P and S⊥ are the nucleon’s four-momentum and polarization vectors,
2 respectively. We choose our coordinate system such that P = p + nMN /2, S⊥ = (0, 1, 0, 0),

and q = ?xB p + νn, where p and n are two light-like vectors de?ned in the last section, MN is the mass of the nucleon, and xB is the Bjorken scaling variable xB = Q2 /(2ν). To perform the k integration in eq. (29), we make a collinear expansion of quark momentum k along p in the fragmentation density matrix, ? ? Mπ (k · np + q, Pπ ) ? ? Mπ (k + q, Pπ ) = Mπ (k · np + q, Pπ ) + (k ? k · np)α + ... ?k α (32)

We temporarily ignore the derivative term, whose contribution will be combined with those from Figs. 2d and 2e to form a gauge-invariant result. The contribution to the k integration from the leading term in eq.(32) is,
a W?ν =

1 4π

d 4 Pπ 2 2πδ(Pπ ? m2 ) π (2π)4

? dxTr MN (xp, S⊥ )γ? Mπ (xp + q, Pπ )γν ,

(33)

where the simpli?ed quark spin-density matrix is MN (xp, S⊥ )αβ = dλ iλx ? e P S⊥ |ψβ (0)ψα (λn)|P S⊥ , 2π (34)

the structure of which has been studied thoroughly in ref. 1. To integrate out the transverse components of Pπ in eq. (33), we make a coordinate transformation to a new system in which Pπ and q have only longitudinal components. If we label momenta in the new system with prime, then, to the order of our interest,
′? ? ′+ ′i Pπ = Pπ , Pπ = 0, Pπ = 0 ? q ′? = Pπ /z, q ′+ = q + , q ′i = 0 i xp′? = 0, xp′+ = xp+ , xp′i = ?Pπ /z

(35)

–10–

In the new system, p′ has non-vanishing transverse components and as a consequence, the spin and fragmentation density matrices in eq. (33) are now linked through transverse-momentum integrations. To decouple them, we Taylor-expand the spin-density matrix, MN (xp′ , S⊥ ) = MN (xp, S⊥ ) ?
i Pπ ?MN (xp, S⊥ ) + ... z ?xpi

(36)

Here we have ignored the transverse components of n′ , whose e?ects are beyond twist-three. The contribution from the derivative term in eq. (36) will be combined with those from Figs. 2b and 2c to form a color gauge-invariance expression, as is shown in eq. (43). And the leading term contribution is,
a W?ν =

1 4π

dz

? dx2πδ((xp + q)2 )Tr MN (xp, S⊥ )γ? Mπ (z, pπ /z)γν

(37)

where the simpli?ed fragmentation density matrix is ? Mπ (z, pπ /z)αβ =
X

dλ ?iλ/z ? e 0|ψα (0)|π(pπ )X π(pπ )X|ψβ (λnπ )|0 2π

(38)

Here we have neglected the pion mass and used two additional light-like vectors pπ and nπ with pπ = zνn and pπ · nπ = 1. Since spin asymmetry is our main interest, we take the transverse-spin-dependent part of the spin-density matrix from ref. 5, MN (x, p, S⊥ ) = 1 1 h1 (x)γ5 S ⊥ / + gT (x)M γ5 S ⊥ +... / p / 2 2 (39)

From eqs. (2) and (3), we have the fragmentation density, ? f1 (z) e1 (z) ? ? + /π p + ... Mπ (z) = M z z Substituting eqs. (39) and (40) into eq. (37) and simplifying the latter, we have,
a W?ν =

(40)

M 2ν +

e2 ha (xB ) a 1
a a e2 gT (xB ) a a

ea? 1 (z) dz i??ναβ pα S⊥β z ?a f1 (z) dz i??ναβ S⊥α pπβ . z

(41)

where the summation runs over di?erent quark ?avors and their charge conjugation, and ea is electric charge of quarks. As it stands, eq. (41) does not satisfy electromagnetic gauge invariance, i.e., W?ν q ? = 0. –11–

We turn to consider the contribution from Fig. 2b, which involves an additional transversely-polarized gluon. After the collinear expansion and coordinate transformation discussed above, we ?nd,
b W?ν =

1 4π

1 )2πδ((q + xp)2 ) z1 i(x / ?(1/z ? 1/z1 ) /π ) p p ? γ M α (z, z1 )γν × Tr MN (xp, S⊥ )iγα 2 ? 1 (xp ? (1/z ? 1/z1 )pπ ) dxdzd(

(42)

where the fragmentation density matrix is, ?α M1 (z, z1 )ρσ = dλ d? ?iλ/z ?i?(1/z1 ?1/z) e e 2π 2π →α ? × 0|i D⊥ (?n)ψρ (0)|π(Pπ )X π(Pπ )X|ψσ (λn)|0

(43)

→α The partial derivative in D⊥ comes from the collinear expansion for Fig. 2a as explained after eq. (36). Because the state |π(Pπ )X is an incoming scattering state which changes to ?α an outgoing scattering state after time reversal, M1 (z, z1 ) do not have a simple hermitian ?α conjugation property. As a consequence, if we make an expansion, M1 = M γ α / E1 (z, z1 )+..., p ? E1 (z, z1 ) is not a real quantity. However, its imaginary part, which we are going to ignore, contributes only to single-spin asymmetry. Its real part is just E(z, z1 )/2, which was de?ned in the last section. Inserting the expansion into eq. (42) and using eq. (8) to eliminate ? E(z, z1 ), we ?nd,
b W?ν =

M 2ν

e2 ha (x) a 1
a

dz?a (z) e

1 1 p? i?ναβγ S⊥α pβ pπγ . π 2 p·p xB z π

(44)

which is just one of the terms required to make W?ν gauge invariant. The contribution of Fig. 2c can be calculated in the same way and the result is complex conjugate of eq. (41) with ?, ν indices interchanged. Combining the h1 (x) term in eq. (41), and eq. (44) and its conjugate, we have the chiral-odd part of the spin-dependent nucleon tensor,
a+b+c W[?ν] =

M 2ν

e2 ha (xB ) a 1
a

dz

ea (z) ?ναβ ?1 i? pα S⊥β z (45)

1 1 1 1 + ip? ?ναβγ S⊥α pβ pπγ ? ipν ??αβγ S⊥α pβ pπγ π zxB p · pπ zxB p · pπ π ea (z) ? M ha (xB ) e2 1 dz 1 = ? i??ναβ qα S⊥β a 2ν a xB z which is explicitly gauge invariant. –12–

Now we consider the contributions from Figs. 2d and 2e. The calculations here parallel these for Figs. 2b and 2c, and the ?nal result for the chiral even part of the nucleon tensor, including the contribution from Fig. 2a, is,
a+d+e W[?ν] = a f1 (z) ?ναβ i? S⊥α pπβ z a xB z ν ?αβγ xB z ? ναβγ ip ? S⊥α pβ pπγ ? ip ? S⊥α pβ pπγ + p · pπ p · pπ M ?a e2 g a (xB ) dz f1 (z) = ? i??ναβ qα S⊥β 2ν a a T

M 2ν

a e2 gT (xB ) a

dz

(46)

Adding eqs. (45) and (46) to the longitudinal-polarization contribution, which is considerably easy to calculate, we have the complete spin-dependent nucleon tensor, W ?ν = ?i??ναβ qα ? ? [(S · n)pβ G1 (x, z) + S⊥β GT (x, z)] ν (47)

The two structure functions are de?ned as, 1 ? G1 (x, z) = 2 1 ? GT (x, z) = 2
a ?a e2 g1 (x)f1 (z) a a

e2 a
a

a ?a gT (x)f1 (z)

ha (x) ea (z) ? + 1 x z

(48)

To isolate the spin-dependent cross section we take the di?erence of cross sections with left-handed and right-hand leptons, α2 E′ d2 ?σ = em ???ν W?ν dE ′ d? Q4 EMN (49)

where Q2 = ?q 2 , k = (E, k) and k ′ = (E ′ , k′ ) are the incident and outgoing momenta of ′ the lepton, and ???ν is the spin-dependent part of the lepton tensor, ???ν = ?1/2Tr[γ ? k / γ ν γ5 k ] = ?2i??ναβ qα kβ . It is convenient to express the cross section in terms of scaling / variables in a frame where the lepton beam de?nes the z-axis and the x ? z plane contains the nucleon polarization vector, which has a polar angle α. In this system, the scattered lepton has polar angles (θ, φ) and therefore the momentum transfer q has polar angles (θ, π ? φ). De?ning a conventional dimensionless variable y = 1 ? E ′ /E, we can write the cross section as 4α2 y ? d4 ?σ em = cos α(1 ? )G1 (x, z) dxdydzdφ Q2 2 y ? ? + cos φ sin α (κ ? 1)(1 ? y) GT (x, z) ? G1 (x, z)(1 ? ) 2 –13–

(50)

where κ = 1 + 4x2 M 2 /Q2 in the second term signals the suppression by a factor of 1/Q ? ? associated with the structure function GT . The existence of of G1 in the same term is due to a small longitudinal polarization of the nucleon when its spin is perpendicular to the lepton beam. Equation (50) is one of our main results. As a check, we multiply by z, integrate over it and sum over all hadron species. Using the well-known momentum sum rule, ?a dzz f1 (z) = 1
hadrons

(51)

and the sum rule, dz?a (z) = 0 e1
hadrons

(52)

which is related to the fact that the chiral condensate vanishes in the perturbative QCD vacuum, we get, 4α2 y d3 ?σ em = cos α(1 ? )g1 (x) dxdydφ Q2 2 y + cos φ sin α (κ ? 1)(1 ? y) gT (x) ? g1 (x)(1 ? ) 2 where g1 (x) =
1 2 2 a a ea g1 (x)

(53)

and gT (x) = g1 (x) + g2 (x) =

1 2

2 a a a ea (g1 (x) + g2 (x))

are the two

conventional spin structure functions. The above result coincides with the same quantity in ref. 12 if one neglects the terms of order 1/Q2 in the latter. The parallelism between the inclusive and semi-inclusive cross sections suggests that the both quantities can be extract from the same set of experiment. In using eq. (50) to analyze experimental data, a lower cut on z must be made to ensure the detected particles emerging from the current fragmentation region. To enhance statistics one can integrate z over a region. By varying φ, we can separate out the following combinations of structure functions, G1 dz = 1 2
a a e2 g1 (x)Nπ a a a a e2 [gT (x)Nπ a a

1 GT dz = 2
a where Nπ =

ha (x) a + 1 Eπ ] x

(54)

a a dzf1 (z) is the pion multiplicity of the quark jet with ?avor a and Eπ =

dzea (z)/z. –14–

2. Spin-1/2 Baryon Production In this subsection we study deep-inelastic scattering of unpolarized lepton beam on transversely polarized nucleon target, focusing on spin-1/2 baryon production from quark fragmentation. The spin e?ects in the scattering can be unravelled through measuring the polarization of the produced baryon. This can be done for an unstable hyperon by measuring angular distribution of its decay product. The process was ?rst studied in ref. 13. Here we include a formula for the spin-dependent cross section in the lab frame. The process can be described as in Fig. 2(a), except the produced pion is replaced here by a spin-1/2 baryon. From eq. (14), we ?nd the spin-dependent piece of the fragmentation density matrix, ? h1 (z) ? γ5 S B⊥ /B +... / p MB (z) = z Thus the spin-dependent nucleon tensor is, W ?ν = ? + 1 2ν e2 ha (x) a 1
a

(55)

where pB = zνn and SB are the momentum and polarization of the baryon, respectively.

? ha (z) ? ν 1 ν ? (S⊥ SB⊥ + S⊥ SB⊥ )p · pB z


(56)

(p? pν B

+

p? pν B

?g

p · pB )S⊥ · SB⊥


Contracting it with the unpolarized lepton tensor, ??ν = 1 Tr[γ? k γν k ], we have, / / 2 ??ν W?ν = ? 4 ν ? e2 ha (x)ha (z) S⊥ · kSB⊥ · kp · pB + k · pk · pB S⊥ · SB⊥ a 1 1
a

(57)

Using the lab coordinate system de?ned in the last subsection to simply (57), we ?nd, ??ν W?ν = ?4Q2 1?y 1 cos(φ + φ′ ) y 2 ? e2 ha (x)ha (z) a 1 1
a

(58)

where φ′ is the azimuthal angle between k′ and SB . This produces the following spindependent cross section, α2 1 ? y 1 d?σ = ?4 em cos(φ + φ′ ) dxdydzdφ Q2 y 2 ? e2 ha (x)ha (z) a 1 1
a

(59)

This expression reaches maximum if S and SB are the mirror imagine of each other with respect to the scattering plane. –15–

3. Vector-Meson and Two-Pion Production Here we consider the same set up for deep-inelastic scattering as in the last subsection, but focusing on vector meson, e.g. ρ, and two-pion production production. Our analysis shows that one can de?ne a single-spin asymmetry sensitive to the nucleon’s transversity distribution at the leading order in Q, however, its magnitude depends also on the unknown ?nal-state interactions between the detected particle(s) and spectators. Similar ideas have also been proposed in refs. 14 and 15. Let us ?rst look at vector meson production. According to our previous discussion on quark fragmentation for a vector meson, there are two twist-two chiral-odd fragmentation ? ?1 functions h1 (z) and h? (z). The former describes the probability of producing vector mesons in vector polarization and the latter in tensor polarization. If one can measure the vector ? polarization, h1 (z) is an ideal choice for coupling with the transversity distribution. However, for the interesting case of ρ meson production, the only way to measure polarization is through ?1 its two-pion decay, which registers only tensor polarization. Thus, it appears that h? (z) is the only choice for coupling with the transversity distribution. However, the size of this fragmentation function depends on unknown ?nal-state interactions. If one is to measure asymmetry associated with inclusive production of two pions, there are other underlying processes which contribute besides the ρ decay, for instance, the interference production of two pions in their relative s and p waves. The contribution depends on the di?erence of the phase shifts. To include all the contributions, we directly introduce quark fragmentation functions for two-pion production, ? M (k, P2π , l) = d4 ξ ?ik·ξ ? e 0|ψ(0)|π(l1)π(l2 )X π(l1)π(l2 )X|ψ(ξ)|0 (2π)4 (60)

where l1 and l2 are momenta of two observed pion and P2π = l1 + l2 and l = (l1 ? l2 )/2 are the total and relative momenta, respectively. The contribution of two-pion fragmentation to the nucleon tensor is, d4 P2π d4 l 2 ? 2πδ(l·P2π /2)2πδ(4l2 +P2π )dxTr MN (xp, S⊥ )γ? M (xp+q, P2π , l)γν (2π)4 (2π)4 (61) where we have neglected the pion mass and made collinear expansion for the intial quark momentum. To proceed further, we make a restriction on the l integrations such that |l2 | < M 2 , where M is a soft scale on the order of ΛQCD . Making a collinear expansion for P2π and neglecting higher-twist contributions, we have, W?ν = 1 4π W?ν = 1 4π dx2πδ((xp + q)2 ) d4 l ? 2πδ(l · p2π /2)dxTr MN (xp, S⊥ )γ? M (p2π /z, l)γν (2π)4 (62) –16–

where p2π = zνn and the fragmentation density simpli?es to, ? M (p2π /z, l) = dλ ?iλ/z ? e 0|ψ(0)|2π(p2π , l)X 2π(p2π , l)X|ψ(λn2π )|0 2π (63)

with n2π = p/(zν). For our purpose, we make the following expansion for the density, ? ? H1 (z, l) F1 (z, l) ? M (p2π /z, l) = γ5 S 2π⊥ /2π + / p /2π +... p z z (64)

α ? where S2π⊥ = ?αβγδ p2πβ n2πγ lδ /|l| and S2π⊥ · p2π = 0. The fragmentation function H1 (z, l) is real according to hermiticity and non-vanishing because of the ?nal-state interactions between π’s and X. Substituting eq. (64) into eq. (62), we have,

W ?ν = ?

1 2ν

e2 ha (x) a 1
a

? H a (z, l) d4 l ? ν ν ? (S⊥ S2π⊥ + S⊥ S2π⊥ )p · p2π 2πδ(l · p2π /2) 1 4 (2π) z

(65)

+ (p? pν + p? pν ? g ?ν p · p2π )S⊥ · S2π⊥ . 2π 2π For this, we can calculated the spin-dependent part of the cross section, d?σ 4α2 1 ? y 1 em = dxdydzdφ Q2 y 2 e2 ha (x) a 1
a

d4 l ?a 2πδ(l · p2π /2)H1 (z, l) sin(φ + φ? ), 2 (2π)

(66)

where φ? is the azimuthal angle between k′ and l. IV. CONCLUSION In this paper, we de?ne a number of low-twist quark fragmentation functions by analyzing the matrix elements of quark bilinears in light-cone separations and expanding them in terms of various Lorentz structures. Some of these fragmentation functions are chiral-odd and polarization-dependent, which are not only interesting phenomenologically, but also useful for describing non-perturbative fragmentation processes. In the examples of using the fragmentation functions, we study measurement of the nucleon’s transversity distribution in deep-inelastic scattering, where chirality conservation selects the class with odd chirality. Fragmentation functions and parton distributions are always coupled in cross sections, thus one can study both in experiments by varying x and z simultaneously. The facilities at CERN, HERA, and SLAC are particularly useful for learning these non-perturbative hadron observables. ACKNOWLEDGMENT I thank Bob Ja?e for his collaboration on parton fragmentation functions and J. Collins for several useful conversations. –17–

REFERENCES 1. R. L. Ja?e and X. Ji, Phy. Rev. Lett. 67 (1991) 552. 2. R. L. Ja?e and X. Ji, Nucl. Phys. B375 (1992) 527. 3. X. Ji, Phys. Lett. B284 (1992) 137. 4. X. Ji, Phys. Lett. B289 (1992) 137. 5. X. Ji, MIT CTP preprint No. 2141, 1992, to be appeared in Nucl. Phys. B. 6. J. P. Ralston and D. E. Soper, Nucl. Phys. B152 (1979) 109. 7. J. C. Collins and D. Soper, Nucl. Phys. B194 (1982) 445. 8. R. P. Feynman, Photon-Hadron Interactions, Benjamin, Reading, MA. 1972. 9. R. L. Ja?e and X. Ji, MIT CTP Preprint No. 2158, 1993. 10. G. Alteralli and G. Parisi, Nucl. Phys. B126 (1977) 278. 11. X. Ji and C. Chou, Phys. Rev. D42 (1990) 3637. 12. R. L. Ja?e, Comm. Nucl. Part. Phys. 14 (1990) 239. 13. X. Artru and M. Mekh?, Z. Phys. C45 (1990) 669. 14. A. V. Efremov, L. Mankiewicz, and N. A. Tornqvist, Phys. Lett. B284 (1992) 394. 15. J. Collins, Nucl. Phys. B396 (1993) 161. FIGURE CAPTIONS [For hard copies of the ?gures for this paper, send email to ereidell@marie.mit.edu.] Fig. 1: Processes in which a quark changes its chirality: a) a generic soft QCD process, b) mass insertion, c) the Drell-Yan scattering, and d) the quark fragmentation. Fig. 2: The twist-two and twist-three cut diagrams for single-pion production in deep-inelastic scattering.

–18–


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