Twist-3 Gluon Fragmentation Contribution to Polarized Hyperon Production in Unpolarized Proton-Proton Collision

Understanding the origin and mechanism of the transverse polarization of hyperons produced in unpolarized proton-proton collision, $pp\to \Lambda^\uparrow X$, has been one of the long-standing issues in high-energy spin physics. In the framework of the collinear factorization applicable to large-$p_T$ hadron productions, this phenomenon is a twist-3 observable which is caused by multi-parton correlations either in the initial protons or in the process of fragmentation into the hyperon. We derive the twist-3 gluon fragmentation function (FF) contribution to this process in the leading order (LO) with respect to the QCD coupling constant. Combined with the known results for the contribution from the twist-3 distribution function and the twist-3 quark FF, this completes the LO twist-3 cross section. We also found that the model independent relations among the twist-3 gluon FFs based on the QCD equation of motion and the Lorentz invariance property of the correlation functions guarantee the color gauge invariance and the frame-independence of the cross section.

For a high-energy collision in which particles with large transverse momentum are produced, the cross section can be computed in the framework of the collinear factorization of perturbative QCD.In this framework, SSAs appear as twist-3 observables to which nonperturbative multiparton correlation functions contribute instead of collinear twist-2 parton distribution functions (PDFs) and/or fragmentation functions (FFs).Through the studies of SSAs, the technique of calculating the twist-3 cross section has made much progress and has been applied to many relevant processes in the leading order (LO) with respect to the QCD coupling.For example, the complete LO cross section for p ↑ p → hX (h = π, D, γ, γ * ) has been derived [27][28][29][30][31][32][33][34][35][36][37][38][39][40], and the RHIC data has been analyzed and interpreted, which suggests the main source of the asymmetry is the twist-3 fragmentation contribution [41,42].
In this paper we study pp → Λ ↑ X in the collinear twist-3 factorization.Two kinds of twist-3 cross section contribute to this process: (i) Twist-3 unpolarized PDF in one of the initial proton convoluted with the twist-2 "transversity" FF for the final hyperon and the twist-2 unpolarized PDF in another proton, and (ii) Twist-3 FFs for the polarized hyperon convoluted with the twist-2 unpolarized PDFs in the initial protons.The complete LO cross section for (i) was derived in [43][44][45].The second one (ii) can be further classified into two, depending on whether the twist-3 FF is of (ii-a) quark-gluon correlation type or of (ii-b) gluon correlation type.The complete LO cross section for (ii-a) was derived in [46], while (ii-b) has not been studied so far.In this paper we focus on the (ii-b) contribution, and derive the corresponding cross section, which completes the LO twist-3 cross section for this process.(A short version of the present work was presented in [47,48].)We develop a formalism for deriving the gauge and frame independent contribution to this twist-3 cross section from the purely gluonic FFs, and present the result for pp → Λ ↑ X.
The remainder of this paper is organized as follows.In section 2, we introduce the complete set of the gluonic FFs for spin-1/2 hadron up to twist-3 defined from correlators of two-and three-gluon field strengths, which are necessary to derive the twist-3 cross section.We also recall from [49] the exact relations among those FFs based on the QCD equation of motion and the Lorentz invariance, which play a crucial role to guarantee the gauge and Lorentz invariance of the cross section 2 .In section 3, we develop a formalism to derive the twist-3 gluon FF contribution to the twist-3 cross section, and present the corresponding LO cross section for pp → Λ ↑ X.We will discuss how the Lorentz invariance of the twist-3 cross section is realized, using the relations introduced in section 2. Gauge invariance of the cross section is discussed in Appendix B. Section 4 is devoted to a brief summary.In other Appendices, we discuss some technical aspects of the actual calculations.
2 Gluon Fragmentation Functions 2.1 Three types of twist-3 gluon fragmentation functions In this section we introduce twist-3 gluon FFs for a spin-1/2 baryon relevant to pp → Λ ↑ X [49,53,54] and summarize their basic properties derived in [49].They are classified into three types; intrinsic, kinematical and dynamical FFs.The intrinsic twist-3 gluon FFs are defined as the Fourier transform of the lightcone correlator of the gluon's field strength F µν a : where N = 3 is the number of colors, |h(P h , S h ) is the baryon state with the four momentum P h (P 2 h = M 2 h ) and the spin vector S h (S 2 h = −M 2 h ), and [λw, ∞w] is the gauge link in the adjoint representation connecting λw and ∞w.For the transversely polarized baryon, we use the spin vector S ⊥ normalized as S 2 ⊥ = −1.In the twist-3 accuracy P h can be regarded as lightlike.For a baryon with large momentum, P h (| P h |, P h ), another lightlike vector w is defined as The kinematical FFs contain the transverse derivative of the correlation functions of the field strengths: where each function is defined to be real.The kinematical FFs are related to the k 2 T /M 2 h -moment of the transverse-momentum-dependent (TMD) FFs [53].Each function in (2) has a support on 0 < z < 1.
To define the dynamical FFs, we introduce the lightcone correlation functions of three field strengths: Three types of the twist-3 gluon FFs defined in the previous subsection are not independent from each other, but are related by the QCD equation-of-motion (EOM relations) and the Lorentz invariance properties of the correlation functions (LIRs).In [49] the complete set of those relations have been derived.Here we recall those relations, which, as we will see in the next section, play a crucial role to guarantee the Lorentz-and gauge-invariance of the twist-3 cross sections for pp → Λ ↑ X.Here we quote from [49] the relevant relations.First the intrinsic FFs ∆ G 3 T (z) can be written in terms of the kinematical FFs and the dynamical ones as (See eq. ( 50) of [49]): where D F T (z) is defined as with C F = N 2 −1 2N and D F T indicates the imaginary part of D F T .The kinematical FFs can be expressed in terms of the dynamical ones as (see eqs. ( 74) and (75) of [49]): From ( 9), (11) and (12), the intrinsic FF ∆ G 3 T (z) is also written by the dynamical ones.For the derivation of the twist-3 cross section for pp → Λ ↑ X, one needs derivatives of the kinematic FFs.From (11) and (12), we can obtain those derivatives in terms of the kinematical FFs themselves and the dynamical ones as and In the next section, we will use ( 9), ( 13) and ( 14) to write the twist-3 gluon FF contribution to pp → Λ ↑ X in a frame-independent form.
3 Twist-3 gluon fragmentation contribution to pp → Λ ↑ X In this section, we develop a formalism for calculating the twist-3 gluon fragmentation contribution to where p, p and P h are the momenta of the particles and S ⊥ is the transverse spin vector of Λ ↑ .We work in Feynman gauge so that one can check the appearance of the gauge invariant FFs explicitly.As in the case of the twist-3 quark FF [46,55], naively T -odd FFs give rise to the cross section as a nonpole contribution.The twist-3 FF contribution to (15) can be written as where S E = (p + p ) 2 is the center-of-mass energy squared, E h = M 2 h + P 2 h is the energy of the hyperon, x, x are the momentum fractions of the partons coming out of the initial nucleons and f 1 (x) is the unpolarized quark or gluon distributions in the nucleon.W q (W g ) is the hadronic tensor representing the partonic hard scattering followed by the fragmentation of a quark (gluon) into the final Λ ↑ .In (16), summation over all possible channels is implied.The LO cross section for W q was derived in [46].Here we focus on the twist-3 gluon fragmentation contribution W g in ( 16) which is diagrammatically shown in Fig. 1.W g consists of three terms corresponding to where the factor 1/2 in front of W ,µνλ L,abc (k, k ) and Γ(1),µνλ R,abc (k, k ) are the corresponding hadronic matrix elements representing fragmentation of partons into h (h = Λ ↑ ).The upper indices (0) and ( 1) represent the number of extra gluon lines compared with the lowest order gluon fragmentation contribution to the cross section.Hadronic matrix elements are defined as (1),µνλ where the gauge coupling g associated with the attachment of the extra gluon line to the hard part is included in Γ(1) L (k, k ) and Γ(1) R (k, k ).Therefore the hard parts S (0) , S L and S R are of O(g 4 ) in the LO calculation.From hermiticity, one has Γ(1),µνλ 17) is real.One can extract the twist-3 effect from (17) by applying the collinear expansion.The collinear expansion of S (0) , S

L and S
(1) R with respect to the parton momenta k µ and k µ around the parent hadron's momentum P µ h reads (1),abc where ,ab µν (z) ≡ S (0),ab µν ,abc L,µνλ (z, z ) ≡ S (1),abc L,µνλ (P h /z, P h /z ), "| c.l. " indicates taking the collinear limit, i.e., k → P h /z and k → P h /z and • • • denotes the contribution of twist-4 or higher.The collinear expansion for S (1) R can be performed similarly.We also decompose the gauge field A µ as Inserting this decomposition into the expression for Γ(0) , Γ(1), L and Γ(1) R , one obtains Γ (1),µνλ L,abc ,www L,abc (1),wwτ L,abc (1),wρw L,abc and likewise for Γ (1),µνλ R,abc .Inserting the above expansion into (17) and keeping the terms with two or three Ω µ ν s in the product of the hard parts S (0) , S L , S R and the fragmentation matrix elements Γ(0) , Γ R , one can obtain the twist-2 and 3 contributions to the cross section.To get a gauge invariant cross section, one needs to fully utilize the following Ward identities for the hard parts (see Appendix A): Here and below we suppress the upper indices (0) and ( 1) from the hard parts S (0) , S L and S R for simplicity and S µν (k) ≡ S µν,ab (k)δ ab .The G-terms appear due to the off-shellness of the parton momenta entering the fragmentation matrix elements.We present the actual forms of those "ghost terms" in Appendix A.Here we only mention that they are proportional to f abc and satisfy the relation We have found that the ghost terms do not contribute to the twist-3 cross section3 .We thus discard them in the following.To get the twist-3 cross section in the gauge invariant form, we use the collinear limit of these identities as well as the first, second and third derivatives with respect to k and k of these Ward identities in the collinear limit.
After very lengthy calculation one eventually obtains the twist-2 and -3 contributions from Fig. 1 (a), (b) and (c) to W g in the following form: Each fragmentation matrix elements Γ's appearing in the above expression are defined as follows: Γ (1)αβ Γ (1)αβ Γ (1)αβγ Γ (1)αβγ Γ (1)αβγ Γ (1)αβγ Γ (1)αβγ Γ (1)αβγ (1)αβγ In the LO calculation of Figs. 1 (a), (b) and (c), only O(1) and O(g) contributions from the hadronic matrix elements are produced.We thus note that, in ( 34)- (36), one can identify the correlation functions of the field strength are the O(g) terms arising from the expansion of the gauge link and the O(g) part of the field strength in Γ(z).These terms contain both twist-2 and intrinsic twist-3 FFs.Likewise the second term in W is the O(g) contribution from ΓF defined in ( 5) and (6).Likewise the third term in W (c) g is associated with ΓF R ∼ ( ΓF ) .This way we have obtained the sum of W g in the color gauge invariant form in terms of the intrinsic, kinematical and dynamical FFs.Inserting these expressions into (16), one can eventually express the twist-3 gluon FF contribution to the cross section as E h dσ(p, p , P h ; S ⊥ ) where we have used the expression (4), and | c.l. implies the collinear limit, k → P h /z.In writing down the contribution from the dynamical FFs in (51), we have interchanged the role of the variables, z and z , from ( 35) and ( 36) for later convenience.Next we substitute (1), ( 2), ( 5), ( 6) into (51).We recall that the q qg-type FFs (8) are related to the purely gluonic FFs, as was shown in Sec.2.2.Therefore we consider the contribution shown in Fig. 2 together.Note also that each hard part contains the factor δ (xp + x p − k) 2 ∼ δ (xp + x p − P h /z) 2 corresponding to the on-shell condition for the final unobserved parton, and its derivative with respect to k causes the derivative of the kinematical FFs by partial integration with respect to 1/z.Separating various contributions based on the z -dependence of the hard cross sections for the dynamical FFs (See Appendix C), we can write the cross section as where Ĥint , ĤNDG , ĤDG , etc. represent the partonic hard cross sections for each FF (after separating the z -dependence for dynamical FFs) and they are the functions of the Mandelstam variables in the parton level, s = (xp + x p , multiplied by kinematic factors with pP h wS ⊥ and p P h wS ⊥ .(See eq.(54) below as an example.)In ( 52), we have used the shorthand notation, in the z and z 2 /z terms in the contribution from N i and O i leads to the following form owing to the exchange symmetry (7) of N 1 and O 1 : Here we remind that the gauge invariance and the frame independence of the cross section (53) are realized in a very nontrivial manner.We show that those properties are guaranteed by the EOM relation and the LIRs introduced in the previous section.The gauge invariance is satisfied by the EOM relation (9), which is discussed in detail in Appendix B.Here we demonstrate how the frame independence of the cross section is achieved.
To make clear the issue of frame dependence, we pick up the hard cross sections ĤN 1 2 − ĤN 1 4 , ĤDG and ĤDH in (53) in the qg → gq-channel, as an example.They can be computed to be We note that each cross section contains the lightlike vector w µ .On the other hand, the physical cross section should be able to be represented in terms of the vectors p, p P h and S ⊥ in a Lorentz invariant form.Since w µ is defined from P µ h , its actual form depends on the frame.One can express the vector w in terms of p, p and P h as [52] which satisfies P h • w = 1 and w 2 = 0.The values of α and β specify the frame we choose, and the above form of the hard cross sections leads to α-and β-dependent cross sections.However, use of the EOM relation, (9), and the LIRs, ( 13) and ( 14), leads to the cross section independent from α and β as will be seen below.In the twist-3 cross section (53), we eliminate the intrinsic FF ∆ G 3 T (z)/z and the derivative of the two kinematical FFs by using those relations.Then the resulting cross section is written in terms of the (nonderivative) kinematical FFs and the dynamical FFs.If we pick up the hard cross section for we have the combination where One should note that the kinematical factor E appearing in this combination is free from α and β, which is written after the last equal sign in (59).We have found that all the coefficient hard cross sections for all FFs with the same z -dependence define the frame-independent cross section with the common kinematic factor E. This shows that the frame dependence has been removed from the twist-3 cross section thanks to the EOM relations and the LIRs.This way we can define the following set of frame independent hard cross sections (in addition to (57)): With these hard cross sections, the manifestly frame-independent twist-3 cross section is given by E h dσ(p, p , P h ; S ⊥ ) Equation ( 79) is the final result for the twist-3 gluon FF contribution to pp → Λ ↑ X.
Below we give the LO Feynman diagrams for the hard part in each channel and present the results for hard cross sections, using the partonic Mandelstam variables, s, t, and u.

Summary and conclusion
In this paper we have studied the transverse polarization of a spin-1/2 hyperon produced in the unpolarized proton-proton collision, pp → Λ ↑ X, within the framework of the collinear twist-3 factorization, which is relevant for the large-p T hyperon production.We focused on the contribution from the twist-3 gluon FFs, which had never been studied in previous studies.To this end we have developed a formalism to include all effects associated with the twist-3 gluon FFs.The twist-3 cross section receives contributions from three types of the gluon FFs, i.e., intrinsic, kinematical and dynamical (purely gluonic and quark-antiquark-gluon type) ones.Applying the formalism we have calculated the LO cross section for pp → Λ ↑ X.This completes the LO twist-3 cross section combined with the known results for the other contributions from the twist-3 distribution in the unpolarized proton and the twist-3 quark FFs for the hyperon.Using the EOM relation and the LIRs for the twist-3 gluon FFs, we have shown that the derived cross section satisfies the color gauge invariance and the frame independence.Since the formalism developed here is a general one, it can be applied to other processes to which the twist-3 gluon FFs contribute.In the RHS, it is implied that the scalar polarized gluon is attached one of the black dots in all possible ways.can be decomposed into several pieces, and some of them cancel each others owing to the on-shell condition of the external lines (See A.3). Taking these facts into account we rearrange each term in the RHS of Fig. 10.
Figure 14: Diagrams for the ghost-term (89) in the q q → gg channel.
This completes the derivation of the Ward identity in the q q → gg channel.

A.1.2 qg → gq channel
Ward identities in this channel are given by ( 83)-(88) except that the ghost terms G µν,abc q q→gg (k 1 , k 2 ) are replaced by G µν,abc qg→gq (k 1 , k 2 ) which takes the following structure: where H e λ (xp, xp + x p − k 2 ) is the quark-line part connected to the left of the cut M ν,bd α (k 2 ) .

A.1.3 gg → gg channel
Ward identities in this channel are given by ( 83)-( 88) in which G µν,abc q q→gg (k 1 , k 2 ) is replaced by the following G µν,abc gg→gg (k 1 , k 2 ): 92) where G 1gg→gg through G 4gg→gg are the ghost terms which occur from four types of diagrams shown in Fig. 15 for the hard scattering amplitudes in this channel.Each term in (92) takes the following structures: and where V β,hf e σρ represents an appropriate three-gluon vertex in each channel.

A.3 Cancellation among vertices
After decomposition of vertices, the following cancellation holds among vertices.

B Color gauge invariance of the twist-3 cross section
Here we prove that the twist-3 cross section in (51) supplemented by the quark-antiquark-gluon contribution shown in Fig. 2 is color gauge invariant owing to the EOM relation (9).We illustrate this property for the qg → gq channel.The proof for other channels is essentially the same.The cross section in this channel can be written as E h dσ(p, p , P h ; S ⊥ )  where G(x ) is the unpolarized gluon distribution in the proton with momentum p , and the Lorentz indices ρ and σ are contracted with g ρσ ⊥ (p ) ≡ g ρσ − p ρ n σ − p σ n ρ where n is the usual lightlike vector satisfying p • n = 1 to extract the contribution from G(x ).From (51), we can read off W ρσ (xp, x p , P h /z) as where the hard part S ab µν,ρσ (k) is related to the hard part S ab µν (k) in (51) by − 1 2 g ρσ ⊥ (p ) S ab µν,ρσ (k) = S ab µν (k) and likewise for S abc µνλ,ρσ (k), and w q qg ρσ (xp, x p , P h /z) represents the contribution from Fig. 2. Then the color gauge invariance of the cross section implies To show (104), we use the EOM relation (9) and eliminate the intrinsic twist-3 FF ∆ G 3 T (z) in the first term of the RHS of (103).Then W ρσ can be decomposed into three pieces: where ρσ represents the contribution from the dynamical 3-body correlation function in ( 5) and ( 6), W (ii) ρσ represents the one from the dynamical q qg-correlation function in (8) and W (iii) ρσ represents the one from the kinematical FFs in (2).To show (104), it suffices to prove that each term of (105) separately satisfies (104).

B.1 Contribution from dynamical FFs: W (i) ρσ
We first show W  21 (c), which is not a part of partonic cross sections for the dynamical FFs.Therefore, after using the EOM relations, we get the following combination of the hard cross section, where the hard part with index L indicates diagrams in Fig. 21 (b) and (c), and that with R indicates their hermitian conjugate diagrams.We define the amplitude M ad ρ,α for qg → gq scattering and M acd ρ,αγ for qg → ggq scattering as shown in Fig. 21.With the physical polarizations for the Lorentz indices α and γ, these amplitudes satisfy the following relations, This implies We thus have The first term in the RHS of (109) reads  ρσ shown in Fig. 24 satisfies (104).Fig. 24 (b)' is defined to include the contribution of the type Fig. 24 (c)' which is not a part of the hard cross section for a quarkantiquark-gluon contribution.Accordingly the corresponding hard cross section is written as Taking into account of the relation (107), we have Therefore we obtain The second term in the RHS of this equation can be written as which is diagrammatically written as Fig. 25.We again find the coefficient of ( 2 ρσ (xp, x p , P h /z) can be written as T (z) + ∆ H
From these relations, we have

C Separation of the 3-body partonic cross sections based on zdependence
Here we discuss separation of the 3-body cross section (52) based on z -dependence.Inserting ( 5) into (51), we write the cross section for N i as E h dσ(p, p , P h ; S ⊥ ) where σ( 1 z , 1 z ) is a partonic hard cross section defined by = δ((xp + x p − P h /z) 2 ) The amplitude in Fig. 26 has two propagators with z -dependence ( 1 ○ and 2 ○).For these two propagators, z -dependence is written as follows: (128) In these two equations, we call each term in the RHS 1 ○-I, 1 ○-II, 2 ○-I and 2 ○-II as shown in the figure.Then, depending on the combination of the propagators, one can separate the zdependence of the cross section as E h dσ(p, p , P h ; S ⊥ ) where each partonic cross section is defined as and we used the relation z /z × 1/(1 − z/z ) = z /z + 1/(1 − z/z ).
Taking into account of the overall factor z /z×1/(1/z−1/z ) and rearranging the z -dependence, one obtains the cross section in the following form: This form is the expression used in (52).One can thus separate different z -dependence of the cross section.

g
takes into account the exchange symmetry of the gluon fields in the fragmentation matrix elements.The four momenta k and k are those of the partons fragmenting into the final Λ. S (0),ab µν (k), S (1),abc L,µνλ (k, k ) and S (1),abc R,µνλ (k, k ) are the partonic hard scattering parts, and Γ(0),µν ab (k), Γ

1 )
contribution from the kinematical twist-3 FFs Γ∂ (z) in (2).The second terms in W (b) g and W (c) g are the O(g) terms arising from the expansion of the gauge link and the O(g) part of the field strength in Γ∂ (z).The third term in W (b) g

Figure 2 :
Figure 2: Diagram for the q qg-type correlation function in pp → Λ ↑ X. Mirror diagram also contributes.

Figure 3 :
Figure 3: Diagrams in the qg → gq channel for the hard part S(k) and S L (z , z) in eq.(51).For S L (z , z), an extra gluon line connecting ⊗ and each of the black dots should be added for each diagram.

Figure 4 :
Figure 4: Diagrams for the hard part in Fig. 2. For the upper middle diagram, a quark loop with the reversed arrow also needs to be included.

Figure 6 :
Figure 6: Diagrams for the hard part in Fig. 2. For the upper left diagram, a quark loop with the reversed arrow also needs to be included.

Figure 7 :
Figure 7: Diagrams in the gg → gg channel for the hard parts S(k) and S L (z , z) in eq.(51).For S L (z , z), an extra gluon line connecting ⊗ and each of the black dots should be added for each diagram.A white circle represents a four-gluon vertex, making clear the difference from the attachment of the extra gluon line.

Figure 8 :
Figure 8: Diagrams contributing to the hard part in Fig. 2 in the gg → gg channel.For all diagrams, a quark loop with the reversed arrow also needs to be included.
S abcL µνλ (k 1 , k 2 ) and the second term represents the attachment of the scalar polarized gluon to the gluon line fragmenting into Λ ↑ .The LO diagrams can be classified into three types as shown in the RHS of this figure.Using the tree level Ward identities (See A.2), each diagram in Fig.10

Figure 10 :
Figure 10: Attachment of the scalar polarized gluon of momentum k 2 − k 1 to qq → gg diagrams.In the RHS, it is implied that the scalar polarized gluon is attached one of the black dots in all possible ways.

Figure 15 :
Figure 15: Diagrammatic representation of the ghost terms in the gg → gg channel.Depending on the position of attachment of the scalar polarized gluon, diagrams can be classified into 4 types.Figure G 3 represents diagrams in which the scalar polarized gluon is attached to an internal propagator.

Figure 16 :
Figure 16: Attachment of a scalar polarized gluon to a quark-gluon vertex corresponding to (97).

Figure 21 :
Figure 21: Diagrams contributing to W (i) ρσ .M ad ρ,α and M acd ρ,αγ , respectively, represent the qg → gq and qg → ggq scattering amplitudes, which constitute the hard cross section as shown in the figure.

Figure 23 :
Figure 23: Diagrammatic representation for x p ρ W (c)L ρσ in the qg → gq channel.
S µν,ρσ (k)x p ρ = 0. (124) This is because the Lorentz indices µ and ν provide physical polarizations for the on-shell gluon with momentum k.By taking the derivative of (124) with respect to k λ and setting k → P h /z, it is easy to see x p ρ W (iii) ρσ (xp, x p , P h /z) = x p σ W (iii) ρσ (xp, x p , P h /z) = 0, which completes the proof.

Figure 26 :
Figure 26: An example of diagrams which contribute to qg → gq channel.Notation pc ≡ P h /z and pc ≡ P h /z are used.