Twist-2 relation and sum rule for tensor-polarized parton distribution functions of spin-1 hadrons

Abstract

Sum rules for structure functions and their twist-2 relations have important roles in constraining their magnitudes and x dependencies and in studying higher-twist effects. The Wandzura-Wilczek (WW) relation and the Burkhardt-Cottingham (BC) sum rule are such examples for the polarized structure functions g1 and g2. Recently, new twist-3 and twist-4 parton distribution functions were proposed for spin-1 hadrons, so that it became possible to investigate spin-1 structure functions including higher-twist ones. We show in this work that an analogous twist-2 relation and a sum rule exist for the tensor-polarized parton distribution functions f1LL and fLT, where f1LL is a twist-2 function and fLT is a twist-3 one. Namely, the twist-2 part of fLT is expressed by an integral of f1LL (or b1) and the integral of the function f2LT = (2/3)fLT− f1LL over x vanishes. If the parton-model sum rule for f1LL (b1) is applied by assuming vanishing tensor-polarized antiquark distributions, another sum rule also exists for fLT itself. These relations should be valuable for studying tensor-polarized distribution functions of spin-1 hadrons and for separating twist-2 components from higher-twist terms, as the WW relation and BC sum rule have been used for investigating x dependence and higher-twist effects in g2. In deriving these relations, we indicate that four twist-3 multiparton distribution functions FLT, GLT, $$ {H}_{LL}^{\perp } $$ H LL ⊥ , and HTT exist for tensor-polarized spin-1 hadrons. These multiparton distribution functions are also interesting to probe multiparton correlations in spin-1 hadrons. In the near future, we expect that physics of spin-1 hadrons will become a popular topic, since there are experimental projects to investigate spin structure of the spin-1 deuteron at the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility, the electron-ion colliders in US and China in 2020’s and 2030’s.

Transverse-momentum-dependent parton distribution functions for spin-1 hadrons

“…F LT (x, y), G LT (x, y), H ⊥ L L (x, y), H T T (x, y), (11) exist in general [16]. The twist-3 matrix element is expressed by F G,LT and G G,LT defined with the gluon field tensor G µν .…”

“…The twist-3 matrix element is expressed by F G,LT and G G,LT defined with the gluon field tensor G µν . Specifying the twist-3 terms, we obtained [16] f LT (x) = 3 2…”

“…In future, there are experimental projects to investigate structure functions of spin-1 hadrons, especially the spin-1 deuteron, at the Jefferson laboratory (JLab) [2,3], Fermilab [4], and Nuclotron-based Ion Collider fAcility (NICA) [5], and electron-ion colliders in US and China. We also have been investigating structure functions of spin-1 hadrons theoretically by considering these experimental projects [6][7][8][9][10][11][12][13][14][15][16]. Therefore, it is a good opportunity and timely to investigate TMDs also for the spin-1 hadrons.…”

“…[15], where new structure functions and sum rules were proposed at twists 3 and 4. In addition, a twist-2 relation and a sum rule were derived for the twist-3 collinear structure function f LT [16]. We explain these works in this paper.…”

Inclusive and semi-inclusive production of spin-3/2 hadrons in $e^+e^-$ annihilation

“…Extending to vector mesons, such as ρ and K * , there are five additional spin states, referred to as rank-2 tensor polarization [44][45][46][47], from the decomposition of the spin density matrix. The measurements of the rank-2 tensor polarization of vector mesons produced in e + e − [48][49][50][51] and pp [52][53][54][55][56][57][58][59][60] collisions have been carried out and the results incite extensive theoretical studies [46,[61][62][63][64][65][66][67][68][69][70][71][72][73][74][75]. A systematic study of the Lorentz decomposition of the quarkquark correlator suggests that the spin alignment of vector mesons is solely determined by a rank-2 tensor polarized FF D 1LL (z) independent of the spin of the fragmenting quark [76] and some parametrizations have been obtained by fitting the LEP data [48][49][50].…”