In this work, we present a complete theoretical framework for analyzing the distribution of polarized hadrons within jets, with and without measuring the transverse momentum relative to the standard jet axis. Using soft-collinear effective theory (SCET), we derive the factorization and provide the theoretical calculation of both semi-inclusive and exclusive fragmenting jet functions (FJFs) under longitudinal and transverse polarization. With the polarized FJFs, one gains access to a variety of new observables that can be used for extracting both collinear and transverse momentum dependent parton distribution functions (PDFs) and fragmentation functions (FFs). As examples, we provide numerical results for the spin asymmetry $$ {A}_{TU,T}^{\cos \left({\phi}_S-{\hat{\phi}}_{S_h}\right)} $$
A
TU
,
T
cos
ϕ
S
−
ϕ
̂
S
h
from polarized semi-inclusive hadron-in-jet production in polarized pp collisions at RHIC kinematics, where a transversely polarized quark would lead to the transverse spin of the final-state hadron inside the jet and is thus sensitive to the transversity fragmentation functions. Similarly, another spin asymmetry, $$ {A}_{TU,L}^{\cos \left({\phi}_q-{\phi}_S\right)} $$
A
TU
,
L
cos
ϕ
q
−
ϕ
S
from polarized exclusive hadron-in-jet production in polarized ep collisions at EIC kinematics would allow us to access the helicity fragmentation functions. These observables demonstrate promising potential in investigating transverse momentum dependent PDFs and FFs and are worthwhile for further measurements.
