Some Aspects of Transversity∗

The specificities of transverse polarization with respect to helicity of ultrarelativistic fermions are pointed out. For massless fermions, a covariant transversity four-vector is defined, up to a kind of gauge transformation. The tranversity distribution of quarks in a nucleon is defined. Its possible connection to the magnetic or electric dipole moment of the baryon is conjectured. Consequences of the approximate chiral invariance on transverse spin asymmetries in hard processes are enumerated. The ”sheared jet effect” introduced by Collins for measuring the transverse polarization of a final quark is presented. 1 What is transversity ? For a massive fermion, there is no problem of defining a transversely polarized particle of momentum ~ p : put it at rest, polarize it in some direction n̂ orthogonal to ~p and then apply the necessary Lorentz boost to give it momentum ~p. For a massless fermion, this definition does not work because there is no rest frame. On the other hand, we know that helicity states exist and form a complete basis. A transversely polarized state should therefore be a linear superposition of helicity ones. To get the coefficients, the most natural method is to give the particle a temporary small mass and let this mass go to zero. Thus, for ~p along the positive z axis, we get |n̂ >= |+ > + e iφ |− > √ 2 (1) where φ is the azimuth of n̂. This receipe would not completely solve the problem in the case of a chiral symmetric world : the relative phase of the |+ > and |− > states would be arbitrary, which would make the azimuth of n̂ ambiguous (a similar ambiguity would exist for linearly polarized photons if the electric-magnetic duality were an exact symmetry). Therefore the observability of transversity is linked to chiral symmetry breaking. In the massive case, any polarization of the fermion can be represented in a noncovariant way by the 3-dimensional vector ~ P = 2 < ~ S > measured in the rest frame (i.e., before boost). Boosting ~ P results in the covariant polarization four-vector S = (0, ~ P⊥) + PL 1 m p̃ = S⊥ + SL (2) where p̃ = (|~ p |, p p̂) and p̂ ≡ ~ p/|~ p |. When m → 0, SL becomes infinite (unless PL is strictly zero). However, the covariant projector u(~ p, s) ū(~ p, s) keeps finite [1] : u(~ p, s) ū(~ p, s) = 1 + γ γ · S 2 (γ · p+m) → 1 + γ 5 γ · S⊥ + PL γ 2 γ · p (3) This equation shows that in the massless case helicity and transversity play very different roles. The later is associated with one more γ matrix and is therefore called a chirality odd quantity [2]. Another interesting feature is the invariance under the ”gauge” transformation of the transversity four-vector S ⊥ ⇒ S μ ⊥ + constant× p ; (4) this invariance makes the application of Lorentz transformations to S ⊥ possible (it would not be the case if we had imposed S ⊥ ≡ 0). The ”gauge” S μ ⊥ = (0, ~ P⊥) is the analogue of the radiation gauge of the photon. One may view the gauge freedom as the relic of an infinitesimal uncertainty of the longitudinal polarization in the limit m → 0. 2 Transversity distribution inside the nucleon In analogy to the quark helicity distribution ∆Lq(x) ≡ 2g1(x) = q(x) − q(x) in a longitudinally polarized nucleon N, we define the quark transversity distribution [3, 4, 2] in a transversely polarized nucleon N : ∆⊥q(x) ≡ 2h1(x) = q(x) + q(x) (5) It obeys the Soffer’s inequality [5] |∆⊥q(x)| ≤ q(x) (6) which is stronger than the trivial one |∆⊥q(x)| ≤ q(x). h1(x) is not the same quantity as g1(x)+g2(x), in spite of the fact that the later is measured with transversely polarized target. Only in a nonrelativistic quark model do they coincide (but such a model is unrealistic for deep inelastic reactions). In the infinite momentum frame,