Structure of parton quasi-distributions and their moments

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Abstract: We discuss the structure of the parton quasi-distributions (quasi-PDFs)

Q (y, P_{3})

outside the "canonical"

- 1 l e q y l e q 1

support region of the usual parton distribution functions (PDFs). Writing the

y^{n}

moments of

Q (y, P_{3})

in terms of the combined

x^{n - 2 l} k_{p} e r p^{2 l}

-moments of the transverse momentum distribution (TMD)

c a l F (x, k_{p} e r p^{2})

, we establish a connection between the large-

| y |

behavior of

Q (y, P_{3})

and large-

k_{p} e r p^{2}

behavior of

c a l F (x, k_{p} e r p^{2})

. In particular, we show that the

1 / k_{p} e r p^{2}

hard tail of TMDs in QCD results in a slowly decreasing

s i m 1 / | y |

behavior of quasi-PDFs for large

| y |

that produces infinite

y^{n}

moments of

Q (y, P_{3})

. We also relate the

s i m 1 / | y |

terms with the

l n z_{3}^{2}

-singulariies of the Ioffe-time pseudo-distributions

m a t h f r a k M (u, z_{3}^{2})

. Converting the operator product expansion for

m a t h f r a k M (u, z_{3}^{2})

into a matching relation between the quasi-PDF

Q (y, P_{3})

and the light-cone PDF

f (x, m u^{2})

, we demonstrate that there is no contradiction between the infinite values of the

y^{n}

moments of

Q (y, P_{3})

and finite values of the

x^{n}

moments of

f (x, m u^{2})

.

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