Transverse Momentum Dependent (TMD)
Parton Distribution Functions
in a Spectator Diquark Model Francesco Conti in collaboration with: Marco Radici (INFN Pavia)
Alessandro Bacchetta (JLAB) Department of Nuclear and Theoretical Physics, University of Pavia
And INFN, Section of Pavia
Transverse Momentum Dependent (TMD)
Parton Distribution Functions
in a Spectator Diquark Model Francesco Conti in collaboration with: Marco Radici (INFN Pavia)
Alessandro Bacchetta (JLAB) Department of Nuclear and Theoretical Physics, University of Pavia
And INFN, Section of Pavia
DIS regime: Nucleon Spin Structure usefulness of an expansion
in powers of 1/Q, besides that in
powers of s (pQCD): TWIST Deep Inelastic Scattering: Leptonic tensor: known
at any order in pQED Hadronic tensor: hadron internal dynamics (low energy non-pert. QCD),
in terms of structure functions, with SCALING properies (Q-INdependence)
PARTON MODEL: incoherent sum of interactions on almost free (on shell) pointlike partons
hard/soft factorization theorems: convolution between hard elementary cross
section and soft (non-pert.) and universal parton distribution functions PDF Asymptotic Freedom /
Confinement Parton distributions = Probability densities of finding a parton with x momentum fraction
in the target hadron (NO intrinsic transverse momentum Collinear factorization)
Nucleon Spin Structure & TMD parton densities The 3 momenta {P,q,Ph} CANNOT be all
collinear ; in T-frame, keeping the cross
section differential in dqT: sensibility to the
parton transverse momenta in the hard
vertex TMD parton densities ! Semi Inclusive Deep Inelastic Scattering: Fragmentation Correlator FFs Hadronic tensor in the Parton Model (tree level, leading twist): Quark-Quark Correlator PDFs TMD hard/soft factorization: Ji, Ma,
Yuan, PRD 71 (04); Collins, Metz,
PRL 93 (04) Diagonal matrix elements of bilocal operators,
built with quark fields, on hadronic states
Nucleon Spin Structure & TMD parton densities (2) Projecting over various Dirac structures, all
leading twist TMD parton distribution functions
can be extracted, with probabilistic interpretation Known x-parametrization,
poorly known pT one (gaussian
and with no flavour dependence;
other possible functional forms!
Connection with orbital L! ) It is of great importance to devise models
showing the ability to predict a non-trivial
pT-dependence for TMD densities!
The Spectator Diquark model The correlator involves matrix elements on bound hadronic states, whose partonic content is
neither known nor computable in pQCD (low energy region!) model calculations required! SPECTATOR DIQUARK model: Simple, Covariant model: analytic results, mainly 3 parameters. Replace the sum over intermediate states in with a
single state of definite mass (on shell) and coloured.
Its quantum numbers are determined by the action of
the quark fields on , so are those of a diquark! (Jakob, Mulders, Rodrigues, A626 (97) 937,
Bacchetta, Schaefer, Yang, P.L. B578 (04) 109)
The Spectator Diquark model (2) Nucleon (N)-quark (q)-diquark (Dq) vertex: Dq Spin = 0 : flavour-singlet [~{ud-du}] Dq Spin = 1 : flavour-triplet [~{dd,ud+du,uu}] Need of Axial-Vector
diquarks in order
to describe d in N! Pointlike: Dipolar: Exponential: N-q-Dq vertex form factors (non-pointlike nature of N and Dq): Virtual S=1 Dq propagator ( real Dq polarization sum): ‘Feynman’:
Bacchetta, Schaefer, Yang, P.L.B578 (04) 109 ‘Covariant’:
Gamberg, Goldstein, Schlegel, arXiv:0708.0324 [hep-ph] ‘Light-Cone’:
Brodsky, Hwang, Ma,
Schmidt, N.P.B593 (01) 311
The Spectator Diquark model (3) Why should we privilege ‘Light-Cone’ (LC) gauge? In DIS process, the exchanged virtual photon
can in in principle probe not only the quark, but
also the diquark, this latter being a charged boson Not only … … but also S=0 diquark contributes to FL only: Adopting LC gauge, the same holds true for S=1 diquark
also, while other gauges give contributions to FT as well! In our model: Systematic calculation of ALL leading twist T-even and T-odd TMD functions
(hence of related PDF also)
Several functional forms for N-q-Dq vertex form factors and S=1 Dq propagator
Moreover, Overlap Representation of all TMD functions in terms of LCWFs!
Overlap representation for T-even TMD following Brodksy, Hwang, Ma, Schmidt, N.P.B593 (01) 311 The light-cone Fock wave-functions (LCWF) are the frame independent interpolating functions
between hadron and quark/gluon degrees of freedom Angular momentum conservation: E.g. : L=0
L=1 L=1 component relativistically enhanced w.r.t.
L=0 one! Spin Crisis as a relativistic effect ?! Non-zero relative orbital angular momentum between q and Dq:
the g.s. of q in N is NOT JP=1/2+; NO SU(4) spin-isospin symmetry for N wave-function!
Overlap representation for T-even TMD For the Unpolarized TMD parton distribution function, e.g. (using LC gauge for axial vector diquark): Besides the Feynman diagram approach, Time-Even TMD densities can be also calculated in terms
of overlaps of our spectator diquark model LCWFs NON-gaussian pT dependence ! Furthermore, using Covariant and Feynman gauges: S=1 diquark contribution
interesting cross-check!
Parameters Fixing SU(4) for |p>: SU(4) for |p>: Jakob, Mulders, Rodrigues,
N.P. A626 (97) 937 (s: S=0, I=0;
a: S=1, I=0) (a’: S=1, I=1) model parameters! Parameters: m=M/3, Ns/a/a’ (fixed from ||f1s/a/a’||=1), Ms/a/a’ , Λs/a/a’ , cs/a/a’ (from a joint fit to data on u & d
unpolarized and polarized PDF: ZEUS for f1 @ Q2=0.3 GeV2, GRSV01 at LO for g1 @ Q2=0.26 GeV2) Hadronic scale
of the model:
Q02 ~ 0.3 GeV2
pT- model dependence Non-monotonic behaviour for small x,
due to L=1 LCWFs, falling linearly with pT2
as pT2 goes to 0! (L=0 LCWFs do not!)
Flavour dependence ! The study of pT-dependence
shed light on the spin/orbital
angular momentum structure
of the Nucleon! ‘+’ combination selects L=0 LCWFs for
S=0 Dq and L=1 LCWFs for S=1 Dq
DGLAP Evolution @ LO using code from
Hirai, Kumano, Miyama,
C.P.C.111 (98) 150 Transversity Change of sign at x=0.5, due to
the negative S=1 Dq contributions,
which become dominant at high x Parametrization: pT- dependence ~ exp[ - pT2 / ]
x- dependence ~ xα(1-x)β …
no change of sign allowed! Anselmino et al.
P.R.D75 (07) 054032 NO TMD Evolution Transverse Spin distribution
Time-Odd TMD distributions T-odd distributions: crucial to explain the evidences of SSA! Their existence is bound to the Gauge
Link operator ( QCD gauge invariance), producing the necessary non-trivial T-odd phases! 1 gluon-loop contribution:
first order approximation
of the Gauge link! Imaginary part: Cutkoski cutting rules! Put on-shell D2 and D4. Analytic results! v: an. mag. mom. of S=1 Dq.
v=1 γWW vertex!
Time-Odd TMD distributions (2) Sivers function appears in the TMD distribution of an unpolarized quark, and
describes the possibility for the latter to be distorted due to the parent Proton
transverse polarization: Boer-Mulders function describes the transverse spin
distribution of a quark in an unpolarized Proton: Both provide crucial information on
partons Orbital Angluar Momentum
contributions to the Proton spin! Sivers Boer-Mulders:
identity for S=0 Dq, simple relation for
S=1 Dq (but only using LC gauge!)
Sivers moments M. Anselmino et al., (2008),
0805.2677. [hep-ph] J. C. Collins et al., (2005),
hep-ph/0510342. No evolution! Signs agreement with
experimental data and also
with lattice calculations! QCDSF, M. Gockeler et al., Phys. Rev.
Lett. 98, 222001 (2007), hep-lat/0612032
: Spin density of unpol. q quark in a
transversely pol. proton Trento conventions
for SIDIS:
Overlap representation for T-odd TMD So far, only results for Sivers function and S=0 diquark Brodsky, Gardner, P.L.B643 (06) 22
Zu, Schmidt, P.R.D75 (07) 073008 Universal FSI operator G !
(using LC gauge for S=1 Dq) Connection with
anomalous magnetic moments:
Conclusions & perspectives Future: calculate observables (SSA) and exploit model LCWFs to compute
other fundamental objects, such as nucleon e.m. form factors and GPDs Why another model for TMD? We actually don’t know much about them!
Why a spectator diquark model? It’s simple, always analytic results! Able to reproduce T-odd effects!
Why including axial-vector diquarks? Needed for down quarks!
What’s new in our work? Systematic calculation of ALL leading twist T-even and T-odd TMDs
Several forms of the N-q-Dq vertex FF and of the S=1 diquark propagator
9 free parameters fixed by fitting available parametrization for f1 and g1
T-even overlap representation: LCWFs with non-zero L, breaking of SU(4)
T-odd overlap representation: universal FSI operator
Generalize relation between Sivers and anomalous magnetic moment
Which are the main results? Interesting pT dependence
Satisfactory agreement with u & d transversity parametrizations
Agreement with lattice on T-odd functions signs for all flavours
Satisfactory agreement with u Sivers mo...
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