Quasi PDF as observables - Los Alamos National Laboratory...Quasi PDF as observables L Del Debbio...

Quasi PDF as observables

L Del Debbio

Higgs Centre for Theoretical PhysicsUniversity of Edinburgh

L Del Debbio Lat PDF Santa Fe, July 2019 1 / 20

in progress/collaboration with K Cichy, T Giani

based on Collins 1980


bilocal operators

F(k+) = k+

∫dy−

2πe−ik

+y− φ(y)φ(0)

FR,MS(k+) =

∫ ∞1

dη

ηK(η, gR, ε)F(ηk+, gR,mR, µ, ε)

=

∫ 1

0

dη

ηK(η, gR, ε)F(k+/η, gR,mR, µ, ε)


parton distribution functions

f(x) = 〈P |F(xP+)|P 〉

fR(x) = 〈P |FR(xP+)|P 〉 =

∫ 1/x

1

dη

ηK(η) f(ηx)

=

∫ 1

x

dη

ηK(η) f(x/η) = K ⊗ f(x)


renormalization at 1-loop

fR (x) =

∫ 1

xdξ ξ−1 f0 (ξ)K (ξ/x, α)

K (η, α) = (1 + ακ) δ (1− η) + αK(1) (η) +O(α2)

f0

(x, µ2, ε

)= z

(µ2, ε

)δ(1− x) + αR (µ) f

(1)0

(x, µ2, ε

)+O

(α2)

K (η, α) =

(1− α

12

1

ε

)δ (1− η)− α 1

ε

η − 1

η2+O

(α2).


deep-inelastic scattering

F (x,Q2) =

∫dy eiqy 〈P |jR(x)jR(0)|P 〉

F (x,Q2) ∼∫ 1/x

1

dη

ηC(η,Q2/µ2, gR) fR(ηx, gR,mR, µ)

= C ⊗ fR(x) + . . .


DIS at 1-loop


IR picture

F (x,Q2) ∼∫ 1

x

dξ

ξf(ξ) F (x/ξ,Q2)

∼∫ 1

x

dξ

ξfR(ξ) C(x/ξ)

where ∫ 1

x

dξ

ξK(x/ξ) C(ξ) = F (x,Q2)

C is IR-finiteL Del Debbio Lat PDF Santa Fe, July 2019 8 / 20

IR picture at 1-loop

F(x,Q2

)=

∫ 1

x

dξ

ξfR (ξ)

(F

(x

ξ,Q2

)− IR

)fR (x) =

∫ 1

xdξ ξ−1 f0 (ξ) K (ξ/x, α)

F(x,Q2

)=

∫ 1

x

dξ

ξfR (ξ) (1− α κ) F

(x

ξ,Q2

)−

− α∫ 1

x

dξ

ξ

∫ 1/ξ

1

dη

ηK(1) (η) fR (ηξ) F

(x

ξ,Q2

)

K (η, α) =

(1− α

12

1

ε

)δ (1− η)− α 1

ε

η − 1

η2= K(η, α)


quasi-PDF

q (x, Pz) =

∫ 1

x

dξ

ξf0 (ξ) q

(x

ξ, Pz

)=

∫ 1

x

dξ

ξfR (ξ)

(q

(x

ξ, Pz

)− IR

)

q (x, Pz) =

∫ 1

x

dξ

ξfR (ξ) (1− ακ) q

(x

ξ,Q2

)−

− α∫ 1

x

dξ

ξ

∫ 1/ξ

1

dη

ηK(1) (η) fR (ηξ) q

(x

ξ, Pz

)

=

∫ 1

x

dξ

ξfR (ξ)

[(1− ακ) q

(x

ξ, Pz

)− αK(1)

(ξ

x

)]


QCD matrix elements

MΓ,A(ζ) = ψ(ζ)ΓλA P exp

(−ig

∫ ζ

0dη A(η)

)ψ(0)

Ioffe time distributions

Mγµ,A(ζ, P ) = 〈P |Mγµ,A(ζ)|P 〉

Lorentz covariance

Mγµ,A(ζ, P ) = Pµhγµ,A(ζ · P, z2) + ζµh′γµ,A(ζ · P, z2)


lattice data as observables

OReγ0

(zPz, z

2)≡ Re

[hγ0,3

(zPz, z

2)]

OImγ0(zPz, z

2)≡ Im

[hγ0,3

(zPz, z

2)]

0 2 4 6 8 10 12 14z

1.0

0.5

0.0

0.5

1.0

Real part

0 2 4 6 8 10 12 14z

1.251.000.750.500.250.000.250.500.75

Imaginary part

[C Alexandrou et al 18]


https://arxiv.org/abs/1803.02685

lattice observables

inverse Fourier transform

OReγ0

(zPz, z

2)

=

∫ ∞−∞

dx cos (xPzz)

∫ +1

−1

dy

|y|C3

(x

y,

µ

|y|Pz

)f3

(y, µ2

)OImγ0(zPz, z

2)

=

∫ ∞−∞

dx sin (xPzz)

∫ +1

−1

dy

|y|C3

(x

y,

µ

|y|Pz

)f3

(y, µ2

)

f3

(x, µ2

)=

{u(x, µ2

)− d

(x, µ2

)if x > 0

−u(−x, µ2

)+ d

(−x, µ2

)if x < 0


factorization formula for MEusing the explicit expressions for C3

OReγ0 (z, µ) =

∫ 1

0dx CRe

3

(x, z,

µ

Pz

)V3 (x, µ) = CRe

3

(z,µ

Pz

)~ V3

(µ2)

OImγ0 (z, µ) =

∫ 1

0dx C Im

3

(x, z,

µ

Pz

)T3 (x, µ) = C Im

3

(z,µ

Pz

)~ T3

(µ2)

where V3 and T3 are the nonsinglet distributions defined by

V3 (x) = u (x)− u (x)−[d (x)− d (x)

]T3 (x) = u (x) + u (x)−

[d (x) + d (x)

]LO : ORe

γ0

(zPz, z

2)

=

∫dx cos(zPzx)V3(x, µ2)


Bjorken scaling of ME

0 50 100 150 200z^2

0.9850

0.9875

0.9900

0.9925

0.9950

0.9975

1.0000

1.0025

zPz = 0.07zPz = 0.14zPz = 0.225zPz = 0.350zPz = 0.650

Real part, LO

0 50 100 150 200z^2

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98 zPz = 0.07zPz = 0.14zPz = 0.225zPz = 0.350zPz = 0.650

Real part, NLO


systematic errors

• cut-off effects

• finite volume effects

• excited states contamination

• truncation effects

• higher-twist terms

• isospin breaking

Scenario Cut-off FVE Excited states TruncationS1 10% 2.5% 5% 10%S2 20% 5% 10% 20%S3 30% e−3+0.062z/a% 15% 30%S4 0.1 0.025 0.05 0.1S5 0.2 0.05 0.1 0.2S6 0.3 e−3+0.062z/a 0.15 0.3


closure test – 1

10 5 10 4 10 3 10 2 10 1 100

x

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xV3(

x)

V3 at 1.6 GeVCT1 (68 c.l.+1 )NNPDF31_nlo_as_0118 (68 c.l.+1 )

0.0 0.2 0.4 0.6 0.8x

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xV3(

x)


10 5 10 4 10 3 10 2 10 1 100

x

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

xT3(

x)

T3 at 1.6 GeVCT1 (68 c.l.+1 )NNPDF31_nlo_as_0118 (68 c.l.+1 )

0.0 0.2 0.4 0.6 0.8x

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xT3(

x)



closure test – 2

10 5 10 4 10 3 10 2 10 1 100

x

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xV3(

x)


0.0 0.2 0.4 0.6 0.8x

0.05

0.10

0.15

0.20

0.25

0.30

0.35

xV3(

x)


10 5 10 4 10 3 10 2 10 1 100

x

1

0

1

2

xT3(

x)


0.0 0.2 0.4 0.6 0.8x

0.0

0.1

0.2

0.3

0.4

0.5

xT3(

x)



fit results

10 5 10 4 10 3 10 2 10 1 100

x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

xV3(

x)

V3 at 1.6 GeVnnpdf31_qpdf_S2 (68 c.l.+1 )nnpdf31_qpdf_S5 (68 c.l.+1 )NNPDF31_nlo_as_0118 (68 c.l.+1 )

0.0 0.2 0.4 0.6 0.8x

0.0

0.2

0.4

0.6

0.8

xV3(

x)

V3 at 1.6 GeVnnpdf31_qpdf_S2 (68 c.l.+1 )nnpdf31_qpdf_S5 (68 c.l.+1 )NNPDF31_nlo_as_0118 (68 c.l.+1 )

10 5 10 4 10 3 10 2 10 1 100

x

5

4

3

2

1

0

1

xT3(

x)

T3 at 1.6 GeV

nnpdf31_qpdf_S2 (68 c.l.+1 )nnpdf31_qpdf_S5 (68 c.l.+1 )NNPDF31_nlo_as_0118 (68 c.l.+1 )

0.0 0.2 0.4 0.6 0.8x

0.0

0.2

0.4

0.6

0.8

1.0

xT3(

x)

T3 at 1.6 GeVnnpdf31_qpdf_S2 (68 c.l.+1 )nnpdf31_qpdf_S5 (68 c.l.+1 )NNPDF31_nlo_as_0118 (68 c.l.+1 )


outlook

• light-cone PDFs + factorization describe the structure of the proton

• necessary input for the exploitation of LHC, HL-LHC

• current extraction from data is very precise + improving

• lattice data provide complementary information, can be included inglobal fits like any other data

• identify the areas where a significant phenomenological impact fromlattice QCD is possible


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