Applications of the WW-type approximation to SIDIS

Kemal Tezgin

University of Connecticut

kemal.tezgin@uconn.edu

Based on: [Arxiv:1807.10606]

Co-authors: S. Bastami, H. Avakian, A. V. Efremov, A. Kotzinian, B. U. Musch,B. Parsamyan, A. Prokudin, M. Schlegel, G. Schnell, P. Schweitzer

DIS 2019, Torino, 8-12 April 2019

April 10, 2019

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 1 / 15

SIDIS Cross Section

Consider the SIDIS process l + N → l ′ + h + X

Θ

����������������������

����������������������

����������

����������

z−axis

hφS

φh

Ph

l’

l

q

HADRON PRODUCTION PLANE

LEPTON SCATTERING PLANE

N

S

S

In single photon exchange approximation, SIDIS cross section can beexpressed by 18 structure functions (SFs) [Kotzinian ’95, Mulders andTangerman ’96, Review: Bacchetta et al. JHEP 02 (2007)]

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 2 / 15

SIDIS Cross Section

TMD factorization for Ph⊥ � Q: Structure functions can be described byconvolutions of TMDs and FFs. Generically

C[ω f D

]= x

∑a

e2a

∫d2k⊥d2P⊥δ(2)(zk⊥+P⊥−Ph⊥) ωf a(x , k2

⊥) Da(z ,P2⊥)

1 8 structure functions → twist-2 (factorization X)2 8 structure functions → twist-3 (factorization assumed)

SIDIS CS can be expressed by1 8 twist-2 TMDs: f1, g1, h1, f

⊥1T , g

⊥1T , h

⊥1T , h

⊥1L, h

⊥1

2 16 twist-3 TMDs: e, eL, eT , f⊥, gT , ...

3 2 twist-2 FFs: D1,H⊥1

4 4 twist-3 FFs: D⊥,G⊥,H,E

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 3 / 15

What is WW Approximation?

First application of EoM [Wandzura and Wilczek ’77]:

gqT (x)︸ ︷︷ ︸

twist−3

= < qq >︸ ︷︷ ︸related to gq

1 (x)

+< qgq >︸ ︷︷ ︸”tilde”term

The approximation is ∣∣∣< qgq >

< qq >

∣∣∣� 1.

Neglecting ”tilde” terms is known to be WW approximation

gqT (x) =

∫ 1

x

dy

ygq

1 (y) + gqT (x)

WW≈∫ 1

x

dy

ygq

1 (y)

Analogously [Jaffe and Ji ’92]

hqL(x) = 2x

∫ 1

x

dy

y2hq1 (y) + hqL(x)

WW≈ 2x

∫ 1

x

dy

y2hq1 (y)

WW approximation is supported by:Instanton model of QCD vacuum [Balla et al. NPB 510 (1998)]Lattice data [Gockeler et al. PRD 63 (2001)]Quark models [Bag, LFCM, Spectator, ...]

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 4 / 15

WW Approximation

Illustration

g2(x) =1

2

∑a

e2a

(g aT (x)− g a

1 (x))

WW≈ d

dx

[x

∫ 1

x

dy

yg1(y)

]

Figure: The structure function xg2(x) in WWapproximation at Q2 = 7.1GeV 2 vs. dataE143[PRD 58 (1998)], E155[PLB 553 (2003)]

Figure: The structure function xg2(x) in WWapproximation at Q2 = 2.4GeV 2 vs. HERMESdata[EPJC 72 (2012)].

holds with 40% or better [Accardi et al. JHEP 11 (2009)]

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 5 / 15

WW-type Approximation

EoM allow us to decompose twist-2 h⊥1L, g⊥1T and all twist-3 TMDs, FFs intoqq and qgq-matrix elements.

Assuming| < qgq > | � | < qq > |

We obtaintwist-2:

g⊥(1)1T (x)

WW−type≈ x

∫ 1

x

dy

yg1(y), h

⊥(1)1L (x)

WW−type≈ −x2

∫ 1

x

dy

y2h1(y).

twist-3:

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 6 / 15

WW-type Approximation

As a result of WW-type approximation, all SIDIS SFs (twist-2 & twist-3)can be expressed in terms of 8 basis functions:

1 6 leading-twist TMDs: f1, g1, h1, f⊥

1T , h⊥1 , h

⊥1T

2 2 leading-twist FFs: D1,H⊥1

Our goal is to check where the WW-type approximation works and where itdoes notWe use state-of-the-art parametrizations for the basis functions [MSTW, DSS,Anselmino et al., Barone et al., Lefky and Prokudin; Gauss Ansatz used, parameters fixedusing lattice data Hagler et al.]

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 7 / 15

WW-type Approximation at Leading Twist

At leading twist, WW-type approximation is useful for 2 structure functions:

Fcos(φh−φS )LT and F

sin(2φh)UL .

Fcos(φh−φS )LT = C

[h·~k⊥MN

g⊥1TD1

].

First application of WW-type approximation [Kotzinian, Parsamyan, Prokudin,

PRD 73 (2006)]Results are compatible with the recent preliminary COMPASS data[Parsamyan, PoS DIS2013 (2013) 231].

-210 -110 1

-0.2

-0.1

0

0.1

0.2preliminaryCOMPASS +h

Proton 2010 data

)Sϕ-

hϕco

s(

LTA

x

PRD73:114017(2006)

arXiv:0806.3804 [hep-ph]PRD79:094012(2009)

PRD73:114017(2006) updated

-210 -110 1

-0.2

-0.1

0

0.1

0.2COMPASS preliminary-h

)S

-ϕh

cos(

ϕLT

Ax

PRD73:114017(2006)PRD73:114017(2006) updatedarXiv:0806.3804 [hep-ph]PRD79:094012(2009)

Proton 2010 data

quark-diquark model [Kotzinian, arXiv:0806.3804] and light-cone quarkconstituent model [Boffi et al., Phys. Rev. D 79, 094012], are also displayed forcomparison.

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 8 / 15

WW-type Approximation at Leading Twist

Similarly, in WW-type approximation: Fsin(2φh)UL = C

[ω h⊥1LH

⊥1

]Results [Avakian et al., PRD 77 (2008)] are compatible with preliminary COMPASS

[Parsamyan, PoS DIS2017 (2018) 259] and JLab π0 [Jawalkar et al., PLB 782 (2018)]data

2−10 1−10

0.05−

0

0.05

+hpreliminaryCOMPASS

HERMES PRL 84(2000) D(y)-rescaledPRD 77, 014023(2008)

hφsi

n 2

UL

A

x2−10 1−10

0.05−

0

0.05

−h > 0.2z

x

π0

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

x

AUL,⟨y⟩

sin(2Φh)

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 9 / 15

WW-type Approximation at Subleading Twist

Subleading twist → Complexity. Example (expression symbolic):

Fcos(φS )LT =

MN

Q

[gT D1 + h1 E + eT H⊥1 + g⊥1T D⊥ + e⊥T H⊥1 + f ⊥1T G⊥

]WW-type approximation → Crucial simplification

Fcos(φS )LT

WW−type≈ MN

Q

[gT D1

]∣∣∣∣gT→g1

Results are compatible with COMPASS data [Parsamyan, PoS DIS2013 (2013)231].

COMPASS preliminaryProton 2010 data

-210 -110 1

-0.2

-0.1

0

0.1

0.2h+

Sco

sϕ

LTA

x

arXiv:0806.3804 [hep-ph]

π+

π-

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-0.06

-0.04

-0.02

0.00

0.02

0.04

x

ALTcos(ΦS)

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 10 / 15

WW-type Approximation at Subleading Twist

Fcos(2φh−φS )LT =

MN

Q

[eT H⊥1 +g⊥1T D⊥+e⊥T H⊥1 +f ⊥1T G⊥+g⊥T D1 +h⊥1T E

]WW-type approximation → Crucial simplification

Fcos(2φh−φS )LT

WW−type≈ MN

Q

[g⊥T D1

]∣∣∣∣g⊥T →g1

Results are not in contradiction with COMPASS data [Parsamyan, PoS DIS2013(2013) 231].

COMPASS preliminaryProton 2010 data

-210 -110 1

-0.2

-0.1

0

0.1

0.2h+

)S

-ϕh

cos(

2ϕLT

A

x

arXiv:0806.3804 [hep-ph]

π+

π-

0.00 0.05 0.10 0.15 0.20 0.25 0.30-0.010

-0.005

0.000

0.005

0.010

x

ALTcos(2Φh-ΦS)

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 11 / 15

WW-type Approximation at Subleading Twist

Fcos(φh)LL =

MN

Q

[eL H⊥1 + g1 D⊥ + g⊥L D1 + h⊥1L E

]In WW-type approximation

F cosφh

LL

WW−type≈ MN

Q

[g⊥L D1

]∣∣∣∣g⊥L →g1

WW-type results [Anselmino et al., Phys. Rev. D74 (2006)] are compatible withCOMPASS preliminary data [Parsamyan, PoS DIS2017 (2018) 259].

2−10 1−10

0.1−

0.05−

0

0.05

0.1 +hpreliminaryCOMPASS

PRD 74, 074015(2006)NPA 945(2016) 153

hφco

s

LL

A

x2−10 1−10

0.1−

0.05−

0

0.05

0.1 −h > 0.1z

x

A model study [Mao et al., Nucl. Phys. A945 (2016)] is also displayed forcomparison.

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 12 / 15

WW-type Approximation at Subleading Twist

F sinφhUL =

MN

Q

[hL H⊥1 + g1 G⊥ + f ⊥L D1 + h⊥1L H

]In WW-type approximation

F sinφh

UL

WW−type≈ MN

Q

[hL H⊥1

]∣∣∣∣hL→h1

We observe discrepancies with HERMES π± [Airapetian et al., Phys. Lett. B622

(2005)] and JLAB π0 [Jawalkar et al., Phys. Lett. B782 (2018)] data(underestimate π+, can not describe π0).

π+

π-

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-0.05

0.00

0.05

x

AUL,<y>

sin(Φh)

π0

0.0 0.2 0.4 0.6 0.8 1.0-0.10

-0.05

0.00

0.05

0.10

x

AUL,⟨y⟩

sin(Φh)

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 13 / 15

WW-type Approximation at Subleading Twist

F sinφh

LU = MQ

[e H⊥1 + f1 G⊥ + g⊥ D1 + h⊥1 E

]WW−type≈ 0

0

0.05

0.1

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6

CLAS, π+

x

Avakian et al, PRD69 (2004) 112004ALUsin φh(x)

Figure: CLAS data shows a non-zero asymmetry

F cosφh

UU = MQ

[hH⊥1 + f1D

⊥+ f ⊥D1 +h⊥1 H]

WW≈ M

Q

[hH⊥1 + f ⊥D1

]∣∣∣∣h→h⊥1 ,f

⊥→f1

Figure: HERMES data [Airapetian et al., Phys. Rev. D87 (2013)]

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 14 / 15

Conclusions

twist-3 PDFs: WW approximation works, supported by experiment for gT ,instanton calculus, lattice QCD

twist-2 TMDs: applicable for gear worms g⊥1T , h⊥1L, in agreement with data

Due to the complexity of SIDIS SFs at subleading-twist, a guideline is muchneeded. Predictions were made for all SIDIS asymmetries at subleadingtwist.

Observations: compatible with data in several cases (e.g. AcosφS

LT , ...)

Not in a agreement with data in some cases (e.g. Asinφh

LU , ....)

More work is needed to reliably test where WW-type approximation works

If we find it works: it will motivate theoretical studies to explain thesuppression mechanism for qgq. If we find it does not work: it will help toidentify large qgq terms, and stimulate studies to find out why they arelarge. We learn in any case!

Mathematica package: https://github.com/prokudin/WW-SIDIS

Kemal Tezgin (University of Connecticut) WW-type approximation to SIDIS April 10, 2019 15 / 15