Matching between TMD and Collinear Factorizations Nobuo Sato - - PowerPoint PPT Presentation

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Matching between TMD and Collinear Factorizations Nobuo Sato - - PowerPoint PPT Presentation

May 11, 2023 150 likes •556 views

Matching between TMD and Collinear Factorizations Nobuo Sato ODU/JLab POETIC 19 LBL 1 / 32 A historical note on DY p T spectrum 2 / 32 3 / 32 pp s = 27 . 4 GeV y = 0 3 / 32 pp s = 27 . 4 GeV y = 0 3 / 32 Fact check... 4 / 32

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SLIDE 1

1 / 32

Matching between TMD and Collinear Factorizations

Nobuo Sato

ODU/JLab

POETIC 19 LBL

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SLIDE 2

2 / 32

A historical note on DY pT spectrum

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SLIDE 3

3 / 32

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SLIDE 4

3 / 32

pp √s = 27.4 GeV y = 0

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SLIDE 5

3 / 32

pp √s = 27.4 GeV y = 0

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SLIDE 6

4 / 32

Fact check...

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SLIDE 7

5 / 32

PDFs from 1978 (PhysRevD.18.3378)

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SLIDE 8

5 / 32

PDFs from 1978 (PhysRevD.18.3378)

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SLIDE 9

5 / 32

PDFs from 1978 (PhysRevD.18.3378)

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SLIDE 10

6 / 32

0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6

xf(x)

u

0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4

d

0.2 0.4 0.6 0.8 0.00 0.25 0.50 0.75

g

0.2 0.4 0.6 0.8

x

1 2 3 4 5

Ratio to CJ15nlo

µ = 5.5 GeV 0.2 0.4 0.6 0.8

x

1 2 3 4 5 HS CJ15nlo 0.2 0.4 0.6 0.8

x

1 2 3 4 5

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SLIDE 11

7 / 32

1 2 3 4 5

qT(GeV)

10−6 10−5 10−4 10−3

dσ/dQ/dy/dp2

T (nb GeV−3)

HS PDFs

Q = 5.5 GeV Q = 7.5 GeV Q = 9.5 GeV

1 2 3 4 5

qT(GeV)

10−6 10−5 10−4 10−3 CJ15nlo PDFs

Q = 5.5 GeV Q = 7.5 GeV Q = 9.5 GeV

1978 2015

Data from Phys.Rev.Lett. 40 (1978) 1117

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SLIDE 12

8 / 32

1 2 3 4 5

qT(GeV)

10−40 10−39 10−38 10−37 10−36

Edσ/d3p (cm2 GeV−2)

HS PDFs

Q = 5.5 GeV Q = 7.5 GeV Q = 9.5 GeV

1 2 3 4 5

qT(GeV)

10−40 10−39 10−38 10−37 10−36 CJ15nlo PDFs

Q = 5.5 GeV Q = 7.5 GeV Q = 9.5 GeV

1978 2015

Data from Phys.Rev. D23 (1981) 604-633

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SLIDE 13

SIDIS

9 / 32 incoming lepton lµ target P µ

  • utgoing lepton l′µ

identified hadron pµ

h

X identified hadron pµ

h

incoming lepton lµ incoming proton P µ

  • utgoing lepton l′µ

exchanged photon q = l − l′ p⊥

h

Lab frame Breit frame

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

Key question : How is p⊥

h generated at

short distances? Different regions are sensitive to distinct physical mechanisms

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SLIDE 14

Nucleon structures accessible in SIDIS

10 / 32

dσ dx dy dΨ dz dφh dP 2

hT

∼

18

  • i=1

Fi(x, z, Q2, P 2

hT )βi

Fi Standard label βi F1 FUU,T 1 F2 FUU,L ε F3 FLL S||λe √ 1 − ε2 F4 F sin(φh+φS)

UT

| S⊥|ε sin(φh + φS) F5 F sin(φh−φS)

UT,T

| S⊥|sin(φh − φS) F6 F sin(φh−φS)

UT,L

| S⊥|ε sin(φh − φS) F7 F cos 2φh

UU

ε cos(2φh) F8 F sin(3φh−ψS)

UT

| S⊥|ε sin(3φh − φS) F9 F cos(φh−φS)

LT

| S⊥|λe √ 1 − ε2 cos(φh − φS) F10 F sin 2φh

UL

S||ε sin(2φh) F11 F cos φS

LT

| S⊥|λe

  • 2ε(1 − ε) cos φS

F12 F cos φh

LL

S||λe

  • 2ε(1 − ε) cos φh

F13 F cos(2φh−φS)

LT

| S⊥|λe

  • 2ε(1 − ε) cos(2φh − φS)

F14 F sin φh

UL

S||

  • 2ε(1 + ε) sin φh

F15 F sin φh

LU

λe

  • 2ε(1 − ε) sin φh

F16 F cos φh

UU

  • 2ε(1 + ε) cos φh

F17 F sin φS

UT

| S⊥|

  • 2ε(1 + ε) sin φS

F18 F sin(2φh−φS)

UT

| S⊥|

  • 2ε(1 + ε) sin(2φh − φS)

Name Symbol meaning

  • upol. PDF

fq

1

  • U. pol. quarks in U. pol. nucleon
  • pol. PDF

gq

1

  • L. pol. quarks in L. pol. nucleon

Transversity hq

1

  • T. pol. quarks in T. pol. nucleon

Sivers f⊥(1)q

1T

  • U. pol. quarks in T. pol. nucleon

Boer-Mulders h⊥(1)q

1

  • T. pol. quarks in U. pol. nucleon

Boer-Mulders h⊥(1)q

1

  • T. pol. quarks in U. pol. nucleon

. . . . . . . . . FF Dq

1

  • U. pol. quarks to U. pol. hadron

Collins H⊥(1)q

1

  • T. pol. quarks to U. pol. hadron

. . . . . . . . .

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SLIDE 15

Regions in SIDIS

11 / 32

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

⊗

incoming quark

  • utgoing

quark detected hadron

small transverse momentum aka W

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SLIDE 16

Regions in SIDIS

12 / 32

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

⊗

incoming quark

  • utgoing

quark detected hadron

large transverse momentum aka FO (=fixed order)

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SLIDE 17

Regions in SIDIS

13 / 32

small transverse momentum

W

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions ⊗

incoming quark

  • utgoing

quark detected hadron

large transverse momentum

FO

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions ⊗

incoming quark

  • utgoing

quark detected hadron

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SLIDE 18

Regions in SIDIS

13 / 32

small transverse momentum

W

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions ⊗

incoming quark

  • utgoing

quark detected hadron

large transverse momentum

FO

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions ⊗

incoming quark

  • utgoing

quark detected hadron

matching region aka ASY (=asymptotic)

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SLIDE 19

What is large or small transverse momentum?

14 / 32

Scale separation qT/Q z = P · ph P · q , qT = p⊥

h /z

Merging factorization theorems dσ dxdQ2dzdp⊥

h

= W + FO − ASY + O(m2/Q2) ∼ W for qT ≪ Q ∼ FO for qT ∼ Q p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

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SLIDE 20

Small transverse momentum Collins, Rogers PRD91 (2015)

15 / 32

W =

  • f

Hf(Q, µ)

d2bT

(2π)2 e−iqT·bTFf/N(x, bT, µ, ζF )Dh/f(z, bT, µ, ζD) + O(q2

T/Q2)

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SLIDE 21

Small transverse momentum Collins, Rogers PRD91 (2015)

15 / 32

W =

  • f

Hf(Q, µ)

d2bT

(2π)2 e−iqT·bTFf/N(x, bT, µ, ζF )Dh/f(z, bT, µ, ζD) + O(q2

T/Q2)

CSS evolution equation ∂ ln Ff/N(x, bT, ζF , µ) ∂ ln √ζF = ˜ K(bT, µ) + Related to vacuum matrix elements of products of Wilson Lines + Independent of flavor, target and spin + Independent of x + Universal across TMDs and processes

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SLIDE 22

Small transverse momentum Collins, Rogers PRD91 (2015)

15 / 32

W =

  • f

Hf(Q, µ)

d2bT

(2π)2 e−iqT·bTFf/N(x, bT, µ, ζF )Dh/f(z, bT, µ, ζD) + O(q2

T/Q2)

CSS evolution equation ∂ ln Ff/N(x, bT, ζF , µ) ∂ ln √ζF = ˜ K(bT, µ) + Related to vacuum matrix elements of products of Wilson Lines + Independent of flavor, target and spin + Independent of x + Universal across TMDs and processes RG equations

d ˜ K(bT, µ) d ln µ = −γK(αS(µ)) d ln Ff/N(bT, µ) d ln µ = γf(αS(µ), 1) − 1 2γK(αS(µ)) ln ζF µ2 d d ln µ ln H(Q, µ) = −2γf(αS(µ), 1) + γK(αS(µ)) ln Q2 µ2

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SLIDE 23

Small transverse momentum Collins, Rogers PRD91 (2015)

16 / 32

W =

  • f

Hf(Q, µ)

d2bT

(2π)2 e−iqT·bT × e−gf/N(x,bT,bmax)

1

x

dˆ x ˆ x ff/N(ˆ x, µb∗) ˜ Cf/p(x/ˆ x, b∗, µ2

b∗, αS(µb∗))

× e−gh/f(z,bT,bmax)

1

z

dˆ z ˆ z3 dh/f(ˆ z, µb∗) ˜ Ch/f(z/ˆ z, b∗, µ2

b∗, αS(µb∗))

×

  • Q2

Q2

−gK(bT,bmax)

Q2 µ2

b∗

˜

K(b∗,µb∗)

× exp

µQ

µb∗

dµ′ µ′

  • 2γ(αS(µ′), 1) − ln Q2

(µ′)2 γK(αS(µ′))

  • Valid for 0 ≤ qT ≪ Q

Quantities in pink are non-perturbative

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SLIDE 24

Large transverse momentum

17 / 32

FO =

  • q

e2

q

1

q2 T Q2 xz 1−z +x

dξ ξ − x H(ξ) fq(ξ, µ) dq(ζ(ξ), µ) + O(α2

S) + O(m2/q2)

+ Attention:

  • q2

T

Q2 xz 1−z + x

  • < ξ < 1

+ large qT probes large ξ in PDFs + Can be useful in collinear global fits Valid for qT ∼ Q

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SLIDE 25

Matching region

18 / 32

ASY ∼ d(z; µ)

1

x

dξ ξ f(ξ; µ)P(x/ξ) + f(x; µ)

1

z

dζ ζ d(ζ; µ)P(z/ζ) + 2CF f(x; µ)d(z; µ)

  • ln
  • Q2

qT

  • − 3

2

  • + Interpolates between W and FO

+ FO − ASY ≡ Y Valid for 0 ≪ qT ≪ Q

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SLIDE 26

Toy example: how we expect the W+FO-ASY to work

19 / 32

0.0 0.5 1.0 1.5 2.0 2.5 3.0

qT (GeV)

10−4 10−3 10−2 10−1 100 Q = 2.0 (GeV) FO |AY| Y |W| W + Y

dσ dxdQ2dzdp⊥

h

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SLIDE 27

Existing phenomenology

20 / 32 Anselmino et al Bacchetta et al

These analyses used only W (Gaussian, CSS) → no FO nor ASY Samples with qT/Q ∼ 1.63 have been included BUT TMDs are only valid for qT/Q ≪ 1 !

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SLIDE 28

FO @ LO predictions (DSS07) Gonzalez, Rogers, NS, Wang PRD98 (2018)

21 / 32 2 4 6 8 10 2 4 6 2 4 6 8 10

Q2 (GeV2) xbj

0.007 0.010 0.016 0.03 0.04 0.07 0.15 0.27 1.3 1.8 3.5 8.3 20.0 2 4 6 2 4 6 8 10

COMPASS 17 h+ data/theory(LO) vs. qT (GeV)

PDF : CJ15 FF : DSS07

qT > Q

2 4 6 2 4 6 2 4 6 8 10

< z >= 0.24 < z >= 0.34 < z >= 0.48 < z >= 0.68

2 4 6 2 4 6 2 4 6 8 10 2 4 6 2 4 6

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

?

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SLIDE 29

Trouble with large transverse momentum

22 / 32

FO =

  • q

e2

q

1

q2 T Q2 xz 1−z +x

dξ ξ − x H(ξ) fq(ξ, µ) dq(ζ(ξ), µ) + O(α2

S) + O(m2/q2)

+ FFs needs to be updated?

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SLIDE 30

FO @ LO predictions (DSS07) Gonzalez, Rogers, NS, Wang PRD98 (2018)

23 / 32 2 4 6 8 10 2 4 6 2 4 6 8 10

Q2 (GeV2) xbj

0.007 0.010 0.016 0.03 0.04 0.07 0.15 0.27 1.3 1.8 3.5 8.3 20.0 2 4 6 2 4 6 8 10

COMPASS 17 h+ data/theory(LO) vs. qT (GeV)

PDF : CJ15 FF : DSS07

qT > Q

2 4 6 2 4 6 2 4 6 8 10

< z >= 0.24 < z >= 0.34 < z >= 0.48 < z >= 0.68

2 4 6 2 4 6 2 4 6 8 10 2 4 6 2 4 6

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

?

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SLIDE 31

FO @ LO predictions (JAM18) Gonzalez, Rogers, NS, Wang PRD98 (2018)

24 / 32 2 4 6 8 10 2 4 6 2 4 6 8 10

Q2 (GeV2) xbj

0.007 0.010 0.016 0.03 0.04 0.07 0.15 0.27 1.3 1.8 3.5 8.3 20.0 2 4 6 2 4 6 8 10

COMPASS 17 h+ data/theory(NLO) vs. qT (GeV)

PDF : JAM18 FF : JAM18

qT > Q

2 4 6 2 4 6 2 4 6 8 10

< z >= 0.24 < z >= 0.34 < z >= 0.48 < z >= 0.68

2 4 6 2 4 6 2 4 6 8 10 2 4 6 2 4 6

data/theory(LO) vs. qT GeV

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

?

slide-32
SLIDE 32

Trouble with large transverse momentum

25 / 32

FO =

  • q

e2

q

1

q2 T Q2 xz 1−z +x

dξ ξ − x H(ξ) fq(ξ, µ) dq(ζ(ξ), µ) + O(α2

S) + O(m2/q2)

+ O(α2

S) corrections might be important

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SLIDE 33
  • rder α2

S corrections to FO

26 / 32

dsp/dpT (pb/GeV)

2 £ Q2 £ 4.5 GeV2 1 10 102

KKP NLO KKP LO K NLO K LO

4.5 £ Q2 £ 15 GeV2 1 10 102

pT (GeV)

15 £ Q2 £70 GeV2 1 10 102 3 4 5 6 7 8 9 10 15

ff ff fi

Daleo,et al. (2005) PRD.71.034013

There are strong indications that order α2

S corrections are

very important An order of magnitude correction at small pT . As a sanity check, we need to have an independent calculation

slide-34
SLIDE 34

O(α2

S) calculation (J. Gonzalez-Hernandes, T.C Rogers, NS, B. Wang - PRD99 (2019))

27 / 32

W µν(P, q, PH) = 1+

x−

dξ ξ 1+

z−

dζ ζ2 ˆ W µν

ij (q, x/ξ, z/ζ)fi/P (ξ)dH/j(ζ)

{Pµν

g

ˆ W (N)

µν ; Pµν P P ˆ

W (N)

µν } ≡

1 (2π)4

  • {|M 2→N

g

|2; |M 2→N

pp

|2} dΠ(N) − Subtractions Born/Virtual Real Generate all 2 → 2 and 2 → 3 squared amplitudes Evaluate 2 → 2 virtual graphs (Passarino-Veltman) Integrate 3-body PS analytically Check cancellation of IR poles

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SLIDE 35

FO @ LO predictions (JAM18)

28 / 32 2 4 6 8 10 2 4 6 2 4 6 8 10

Q2 (GeV2) xbj

0.007 0.010 0.016 0.03 0.04 0.07 0.15 0.27 1.3 1.8 3.5 8.3 20.0 2 4 6 2 4 6 8 10

COMPASS 17 h+ data/theory(NLO) vs. qT (GeV)

PDF : JAM18 FF : JAM18

qT > Q

2 4 6 2 4 6 2 4 6 8 10

< z >= 0.24 < z >= 0.34 < z >= 0.48 < z >= 0.68

2 4 6 2 4 6 2 4 6 8 10 2 4 6 2 4 6

data/theory(LO) vs. qT GeV

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

?

slide-36
SLIDE 36

FO @ NLO (JAM18)

29 / 32 2 4 6 8 10 2 4 6 2 4 6 8 10

Q2 (GeV2) xbj

0.007 0.010 0.016 0.03 0.04 0.07 0.15 0.27 1.3 1.8 3.5 8.3 20.0 2 4 6 2 4 6 8 10

COMPASS 17 h+ data/theory(NLO) vs. qT (GeV)

PDF : JAM18 FF : JAM18

qT > Q

2 4 6 2 4 6 2 4 6 8 10

< z >= 0.24 < z >= 0.34 < z >= 0.48 < z >= 0.68

2 4 6 2 4 6 2 4 6 8 10 2 4 6 2 4 6

p⊥

h

yh

Current fragmentation TMD factorization Current fragmentation Collinear factorization Soft region ???? Target region Fracture functions

?

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SLIDE 37

Understanding the large x

(J. Gonzalez-Hernandes, T.C Rogers, NS, B. Wang - PRD99 (2019))

30 / 32

2 4 6

F NLO

1

/F LO

1

z = 0.2 qT = Q z = 0.8 qT = Q

0.01 0.1 2 4 6

z = 0.2 qT = 2Q

0.01 0.1

x

z = 0.8 qT = 2Q

Q = 2 GeV Q = 20 GeV

Large corrections threshold corrections are observed The x at the minimum can be used as an indicator of where such corrections are expected to be large

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SLIDE 38

Understanding the large x

(J. Gonzalez-Hernandes, T.C Rogers, NS, B. Wang - PRD99 (2019))

31 / 32

COMPASS kinematics

0.01 0.1

x

1 2 3 4 5 6 7

qT/Q

< z >= 0.24

0.01 0.1

x

1 2 3 4 5 6 7

< z >= 0.48

0.01 0.1

x

1 2 3 4 5 6 7

< z >= 0.69

x > x0 x ≤ x0

The blue region might receive large threshold corrections This can potential explain why the O(α2

S) fail to describe the data at

large x

slide-39
SLIDE 39

Summary and outlook (the next steps)

32 / 32

A new combined PDF/FF (charged hadrons) global analysis with the inclusion of COMPASS qT data is on the way (JAM19+) Need to identify regions of kinematics where FO and data are compatible Explore/combined alternative approaches based on power corrections → Liu, Qiu (arXiv:1907.06136) SIDIS region pheno analysis based on tools developed at Boglione, et al. (arXiv:1904.12882)