Saturation Physics on the Energy Frontier arxiv:1505.05183 (to appear - - PowerPoint PPT Presentation

SLIDE 1

Saturation Physics on the Energy Frontier

arxiv:1505.05183 (to appear in Phys. Rev. D)

David Zaslavsky

with Kazuhiro Watanabe, Bo-Wen Xiao, Feng Yuan

Central China Normal University

APS DPF Meeting — August 6, 2015

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1

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Saturation and pA Collisions

Saturation

x = 1 ln 1

x

small x small Q ln Q2

Q2

large Q saturation

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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2

• f 16

Saturation and pA Collisions

Advantages of pA

p A hadron X

x = 1 ln 1

x

small x small Q ln Q2

Q2

large Q saturation

Q2 Q2

s = cA1/3Q2

x0 x λ Heavy target: large A Light projectile: no medium

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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2

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Saturation and pA Collisions

Advantages of pA

p A hadron X

x = 1 ln 1

x

small x small Q ln Q2

Q2

large Q saturation

Q2 Q2

s = cA1/3Q2

x0 x λ Heavy target: large A Light projectile: no medium

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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3

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Saturation and pA Collisions

Hybrid Model

Cross section in the hybrid formalism: d3σ dY d2 p⊥ =

• i

dz z2 dx x xfi(x, µ)Dh/i(z, µ)F

• x, p⊥

z

• P(ξ)(. . .)

Parton distribution (initial state projectile) Dipole gluon distribution (initial state target) Fragmentation function (final state) Perturbative factors

z p xppp k ph h X A xgpA z

figure adapted from Dominguez 2011. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Inclusive Cross Section

History of the pA Calculation

Dumitru and Jalilian-Marian (2002) Dumitru, Hayashigaki, et al. (2006) Fujii et al. (2011) Albacete et al. (2013) Rezaeian (2013) Sta´ sto, Xiao, and Zaslavsky (2014) Kang et al. (2014) Sta´ sto, Xiao, Yuan, et al. (2014) Altinoluk et al. (2014) Watanabe et al. (2015)

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 7

5

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Inclusive Cross Section

First Calculation

Dumitru and Jalilian-Marian (2002)

p A

No numerical results

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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6

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Inclusive Cross Section

First Numerical Results

Dumitru, Hayashigaki, et al. (2006)

1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 10 0.5 1 1.5 2 2.5 3 3.5 4 dN/dy d2pt [GeV-2] pt [GeV] Minimum bias, K = 1.6 dAu BRAHMS min. bias data (h-) at y=3.2 x- and DGLAP-evolution MV model No DGLAP-evolution

Prefactor K = 1.6

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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7

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Inclusive Cross Section

Inelastic Diagrams

Leading:

p A

Next-to-leading:

p A p A p A p A

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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8

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Inclusive Cross Section

Inelastic NLO Terms

Albacete et al. (2013)

1 2 3 4 5 0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 BRAHMS η=2.2 h± (x200). K-factor=1 BRAHMS η=3.2 h± (x50). K-factor=1 STAR η=4 π'0. K-factor=0.4

• nly elastic

elas+inelas α=0.1 elas+inelas α(Q=pt)

pt (GeV) dN/dη/d2pt (GeV-2)

dAu @ 200 GeV

g=1.119 i.c

Prefactor K = 1 for charged hadrons K = 0.4 for neutral hadrons

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Inclusive Cross Section

NLO Diagrams

Leading:

p A

Next-to-leading:

p A p A p A p A p A p A p A p A

Chirilli et al. 2012, 1203.6139. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Inclusive Cross Section

NLO Numerical Result

Sta´ sto, Xiao, and Zaslavsky (2014)

1 2 3 10−7 10−5 10−3 10−1 101 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

BRAHMS η = 3.2 LO NLO data

Includes virtual corrections K = 1

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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11

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Inclusive Cross Section

Kinematical Constraint

Watanabe et al. (2015) p A 1 − ξ First LHC numerical results

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Alternate derivation: Altinoluk et al. 2014, 1411.2869. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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12

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Inclusive Cross Section

Challenges for Numerical Calculation

Singularities

1

τ

dz 1

τ z

dξ Fs(z, ξ) (1 − ξ)+ + Fn(z, ξ) + Fd(z, ξ)δ(1 − ξ)

• Watanabe et al. 2015, 1505.05183.

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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12

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Inclusive Cross Section

Challenges for Numerical Calculation

Singularities

1

τ

dz 1

τ z

dξ Fs(z, ξ) (1 − ξ)+ + Fn(z, ξ) + Fd(z, ξ)δ(1 − ξ)

• Fourier integrals
• d2

s⊥d2 t⊥ei

l⊥· s⊥ei l′

⊥·

t⊥(. . .)

Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Inclusive Cross Section

Challenges for Numerical Calculation

Singularities

1

τ

dz 1

τ z

dξ Fs(z, ξ) (1 − ξ)+ + Fn(z, ξ) + Fd(z, ξ)δ(1 − ξ)

• Fourier integrals
• d2

s⊥d2 t⊥ei

l⊥· s⊥ei l′

⊥·

t⊥(. . .)

Leading Order Cancellations

O

• k−2

⊥

• − O
• k−2

⊥

• → O
• k−4

⊥

• ...plus Monte Carlo statistical error

Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Results

RHIC Results

0.5 1 1.5 2 2.5 3 10−7 10−5 10−3 10−1 101 y = 2.2 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg BRAHMS 0.5 1 1.5 2 2.5 3 y = 2.2 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg BRAHMS

New terms improve matching at low p⊥

data: Arsene et al. 2004, nucl-ex/0403005. plots: Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Results

RHIC Results

0.5 1 1.5 2 2.5 3 10−7 10−5 10−3 10−1 101 y = 3.2 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

LO +NLO +Lq + Lg BRAHMS 0.5 1 1.5 2 2.5 3 y = 3.2 p⊥[GeV] LO +NLO +Lq + Lg BRAHMS

New terms improve matching at low p⊥

data: Arsene et al. 2004, nucl-ex/0403005. plots: Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Results

LHC Results

1 2 3 4 5 6 10−6 10−5 10−4 10−3 10−2 10−1 100 101 y = 1.75 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg ATLAS 1 2 3 4 5 6 y = 1.75 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg ATLAS

rcBK calculation matches neatly up to p⊥ ≈ 6 GeV

data: Milov 2014, 1403.5738. plots: Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Results

Importance of Higher Rapidity

1 2 3 4 5 6 10−6 10−5 10−4 10−3 10−2 10−1 100 101 y = 1.75 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg ATLAS y = 1.75 1 2 3 4 5 6 y = 1.75 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg ATLAS y = 1.75

Higher rapidity alters low-p⊥ result

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Results

Importance of Higher Rapidity

1 2 3 4 5 6 10−6 10−5 10−4 10−3 10−2 10−1 100 101 y = 2.5 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg ATLAS y = 1.75 1 2 3 4 5 6 y = 2.5 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg ATLAS y = 1.75

Higher rapidity alters low-p⊥ result

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Results

Importance of Higher Rapidity

1 2 3 4 5 6 10−6 10−5 10−4 10−3 10−2 10−1 100 101 y = 3.5 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg ATLAS y = 1.75 1 2 3 4 5 6 y = 3.5 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg ATLAS y = 1.75

Higher rapidity alters low-p⊥ result

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Conclusion

Summary

d5σpA→hX dY d2p⊥d2b⊥ = dzdx z2 q(x, Q2

f)Dq/h(z, Q2 f)

d5σtot

qA

dYqd2q⊥d2b⊥ 0.5 1 1.5 2 2.5 3 10−7 10−5 10−3 10−1 101 y = 3.2 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

Complete numerical implementation of NLO pA → h + X

Critical step

More forward-rapidity data from LHC experiments

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 24

Supplemental Slides

full expressions additional history

rcBK rapidity divergence collinear matching Ioffe time

numerical challenges

singularities Fourier integrals new Fourier transforms

• ther numerical errors

sources of negativity kinematical constraint beam direction LHC results

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Hard Factors

Full NLO Cross Section

Complete NLO corrections to the cross section for pA → h + X: d3σ dY d2 p⊥ = dzdξ z2

• xqi(x, µ)

xg(x, µ) Sqq Sqg Sgq Sgg Dh/qi(z, µ) Dh/g(z, µ)

• Sjk =
• d2

r⊥ (2π)2 S(2)

Y (r⊥)H(0) 2jk

LO dipole + αs 2π

• d2

r⊥ (2π)2 S(2)

Y (r⊥)H(1) 2jk

NLO dipole + αs 2π d2 s⊥d2 t⊥ (2π)2 S(4)

Y (r⊥, s⊥, t⊥)H(1) 4jk

NLO quadrupole + · · · etc.

Note: we also use S(4)

Y

(r⊥, s⊥, t⊥) → S(2)

Y

(s⊥)S(2)

Y

(t⊥) Chirilli et al. 2012, 1203.6139. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Hard Factors

Quark-Quark Channel

Sqq =

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)H(0)

2qq +

αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)H(1)

2qq

+ αs 2π

• d2

s⊥d2 t⊥ (2π)2 S(4)

Y

(r⊥, s⊥, t⊥)H(1)

4qq

H(0)

2qq = e−i k⊥· r⊥ δ(1 − ξ)

H(1)

2qq = CF Pqq(ξ)

• e−i

k⊥· r⊥ +

1 ξ2 e−i

k⊥· r⊥/ξ

• ln

c2 r2

⊥µ2 − 3CF e−i k⊥· r⊥ δ(1 − ξ) ln

c2 r2

⊥k2 ⊥

H(1)

4qq = −4πNce−ik⊥·r⊥

• e

−i 1−ξ ξ k⊥·(x⊥−b⊥)

1 + ξ2 (1 − ξ)+ 1 ξ x⊥ − b⊥ (x⊥ − b⊥)2 · y⊥ − b⊥ (y⊥ − b⊥)2 − δ(1 − ξ) 1 dξ′ 1 + ξ′2 (1 − ξ′)+ e−i(1−ξ)k⊥·(y⊥−b⊥) (b⊥ − y⊥)2 − δ(2)(b⊥ − y⊥)

• d2r′

⊥

e−ik⊥·r′

⊥

r′2

⊥

• where

Pqq(ξ) = 1 + ξ2 (1 − ξ)+ + 3 2 δ(1 − ξ) Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Hard Factors

Gluon-Gluon Channel

Sgg =

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)S(2)

Y

(r⊥)H(0)

2gg +

αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)H(1)

2gg +

αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)H(1)

2q¯ q

+ αs 2π

• d2

s⊥d2 t⊥ (2π)2 S(2)

Y

(r⊥)S(2)

Y

(s⊥)S(2)

Y

(t⊥)H(1)

6gg

H(0)

2gg = e−i k⊥· r⊥ δ(1 − ξ)

H(1)

2gg = Nc

• 2ξ

(1 − ξ)+ + 2(1 − ξ) ξ + 2ξ(1 − ξ) + 11 6 − 2Nf TR 3Nc

• δ(1 − ξ)
• × ln

c2 µ2r2

⊥

• e−i

k⊥· r⊥ +

1 ξ2 e

−i

• k⊥

ξ · r⊥

• −

11 3 − 4Nf TR 3Nc

• Ncδ(1 − ξ)e−i

k⊥· r⊥ ln

c2 r2

⊥k2 ⊥

H(1)

2q¯ q = 8πNf TRe−i k⊥·( y⊥− b⊥)δ(1 − ξ)

× 1 dξ′[ξ′2 + (1 − ξ′)2] e−iξ′

k⊥·( x⊥− y⊥)

( x⊥ − y⊥)2 − δ(2)( x⊥ − y⊥)

• d2

r′

⊥

ei

k⊥· r′ ⊥

r′2

⊥

• Saturation Physics on the Energy Frontier

David Zaslavsky — CCNU

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3

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Hard Factors

Gluon-Gluon Channel

Sgg =

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)S(2)

Y

(r⊥)H(0)

2gg +

αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)H(1)

2gg +

αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)H(1)

2q¯ q

+ αs 2π

• d2

s⊥d2 t⊥ (2π)2 S(2)

Y

(r⊥)S(2)

Y

(s⊥)S(2)

Y

(t⊥)H(1)

6gg

H(1)

6gg = −16πNce−i k⊥· r⊥

• e

−i

• k⊥

ξ ·( y⊥− b⊥) [1 − ξ(1 − ξ)]2

(1 − ξ)+ 1 ξ2

• x⊥ −

y⊥ ( x⊥ − y⊥)2 ·

• b⊥ −

y⊥ ( b⊥ − y⊥)2 − δ(1 − ξ) 1 dξ′

• ξ′

(1 − ξ′)+ + 1 2 ξ′(1 − ξ′)

• e−iξ′

k⊥·( y⊥− b⊥)

( b⊥ − y⊥)2 − δ(2)( b⊥ − y⊥)

• d2

r′

⊥

ei

k⊥· r′ ⊥

r′2

⊥

• Saturation Physics on the Energy Frontier

David Zaslavsky — CCNU

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Hard Factors

Quark-Gluon Channel

Sgq = αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)[H(1,1)

2gq

+ S(2)

Y

(r⊥)H(1,2)

2gq ]

+ αs 2π

• d2

s⊥d2 t⊥ (2π)2 S(4)

Y

(r⊥, s⊥, t⊥)H(1)

4gq

H(1,1)

2gq

= Nc 2 1 ξ2 e−i

k⊥· r⊥/ξ 1

ξ

• 1 + (1 − ξ)2

ln c2 r2

⊥µ2

H(1,2)

2gq

= Nc 2 e−i

k⊥· r⊥ 1

ξ

• 1 + (1 − ξ)2

ln c2 r2

⊥µ2

H(1)

4gq = 4πNce−i k⊥· r⊥/ξ−i k⊥· t⊥ 1

ξ

• 1 + (1 − ξ)2

r⊥ r2

⊥

·

• t⊥

t2

⊥

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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5

• f 20

Hard Factors

Gluon-Quark Channel

Sqg = αs 2π

• d2

r⊥ (2π)2 S(2)

Y

(r⊥)[H(1,1)

2qg

+ S(2)

Y

(r⊥)H(1,2)

2qg ]

+ αs 2π

• d2

s⊥d2 t⊥ (2π)2 S(4)

Y

(r⊥, s⊥, t⊥)H(1)

4qg

H(1,1)

2qg

= 1 2 e−i

k⊥· r⊥

(1 − ξ)2 + ξ2 ln c2 r2

⊥µ2 − 1

• H(1,2)

2qg

= 1 2ξ2 e−i

k⊥· r⊥/ξ

(1 − ξ)2 + ξ2 ln c2 r2

⊥µ2 − 1

• H(1)

4qg = 4πe−i k⊥· r⊥−i k⊥· t⊥/ξ (1 − ξ)2 + ξ2

ξ

• r⊥

r2

⊥

·

• t⊥

t2

⊥

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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Additional History

Incorporating rcBK

Fujii et al. (2011)

1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 1 2 3 4 5 dN/dηd2pT [GeV–2] pT[GeV/c] BRAHMS h– η=2.2 (x20) BRAHMS h– η=3.2 (x4) STAR π0 η=4 param h rcMV

dAu

Prefactor K = 1.5 for charged particles K = 0.5 for neutral particles

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 32

7

• f 20

Additional History

Rapidity Divergence

Rapidity divergence in gluon distribution1 F(xg, k⊥) = F(0)(xg, k⊥) − αsNc 2π2 1 dξ 1 − ξ × d2 x⊥d2 y⊥d2 b⊥ (2π)2 e−i

k⊥·( x⊥− y⊥)

( x⊥ − y⊥)2 ( x⊥ − b⊥)2( y⊥ − b⊥)2 ×

• S(2)

Y (

x⊥, y⊥) − S(4)

Y (

x⊥, b⊥, y⊥)

• Upper limit:

ξmax = 1 in √s → ∞ limit ξmax = 1 − k⊥

√se−y = 1 − e−Y in exact kinematics

ξmax = 1 − e−Y0 with rapidity cutoff?2

1Chirilli et al. 2012, 1203.6139, eq. (21). 2Kang et al. 2014, 1403.5221.

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 33

8

• f 20

Additional History

Rapidity Correction

Kang et al. (2014)

10

• 6

10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

1 0.5 1 1.5 2 2.5 3 3.5 4

BRAHMS h- y=3.2 LO NLO (without ∆HY) NLO (with ∆HY) µ2=10 GeV2

p⊥ (GeV) dN/dyd2p⊥ (GeV-2)

Rapidity correction (believed unphysical) (by us)

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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9

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Additional History

Matching to Collinear

Sta´ sto, Xiao, Yuan, et al. (2014)

1 2 3 10−7 10−5 10−3 10−1 101 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

BRAHMS η = 3.2 LO NLO exact data

Primitive kinematical constraint

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Beuf:2014uia. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 35

10

• f 20

Additional History

Ioffe Time

Altinoluk et al. (2014) p τ 2(1 − ξ)ξxgP + k2

⊥

> τ No numerical results

GBW MV/AAMQS LO BK rcBK b-CGC NLO BK LO inel NLO

• ther NLO

rapidity NLO splitting NLO

Gluon dist Cross section

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 36

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Analysis of Negativity

Breakdown by Channel

0.5 1 1.5 2 2.5 3 3.5 10−6 10−5 10−4 10−3 10−2 10−1 100 LO qq and gg p⊥[GeV]

• d3N

dηd2p⊥

• GeV−2

Plot shows magnitude of channel contribution Coloring indicates where value is Negative Positive Negativity comes from NLO diagonal channels: qq and gg

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

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11

• f 20

Analysis of Negativity

Breakdown by Channel

0.5 1 1.5 2 2.5 3 3.5 10−6 10−5 10−4 10−3 10−2 10−1 100 NLO qq p⊥[GeV]

• d3N

dηd2p⊥

• GeV−2

Plot shows magnitude of channel contribution Coloring indicates where value is Negative Positive Negativity comes from NLO diagonal channels: qq and gg

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 38

11

• f 20

Analysis of Negativity

Breakdown by Channel

0.5 1 1.5 2 2.5 3 3.5 10−6 10−5 10−4 10−3 10−2 10−1 100 NLO gg p⊥[GeV]

• d3N

dηd2p⊥

• GeV−2

Plot shows magnitude of channel contribution Coloring indicates where value is Negative Positive Negativity comes from NLO diagonal channels: qq and gg

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 39

11

• f 20

Analysis of Negativity

Breakdown by Channel

0.5 1 1.5 2 2.5 3 3.5 10−6 10−5 10−4 10−3 10−2 10−1 100 NLO gq p⊥[GeV]

• d3N

dηd2p⊥

• GeV−2

Plot shows magnitude of channel contribution Coloring indicates where value is Negative Positive Negativity comes from NLO diagonal channels: qq and gg

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 40

11

• f 20

Analysis of Negativity

Breakdown by Channel

0.5 1 1.5 2 2.5 3 3.5 10−6 10−5 10−4 10−3 10−2 10−1 100 NLO qg p⊥[GeV]

• d3N

dηd2p⊥

• GeV−2

Plot shows magnitude of channel contribution Coloring indicates where value is Negative Positive Negativity comes from NLO diagonal channels: qq and gg

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 41

12

• f 20

Analysis of Negativity

Breakdown by Term

Negativity comes from plus prescription 1

τ/z

dξ (1 − ξ)+ f(ξ) = 1

τ/z

dξ f(ξ) − f(1) 1 − ξ + f(1) ln

• 1 − τ

z

• First term negative because f(ξ) < f(1)

Second term negative because τ

z < 1

qq f(ξ) ∼ 1 + ξ2 gg f(ξ) ∼ ξ gg f(ξ) ∼ (1 − ξ + ξ2)2

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 42

13

• f 20

Numerical Challenges

Removing Singularities

Eliminate delta functions and plus prescriptions 1

τ

dz 1

τ z

dξ Fs(z, ξ) (1 − ξ)+ + Fn(z, ξ) + Fd(z, ξ)δ(1 − ξ)

• =

1

τ

dz 1

τ

dy z − τ z(1 − τ) Fs(z, ξ) − Fs(z, 1) 1 − ξ + Fn(z, ξ)

• +

1

τ

dz

• Fs(z, 1) ln
• 1 − τ

z

• + Fd(z, 1)
• δ2(

r⊥) d2 r′

⊥

r′2

⊥

ei

k⊥· r′

⊥ − 1

r2

⊥

e−iξ′

k⊥· r⊥

= 1 4π

• d2

k′

⊥e−i k′

⊥·

r⊥ ln (

k′

⊥ − ξ′

k⊥)2 k2

⊥

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 43

14

• f 20

Numerical Challenges

Fourier Integrals

Fourier integrals are highly imprecise

• d2

r⊥S(2)

Y (r⊥)ei k⊥· r⊥(. . .)

• d2

s⊥S(4)

Y (r⊥, s⊥, t⊥)ei k⊥· r⊥(. . .)

Easiest solution: transform to momentum space F(k⊥) = 1 (2π)2

• d2

r⊥S(2)

Y (r⊥)ei k⊥· r⊥

= 1 2π ∞ dr⊥S(2)

Y (r⊥)J0(k⊥r⊥)

and compute F directly

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 44

14

• f 20

Numerical Challenges

Fourier Integrals

Fourier integrals are highly imprecise

• d2

r⊥S(2)

Y (r⊥)ei k⊥· r⊥(. . .)

• d2

s⊥S(4)

Y (r⊥, s⊥, t⊥)ei k⊥· r⊥(. . .)

Alternate solution: algorithms for direct evaluation of multidimen- sional Fourier integrals (not explored)

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 45

15

• f 20

Numerical Challenges

New Fourier Transforms

d2x⊥ (2π)2 S(x⊥) ln c2 x2

⊥µ2 e−ik⊥·x⊥

= 1 π d2 l⊥ l2

⊥

• F(

k⊥ + l⊥) − J0 c0 µ l⊥

• F(k⊥)
• d2r⊥

(2π)2 S(r⊥)

• ln r2

⊥k2 ⊥

c2 2 e−ik⊥·r⊥ = 2 π d2 l⊥ l2

⊥

ln k2

⊥

l2

⊥

• θ(k⊥ − l⊥)F(k⊥) − F(

k⊥ + l⊥)

• Watanabe et al. 2015, 1505.05183.

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 46

16

• f 20

Numerical Challenges

Remaining Evaluation Errors

Inaccuracy of Fourier integrals Monte Carlo statistical error Cancellation of large terms Multiple runs to improve statistics

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 47

16

• f 20

Numerical Challenges

Remaining Evaluation Errors

Inaccuracy of Fourier integrals Monte Carlo statistical error Cancellation of large terms Two parallel implementations

• f selected parts:

Mathematica, for rapid prototyping C++, for execution speed

2 4 10−4 10−2 100 Lq(k⊥) MMA pos Lq(k⊥) MMA mom Lq(k⊥) C++ mom F(k⊥) (GBW) αsNc/(π2k4

⊥)

2 4 1 2 ·10−4 abs diff from pos space 2 4 −5 5 ·10−2 rel diff from pos space k⊥/Qs

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 48

17

• f 20

Kinematical Constraint

Derivation of the Kinematical Constraint

p

• xpP +, 0, 0⊥
• ξxpP +, k−, k⊥
• A
• 0, xgP −, k⊥ + l⊥
• (1 − ξ)xpP +, l−, l⊥
• xgP − =

l2

⊥

2(1 − ξ)xpP + + k2

⊥

2ξxpP + ≤ P − xg ≤ 1 ξ ≤ 1 − l2

⊥

xps

figure adapted from Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 49

18

• f 20

Beam Direction

The Beam Direction Problem

forward hadron production y = 0 (LHC) y > 0 y < 0 p, d A

figure adapted from Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 50

19

• f 20

Additional LHC Results

LHC Results at Central Rapidity

1 2 3 4 5 6 10−6 10−5 10−4 10−3 10−2 10−1 100 101 y = 0 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg ATLAS ALICE 1 2 3 4 5 6 y = 0 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg ATLAS ALICE

ALICE:2012mj; data: Milov 2014, 1403.5738. plots: Watanabe et al. 2015, 1505.05183. Saturation Physics on the Energy Frontier David Zaslavsky — CCNU

SLIDE 51

20

• f 20

Additional LHC Results

LHC Predictions for Run II

1 2 3 4 5 6 10−6 10−5 10−4 10−3 10−2 10−1 100 101 y = 1.75 p⊥[GeV]

d3N dηd2p⊥

• GeV−2

GBW LO +NLO +Lq + Lg 1 2 3 4 5 6 y = 1.75 p⊥[GeV] rcBK Λ2

QCD = 0.01

LO +NLO +Lq + Lg

Saturation Physics on the Energy Frontier David Zaslavsky — CCNU