second electron-ion collider workshop jefferson lab, march ... · jefferson lab, march 15th 2004...
TRANSCRIPT
![Page 1: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/1.jpg)
Second Electron-Ion Collider WorkshopJefferson Lab, March 15th 2004
Evolution Equations of QCDin the Non-Linear Saturation Region
Nestor Armesto
Department of Physics, Theory Division, CERN
1. Introduction.
2. Formalism.
3. The Balitsky-Kovchegov equation: features of the solutions.
4. Consequences in eA and HIC.
5. Summary.
Also talks by M. Baker, V. Guzey, J. Jalilian-Marian, A. H. Mueller and K. Tuchin.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.1
![Page 2: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/2.jpg)
1. Introduction (I): Néstor Armesto
• Experimentally xG(x, Q2) ∝ x−λ, λ ∼ 0.3 =⇒ at small x (largeenergies), high partonic densities =⇒ saturation: parton densitiescannot grow forever (parton fusion balances parton splitting if Aµ ∝ 1/g⇒ non-linear process).
• Appeal:1) New regime of QCD, with old ideas (pomeron interaction, multiplescattering in Reggeon Field Theory, geometrical cross sections,coherence arguments,. . .) in a new (QCD) language.2) Link of small x Physics and ultrarelativistic nuclear collisions (initialstage of a nuclear collision, enhancement due to nuclear length).
• Basic aim: to study the formulation and consequences of unitarity interms of the QCD fields.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.2
![Page 3: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/3.jpg)
1. Introduction (I): Néstor Armesto
• Experimentally xG(x, Q2) ∝ x−λ, λ ∼ 0.3 =⇒ at small x (largeenergies), high partonic densities =⇒ saturation: parton densitiescannot grow forever (parton fusion balances parton splitting if Aµ ∝ 1/g⇒ non-linear process).
• Appeal:
1) New regime of QCD, with old ideas (pomeron interaction, multiplescattering in Reggeon Field Theory, geometrical cross sections,coherence arguments,. . .) in a new (QCD) language.2) Link of small x Physics and ultrarelativistic nuclear collisions (initialstage of a nuclear collision, enhancement due to nuclear length).
• Basic aim: to study the formulation and consequences of unitarity interms of the QCD fields.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.2
![Page 4: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/4.jpg)
1. Introduction (I): Néstor Armesto
• Experimentally xG(x, Q2) ∝ x−λ, λ ∼ 0.3 =⇒ at small x (largeenergies), high partonic densities =⇒ saturation: parton densitiescannot grow forever (parton fusion balances parton splitting if Aµ ∝ 1/g⇒ non-linear process).
• Appeal:1) New regime of QCD, with old ideas (pomeron interaction, multiplescattering in Reggeon Field Theory, geometrical cross sections,coherence arguments,. . .) in a new (QCD) language.
2) Link of small x Physics and ultrarelativistic nuclear collisions (initialstage of a nuclear collision, enhancement due to nuclear length).
• Basic aim: to study the formulation and consequences of unitarity interms of the QCD fields.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.2
![Page 5: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/5.jpg)
1. Introduction (I): Néstor Armesto
• Experimentally xG(x, Q2) ∝ x−λ, λ ∼ 0.3 =⇒ at small x (largeenergies), high partonic densities =⇒ saturation: parton densitiescannot grow forever (parton fusion balances parton splitting if Aµ ∝ 1/g⇒ non-linear process).
• Appeal:1) New regime of QCD, with old ideas (pomeron interaction, multiplescattering in Reggeon Field Theory, geometrical cross sections,coherence arguments,. . .) in a new (QCD) language.2) Link of small x Physics and ultrarelativistic nuclear collisions (initialstage of a nuclear collision, enhancement due to nuclear length).
• Basic aim: to study the formulation and consequences of unitarity interms of the QCD fields.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.2
![Page 6: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/6.jpg)
1. Introduction (I): Néstor Armesto
• Experimentally xG(x, Q2) ∝ x−λ, λ ∼ 0.3 =⇒ at small x (largeenergies), high partonic densities =⇒ saturation: parton densitiescannot grow forever (parton fusion balances parton splitting if Aµ ∝ 1/g⇒ non-linear process).
• Appeal:1) New regime of QCD, with old ideas (pomeron interaction, multiplescattering in Reggeon Field Theory, geometrical cross sections,coherence arguments,. . .) in a new (QCD) language.2) Link of small x Physics and ultrarelativistic nuclear collisions (initialstage of a nuclear collision, enhancement due to nuclear length).
• Basic aim: to study the formulation and consequences of unitarity interms of the QCD fields.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.2
![Page 7: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/7.jpg)
1. Introduction (II): Néstor Armesto
LOW DENSITY
HIGHDENSITY
CO
NFI
NE
ME
NT
BFKL
DGLAP
Q (x)2
s
DLL
GLRMQ
ln 1/x
lnQ
BK
• Linear evolution equations: Dokshitzer-Gribov-Lipatov-Altarelli-Parisiin lnQ2, Balistky-Fadin-Kuraev-Lipatov in ln (1/x).• First (twist-four) non-linear correction:Gribov-Levin-Ryskin-Mueller-Qiu in lnQ2 (GLR, PR100(83)1; MQ, NPB268(86)427).• Non-linear, all-twist evolution equation in the saturation region:Balitsky-Kovchegov in ln (1/x) (B, NPB463(96)99; K, PRD60(99)034008).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.3
![Page 8: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/8.jpg)
1. Introduction (II): Néstor Armesto
LOW DENSITY
HIGHDENSITY
CO
NFI
NE
ME
NT
BFKL
DGLAP
Q (x)2
s
DLL
GLRMQ
ln 1/x
lnQ
BK
• Linear evolution equations: Dokshitzer-Gribov-Lipatov-Altarelli-Parisiin lnQ2, Balistky-Fadin-Kuraev-Lipatov in ln (1/x).
• First (twist-four) non-linear correction:Gribov-Levin-Ryskin-Mueller-Qiu in lnQ2 (GLR, PR100(83)1; MQ, NPB268(86)427).• Non-linear, all-twist evolution equation in the saturation region:Balitsky-Kovchegov in ln (1/x) (B, NPB463(96)99; K, PRD60(99)034008).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.3
![Page 9: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/9.jpg)
1. Introduction (II): Néstor Armesto
LOW DENSITY
HIGHDENSITY
CO
NFI
NE
ME
NT
BFKL
DGLAP
Q (x)2
s
DLL
GLRMQ
ln 1/x
lnQ
BK
• Linear evolution equations: Dokshitzer-Gribov-Lipatov-Altarelli-Parisiin lnQ2, Balistky-Fadin-Kuraev-Lipatov in ln (1/x).• First (twist-four) non-linear correction:Gribov-Levin-Ryskin-Mueller-Qiu in lnQ2 (GLR, PR100(83)1; MQ, NPB268(86)427).
• Non-linear, all-twist evolution equation in the saturation region:Balitsky-Kovchegov in ln (1/x) (B, NPB463(96)99; K, PRD60(99)034008).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.3
![Page 10: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/10.jpg)
1. Introduction (II): Néstor Armesto
LOW DENSITY
HIGHDENSITY
CO
NFI
NE
ME
NT
BFKL
DGLAP
Q (x)2
s
DLL
GLRMQ
ln 1/x
lnQ
BK
• Linear evolution equations: Dokshitzer-Gribov-Lipatov-Altarelli-Parisiin lnQ2, Balistky-Fadin-Kuraev-Lipatov in ln (1/x).• First (twist-four) non-linear correction:Gribov-Levin-Ryskin-Mueller-Qiu in lnQ2 (GLR, PR100(83)1; MQ, NPB268(86)427).• Non-linear, all-twist evolution equation in the saturation region:Balitsky-Kovchegov in ln (1/x) (B, NPB463(96)99; K, PRD60(99)034008).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.3
![Page 11: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/11.jpg)
2. Formalism (I): Néstor Armesto
• The McLerran-Venugopalan model (MV, PRD49(94)2233; 3354) treatsclassical radiation from color sources moving ultrarelativistically througha large nucleus. With a form for the color correlators in the target, agluon distribution saturated at small k⊥ is obtained: initial condition.
• Later on, gluon radiation (quantum evolution) of the color sources wasintroduced, which leads to an evolution equation: Color GlassCondensate (JKMW, PRD55(97)5414; ILM, PLB510(01)133). The rescattering of theprojectile in the nucleus is described through Wilson lines whoseaverage on target configurations gives the S-matrix (KW, PRD64(01)114002).
• Key point: fields (occupation numbers) become large (Fµν , fg ∝ 1/g2)⇒ classical arguments, but coupling small ⇒ perturbative methods.• The coupled equations for n-gluon correlators decouple inthe large Nc limit: the BK equa-tion appears for n = 2; also de-duced in BFKL (B, EPJC16(00)337) asa sum of fan diagrams in LL1/x(αs is fixed).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.4
![Page 12: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/12.jpg)
2. Formalism (I): Néstor Armesto
• The McLerran-Venugopalan model (MV, PRD49(94)2233; 3354) treatsclassical radiation from color sources moving ultrarelativistically througha large nucleus. With a form for the color correlators in the target, agluon distribution saturated at small k⊥ is obtained: initial condition.
• Later on, gluon radiation (quantum evolution) of the color sources wasintroduced, which leads to an evolution equation: Color GlassCondensate (JKMW, PRD55(97)5414; ILM, PLB510(01)133). The rescattering of theprojectile in the nucleus is described through Wilson lines whoseaverage on target configurations gives the S-matrix (KW, PRD64(01)114002).
• Key point: fields (occupation numbers) become large (Fµν , fg ∝ 1/g2)⇒ classical arguments, but coupling small ⇒ perturbative methods.• The coupled equations for n-gluon correlators decouple inthe large Nc limit: the BK equa-tion appears for n = 2; also de-duced in BFKL (B, EPJC16(00)337) asa sum of fan diagrams in LL1/x(αs is fixed).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.4
![Page 13: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/13.jpg)
2. Formalism (I): Néstor Armesto
• The McLerran-Venugopalan model (MV, PRD49(94)2233; 3354) treatsclassical radiation from color sources moving ultrarelativistically througha large nucleus. With a form for the color correlators in the target, agluon distribution saturated at small k⊥ is obtained: initial condition.
• Later on, gluon radiation (quantum evolution) of the color sources wasintroduced, which leads to an evolution equation: Color GlassCondensate (JKMW, PRD55(97)5414; ILM, PLB510(01)133). The rescattering of theprojectile in the nucleus is described through Wilson lines whoseaverage on target configurations gives the S-matrix (KW, PRD64(01)114002).
• Key point: fields (occupation numbers) become large (Fµν , fg ∝ 1/g2)⇒ classical arguments, but coupling small ⇒ perturbative methods.
• The coupled equations for n-gluon correlators decouple inthe large Nc limit: the BK equa-tion appears for n = 2; also de-duced in BFKL (B, EPJC16(00)337) asa sum of fan diagrams in LL1/x(αs is fixed).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.4
![Page 14: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/14.jpg)
2. Formalism (I): Néstor Armesto
• The McLerran-Venugopalan model (MV, PRD49(94)2233; 3354) treatsclassical radiation from color sources moving ultrarelativistically througha large nucleus. With a form for the color correlators in the target, agluon distribution saturated at small k⊥ is obtained: initial condition.
• Later on, gluon radiation (quantum evolution) of the color sources wasintroduced, which leads to an evolution equation: Color GlassCondensate (JKMW, PRD55(97)5414; ILM, PLB510(01)133). The rescattering of theprojectile in the nucleus is described through Wilson lines whoseaverage on target configurations gives the S-matrix (KW, PRD64(01)114002).
• Key point: fields (occupation numbers) become large (Fµν , fg ∝ 1/g2)⇒ classical arguments, but coupling small ⇒ perturbative methods.• The coupled equations for n-gluon correlators decouple inthe large Nc limit: the BK equa-tion appears for n = 2; also de-duced in BFKL (B, EPJC16(00)337) asa sum of fan diagrams in LL1/x(αs is fixed).
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.4
![Page 15: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/15.jpg)
2. Formalism (II): Néstor Armesto
q_ q
_q_
q_
q_
x1 x1 x1
x1x1
2x 2x 2x
2x 2x
+
+=
q q
q
q
q
z
zz
z
boost
large Nc
BK equation
Y = ln (x0/x), y = αsNcY/π, r = x1 − x2, b = (x1 + x2)/2;
WF (x1, z+ = 0) = Peig∫
dz−
T aA+a (x1,z
−);
Nx1,x2= N−1
c
⟨
tr[
1 − W †F (x1)WF (x2)
]⟩
t, φ(k, b) =
∫
d2r
2πr2eik·rNx1,x2
;
∂Nx1,x2
∂Y=
αsNc
π
∫
d2z
2π
(x1 − x2)2
(x1 − z)2(z − x2)2[Nx1,z + Nz,x2
− Nx1,x2
−Nx1,zNz,x2]−→
(
∂
∂y+ HBFKL
)
φ(k, b) = −φ2(k, b) for b � r.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.5
![Page 16: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/16.jpg)
2. Formalism (II): Néstor Armesto
q_ q
_q_
q_
q_
x1 x1 x1
x1x1
2x 2x 2x
2x 2x
+
+=
q q
q
q
q
z
zz
z
boost
large Nc
BK equation
Y = ln (x0/x), y = αsNcY/π, r = x1 − x2, b = (x1 + x2)/2;
WF (x1, z+ = 0) = Peig∫
dz−
T aA+a (x1,z
−);
Nx1,x2= N−1
c
⟨
tr[
1 − W †F (x1)WF (x2)
]⟩
t, φ(k, b) =
∫
d2r
2πr2eik·rNx1,x2
;
∂Nx1,x2
∂Y=
αsNc
π
∫
d2z
2π
(x1 − x2)2
(x1 − z)2(z − x2)2[Nx1,z + Nz,x2
− Nx1,x2
−Nx1,zNz,x2]−→
(
∂
∂y+ HBFKL
)
φ(k, b) = −φ2(k, b) for b � r.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.5
![Page 17: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/17.jpg)
2. Formalism (II): Néstor Armesto
q_ q
_q_
q_
q_
x1 x1 x1
x1x1
2x 2x 2x
2x 2x
+
+=
q q
q
q
q
z
zz
z
boost
large Nc
BK equation
Y = ln (x0/x), y = αsNcY/π, r = x1 − x2, b = (x1 + x2)/2;
WF (x1, z+ = 0) = Peig∫
dz−
T aA+a (x1,z
−);
Nx1,x2= N−1
c
⟨
tr[
1 − W †F (x1)WF (x2)
]⟩
t, φ(k, b) =
∫
d2r
2πr2eik·rNx1,x2
;
∂Nx1,x2
∂Y=
αsNc
π
∫
d2z
2π
(x1 − x2)2
(x1 − z)2(z − x2)2[Nx1,z + Nz,x2
− Nx1,x2
−Nx1,zNz,x2]−→
(
∂
∂y+ HBFKL
)
φ(k, b) = −φ2(k, b) for b � r.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.5
![Page 18: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/18.jpg)
2. Formalism (II): Néstor Armesto
q_ q
_q_
q_
q_
x1 x1 x1
x1x1
2x 2x 2x
2x 2x
+
+=
q q
q
q
q
z
zz
z
boost
large Nc
BK equation
Y = ln (x0/x), y = αsNcY/π, r = x1 − x2, b = (x1 + x2)/2;
WF (x1, z+ = 0) = Peig∫
dz−
T aA+a (x1,z
−);
Nx1,x2= N−1
c
⟨
tr[
1 − W †F (x1)WF (x2)
]⟩
t, φ(k, b) =
∫
d2r
2πr2eik·rNx1,x2
;
∂Nx1,x2
∂Y=
αsNc
π
∫
d2z
2π
(x1 − x2)2
(x1 − z)2(z − x2)2[Nx1,z + Nz,x2
− Nx1,x2
−Nx1,zNz,x2]−→
(
∂
∂y+ HBFKL
)
φ(k, b) = −φ2(k, b) for b � r.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.5
![Page 19: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/19.jpg)
2. Formalism (II): Néstor Armesto
q_ q
_q_
q_
q_
x1 x1 x1
x1x1
2x 2x 2x
2x 2x
+
+=
q q
q
q
q
z
zz
z
boost
large Nc
BK equation
Y = ln (x0/x), y = αsNcY/π, r = x1 − x2, b = (x1 + x2)/2;
WF (x1, z+ = 0) = Peig∫
dz−
T aA+a (x1,z
−);
Nx1,x2= N−1
c
⟨
tr[
1 − W †F (x1)WF (x2)
]⟩
t, φ(k, b) =
∫
d2r
2πr2eik·rNx1,x2
;
∂Nx1,x2
∂Y=
αsNc
π
∫
d2z
2π
(x1 − x2)2
(x1 − z)2(z − x2)2[Nx1,z + Nz,x2
− Nx1,x2
−Nx1,zNz,x2]
−→
(
∂
∂y+ HBFKL
)
φ(k, b) = −φ2(k, b) for b � r.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.5
![Page 20: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/20.jpg)
2. Formalism (II): Néstor Armesto
q_ q
_q_
q_
q_
x1 x1 x1
x1x1
2x 2x 2x
2x 2x
+
+=
q q
q
q
q
z
zz
z
boost
large Nc
BK equation
Y = ln (x0/x), y = αsNcY/π, r = x1 − x2, b = (x1 + x2)/2;
WF (x1, z+ = 0) = Peig∫
dz−
T aA+a (x1,z
−);
Nx1,x2= N−1
c
⟨
tr[
1 − W †F (x1)WF (x2)
]⟩
t, φ(k, b) =
∫
d2r
2πr2eik·rNx1,x2
;
∂Nx1,x2
∂Y=
αsNc
π
∫
d2z
2π
(x1 − x2)2
(x1 − z)2(z − x2)2[Nx1,z + Nz,x2
− Nx1,x2
−Nx1,zNz,x2]−→
(
∂
∂y+ HBFKL
)
φ(k, b) = −φ2(k, b) for b � r.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.5
![Page 21: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/21.jpg)
Néstor Armesto
3. The Balitsky-Kovchegov equation:features of the solutions:
• Asymptotic integrability.
• Scaling.
• Froissart bound.
• Running coupling.
• Beyond BK.
The analytical solution is unknown: numerical resultswill be shown and analytical estimations commented.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.6
![Page 22: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/22.jpg)
Asymptotic integrability: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• Solutions tend to a universal shape independent of the initialcondition; for h = k2∇2
kφ,∫
d2k2πk2 h(k) = 1, a soliton-like behavior
appears (B, EPJC16(00)337).• Integrability (for y → ∞) has been proved (MP, PRL91(03)232001;
hep-ph/0401215).• Solution to the IR diffusion problem of BFKL (GBMS, PRD65(02)074037).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.7
![Page 23: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/23.jpg)
Asymptotic integrability: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• Solutions tend to a universal shape independent of the initialcondition; for h = k2∇2
kφ,∫
d2k2πk2 h(k) = 1, a soliton-like behavior
appears (B, EPJC16(00)337).
• Integrability (for y → ∞) has been proved (MP, PRL91(03)232001;
hep-ph/0401215).• Solution to the IR diffusion problem of BFKL (GBMS, PRD65(02)074037).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.7
![Page 24: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/24.jpg)
Asymptotic integrability: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• Solutions tend to a universal shape independent of the initialcondition; for h = k2∇2
kφ,∫
d2k2πk2 h(k) = 1, a soliton-like behavior
appears (B, EPJC16(00)337).• Integrability (for y → ∞) has been proved (MP, PRL91(03)232001;
hep-ph/0401215).
• Solution to the IR diffusion problem of BFKL (GBMS, PRD65(02)074037).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.7
![Page 25: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/25.jpg)
Asymptotic integrability: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• Solutions tend to a universal shape independent of the initialcondition; for h = k2∇2
kφ,∫
d2k2πk2 h(k) = 1, a soliton-like behavior
appears (B, EPJC16(00)337).• Integrability (for y → ∞) has been proved (MP, PRL91(03)232001;
hep-ph/0401215).• Solution to the IR diffusion problem of BFKL (GBMS, PRD65(02)074037).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.7
![Page 26: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/26.jpg)
Scaling: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• A universal function of k/Qs appears (AB, EPJC20(01)517; L, EPJC21(01)513).• h(k = Qs) = hmax; Q2
s(y) ∝ A∼1/3 (as expected), and ∝ exp (∆y) with4 (DLL) < ∆ < 4.88 (BFKL) (LT, NPB573(00)833; LL, NPA696(01)833).• Scaling observed in l-p (SGBK, PRL86(01)596), l-A? (FRWS, PRL90(03)222002).• For Qs < k < ksw(Qs), log corrected shape (MT, NPB640(02)331) favored(AAKSW, PRL92(04)082001) over pure power (IIM, NPA708(02)327).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.8
![Page 27: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/27.jpg)
Scaling: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• A universal function of k/Qs appears (AB, EPJC20(01)517; L, EPJC21(01)513).
• h(k = Qs) = hmax; Q2s(y) ∝ A∼1/3 (as expected), and ∝ exp (∆y) with
4 (DLL) < ∆ < 4.88 (BFKL) (LT, NPB573(00)833; LL, NPA696(01)833).• Scaling observed in l-p (SGBK, PRL86(01)596), l-A? (FRWS, PRL90(03)222002).• For Qs < k < ksw(Qs), log corrected shape (MT, NPB640(02)331) favored(AAKSW, PRL92(04)082001) over pure power (IIM, NPA708(02)327).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.8
![Page 28: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/28.jpg)
Scaling: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• A universal function of k/Qs appears (AB, EPJC20(01)517; L, EPJC21(01)513).• h(k = Qs) = hmax; Q2
s(y) ∝ A∼1/3 (as expected), and ∝ exp (∆y) with4 (DLL) < ∆ < 4.88 (BFKL) (LT, NPB573(00)833; LL, NPA696(01)833).
• Scaling observed in l-p (SGBK, PRL86(01)596), l-A? (FRWS, PRL90(03)222002).• For Qs < k < ksw(Qs), log corrected shape (MT, NPB640(02)331) favored(AAKSW, PRL92(04)082001) over pure power (IIM, NPA708(02)327).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.8
![Page 29: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/29.jpg)
Scaling: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• A universal function of k/Qs appears (AB, EPJC20(01)517; L, EPJC21(01)513).• h(k = Qs) = hmax; Q2
s(y) ∝ A∼1/3 (as expected), and ∝ exp (∆y) with4 (DLL) < ∆ < 4.88 (BFKL) (LT, NPB573(00)833; LL, NPA696(01)833).• Scaling observed in l-p (SGBK, PRL86(01)596), l-A? (FRWS, PRL90(03)222002).
• For Qs < k < ksw(Qs), log corrected shape (MT, NPB640(02)331) favored(AAKSW, PRL92(04)082001) over pure power (IIM, NPA708(02)327).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.8
![Page 30: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/30.jpg)
Scaling: Néstor Armesto
Solutions of BK for y = 0, 5, 10 for GBW (solid), and MV with Qs = 2 GeV (dashed) and 10 GeV
(dotted) (AAKSW, PRL92(04)082001).
• A universal function of k/Qs appears (AB, EPJC20(01)517; L, EPJC21(01)513).• h(k = Qs) = hmax; Q2
s(y) ∝ A∼1/3 (as expected), and ∝ exp (∆y) with4 (DLL) < ∆ < 4.88 (BFKL) (LT, NPB573(00)833; LL, NPA696(01)833).• Scaling observed in l-p (SGBK, PRL86(01)596), l-A? (FRWS, PRL90(03)222002).• For Qs < k < ksw(Qs), log corrected shape (MT, NPB640(02)331) favored(AAKSW, PRL92(04)082001) over pure power (IIM, NPA708(02)327).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.8
![Page 31: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/31.jpg)
Froissart bound: Néstor Armesto
Solution of BK with b-dependence, for fixed r and angle between r and b (GBS, NPB668(03)345).
10-9
10-7
10-5
10-3
10-1
10
10-1
1 10
Cosθ = 0.0r = 0.1
InputY=0.1
Y=2.0
Y=5.0
Y=8.0
Y=11.0
b
N(r
,b,θ
;Y)
• Froissart bound, σ ∝ ln2 s, recovered by non-linear effects? (FIIM,
NPA710(01)373; KW, PRD66(02)051502; PLB551(03)311): b-dependence.• No mass gap ⇒ no Froissart expected; initial condition ∝ e−Ab for b �but a tail ∝ b−4 develops ⇒ regulation of the kernel is demanded.• Relevant for soft physics and to really determine Qs(b).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.9
![Page 32: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/32.jpg)
Froissart bound: Néstor Armesto
Solution of BK with b-dependence, for fixed r and angle between r and b (GBS, NPB668(03)345).
10-9
10-7
10-5
10-3
10-1
10
10-1
1 10
Cosθ = 0.0r = 0.1
InputY=0.1
Y=2.0
Y=5.0
Y=8.0
Y=11.0
b
N(r
,b,θ
;Y)
• Froissart bound, σ ∝ ln2 s, recovered by non-linear effects? (FIIM,
NPA710(01)373; KW, PRD66(02)051502; PLB551(03)311): b-dependence.
• No mass gap ⇒ no Froissart expected; initial condition ∝ e−Ab for b �but a tail ∝ b−4 develops ⇒ regulation of the kernel is demanded.• Relevant for soft physics and to really determine Qs(b).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.9
![Page 33: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/33.jpg)
Froissart bound: Néstor Armesto
Solution of BK with b-dependence, for fixed r and angle between r and b (GBS, NPB668(03)345).
10-9
10-7
10-5
10-3
10-1
10
10-1
1 10
Cosθ = 0.0r = 0.1
InputY=0.1
Y=2.0
Y=5.0
Y=8.0
Y=11.0
b
N(r
,b,θ
;Y)
• Froissart bound, σ ∝ ln2 s, recovered by non-linear effects? (FIIM,
NPA710(01)373; KW, PRD66(02)051502; PLB551(03)311): b-dependence.• No mass gap ⇒ no Froissart expected; initial condition ∝ e−Ab for b �but a tail ∝ b−4 develops ⇒ regulation of the kernel is demanded.
• Relevant for soft physics and to really determine Qs(b).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.9
![Page 34: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/34.jpg)
Froissart bound: Néstor Armesto
Solution of BK with b-dependence, for fixed r and angle between r and b (GBS, NPB668(03)345).
10-9
10-7
10-5
10-3
10-1
10
10-1
1 10
Cosθ = 0.0r = 0.1
InputY=0.1
Y=2.0
Y=5.0
Y=8.0
Y=11.0
b
N(r
,b,θ
;Y)
• Froissart bound, σ ∝ ln2 s, recovered by non-linear effects? (FIIM,
NPA710(01)373; KW, PRD66(02)051502; PLB551(03)311): b-dependence.• No mass gap ⇒ no Froissart expected; initial condition ∝ e−Ab for b �but a tail ∝ b−4 develops ⇒ regulation of the kernel is demanded.• Relevant for soft physics and to really determine Qs(b).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.9
![Page 35: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/35.jpg)
Running coupling: Néstor Armesto
Solutions of BK with (two upper curves on the left, lower one on the right), and without running coupling
for different IR regularizations (B, PLB576(03)115); here y is real rapidity.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.01 0.1 1 10 100
h(y,
k)
k/Qs(y)
y=6 and 10
0.1
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
1 2 3 4 5 6 7 8 9 10
Qs(
y)
y
• A scaling function of k/Qs still appears.
• Better IR and UV behavior (h = k2∇2kφ).
• With Q2s(y) ∝ exp (∆y), ∆rc ∼ ∆fc/2 (T, 648(03)293; RW, hep-ph/0309306): much
closer to the value ' 0.3 (for αs ∼ 0.2) found in data (GBW, PRD59(99)014017).
• Behavior of Qs with A in this case? (M, NPA724(03)223); b-dependence?(GKLMN, hep-ph/0401021).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.10
![Page 36: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/36.jpg)
Running coupling: Néstor Armesto
Solutions of BK with (two upper curves on the left, lower one on the right), and without running coupling
for different IR regularizations (B, PLB576(03)115); here y is real rapidity.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.01 0.1 1 10 100
h(y,
k)
k/Qs(y)
y=6 and 10
0.1
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
1 2 3 4 5 6 7 8 9 10
Qs(
y)
y
• A scaling function of k/Qs still appears.
• Better IR and UV behavior (h = k2∇2kφ).
• With Q2s(y) ∝ exp (∆y), ∆rc ∼ ∆fc/2 (T, 648(03)293; RW, hep-ph/0309306): much
closer to the value ' 0.3 (for αs ∼ 0.2) found in data (GBW, PRD59(99)014017).
• Behavior of Qs with A in this case? (M, NPA724(03)223); b-dependence?(GKLMN, hep-ph/0401021).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.10
![Page 37: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/37.jpg)
Running coupling: Néstor Armesto
Solutions of BK with (two upper curves on the left, lower one on the right), and without running coupling
for different IR regularizations (B, PLB576(03)115); here y is real rapidity.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.01 0.1 1 10 100
h(y,
k)
k/Qs(y)
y=6 and 10
0.1
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
1 2 3 4 5 6 7 8 9 10
Qs(
y)
y
• A scaling function of k/Qs still appears.
• Better IR and UV behavior (h = k2∇2kφ).
• With Q2s(y) ∝ exp (∆y), ∆rc ∼ ∆fc/2 (T, 648(03)293; RW, hep-ph/0309306): much
closer to the value ' 0.3 (for αs ∼ 0.2) found in data (GBW, PRD59(99)014017).
• Behavior of Qs with A in this case? (M, NPA724(03)223); b-dependence?(GKLMN, hep-ph/0401021).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.10
![Page 38: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/38.jpg)
Running coupling: Néstor Armesto
Solutions of BK with (two upper curves on the left, lower one on the right), and without running coupling
for different IR regularizations (B, PLB576(03)115); here y is real rapidity.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.01 0.1 1 10 100
h(y,
k)
k/Qs(y)
y=6 and 10
0.1
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
1 2 3 4 5 6 7 8 9 10
Qs(
y)
y
• A scaling function of k/Qs still appears.
• Better IR and UV behavior (h = k2∇2kφ).
• With Q2s(y) ∝ exp (∆y), ∆rc ∼ ∆fc/2 (T, 648(03)293; RW, hep-ph/0309306): much
closer to the value ' 0.3 (for αs ∼ 0.2) found in data (GBW, PRD59(99)014017).
• Behavior of Qs with A in this case? (M, NPA724(03)223); b-dependence?(GKLMN, hep-ph/0401021).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.10
![Page 39: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/39.jpg)
Running coupling: Néstor Armesto
Solutions of BK with (two upper curves on the left, lower one on the right), and without running coupling
for different IR regularizations (B, PLB576(03)115); here y is real rapidity.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.01 0.1 1 10 100
h(y,
k)
k/Qs(y)
y=6 and 10
0.1
1
10
100
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
1 2 3 4 5 6 7 8 9 10
Qs(
y)
y
• A scaling function of k/Qs still appears.
• Better IR and UV behavior (h = k2∇2kφ).
• With Q2s(y) ∝ exp (∆y), ∆rc ∼ ∆fc/2 (T, 648(03)293; RW, hep-ph/0309306): much
closer to the value ' 0.3 (for αs ∼ 0.2) found in data (GBW, PRD59(99)014017).
• Behavior of Qs with A in this case? (M, NPA724(03)223); b-dependence?(GKLMN, hep-ph/0401021).
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.10
![Page 40: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/40.jpg)
Beyond BK: Néstor Armesto
Difference between the average of the correlator of 4 Wilson lines and the factorized ansatz (RW,
hep-ph/0309306); here 0 < Y < 2.5 and U = WF previously defined.
1 10 100
r = |x| = |y|
0
0.005
0.01
0.015
0.02
⟨- 91 Tr
(1 -
U+ 0U
x) T
r (1
- U
+ 0Uy)
⟩ - f
acto
rize
d
5122 Gaussian initial state with R
0=66
•
⟨
W †F (x1)WF (z)W †
F (z)WF (x2)⟩
→
⟨
W †F (x1)WF (z)
⟩ ⟨
W †F (z)WF (x2)
⟩
as in BK implies a loss of fluctuations (IM, NPA730(04)494; MS, hep-ph/0402193).• Better use full CGC (JIMWLK equation): far more involved.• This factorization does not seem a large effect (compared to thetypical size ∼ 1) but maybe crucial to get the correct small-r behavior.
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.11
![Page 41: Second Electron-Ion Collider Workshop Jefferson Lab, March ... · Jefferson Lab, March 15th 2004 Evolution Equations of QCD in the Non-Linear Saturation Region Nestor´ Armesto Department](https://reader034.vdocuments.us/reader034/viewer/2022051814/6037119b50559f79d84b65b1/html5/thumbnails/41.jpg)
Beyond BK: Néstor Armesto
Difference between the average of the correlator of 4 Wilson lines and the factorized ansatz (RW,
hep-ph/0309306); here 0 < Y < 2.5 and U = WF previously defined.
1 10 100
r = |x| = |y|
0
0.005
0.01
0.015
0.02
⟨- 91 Tr
(1 -
U+ 0U
x) T
r (1
- U
+ 0Uy)
⟩ - f
acto
rize
d
5122 Gaussian initial state with R
0=66
•
⟨
W †F (x1)WF (z)W †
F (z)WF (x2)⟩
→
⟨
W †F (x1)WF (z)
⟩ ⟨
W †F (z)WF (x2)
⟩
as in BK implies a loss of fluctuations (IM, NPA730(04)494; MS, hep-ph/0402193).
• Better use full CGC (JIMWLK equation): far more involved.• This factorization does not seem a large effect (compared to thetypical size ∼ 1) but maybe crucial to get the correct small-r behavior.
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.11
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Beyond BK: Néstor Armesto
Difference between the average of the correlator of 4 Wilson lines and the factorized ansatz (RW,
hep-ph/0309306); here 0 < Y < 2.5 and U = WF previously defined.
1 10 100
r = |x| = |y|
0
0.005
0.01
0.015
0.02
⟨- 91 Tr
(1 -
U+ 0U
x) T
r (1
- U
+ 0Uy)
⟩ - f
acto
rize
d
5122 Gaussian initial state with R
0=66
•
⟨
W †F (x1)WF (z)W †
F (z)WF (x2)⟩
→
⟨
W †F (x1)WF (z)
⟩ ⟨
W †F (z)WF (x2)
⟩
as in BK implies a loss of fluctuations (IM, NPA730(04)494; MS, hep-ph/0402193).• Better use full CGC (JIMWLK equation): far more involved.
• This factorization does not seem a large effect (compared to thetypical size ∼ 1) but maybe crucial to get the correct small-r behavior.
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.11
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Beyond BK: Néstor Armesto
Difference between the average of the correlator of 4 Wilson lines and the factorized ansatz (RW,
hep-ph/0309306); here 0 < Y < 2.5 and U = WF previously defined.
1 10 100
r = |x| = |y|
0
0.005
0.01
0.015
0.02
⟨- 91 Tr
(1 -
U+ 0U
x) T
r (1
- U
+ 0Uy)
⟩ - f
acto
rize
d
5122 Gaussian initial state with R
0=66
•
⟨
W †F (x1)WF (z)W †
F (z)WF (x2)⟩
→
⟨
W †F (x1)WF (z)
⟩ ⟨
W †F (z)WF (x2)
⟩
as in BK implies a loss of fluctuations (IM, NPA730(04)494; MS, hep-ph/0402193).• Better use full CGC (JIMWLK equation): far more involved.• This factorization does not seem a large effect (compared to thetypical size ∼ 1) but maybe crucial to get the correct small-r behavior.
Evolution Equations of QCD in the Non-Linear Saturation Region: 3. BK. – p.11
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Néstor Armesto
4. Consequences in eA and HIC:
• eA: F2 and heavy flavor production.
• HIC: Cronin effect.
Just two aspects touched, a huge range of applicationsis available, see the other mentioned talks.
Note: x is usually assumed to be small enough,< (2mNRA)−1.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.12
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eA: F2 and heavy flavor production: Néstor Armesto
Results of BK for F2Pb and charm production in eA collisions (AB, EPJC22(01)351).
• F2A: slower increase with 1/x, ∝ ln2(1/x) (but no b-dependence, sothe initial condition determines the b-dependence).
• If F2A = Aα(x,Q2)F2p, α ∼ 0.9 for x ∼ 10−3 and 0.7 ÷ 0.8 for x ∼ 10−6.
• Heavy quark production larger than in collinear factorization (GM,
EPJC30(03)387); transverse momenta � than the mass are not negligible.
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.13
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eA: F2 and heavy flavor production: Néstor Armesto
Results of BK for F2Pb and charm production in eA collisions (AB, EPJC22(01)351).
• F2A: slower increase with 1/x, ∝ ln2(1/x) (but no b-dependence, sothe initial condition determines the b-dependence).
• If F2A = Aα(x,Q2)F2p, α ∼ 0.9 for x ∼ 10−3 and 0.7 ÷ 0.8 for x ∼ 10−6.
• Heavy quark production larger than in collinear factorization (GM,
EPJC30(03)387); transverse momenta � than the mass are not negligible.
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.13
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eA: F2 and heavy flavor production: Néstor Armesto
Results of BK for F2Pb and charm production in eA collisions (AB, EPJC22(01)351).
• F2A: slower increase with 1/x, ∝ ln2(1/x) (but no b-dependence, sothe initial condition determines the b-dependence).
• If F2A = Aα(x,Q2)F2p, α ∼ 0.9 for x ∼ 10−3 and 0.7 ÷ 0.8 for x ∼ 10−6.
• Heavy quark production larger than in collinear factorization (GM,
EPJC30(03)387); transverse momenta � than the mass are not negligible.
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.13
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eA: F2 and heavy flavor production: Néstor Armesto
Results of BK for F2Pb and charm production in eA collisions (AB, EPJC22(01)351).
• F2A: slower increase with 1/x, ∝ ln2(1/x) (but no b-dependence, sothe initial condition determines the b-dependence).
• If F2A = Aα(x,Q2)F2p, α ∼ 0.9 for x ∼ 10−3 and 0.7 ÷ 0.8 for x ∼ 10−6.
• Heavy quark production larger than in collinear factorization (GM,
EPJC30(03)387); transverse momenta � than the mass are not negligible.
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.13
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HIC: Cronin effect (I): Néstor Armesto
RpA =
dNpA
dyd2p d2b
A1/3 dNpp
dyd2p d2b
, RAA =
dNAA
dyd2p d2b
A4/3 dNpp
dyd2p d2b
.
• Gluon spectra in pp, pA or AB are computed using some factorization,Ng ∝
∫
fA fB (GLR, PR100(83)1; KM, NPB529(98)451; B, PLB483(00)105; KKT,
PRD68(03)094013; BGV, hep-ph/0402256), or classical field simulation (KV,
PRL84(00)4307; KNV, NPA727(03)427; L, PRC67(03)054903); hadronization or LPHD isthen used. In this way, bulk properties like multiplicities are understoodwithin saturation (KL, PLB523(01)79).
• It has been argued (KLM, PLB561(03)93) that large transverse momentumsuppression in AuAu at RHIC can be described within saturation. But, ifso, it should also be present in dAu at the same rapidity, which is not thecase =⇒ final state effects in AuAu.• The presence (Cronin) or absence of enhancement in the ratios is dueto the behavior of gluon distributions just above Qs (BKW, PRD68(03)054009);other ingredients (e.g. factorization, hadronization,. . .) do not changethis qualitative behavior.• Initial conditions based on multiple scattering (e.g. Glauber-Mueller orMV) lead to enhancement (JNV, PLB577(03)54).
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.14
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HIC: Cronin effect (I): Néstor Armesto
RpA =
dNpA
dyd2p d2b
A1/3 dNpp
dyd2p d2b
, RAA =
dNAA
dyd2p d2b
A4/3 dNpp
dyd2p d2b
.
• Gluon spectra in pp, pA or AB are computed using some factorization,Ng ∝
∫
fA fB (GLR, PR100(83)1; KM, NPB529(98)451; B, PLB483(00)105; KKT,
PRD68(03)094013; BGV, hep-ph/0402256), or classical field simulation (KV,
PRL84(00)4307; KNV, NPA727(03)427; L, PRC67(03)054903); hadronization or LPHD isthen used. In this way, bulk properties like multiplicities are understoodwithin saturation (KL, PLB523(01)79).• It has been argued (KLM, PLB561(03)93) that large transverse momentumsuppression in AuAu at RHIC can be described within saturation. But, ifso, it should also be present in dAu at the same rapidity, which is not thecase =⇒ final state effects in AuAu.
• The presence (Cronin) or absence of enhancement in the ratios is dueto the behavior of gluon distributions just above Qs (BKW, PRD68(03)054009);other ingredients (e.g. factorization, hadronization,. . .) do not changethis qualitative behavior.• Initial conditions based on multiple scattering (e.g. Glauber-Mueller orMV) lead to enhancement (JNV, PLB577(03)54).
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.14
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HIC: Cronin effect (I): Néstor Armesto
RpA =
dNpA
dyd2p d2b
A1/3 dNpp
dyd2p d2b
, RAA =
dNAA
dyd2p d2b
A4/3 dNpp
dyd2p d2b
.
• Gluon spectra in pp, pA or AB are computed using some factorization,Ng ∝
∫
fA fB (GLR, PR100(83)1; KM, NPB529(98)451; B, PLB483(00)105; KKT,
PRD68(03)094013; BGV, hep-ph/0402256), or classical field simulation (KV,
PRL84(00)4307; KNV, NPA727(03)427; L, PRC67(03)054903); hadronization or LPHD isthen used. In this way, bulk properties like multiplicities are understoodwithin saturation (KL, PLB523(01)79).• It has been argued (KLM, PLB561(03)93) that large transverse momentumsuppression in AuAu at RHIC can be described within saturation. But, ifso, it should also be present in dAu at the same rapidity, which is not thecase =⇒ final state effects in AuAu.• The presence (Cronin) or absence of enhancement in the ratios is dueto the behavior of gluon distributions just above Qs (BKW, PRD68(03)054009);other ingredients (e.g. factorization, hadronization,. . .) do not changethis qualitative behavior.
• Initial conditions based on multiple scattering (e.g. Glauber-Mueller orMV) lead to enhancement (JNV, PLB577(03)54).
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.14
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HIC: Cronin effect (I): Néstor Armesto
RpA =
dNpA
dyd2p d2b
A1/3 dNpp
dyd2p d2b
, RAA =
dNAA
dyd2p d2b
A4/3 dNpp
dyd2p d2b
.
• Gluon spectra in pp, pA or AB are computed using some factorization,Ng ∝
∫
fA fB (GLR, PR100(83)1; KM, NPB529(98)451; B, PLB483(00)105; KKT,
PRD68(03)094013; BGV, hep-ph/0402256), or classical field simulation (KV,
PRL84(00)4307; KNV, NPA727(03)427; L, PRC67(03)054903); hadronization or LPHD isthen used. In this way, bulk properties like multiplicities are understoodwithin saturation (KL, PLB523(01)79).• It has been argued (KLM, PLB561(03)93) that large transverse momentumsuppression in AuAu at RHIC can be described within saturation. But, ifso, it should also be present in dAu at the same rapidity, which is not thecase =⇒ final state effects in AuAu.• The presence (Cronin) or absence of enhancement in the ratios is dueto the behavior of gluon distributions just above Qs (BKW, PRD68(03)054009);other ingredients (e.g. factorization, hadronization,. . .) do not changethis qualitative behavior.• Initial conditions based on multiple scattering (e.g. Glauber-Mueller orMV) lead to enhancement (JNV, PLB577(03)54).
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.14
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HIC: Cronin effect (II): Néstor Armesto
• Evolution erases the Cronineffect present in the initial con-dition (AAKSM, PRL92(04)082001; KKT,
PRD68(03)094013).
• Thus, it predicts its disap-pearance at forward rapidities atRHIC, seen in data (BRAHMS Coll.,
nucl-ex/0403005).• No attempt to reproduce data;uncertainties (e.g. fc vs rc, finiteenergy effects,. . .) still large.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x
Non-Linear Evolution of Cronin Enhancement2=2 GeV2
SAuQ2=0.1 GeV2
SdQ
sαrunning =0.5sαfixed
tp (GeV/c)
dAu
R
Y=0
Y=1
Y=3
Albacete, Armesto, Kovner, Salgado, Wiedemann,hep-ph/0307179, PRL to appear.
Braun, PLB576 (2003) 115.
[GeV/c]Tp1 2 3 4 5
CP
R0
0.5
1
1.5
2
= 3.2η -h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
= 2.2η -h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
= 1η2
-+h+h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
0-20%/60-80%30-50%/60-80%
= 0η2
-+h+h
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.15
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HIC: Cronin effect (II): Néstor Armesto
• Evolution erases the Cronineffect present in the initial con-dition (AAKSM, PRL92(04)082001; KKT,
PRD68(03)094013).• Thus, it predicts its disap-pearance at forward rapidities atRHIC, seen in data (BRAHMS Coll.,
nucl-ex/0403005).
• No attempt to reproduce data;uncertainties (e.g. fc vs rc, finiteenergy effects,. . .) still large.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x
Non-Linear Evolution of Cronin Enhancement2=2 GeV2
SAuQ2=0.1 GeV2
SdQ
sαrunning =0.5sαfixed
tp (GeV/c)
dAu
R
Y=0
Y=1
Y=3
Albacete, Armesto, Kovner, Salgado, Wiedemann,hep-ph/0307179, PRL to appear.
Braun, PLB576 (2003) 115.
[GeV/c]Tp1 2 3 4 5
CP
R0
0.5
1
1.5
2
= 3.2η -h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
= 2.2η -h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
= 1η2
-+h+h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
0-20%/60-80%30-50%/60-80%
= 0η2
-+h+h
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.15
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HIC: Cronin effect (II): Néstor Armesto
• Evolution erases the Cronineffect present in the initial con-dition (AAKSM, PRL92(04)082001; KKT,
PRD68(03)094013).• Thus, it predicts its disap-pearance at forward rapidities atRHIC, seen in data (BRAHMS Coll.,
nucl-ex/0403005).• No attempt to reproduce data;uncertainties (e.g. fc vs rc, finiteenergy effects,. . .) still large.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x
Non-Linear Evolution of Cronin Enhancement2=2 GeV2
SAuQ2=0.1 GeV2
SdQ
sαrunning =0.5sαfixed
tp (GeV/c)
dAu
R
Y=0
Y=1
Y=3
Albacete, Armesto, Kovner, Salgado, Wiedemann,hep-ph/0307179, PRL to appear.
Braun, PLB576 (2003) 115.
[GeV/c]Tp1 2 3 4 5
CP
R0
0.5
1
1.5
2
= 3.2η -h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
= 2.2η -h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
= 1η2
-+h+h
[GeV/c]Tp1 2 3 4 5
CP
R
0
0.5
1
1.5
2
0-20%/60-80%30-50%/60-80%
= 0η2
-+h+h
Evolution Equations of QCD in the Non-Linear Saturation Region: 4. eA and HIC. – p.15
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5. Summary: Néstor Armesto
• Evolution equations in the high parton density, non-linear regime ofQCD are available. They are, together with factorization theorems, thekey ingredient to compute observables in high-energy collisionsinvolving nuclei.
• The main features of the solutions of the most widely used tool, the BKequation, are known: integrability, scaling and good IR behavior.Analytical estimates and numerical solutions are available.
• The inclusion of running coupling and effects beyond BK is understudy.
• These works have already been crucial to establish the largetransverse momentum suppression measured in AuAu at RHIC as afinal state effect.
• Many predictions are available. An eA collider is the cleanest place tostudy these high parton density phenomena.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.16
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5. Summary: Néstor Armesto
• Evolution equations in the high parton density, non-linear regime ofQCD are available. They are, together with factorization theorems, thekey ingredient to compute observables in high-energy collisionsinvolving nuclei.
• The main features of the solutions of the most widely used tool, the BKequation, are known: integrability, scaling and good IR behavior.Analytical estimates and numerical solutions are available.
• The inclusion of running coupling and effects beyond BK is understudy.
• These works have already been crucial to establish the largetransverse momentum suppression measured in AuAu at RHIC as afinal state effect.
• Many predictions are available. An eA collider is the cleanest place tostudy these high parton density phenomena.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.16
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5. Summary: Néstor Armesto
• Evolution equations in the high parton density, non-linear regime ofQCD are available. They are, together with factorization theorems, thekey ingredient to compute observables in high-energy collisionsinvolving nuclei.
• The main features of the solutions of the most widely used tool, the BKequation, are known: integrability, scaling and good IR behavior.Analytical estimates and numerical solutions are available.
• The inclusion of running coupling and effects beyond BK is understudy.
• These works have already been crucial to establish the largetransverse momentum suppression measured in AuAu at RHIC as afinal state effect.
• Many predictions are available. An eA collider is the cleanest place tostudy these high parton density phenomena.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.16
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5. Summary: Néstor Armesto
• Evolution equations in the high parton density, non-linear regime ofQCD are available. They are, together with factorization theorems, thekey ingredient to compute observables in high-energy collisionsinvolving nuclei.
• The main features of the solutions of the most widely used tool, the BKequation, are known: integrability, scaling and good IR behavior.Analytical estimates and numerical solutions are available.
• The inclusion of running coupling and effects beyond BK is understudy.
• These works have already been crucial to establish the largetransverse momentum suppression measured in AuAu at RHIC as afinal state effect.
• Many predictions are available. An eA collider is the cleanest place tostudy these high parton density phenomena.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.16
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5. Summary: Néstor Armesto
• Evolution equations in the high parton density, non-linear regime ofQCD are available. They are, together with factorization theorems, thekey ingredient to compute observables in high-energy collisionsinvolving nuclei.
• The main features of the solutions of the most widely used tool, the BKequation, are known: integrability, scaling and good IR behavior.Analytical estimates and numerical solutions are available.
• The inclusion of running coupling and effects beyond BK is understudy.
• These works have already been crucial to establish the largetransverse momentum suppression measured in AuAu at RHIC as afinal state effect.
• Many predictions are available. An eA collider is the cleanest place tostudy these high parton density phenomena.
Evolution Equations of QCD in the Non-Linear Saturation Region. – p.16