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Analytical approach to gluon saturation anddescription of DIS data
Andrey Kormilitzin
Tel Aviv University
EMMI workshopGSI, Darmstadt, November 22-24, 2010
Outline
I Brief overview of saturation models
I Analytical solution to BK equation in saturation region
I The model
I Fit results
I Description of data
I Conclusions
Outline
I Brief overview of saturation models
I Analytical solution to BK equation in saturation region
I The model
I Fit results
I Description of data
I Conclusions
Outline
I Brief overview of saturation models
I Analytical solution to BK equation in saturation region
I The model
I Fit results
I Description of data
I Conclusions
Outline
I Brief overview of saturation models
I Analytical solution to BK equation in saturation region
I The model
I Fit results
I Description of data
I Conclusions
Outline
I Brief overview of saturation models
I Analytical solution to BK equation in saturation region
I The model
I Fit results
I Description of data
I Conclusions
Outline
I Brief overview of saturation models
I Analytical solution to BK equation in saturation region
I The model
I Fit results
I Description of data
I Conclusions
Brief overview of saturation models
Eikonal Glauber-type gluon saturation model in dipole framework(K. J. Golec-Biernat and M. Wusthoff; Phys.Rev.D 59 (1998) 014017; Phys.Rev. D 60 (1999), 114023; )
γ* γ*
proton
r
σ(x ,Q2) =
∫d2r
∫dz |Ψ(r , z)|2 σ(r , x)
σ(r , x) = σ0
(1− e−
r2 Q2s
4
)
Q2s (x , λ) = Q2
0
( x0
x
)λI Bjorken-x defined as x = Q2
Q2+W 2 , W 2 = (p2 + q2)
I r - dipole transverse size
I |Ψ(r , z)|2 is the squared photon wave function
I σ(r , x) is the dipole cross section which is modeled
I Q2s (x , λ) is the saturation scale (with Q2
0 = 1GeV 2)
I σ0, x0, λ - are free parameters of the model
Brief overview of saturation models
Eikonal Glauber-type gluon saturation model in dipole framework(K. J. Golec-Biernat and M. Wusthoff; Phys.Rev.D 59 (1998) 014017; Phys.Rev. D 60 (1999), 114023; )
γ* γ*
proton
r σ(x ,Q2) =
∫d2r
∫dz |Ψ(r , z)|2 σ(r , x)
σ(r , x) = σ0
(1− e−
r2 Q2s
4
)
Q2s (x , λ) = Q2
0
( x0
x
)λ
I Bjorken-x defined as x = Q2
Q2+W 2 , W 2 = (p2 + q2)
I r - dipole transverse size
I |Ψ(r , z)|2 is the squared photon wave function
I σ(r , x) is the dipole cross section which is modeled
I Q2s (x , λ) is the saturation scale (with Q2
0 = 1GeV 2)
I σ0, x0, λ - are free parameters of the model
Brief overview of saturation models
Eikonal Glauber-type gluon saturation model in dipole framework(K. J. Golec-Biernat and M. Wusthoff; Phys.Rev.D 59 (1998) 014017; Phys.Rev. D 60 (1999), 114023; )
γ* γ*
proton
r σ(x ,Q2) =
∫d2r
∫dz |Ψ(r , z)|2 σ(r , x)
σ(r , x) = σ0
(1− e−
r2 Q2s
4
)
Q2s (x , λ) = Q2
0
( x0
x
)λI Bjorken-x defined as x = Q2
Q2+W 2 , W 2 = (p2 + q2)
I r - dipole transverse size
I |Ψ(r , z)|2 is the squared photon wave function
I σ(r , x) is the dipole cross section which is modeled
I Q2s (x , λ) is the saturation scale (with Q2
0 = 1GeV 2)
I σ0, x0, λ - are free parameters of the model
Brief overview of saturation models
A Modification of the Saturation Model: DGLAP Evolution(J. Bartels, K. Golec-Biernat, H. Kowalski; Phys.Rev. D 66 (2002) 014001)
In order to get a better description of DIS data at high values of photon virtuality Q2,one has to incorporate DGLAP evolution.
Key idea: in the small-r region the dipole cross section is related to the gluon density
σ(x , r) 'π
3r2αsxg(x , µ2), µ2 =
C
r2+ µ2
0
with initial condition
xg(x ,Q20 ) = Agx
−λg (1− x)5.6, (Q20 = 1GeV 2)
The modified model:
σ(r , x) = σ0
(1− e
− 14r2Q2
0
(xx0
)−λ)
V σ(r , x) = σ0
{1− e
− π3σ0
r2αs (µ2) xg(x,µ2)}
Brief overview of saturation models
A Modification of the Saturation Model: DGLAP Evolution(J. Bartels, K. Golec-Biernat, H. Kowalski; Phys.Rev. D 66 (2002) 014001)
In order to get a better description of DIS data at high values of photon virtuality Q2,one has to incorporate DGLAP evolution.
Key idea: in the small-r region the dipole cross section is related to the gluon density
σ(x , r) 'π
3r2αsxg(x , µ2), µ2 =
C
r2+ µ2
0
with initial condition
xg(x ,Q20 ) = Agx
−λg (1− x)5.6, (Q20 = 1GeV 2)
The modified model:
σ(r , x) = σ0
(1− e
− 14r2Q2
0
(xx0
)−λ)
V σ(r , x) = σ0
{1− e
− π3σ0
r2αs (µ2) xg(x,µ2)}
Brief overview of saturation models
A Modification of the Saturation Model: DGLAP Evolution(J. Bartels, K. Golec-Biernat, H. Kowalski; Phys.Rev. D 66 (2002) 014001)
In order to get a better description of DIS data at high values of photon virtuality Q2,one has to incorporate DGLAP evolution.
Key idea: in the small-r region the dipole cross section is related to the gluon density
σ(x , r) 'π
3r2αsxg(x , µ2), µ2 =
C
r2+ µ2
0
with initial condition
xg(x ,Q20 ) = Agx
−λg (1− x)5.6, (Q20 = 1GeV 2)
The modified model:
σ(r , x) = σ0
(1− e
− 14r2Q2
0
(xx0
)−λ)
V σ(r , x) = σ0
{1− e
− π3σ0
r2αs (µ2) xg(x,µ2)}
Brief overview of saturation models
An Impact Parameter Dipole Saturation Model(H. Kowalski, D. Teaney; Phys.Rev. D 68 (2003) 114005)
In order to improve previously presented saturation models, the impact parameterdependence is introduced.
The dipole-targe cross section at a given impact parameter b is
d2σqq
d2b= 2
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
with the proton profile function
S(b) =1
2πBGe− b2
2BG
and thus
σ(r , x) = 2
∫d2b
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
Brief overview of saturation models
An Impact Parameter Dipole Saturation Model(H. Kowalski, D. Teaney; Phys.Rev. D 68 (2003) 114005)
In order to improve previously presented saturation models, the impact parameterdependence is introduced.
The dipole-targe cross section at a given impact parameter b is
d2σqq
d2b= 2
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
with the proton profile function
S(b) =1
2πBGe− b2
2BG
and thus
σ(r , x) = 2
∫d2b
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
Brief overview of saturation models
An Impact Parameter Dipole Saturation Model(H. Kowalski, D. Teaney; Phys.Rev. D 68 (2003) 114005)
In order to improve previously presented saturation models, the impact parameterdependence is introduced.
The dipole-targe cross section at a given impact parameter b is
d2σqq
d2b= 2
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
with the proton profile function
S(b) =1
2πBGe− b2
2BG
and thus
σ(r , x) = 2
∫d2b
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
Brief overview of saturation models
An Impact Parameter Dipole Saturation Model(H. Kowalski, D. Teaney; Phys.Rev. D 68 (2003) 114005)
In order to improve previously presented saturation models, the impact parameterdependence is introduced.
The dipole-targe cross section at a given impact parameter b is
d2σqq
d2b= 2
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
with the proton profile function
S(b) =1
2πBGe− b2
2BG
and thus
σ(r , x) = 2
∫d2b
(1− e
− π2Nc
r2αs (µ2) xg(x,µ2) S(b))
Brief overview of saturation models
Saturation and BFKL dynamics in the HERA data at small x(E. Iancu, K. Itakura, S. Munier; Phys.Lett. B 590:199-208 (2004))
In this approach, the total cross section is written as σ(x , r) = 2πR2 N(x , r)where N(x , r) given by the solution to the homogeneous version of the non-linearevolution equation.
The function N(r , x) constructed by smoothly interpolating between two limitingbehaviors:
I the solution to the BFKL equation for small dipole sizes r � 1/Qs(x)
I the Levin-Tuchin result for larger dipoles r � 1/Qs(x).
To summarize, the scattering amplitude used in the CGC model
N(rQs ,Y ) =
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−A ln2(B rQs ) for rQs > 2
where A,B satisfy the continuity condition, κ = χ′′(γs)/χ′(γs), Y = ln(1/x),
Qs(x) = Q20
( x0x
)λ
Brief overview of saturation models
Saturation and BFKL dynamics in the HERA data at small x(E. Iancu, K. Itakura, S. Munier; Phys.Lett. B 590:199-208 (2004))
In this approach, the total cross section is written as σ(x , r) = 2πR2 N(x , r)where N(x , r) given by the solution to the homogeneous version of the non-linearevolution equation.
The function N(r , x) constructed by smoothly interpolating between two limitingbehaviors:
I the solution to the BFKL equation for small dipole sizes r � 1/Qs(x)
I the Levin-Tuchin result for larger dipoles r � 1/Qs(x).
To summarize, the scattering amplitude used in the CGC model
N(rQs ,Y ) =
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−A ln2(B rQs ) for rQs > 2
where A,B satisfy the continuity condition, κ = χ′′(γs)/χ′(γs), Y = ln(1/x),
Qs(x) = Q20
( x0x
)λ
Brief overview of saturation models
Saturation and BFKL dynamics in the HERA data at small x(E. Iancu, K. Itakura, S. Munier; Phys.Lett. B 590:199-208 (2004))
In this approach, the total cross section is written as σ(x , r) = 2πR2 N(x , r)where N(x , r) given by the solution to the homogeneous version of the non-linearevolution equation.
The function N(r , x) constructed by smoothly interpolating between two limitingbehaviors:
I the solution to the BFKL equation for small dipole sizes r � 1/Qs(x)
I the Levin-Tuchin result for larger dipoles r � 1/Qs(x).
To summarize, the scattering amplitude used in the CGC model
N(rQs ,Y ) =
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−A ln2(B rQs ) for rQs > 2
where A,B satisfy the continuity condition, κ = χ′′(γs)/χ′(γs), Y = ln(1/x),
Qs(x) = Q20
( x0x
)λ
Brief overview of saturation models
Saturation and BFKL dynamics in the HERA data at small x(E. Iancu, K. Itakura, S. Munier; Phys.Lett. B 590:199-208 (2004))
In this approach, the total cross section is written as σ(x , r) = 2πR2 N(x , r)where N(x , r) given by the solution to the homogeneous version of the non-linearevolution equation.
The function N(r , x) constructed by smoothly interpolating between two limitingbehaviors:
I the solution to the BFKL equation for small dipole sizes r � 1/Qs(x)
I the Levin-Tuchin result for larger dipoles r � 1/Qs(x).
To summarize, the scattering amplitude used in the CGC model
N(rQs ,Y ) =
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−A ln2(B rQs ) for rQs > 2
where A,B satisfy the continuity condition, κ = χ′′(γs)/χ′(γs), Y = ln(1/x),
Qs(x) = Q20
( x0x
)λ
Brief overview of saturation models
Saturation and BFKL dynamics in the HERA data at small x(E. Iancu, K. Itakura, S. Munier; Phys.Lett. B 590:199-208 (2004))
In this approach, the total cross section is written as σ(x , r) = 2πR2 N(x , r)where N(x , r) given by the solution to the homogeneous version of the non-linearevolution equation.
The function N(r , x) constructed by smoothly interpolating between two limitingbehaviors:
I the solution to the BFKL equation for small dipole sizes r � 1/Qs(x)
I the Levin-Tuchin result for larger dipoles r � 1/Qs(x).
To summarize, the scattering amplitude used in the CGC model
N(rQs ,Y ) =
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−A ln2(B rQs ) for rQs > 2
where A,B satisfy the continuity condition, κ = χ′′(γs)/χ′(γs), Y = ln(1/x),
Qs(x) = Q20
( x0x
)λ
Brief overview of saturation models
Impact parameter dependent colour glass condensate dipole model(G. Watt, H. Kowalski; Phys.Rev. D 78 (2004) 014016)
The impact parameter b dependance is introduced into CGC model in the followingway:
d2σ
d2b= 2N(rQs ,Y ) = 2×
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−A ln2(B rQs ) for rQs > 2
where A,B satisfy the continuity condition, κ = χ′′(γs)/χ′(γs), Y = ln(1/x),
Qs(x) = Q20
( x0x
)λThe b dependent saturation scale defined as
Qs(x , b) =( x0
x
)λ2
[exp
(−
b2
2BCGC
)] 12γs
Analytical solution to BK equation in saturation region
Solution to the evolution equation for high parton density QCD(E. Levin, K. Tuchin; Nucl.Phys. B 573 (2000) 833-852)
Analytical solution to non-linear Balitsky-Kovchegov equation with the simplifiedkernel function is obtained.
The BK equation (written in momentum space)
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
where χ(γ(k)) is an operator
γ(k) = 1 +∂
∂ ln k2
andχ(γ) = 2ψ(1)− ψ(1− γ)− ψ(γ)
which corresponds to the eigenvalue of BFKL equation.
Analytical solution to BK equation in saturation region
Solution to the evolution equation for high parton density QCD(E. Levin, K. Tuchin; Nucl.Phys. B 573 (2000) 833-852)
Analytical solution to non-linear Balitsky-Kovchegov equation with the simplifiedkernel function is obtained.
The BK equation (written in momentum space)
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
where χ(γ(k)) is an operator
γ(k) = 1 +∂
∂ ln k2
andχ(γ) = 2ψ(1)− ψ(1− γ)− ψ(γ)
which corresponds to the eigenvalue of BFKL equation.
Analytical solution to BK equation in saturation region
Solution to the evolution equation for high parton density QCD(E. Levin, K. Tuchin; Nucl.Phys. B 573 (2000) 833-852)
Analytical solution to non-linear Balitsky-Kovchegov equation with the simplifiedkernel function is obtained.
The BK equation (written in momentum space)
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
where χ(γ(k)) is an operator
γ(k) = 1 +∂
∂ ln k2
andχ(γ) = 2ψ(1)− ψ(1− γ)− ψ(γ)
which corresponds to the eigenvalue of BFKL equation.
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
The model for the kernel:
I γ → 0 corresponds to the double logarithmic approximation to the BFKLequation (DGLAP)
I γ → 1 corresponds to the saturation region
Key-idea: take a limit of γ → 1 and obtain a solution in the saturation region.
Solution to BK equation in saturation domain:
1. Expand χ(γ)|γ=1 ⇒ χ(γ) = 11−γ (leading twist)
2. Plug in
∂N(k,Y )
∂Y= αs χ(γ(k)) N(k,Y )− αs N
2(k,Y )
3. Solve
Analytical solution to BK equation in saturation region
After a lot of tedious algebra one arrives at:
Nsat(z) = 1− e−φ(z)
where
I z = 2 ln(
rQs2
)I φ(z) is obtained from
z =√
2
∫ φ
φ0
dφ′√φ′ + e−φ′ − 1
with the initial condition φ0
Analytical solution to BK equation in saturation region
After a lot of tedious algebra one arrives at:
Nsat(z) = 1− e−φ(z)
where
I z = 2 ln(
rQs2
)I φ(z) is obtained from
z =√
2
∫ φ
φ0
dφ′√φ′ + e−φ′ − 1
with the initial condition φ0
Analytical solution to BK equation in saturation region
After a lot of tedious algebra one arrives at:
Nsat(z) = 1− e−φ(z)
where
I z = 2 ln(
rQs2
)I φ(z) is obtained from
z =√
2
∫ φ
φ0
dφ′√φ′ + e−φ′ − 1
with the initial condition φ0
Analytical solution to BK equation in saturation region
After a lot of tedious algebra one arrives at:
Nsat(z) = 1− e−φ(z)
where
I z = 2 ln(
rQs2
)
I φ(z) is obtained from
z =√
2
∫ φ
φ0
dφ′√φ′ + e−φ′ − 1
with the initial condition φ0
Analytical solution to BK equation in saturation region
After a lot of tedious algebra one arrives at:
Nsat(z) = 1− e−φ(z)
where
I z = 2 ln(
rQs2
)I φ(z) is obtained from
z =√
2
∫ φ
φ0
dφ′√φ′ + e−φ′ − 1
with the initial condition φ0
The model
Summarizing, the scattering amplitude based on analytical solution to BK equationwith the simplified kernel is
N(r ,Y ) = 2×
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−φ(2 ln
(rQs
2
))
for rQs > 2
The main features of the model:
1. the solution allows to take into account the impact parameter dependencewhich can be absorbed in b-dependence of the saturation scale.
Qs(x , b) =( x0
x
)λ2
[exp
(−
b2
2B
)] 12γs
2. the exact form of the function φ(z) allows a proper description of the entiresaturation domain and not only asymptotically (z ≈ 0 or z � 0)
I φ0, λ, x0, B - are free parameters of the model
The model
Summarizing, the scattering amplitude based on analytical solution to BK equationwith the simplified kernel is
N(r ,Y ) = 2×
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−φ(2 ln
(rQs
2
))
for rQs > 2
The main features of the model:
1. the solution allows to take into account the impact parameter dependencewhich can be absorbed in b-dependence of the saturation scale.
Qs(x , b) =( x0
x
)λ2
[exp
(−
b2
2B
)] 12γs
2. the exact form of the function φ(z) allows a proper description of the entiresaturation domain and not only asymptotically (z ≈ 0 or z � 0)
I φ0, λ, x0, B - are free parameters of the model
The model
Summarizing, the scattering amplitude based on analytical solution to BK equationwith the simplified kernel is
N(r ,Y ) = 2×
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−φ(2 ln
(rQs
2
))
for rQs > 2
The main features of the model:
1. the solution allows to take into account the impact parameter dependencewhich can be absorbed in b-dependence of the saturation scale.
Qs(x , b) =( x0
x
)λ2
[exp
(−
b2
2B
)] 12γs
2. the exact form of the function φ(z) allows a proper description of the entiresaturation domain and not only asymptotically (z ≈ 0 or z � 0)
I φ0, λ, x0, B - are free parameters of the model
The model
Summarizing, the scattering amplitude based on analytical solution to BK equationwith the simplified kernel is
N(r ,Y ) = 2×
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−φ(2 ln
(rQs
2
))
for rQs > 2
The main features of the model:
1. the solution allows to take into account the impact parameter dependencewhich can be absorbed in b-dependence of the saturation scale.
Qs(x , b) =( x0
x
)λ2
[exp
(−
b2
2B
)] 12γs
2. the exact form of the function φ(z) allows a proper description of the entiresaturation domain and not only asymptotically (z ≈ 0 or z � 0)
I φ0, λ, x0, B - are free parameters of the model
The model
Summarizing, the scattering amplitude based on analytical solution to BK equationwith the simplified kernel is
N(r ,Y ) = 2×
N0
(rQs
2
)2{γs+ 1
κλYln(
2rQs
)}for rQs ≤ 2
1− e−φ(2 ln
(rQs
2
))
for rQs > 2
The main features of the model:
1. the solution allows to take into account the impact parameter dependencewhich can be absorbed in b-dependence of the saturation scale.
Qs(x , b) =( x0
x
)λ2
[exp
(−
b2
2B
)] 12γs
2. the exact form of the function φ(z) allows a proper description of the entiresaturation domain and not only asymptotically (z ≈ 0 or z � 0)
I φ0, λ, x0, B - are free parameters of the model
Fit results
In order to fix the free parameters, we expressed a proton structure function F2 interms of the model amplitude N(r , b, x)
F2(x ,Q2) =Q2
4π2αem
∑T ,L
∫d2b
∫d2r
∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))
and fitted to the experimental data on DIS.
The fit was performed with Nf = 4 flavors with mu,d,s = 0.140GeV 2 and charmquark with mc = 1.4GeV 2
Warming up, the model was fitted to previous data on DIS from H1 collaboration(x < 0.01, Q2 : 0.045− 150GeV 2)
The result:
x0 λ BCGC/GeV2 φ0 χ2/p.d .f
2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81
Fit results
In order to fix the free parameters, we expressed a proton structure function F2 interms of the model amplitude N(r , b, x)
F2(x ,Q2) =Q2
4π2αem
∑T ,L
∫d2b
∫d2r
∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))
and fitted to the experimental data on DIS.
The fit was performed with Nf = 4 flavors with mu,d,s = 0.140GeV 2 and charmquark with mc = 1.4GeV 2
Warming up, the model was fitted to previous data on DIS from H1 collaboration(x < 0.01, Q2 : 0.045− 150GeV 2)
The result:
x0 λ BCGC/GeV2 φ0 χ2/p.d .f
2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81
Fit results
In order to fix the free parameters, we expressed a proton structure function F2 interms of the model amplitude N(r , b, x)
F2(x ,Q2) =Q2
4π2αem
∑T ,L
∫d2b
∫d2r
∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))
and fitted to the experimental data on DIS.
The fit was performed with Nf = 4 flavors with mu,d,s = 0.140GeV 2 and charmquark with mc = 1.4GeV 2
Warming up, the model was fitted to previous data on DIS from H1 collaboration(x < 0.01, Q2 : 0.045− 150GeV 2)
The result:
x0 λ BCGC/GeV2 φ0 χ2/p.d .f
2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81
Fit results
In order to fix the free parameters, we expressed a proton structure function F2 interms of the model amplitude N(r , b, x)
F2(x ,Q2) =Q2
4π2αem
∑T ,L
∫d2b
∫d2r
∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))
and fitted to the experimental data on DIS.
The fit was performed with Nf = 4 flavors with mu,d,s = 0.140GeV 2 and charmquark with mc = 1.4GeV 2
Warming up, the model was fitted to previous data on DIS from H1 collaboration(x < 0.01, Q2 : 0.045− 150GeV 2)
The result:
x0 λ BCGC/GeV2 φ0 χ2/p.d .f
2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81
Fit results
In order to fix the free parameters, we expressed a proton structure function F2 interms of the model amplitude N(r , b, x)
F2(x ,Q2) =Q2
4π2αem
∑T ,L
∫d2b
∫d2r
∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))
and fitted to the experimental data on DIS.
The fit was performed with Nf = 4 flavors with mu,d,s = 0.140GeV 2 and charmquark with mc = 1.4GeV 2
Warming up, the model was fitted to previous data on DIS from H1 collaboration(x < 0.01, Q2 : 0.045− 150GeV 2)
The result:
x0 λ BCGC/GeV2 φ0 χ2/p.d .f
2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81
Fit results
In order to fix the free parameters, we expressed a proton structure function F2 interms of the model amplitude N(r , b, x)
F2(x ,Q2) =Q2
4π2αem
∑T ,L
∫d2b
∫d2r
∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))
and fitted to the experimental data on DIS.
The fit was performed with Nf = 4 flavors with mu,d,s = 0.140GeV 2 and charmquark with mc = 1.4GeV 2
Warming up, the model was fitted to previous data on DIS from H1 collaboration(x < 0.01, Q2 : 0.045− 150GeV 2)
The result:
x0 λ BCGC/GeV2 φ0 χ2/p.d .f
2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
Armed with this success, the model was fitted to recent compilation of combinedHERA data (from H1 and ZEUS collaborations).
preliminary result for χ2/p.d .f . : 671/234 = 2.86
in order to improve the fit, a sieve-procedure (eliminating the ”outliers”) was applied(M. M. Block, ”Sifting data in the real world”, Nucl. Instrum. Meth. A 556 (2006) 308324)
The ”sieve-procedure” states:
1. make a ”robust” fit to all data
2. using the parameters from the fit, evaluate ∆χ2i for each of the N experimental
points
3. eliminate points which satisfy ∆χ2i > ∆χ2
max = 9 and perform a fit to theremained points
4. estimate χ2sieve after rejection of ”outliers” and renormalize:
χ2new = R(∆χ2
max )χ2sieve
5. if χ2new is acceptable (∼ 1), stop. If not, return to step 2 and repeat all the
steps with ∆χ2max = 6, 4, 2 and appropriate renormalization factor R(χ2
max )
*R(9) = 1.01; R(6) = 1.06; R(4) = 1.19; R(2) = 1.97;
Fit results
The fit:
φ0 λ B/GeV 2 x0 R × χ2/p.d .f
robust 0.10886 0.110553 3.59455 9.47162× 10−8 2.86
∆χ2max = 9 0.107447 0.109086 3.60267 1.06802× 10−7 2.17
∆χ2max = 6 0.104895 0.108138 3.58692 1.39792× 10−7 1.63
∆χ2max = 4 0.105089 0.107562 3.58494 1.425× 10−7 1.19
∆χ2max = 2 0.105325 0.106371 3.54161 1.60654× 10−7 1.06
Fit results
The fit:
φ0 λ B/GeV 2 x0 R × χ2/p.d .f
robust 0.10886 0.110553 3.59455 9.47162× 10−8 2.86
∆χ2max = 9 0.107447 0.109086 3.60267 1.06802× 10−7 2.17
∆χ2max = 6 0.104895 0.108138 3.58692 1.39792× 10−7 1.63
∆χ2max = 4 0.105089 0.107562 3.58494 1.425× 10−7 1.19
∆χ2max = 2 0.105325 0.106371 3.54161 1.60654× 10−7 1.06
Fit results
The fit:
φ0 λ B/GeV 2 x0 R × χ2/p.d .f
robust 0.10886 0.110553 3.59455 9.47162× 10−8 2.86
∆χ2max = 9 0.107447 0.109086 3.60267 1.06802× 10−7 2.17
∆χ2max = 6 0.104895 0.108138 3.58692 1.39792× 10−7 1.63
∆χ2max = 4 0.105089 0.107562 3.58494 1.425× 10−7 1.19
∆χ2max = 2 0.105325 0.106371 3.54161 1.60654× 10−7 1.06
Fit results
The fit:
φ0 λ B/GeV 2 x0 R × χ2/p.d .f
robust 0.10886 0.110553 3.59455 9.47162× 10−8 2.86
∆χ2max = 9 0.107447 0.109086 3.60267 1.06802× 10−7 2.17
∆χ2max = 6 0.104895 0.108138 3.58692 1.39792× 10−7 1.63
∆χ2max = 4 0.105089 0.107562 3.58494 1.425× 10−7 1.19
∆χ2max = 2 0.105325 0.106371 3.54161 1.60654× 10−7 1.06
Fit results
The fit:
φ0 λ B/GeV 2 x0 R × χ2/p.d .f
robust 0.10886 0.110553 3.59455 9.47162× 10−8 2.86
∆χ2max = 9 0.107447 0.109086 3.60267 1.06802× 10−7 2.17
∆χ2max = 6 0.104895 0.108138 3.58692 1.39792× 10−7 1.63
∆χ2max = 4 0.105089 0.107562 3.58494 1.425× 10−7 1.19
∆χ2max = 2 0.105325 0.106371 3.54161 1.60654× 10−7 1.06
Figure: Description of proton structure function F2(x,Q2). Data points are taken from the recent compilation
of combined HERA data. (F. D. Aaron et al.; JHEP 1001 (2010) 109 ); χ2/d .o.f . = 1.06
0
0.5
1
1.5
2
2.5
Q2 = 0.1 GeV2
Q2 = 0.15 GeV2
Q2 = 0.2 GeV2
Q2 = 0.25 GeV2
0
0.5
1
1.5
2
2.5
Q2 = 0.35 GeV2
0
0.5
1
1.5
2
Q2 = 0.4 GeV2
Q2 = 0.5 GeV2
Q2 = 0.65 GeV2
Q2 = 0.85 GeV2
0
0.5
1
1.5
2
Q2 = 1.2 GeV2
0
0.5
1
1.5
2
Q2 = 1.5 GeV2
Q2 = 2 GeV2
Q2 = 2.7 GeV2
Q2 = 3.5 GeV2
0
0.5
1
1.5
2
Q2 = 4.5 GeV2
0
0.5
1
1.5
2
Q2 = 6.5 GeV2
Q2 = 8.5 GeV2
Q2 = 10 GeV2
Q2 = 12 GeV2
0
0.5
1
1.5
2
Q2 = 15 GeV2
0
0.5
1
1.5
2
Q2 = 18 GeV2
Q2 = 22 GeV2
Q2 = 27 GeV2
Q2 = 35 GeV2
0
0.5
1
1.5
2
Q2 = 45 GeV2
10−6
10−5
10−4
10−3
10−20
0.5
1
1.5
2
Q2 = 60 GeV2
10−6
10−5
10−4
10−3
10−2
Q2 = 70 GeV2
10−6
10−5
10−4
10−3
10−2
Q2 = 90 GeV2
10−6
10−5
10−4
10−3
10−2
Q2 = 120 GeV2
10−6
10−5
10−4
10−3
10−2 0
0.5
1
1.5
2
Q2 = 150 GeV2
10−6
10−5
10−4
10−3
10−20
0.1
0.2
0.1
10−6
10−5
10−4
10−3
10−20
0.2
0.4
0.15
10−6
10−5
10−4
10−3
10−20.1
0.2
0.3
0.4
0.2
10−6
10−5
10−4
10−3
10−20
0.2
0.4
0.25
10−6
10−5
10−4
10−3
10−20
0.5
1
0.35
10−6
10−5
10−4
10−3
10−20
0.5
1
0.4
10−6
10−5
10−4
10−3
10−20
0.5
1
0.5
10−6
10−5
10−4
10−3
10−20.2
0.4
0.6
0.8
0.65
10−6
10−5
10−4
10−3
10−20
0.5
1
0.85
10−6
10−5
10−4
10−3
10−20
0.5
1
1.2
10−6
10−5
10−4
10−3
10−20
0.5
1
1.5
10−6
10−5
10−4
10−3
10−20
0.5
1
1.5
2
10−6
10−5
10−4
10−3
10−20
0.5
1
1.5
2.7
10−6
10−5
10−4
10−3
10−20
1
2
3.5
10−6
10−5
10−4
10−3
10−20
1
2
4.5
10−6
10−5
10−4
10−3
10−20
1
2
3
6.5
10−6
10−5
10−4
10−3
10−20
1
2
3
8.5
10−6
10−5
10−4
10−3
10−20
1
2
3
10
10−6
10−5
10−4
10−3
10−20
1
2
3
12
10−6
10−5
10−4
10−3
10−20
2
4
15
10−6
10−5
10−4
10−3
10−20
2
4
18
10−6
10−5
10−4
10−3
10−20
2
4
22
10−6
10−5
10−4
10−3
10−20
2
4
27
10−6
10−5
10−4
10−3
10−20
2
4
6
35
10−6
10−5
10−4
10−3
10−20
2
4
6
45
10−6
10−5
10−4
10−3
10−20
2
4
6
60
10−6
10−5
10−4
10−3
10−20
2
4
6
70
10−6
10−5
10−4
10−3
10−20
5
10
90
Conclusions
Concluding this work:
I This model is a direct consequence of the analytical solutionto the non linear evolution equation with the simplified kernel
I The form of φ(z) gives a proper description inside thesaturation region for all values of z and not onlyasymptotically z → 0 or z � 0
I The overall fit is good and the model can provide a reliableprediction
I The model is impact parameter dependent and it isintroduced via the saturation scale
Conclusions
Concluding this work:
I This model is a direct consequence of the analytical solutionto the non linear evolution equation with the simplified kernel
I The form of φ(z) gives a proper description inside thesaturation region for all values of z and not onlyasymptotically z → 0 or z � 0
I The overall fit is good and the model can provide a reliableprediction
I The model is impact parameter dependent and it isintroduced via the saturation scale
Conclusions
Concluding this work:
I This model is a direct consequence of the analytical solutionto the non linear evolution equation with the simplified kernel
I The form of φ(z) gives a proper description inside thesaturation region for all values of z and not onlyasymptotically z → 0 or z � 0
I The overall fit is good and the model can provide a reliableprediction
I The model is impact parameter dependent and it isintroduced via the saturation scale
Conclusions
Concluding this work:
I This model is a direct consequence of the analytical solutionto the non linear evolution equation with the simplified kernel
I The form of φ(z) gives a proper description inside thesaturation region for all values of z and not onlyasymptotically z → 0 or z � 0
I The overall fit is good and the model can provide a reliableprediction
I The model is impact parameter dependent and it isintroduced via the saturation scale
Conclusions
Concluding this work:
I This model is a direct consequence of the analytical solutionto the non linear evolution equation with the simplified kernel
I The form of φ(z) gives a proper description inside thesaturation region for all values of z and not onlyasymptotically z → 0 or z � 0
I The overall fit is good and the model can provide a reliableprediction
I The model is impact parameter dependent and it isintroduced via the saturation scale
Thank you! :)