Analytical approach to gluon saturation and description of ... · Analytical approach to gluon...

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 The model

I Fit results

I Conclusions

Outline

I The model

I Fit results

I Conclusions

Outline

I The model

I Fit results

I Conclusions

Outline

I The model

I Fit results

I Conclusions

Outline

I The model

I Fit results

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

σ(x ,Q2) =

∫d2r

∫dz |Ψ(r , z)|2 σ(r , x)

σ(r , x) = σ0

(1− e−

r2 Q2s

Q2s (x , λ) = Q2

)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

γ* γ*

proton

r σ(x ,Q2) =

∫d2r

∫dz |Ψ(r , z)|2 σ(r , x)

σ(r , x) = σ0

(1− e−

r2 Q2s

Q2s (x , λ) = Q2

I Bjorken-x defined as x = Q2

Q2+W 2 , W 2 = (p2 + q2)

0 = 1GeV 2)

γ* γ*

proton

r σ(x ,Q2) =

∫d2r

∫dz |Ψ(r , z)|2 σ(r , x)

σ(r , x) = σ0

(1− e−

r2 Q2s

Q2s (x , λ) = Q2

)λI Bjorken-x defined as x = Q2

Q2+W 2 , W 2 = (p2 + q2)

0 = 1GeV 2)

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 =

r2+ µ2

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

)−λ)

V σ(r , x) = σ0

{1− e

− π3σ0

r2αs (µ2) xg(x,µ2)}

σ(x , r) 'π

3r2αsxg(x , µ2), µ2 =

r2+ µ2

xg(x ,Q20 ) = Agx

−λg (1− x)5.6, (Q20 = 1GeV 2)

The modified model:

σ(r , x) = σ0

(1− e

− 14r2Q2

)−λ)

V σ(r , x) = σ0

{1− e

− π3σ0

σ(x , r) 'π

3r2αsxg(x , µ2), µ2 =

r2+ µ2

xg(x ,Q20 ) = Agx

−λg (1− x)5.6, (Q20 = 1GeV 2)

The modified model:

σ(r , x) = σ0

(1− e

− 14r2Q2

)−λ)

V σ(r , x) = σ0

{1− e

− π3σ0

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

and thus

σ(r , x) = 2

∫d2b

(1− e

− π2Nc

d2σqq

d2b= 2

(1− e

− π2Nc

S(b) =1

2πBGe− b2

and thus

σ(r , x) = 2

∫d2b

(1− e

− π2Nc

d2σqq

d2b= 2

(1− e

− π2Nc

S(b) =1

2πBGe− b2

and thus

σ(r , x) = 2

∫d2b

(1− e

− π2Nc

d2σqq

d2b= 2

(1− e

− π2Nc

S(b) =1

2πBGe− b2

and thus

σ(r , x) = 2

∫d2b

(1− e

− π2Nc

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 ) =

)2{γs+ 1

κλYln(

)}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

N(rQs ,Y ) =

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

Qs(x) = Q20

N(rQs ,Y ) =

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

Qs(x) = Q20

N(rQs ,Y ) =

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

Qs(x) = Q20

N(rQs ,Y ) =

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

Qs(x) = Q20

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:

d2b= 2N(rQs ,Y ) = 2×

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

Qs(x) = Q20

)λThe b dependent saturation scale defined as

Qs(x , b) =( x0

)] 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.

∂N(k,Y )

2(k,Y )

γ(k) = 1 +∂

∂ ln k2

andχ(γ) = 2ψ(1)− ψ(1− γ)− ψ(γ)

∂N(k,Y )

2(k,Y )

γ(k) = 1 +∂

∂ ln k2

andχ(γ) = 2ψ(1)− ψ(1− γ)− ψ(γ)

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 )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

2. Plug in

∂N(k,Y )

2(k,Y )

3. Solve

After a lot of tedious algebra one arrives at:

Nsat(z) = 1− e−φ(z)

I z = 2 ln(

)I φ(z) is obtained from

z =√

∫ φ

dφ′√φ′ + e−φ′ − 1

with the initial condition φ0

I z = 2 ln(

z =√

∫ φ

dφ′√φ′ + e−φ′ − 1

I z = 2 ln(

z =√

∫ φ

dφ′√φ′ + e−φ′ − 1

I z = 2 ln(

I φ(z) is obtained from

z =√

∫ φ

dφ′√φ′ + e−φ′ − 1

I z = 2 ln(

z =√

∫ φ

dφ′√φ′ + e−φ′ − 1

The model

Summarizing, the scattering amplitude based on analytical solution to BK equationwith the simplified kernel is

N(r ,Y ) = 2×

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

1− e−φ(2 ln

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

)] 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

N(r ,Y ) = 2×

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

1− e−φ(2 ln

for rQs > 2

Qs(x , b) =( x0

)] 12γs

The model

N(r ,Y ) = 2×

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

1− e−φ(2 ln

for rQs > 2

Qs(x , b) =( x0

)] 12γs

The model

N(r ,Y ) = 2×

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

1− e−φ(2 ln

for rQs > 2

Qs(x , b) =( x0

)] 12γs

The model

N(r ,Y ) = 2×

)2{γs+ 1

κλYln(

)}for rQs ≤ 2

1− e−φ(2 ln

for rQs > 2

Qs(x , b) =( x0

)] 12γs

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

F2(x ,Q2) =Q2

4π2αem

∑T ,L

∫d2b

∫d2r

∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))

The result:

2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81

Fit results

F2(x ,Q2) =Q2

4π2αem

∑T ,L

∫d2b

∫d2r

∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))

The result:

2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81

Fit results

F2(x ,Q2) =Q2

4π2αem

∑T ,L

∫d2b

∫d2r

∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))

The result:

2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81

Fit results

F2(x ,Q2) =Q2

4π2αem

∑T ,L

∫d2b

∫d2r

∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))

The result:

2.63× 10−7 0.105 3.34 0.106 142/174 = 0.81

Fit results

F2(x ,Q2) =Q2

4π2αem

∑T ,L

∫d2b

∫d2r

∫dz|Ψ(z,Q2, r)T ,L|2 N(r , ln(1/x))

The result:

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