The universal function in color dipole model

Physics Letters B 773 (2017) 455–461

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Physics Letters B www.elsevier.com/locate/physletb

The universal function in color dipole model Z. Jalilian, G.R. Boroun ∗ a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 July 2017 Accepted 31 August 2017 Available online 7 September 2017 Editor: J. Hisano

In this work we review color dipole model and recall properties of the saturation and geometrical scaling in this model. Our primary aim is determining the exact universal function in terms of the introduced scaling variable in different distance than the saturation radius. With inserting the mass in calculation we compute numerically the contribution of heavy productions in small x from the total structure function by the fraction of universal functions and show the geometrical scaling is established due to our scaling variable in this study. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. Introduction The color dipole model at the small-x limit of quantum chromodynamics was first derived by Nikolaev and Zaharov [1,2]. Recently this picture as an important tool confirms a different approach in investigation of the deep inelastic scattering for physicists to plan models for justifying the cross section and structure function, which has interesting applications in the low-x region [3–6]. In color dipole picture, when x → 0, in the rest frame of the proton, virtual photon transforms into a fixed-size quark–antiquark pair which remains unchanged during the scattering. The absorption cross section by quantum mechanics averaging over the effective dipole cross section σ (x, r 2 ) and according to the photon polarization is formulized as [7] γP

1

σL,T =

 dz

d2 r | L , T ( z, r )|2 σ (x, r 2 ).

(1)

0

| T , L (z, r )|2 introduces the probability of the transverse ( T ) and longitudinal ( L ) polarized photon that depends on photon virtuality, Q 2 , and the ratio of the momentum carrying by quark, z, and is given by

| T , L ( z, r )|2 =  ×

*

3αem  2π 2

eq2

q

[ z2 + (1 − z)2 ]ε 2 K 12 (εr ) + mq2 K 02 (εr ), ( T ) 4Q 2 z2 (1 − z)2 K 02 (εr ).

(L)

Corresponding author. E-mail addresses: [email protected] (Z. Jalilian), [email protected], [email protected] (G.R. Boroun).

(2)

In above expression summation is performed over quark flavors, with the charge eq and the mass mq , also ε 2 = z(1 − z) Q 2 + (mq )2 and for εr < 1 must be K 1 (εr ) ∼ θ(1 − εr )/εr and K (εr ) ∼ θ(1 − εr ), θ(1 − εr ) is be estimated by Heaviside function [8]. In very small-x region the contribution of valance quarks is negligible but the gluon distribution rises quickly and occurs a kind of the saturation called the small-x saturation [9,10]. In color 1 dipole model, in x < 0.01, gluons with the size of start to Q recombine and the effect of this mechanism appears directly in the dedicated dipole-proton cross section, σ (x, r 2 ). A suitable form of the dipole cross section has been estimated by Bartels, GolecBierant and Kowalski, BGK model, with the help of the DGLAP analysis of the gluon distribution which is expressed by [11]

σ (x, r 2 ) ∼ =

π2 3

r 2 αs ( Q 2 )xg (x, Q 2 ).

(3)

An important schema of the saturation picture leads to definition of the geometrical scaling. The geometrical scaling allows us to rewrite the total cross section to form the product of the saturation cross section in a function of the dimensionless scaling variable, universal function [12,13]. A universal function doesn’t have a fixed form but it is important in determining the behavior of the structure function in low-x. The aim of this paper is investigating and computing this function in color dipole model therefore we consider the following strategy. We introduce In section 2 the relevant geometrical scaling variable. In section 3 by calculating the total cross section according to the scaling variable we will obtain the index universal function for both the small size and the large size pairs. Since creation of the heavy pairs in high energy for discovery of the new effects in deep inelastic scattering is a significant concept, we proceed to compute

http://dx.doi.org/10.1016/j.physletb.2017.08.076 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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Z. Jalilian, G.R. Boroun / Physics Letters B 773 (2017) 455–461

this contribution in section 4. Finally in section 5 results will be summarized. 2. The geometrical scaling variable The saturation is a property of gluonic states therefore understanding the gluon density is very important. In x < 0.01 gluons start to interact to each other and it leads to the presence of the non-linear parton distribution dependent on gluon distribution, xg (x, Q 2 ), in the evolution equations. Indeed, the evolved quantum chromodynamics including the gluon distribution determines nearly precise the cross section in different physics processes. A simple approach for defining this distribution function in phenomenology has been developed by Kharzev, Lcvin and Nardi as the following [14]

 2

xg (x, Q ) =

k0

αs ( Q s2 )

S Q 2 (1 − x)4 ,

(Q < Q S )

S Q s2 (1 − x)4 .

(Q > Q S )

k0

αs ( Q s2 )

(4)

In this expression αs = 12/[25π log( Q 2 /0.224)] introduces the 1 running coupling constant, k0 S is adjusted by requiring 0 xg (x, Q 2 ) = p by GRV94 gluon distribution introduced in Ref. [15] and Q s , the saturation scale, controls the parton density growth and is defined by

Q s2 = Q 02 (

x −λ ) .

(5)

x0

The idea of the saturation focuses on transferring from the hard physics to the soft physics. At small enough x values the saturation effect leads us to the geometrical scaling as the dense gluon system. The geometrical scaling acts in both forms of evaluation equations, DGLAP and BFKL. Since this property reflects interesting details about the symmetry for discovery of this variable has been spent much attention to be acceptable in the theory and the phenomenology [16–18]. Namely, the underlying assumption of the geometrical scaling is based on writing the total cross section as a function of the dimensionless variable τ as [12] γP

2

σtot (x, Q ) = σ0 f (τ ).

(6)

In this investigation we choose the scaling variable as

τ=

σ0 ∼ =

R 0 (x)

3

αs ( Q 2 )

xg (x, r 2 ) =

k0

S

1

αs ( Q s2 ) r 2 k0

α

2 s(Q s )

S

1

σT =

3αem

σ0

2π 2



d2 r ε 2 (

× 1 + =

 σ0 , 2

σ (x, r ) = with

σ0

r

r > R0

2

, r < R0 2

R0

(8)

ε2r

) 2

d2 rmq2

dz 0

3αem

σ0

2π

dz(2z2 − 2z + 1)

r2 R 20 (x) r2

R 20 (x)

r2



R 20 (x)

q

 ,

eq2

2 3

+

mq2 r 2

,

2

(10)

and

σL =

3αem 2π 2

σ0



3αem 2π

eq2

q

d2 r 0

=

Therefore with substituting current expression in Eq. (3) we can convert the color dipole cross section to

1

r 2

0

 1 0

0

(7)

R0

eq2

q

r 2

×

(1 − x)4 . r < R 0 2

(9)

After determining the constant value, σ0 , for calculating the dimensionless universal function, according to Eq. (6), we need to obtain the physical cross section from summation the longitudinal and transverse cross section contributions. Since the qq pair with 1 1 the size of r 2 ∼ 2 ∼ makes the dominant contribu= 2 ε Q z(1 − z) ∗ P 2 tion of σ L , T (x, Q ), the integration in Eq. (1) have to be performed for εr < 1. 0 ≤ z ≤ 1 is the momentum fraction of the quark with two special cases: (1) z ∼ = 1/2 forms symmetric pairs, small size dipoles with r ≤ 1/ Q . (2) On the contrary, z ∼ = 0 or z ∼ = 1 makes asymmetric pairs, large size dipoles with r > 1/ Q . In the beginning we consider the small size pair where the dipole size is small in comparison to the saturation radius, r < R 0 , and find by putting relations (2) and (8) in Eq. (1) the contribution of transverse and longitudinal cross sections respectively

r 2

(1 − x)4 , r > R 0

(1 − x)4 .

3. The universal function

we prefer to rewrite the KLN gluon distribution as



k0 S

αs ( Q s2 )

σ0 is a normalization factor independent of x and Q 2 simultaneous associated with the saturation cross section. Indeed, when the transverse separation between the quark and antiquark is been 1 more than the saturation scale as r > , the dipole cross section Qs is limited to a black disk damps the growth of the gluon density at low x. With the help of the gluon distribution introduced in Ref. [15] we obtain this element about 23mb that agrees with experimental data proposed by H1 and ZEUS in [8,19].

R 20 (x)

r2 that R 0 (x), the x-dependence saturation radius, is defined as 1 x λ/2 R 0 (x) = ( ) . Variables Q 0 , x0 and λ > 0 are constant valQ 0 x0 ues to be obtained by fitting done by Golec-Bierant and Wuthoff with H1 and ZEUS data [10,19]. It is clear Q s2 = 21 , from Eq. (5) and the saturation radius, so

π2

r

2

R 20 (x)

σ0

 1

dz4Q 2 z2 (1 − z)2

0

 ,

r2



R 20 (x)

q

eq2

2 30



Q 2r2 .

(11)

Therefore the universal function by attention to the color dipole size of r 2 ∼ = 1/ Q 2 can be written as

f (τ ) =

1

(σT + σ L ), σ0 mq2 r 2 3αem 1  2  11 . eq + = 2π τ q 15 2

(12)

Z. Jalilian, G.R. Boroun / Physics Letters B 773 (2017) 455–461

457

Afterwards we assume small pair that its size is larger than the saturation radius, r > R 0 , and obtain intended qualities as the following

σT =

3αem

σ0

2π 2



 1

q

R 20



d2 r ε 2 (

×

eq2

0

1

ε2r

) 2

0

r2 R 20 (x)

2

1 +

dz(2z2 − 2z + 1)

R 0

r2

d2 rmq2

dz 0

R 20 (x)

0

1

r 2 2

0

1

ε2r 2

)

R 20

1 +

r

2

dz 0

=

d2 r ε 2 (

dz(2z − 2z + 1)

+

d

2

rmq2

Fig. 1. The ratio

f (τ )

αem

for small size pairs in x rang 10−6 –10−2 to dipole size,

r 2 (GeV−2 ), for light flavors.

 ,

R 20

3αem

σ0

2π



eq2

q

2 3

(1 + log (

r2

R 20

R0

2r 2

) + mq2 r 2 (1 − 2

) , (13)

and

σL =

3αem 2π 2

σ0



d2 r 0

eq2

q

R 20

×



 1

dz4Q 2 z2 (1 − z)2

0

r2 R 20 (x)

1

r 2 2 2

2

0

=



d2 r ,

dz4Q z (1 − z)

+

Fig. 2. The ratio

R 20

3αem 2π

σ0



eq2

q

2 3

Q 2 (r 2 −

light flavors.

R 20 2

1

(σT + σ L ), σ0 3αem  2  2 τ 4 eq + (1 − )(mq2 r 2 + ) . = 2π q 3 2 30

In the limitation of

αem  2 f (τ ) ∼ e . = π q q

2

(15)

τ → 0 we will have

Now we plot the ratio

(16) f (τ )

αem

αem

for small size pairs in x rang 10−6 –10−2 to

τ variable for

(14)

) .

The behavior of the universal function in this case is exhibited by

f (τ ) =

f (τ )

for universal functions (12) and (15)

in terms of r variable in Fig. 1 in different x values only for light flavors with mq  140 MeV. According to these diagrams r 2 = R 20 acts as a critical element in determining the saturation point and f (τ ) changes its position with changing x value. In Fig. 2 has been

αem

plotted in terms of the scaling variable τ in several different x. We note the situation of τ = 1 divides the plane to two distinct areas in all of diagrams and is independent of x. Briefly when r changes from r < R 0 to r > R 0 the universal function f (τ ) in Fig. 2 behaves as the following

f (τ ) ∼

1

τ

−→ f (τ ) ∼ 1,

(17)

this behavior corresponds to a smooth transition from the scaling region for τ ≥ 1 to the saturation region when τ changes from 1 to 0. So when τ ≥ 1 weak interactions occurs and when 0 < τ < 1 the system will be heavily absorbed. To continue we perform a similar analysis for large size pair. In 1 this case the condition εr < 1 is fulfilled only if z < 2 2 so the r Q integration over z in Eq. (1) has to be done before integrating over r 2 . Also for the large size dipole there must be a cut-off such as μ on energy with μ2  4mq2 . In this case for r < R 0 we will obtain required contributions as the following

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Z. Jalilian, G.R. Boroun / Physics Letters B 773 (2017) 455–461

σT =

3αem 2π 2

σ0



eq2

2  r

q

2

d rε (

1

1

2

ε2r 2

r

)

d2 r ε 2 (

+

1

μ2

1

ε2r 2

0

1

1 dz(2z2 − 2z + 1)

× 0

Q2 r 2

r 2 d

2

rmq2

r2

dz

1

0

= , +

2

μ 3αem r2  2 2 1 1 = σ0 2 eq + mq2 ( 2 − 2 ) 2π 3 Q μ R 0 (x) q μ2

μ2

Q

Q2

) + (− 2

+

σ0

2π



σL =

2π

2

σ0



eq2

(18)

) , 4

d r

2π

r2



R 20 (x)

q

R0

μ

Q

σL =

3αem 2π 2

σ0



eq2

 −9 15

r2 R 20 (x)

+(

 ,

σT =

2π 2

σ0

R 20



d2 r ε 2 (

q

4μ2 3Q 2

1

ε2r 2

1

1 dz(2z2 − 2z + 1) 0

r2 R 20 (x)

1 Q2 r 2



d2 rmq2

+ 1 2

μ

dz 0

d2 r ×

dz

1

0

r2 R 20 (x)

−

μ4 Q4

+

=

4μ6 15Q

) . (19) 6

)



dz(4Q 2 z2 (1 − z)2 ) , 0

3αem 2π

σ0





eq2 −

q

9 15

+(

4μ6 15Q 6

−

μ4 Q4

+

4μ2 3Q 2

) .

(22)

Here f (τ ) is given by

f (τ ) =

(20)

1

(σT + σ L ), σ0 mq2 3αem  2  2 r2 eq + 2 (1 + log ( 2 )) . = 2π q 30 Q R0

In the limitation of

f (τ ) ∼ =

(23)

τ −→ 0 we get the familiar relation

αem  2 e . π q q

(24)

Results for large size pairs eventually is the same as for small pairs. f (τ ) Similarly to Fig. 2 we consider this time for large pairs in

αem

Q2 r 2

R 20

eq2

terms of the scaling variable in Fig. 3 for light flavors. Again we result when τ changes from large to small value there is a re1 markable transfer from forming to the establishment the unitary

μ2

×

(21)

Q2 r 2



Q2 r 2

R 20

Finally for the large size dipole where its size is larger than the saturation radius, r > R 0 , we will have

eq2

) + log ( 2

R 20

q

d2 r

+

(σT + σ L ), σ0 3αem 1  2  1 μ2 1 1 eq + log ( 2 ) + mq2 ( 2 − 2 ) . = 2π τ q 15 Q Q μ



 ) , 2

Q

)− 2

1

1

3αem

)

μ2

So we get

f (τ ) =

Q2

Q2

r 2

dz(4Q 2 z2 (1 − z)2 )

σ0

μ2

r2

(1 + log ( 2

( 4Q 2 z2 (1 − z)2 )

1

3αem

−

μ 2

Q2 r 2

=

3Q

4

mq2

1

0

μ4

1

μ2

×

+(

2

2  r

q

6 9

q

and

3αem

eq2

and

μ4 Q

3αem

 dz ,

0

R 20



R 20 (x)

d2 rmq2

+

R 20 (x)

r2

Q2 r 2

r 2

1

+ log (

dz(2z2 − 2z + 1)

)

R 20

Q2 r 2

+

Q2 r 2

2

r2 R 20 (x)

τ

indication and τ = 1 is intermediate these two states. Up to now we conclude although the universal function is not a bounded function and does not a fixed formula, it depends only on inherent properties of dipole. 4. The contribution of heavy flavors The research on the heavy flavors creates ways for checking exactly QCD. A method for study on heavy production is the ratio

Z. Jalilian, G.R. Boroun / Physics Letters B 773 (2017) 455–461

f (τ )

Fig. 3. The ratio

for large size pairs in x rang 10−6 –10−2 to

αem

light flavors.

τ variable for

Fig. 4. The ratio

f (τ )

αem

for small size pairs in x rang 10−6 –10−2 to

459

τ variable for

light and charm flavors.

method. This method is used in order to introduced geometrical scaling variable. In [18] T. Steble has presented this variable for massless quarks as

τ0 = R 20 Q 2 ,

(25)

and has introduced for heavy ones with q-flavor the following formation

4mq2

τq = τ0 (1 +

Q2

)1+λ .

(26)

Q 2  4mq2 is requisite for creating the heavy productions. Author in his review on the scaling variable has only considered the fixed 1 case z = whereas the exact calculation of the cross section is 2 contain the integration over 0 ≤ z ≤ 1. In previous section we performed quantitative analysis for all amounts of z. In heavy productions the mass improves the agreement with the experimental data therefore we choose the standard R2 scaling form τ = 20 and insert the mass in present coefficients in r universal functions that here we introduce them as weight factors, W α . α index denotes to the relation between the dipole size and saturation radius. These weight coefficients for small pairs are

W r
2 3

+

2

2

30

2

r Q +

mq2 r 2 2

,

(27)

and

R2 2 W r > R 0 ∼ ( Q 2 + mq2 )(r 2 − 0 ). 3 2

(28)

Thus the behavior of the physical cross sections in these cases are

σrtot
r2 R 20

=

1

τ

,

(29)

and, in saturation region,

σrtot > R 0 ∼ O (1).

(30)

Corresponding weight factors for large dipoles are expressed as the following

W r
1 15

+ mq2 r 2 ,

(31)

Fig. 5. The ratio

f (τ )

αem

for small size pairs in x rang 10−6 –10−3 to

τ variable for

light, charm and bottom flavors.

and

R2 2 W r > R 0 ∼ ( Q 2 + mq2 )(r 2 − 0 ), 3 2

(32)

and the physical cross sections in these cases behaves similarly to Eq. (29) and Eq. (30) respectively. For determining the impression of heavy flavors in this work we will analyze following steps. The first we consider only light quarks and gluons as active partons and plot desirable universal functions that we did it in Fig. 2 and Fig. 3. The second we also consider charm and bottom quarks as active flavors and plot corresponding diagrams. Charm and bottom productions have been seen in the form of the small size we consider, based on this, f (τ ) by Eq. (12) in Fig. 4 for four active flavors and in Fig. 5 for

αem

five active quarks. These diagrams have the similar behavior with ones in Fig. 2. In light flavors due to the mass is insignificant we r2 1 can replace r 2 with 2 = so that the scaling and saturation are τ R0 mq 2 established. In heavy productions the element ( ) becomes sigQ

460

Z. Jalilian, G.R. Boroun / Physics Letters B 773 (2017) 455–461

Fig. 6. The contribution of the charm distribution function in proton structure function in three different Q 2 values.

nificant in calculations but we see the scaling is achieved again and there is only some increase in the value in saturation region. According to H1 and ZEUS reports charm component of the proton structure function, F 2c (x, Q 2 ), includes a significant fraction of the proton structure function, F 2 (x, Q 2 ) [18,19]. The contribution of this flavor according to universal function is written by

F 2c (x, Q 2 ) F 2 (x, Q 2 )

=

f c (τ ) f (τ )

δqc =

f (τ )

σ (x, Q 2 ) =

(33)

,

f (τ )

4π 2 αem

F 2c (x, Q 2 ). cc pair is dominant in small Q2 size dipoles with r < R 0 therefore by Eq. (12) we get

F 2c (x, Q 2 ) F 2 (x, Q 2 )

= ec2 (

11 ec2 (

15

+

mc2 r 2 2

) + (e 2u

11 15

+

+ ed2

mc2 r 2 2

)

11 + e 2s )(

15

+

(0.140)2 r 2 2

.

(34)

)

Since 0.037 ≤ rc ≤ 0.13 fm, (0.188 ≤ rc ≤ 0.66 GeV−1 ), and mc = 1.5 GeV, it is obvious the average of the charm production in the proton structure function in this region is about 40 percent that it is seen from Fig. 6. According to these diagrams we see this fraction is independent of x and Q 2 values and all curves almost fall on one, in other words the geometrical scaling is confirmed in charm production. F b (x, Q 2 ) Also we similarly calculate 2 for estimating the perF 2 (x, Q 2 ) ¯ centage of bottom production bb

F 2b (x,

Q 2)

F 2 (x, Q

2)

=

f b (τ ) f (τ )



δqb f (τ ) =

f (τ )

F 2b (x, Q 2 ) F 2 (x, Q 2 )

= eb2 (

11 eb2 (

15

+

mb2 r 2 2

) + ec2 (

11

+

mb2 r 2

15 2 11 mc2 r 2 15

+

2

) 2

11

3

15

) + ( )(

+

(0.140)2 r 2 2

, ) (36)

δqc is delta function that chooses charm of all active quarks and we used

Fig. 7. The contribution of the bottom distribution function in proton structure function in three different Q 2 values.

,

(35)

that δqb selects bottom quark of all five active flavors [20]. There is the most possibility for finding the bottom production with mb = 4.75 GeV in the range 0.014 ≤ rb ≤ 0.043 fm, (0.071 ≤ rc ≤ 0.22 GeV−1 ), therefore by Eq. (12) we will have

that is about 10 percent according to Fig. 7. All of these curves also fall on one that means the geometrical scaling is established for the bottom production and independent of x and Q 2 . It is obvious from Eqs. (34) and (36) the contribution of charm and bottom flavors in total structure function only relates to their mass, charge and size. 5. Summary The selection of color dipoles in high energy as freedom degrees is useful to study on the cross section and structure function. The dipole formalism enable us to investigate such phenomenon as the saturation and geometrical scaling in low-x through the gluon distribution. In x → 0 the gluon distribution directly appears in the universal dipole cross section in the target rest frame no need for non-linear distribution for describing the structure function behavior in this limitation. We in this work obtain the universal function as a striking finding related to establish the geometrical scaling without any restriction. Form of this function depends on the dipole size and mass and changes whenever the transverse separation of quark– antiquark changes to the saturation radius. It is clear from Fig. 1 transfer from scaling to saturation region depends on x and is done in r = R 0 (x) while according to Figs. 2 and 3 this transition occurs in τ = 1 independent of Bjorken variable x. Figs. 4 and 5 in comparison to Fig. 2 show in enough energy the mass of heavy productions plays the role of a regulator in calculation and there is only a slight increase in the value compared to a light production. Also all of diagrams appear a smooth transfer from color transparency to saturation when the scaling variable changes from τ > 1 to τ < 1. We know a usual method for checking the geometrical scaling is the ratio method of structure functions that we extend it to the ratio universal functions. Figs. 2–5 show clearly this property for both small and large size pairs not only in light flavors but also

Z. Jalilian, G.R. Boroun / Physics Letters B 773 (2017) 455–461

for heavy ones is available i.e. in every figure all curves fall on one unique curve. Charm production cc originates directly from virtual photon if Q 2 ≥ 4mc2 according to the experimental data the ratio of F 2c (x, Q 2 )

in large x, x > 0.1, is the times 10−2 percentage while F 2 (x, Q 2 ) for x < 0.01 we observed this amount in color dipole frame is about 40 percent. This impressive contribution introduces the importance of the charm cross section in collides in high energy. The corresponding ratio for bottom production,

F 2b (x, Q 2 )

, because F 2 (x, Q 2 ) of its small size and the large mass into other active flavors, in Q 2 ≥ 4mb2 , is closed to about 10 percent. Both of the obtained magnitudes are linking with the mass and the charge of active quarks involved in these processes and according to Figs. 6 and 7 are independent of x and Q 2 . In conclusion, the universal function, f (τ ), in extreme states matches with results presented in literature. According to this function we can calculate the contribution of heavy quarks in proton that is comparable to experimental data. References [1] N.N. Nikolaev, B.G. Zakharov, Z. Phys. C 49 (1991) 607. [2] N.N. Nikolaev, B.G. Zakharov, Z. Phys. C 53 (1992) 331. [3] C. Ewerz, O. Nachtmann, Towards a nonperturbative foundation of the dipole picture: I. Functional methods, Ann. Phys. 322 (2007) 1635, arXiv:hep-ph/ 0404254.

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