Few-Body Syst (2014) 55:85–100 DOI 10.1007/s00601-013-0733-y

M. Modarres · M. Rasti · M. M. Yazdanpanah

The Structure Functions of 3 H e and 3 H Nuclei in the Constituent Quark Exchange Model

Received: 17 October 2012 / Accepted: 31 August 2013 / Published online: 18 September 2013 © Springer-Verlag Wien 2013

Abstract The valence quark exchange (Q E) model for the nuclear system and the constituent quark (C Q) model in which the quarks are assumed to be the complex objects, are used to calculate the parton distributions and the structure functions (S F) of 3 H e and 3 H mirror nuclei. The effect and the role of sea quarks and gluons, and their contributions to the S F of 3 H e and 3 H nuclei are analyzed. Specifically, for small x regions a more “realistic” result is found with respect to our previous works, in which only the valence quarks have been considered. By using the DG L A P evolution equations, the resulting parton distributions and S F are evolved to the high-energy scales and compared with available data. Finally, the ratios of 3 H e to 3 H nuclei S F with the isospin symmetry assumption is compared to the results of the deep inelastic electron experiments on the 3 H e and 3 H nuclei, especially, those which have been extracted from the kinematic region of the proposed 11 GeV upgraded beam experiment of Jefferson laboratory, and a reasonable agreement is found.

1 Introduction Since the hadrons are made of the partons [1–5], one of the today’s main physics issue, is the understanding of the nuclear matter structure in terms of the partons i.e. the valence quarks, the sea quarks and the gluons. In the recent years, in order to reveal more information about the partons, many deep-inelastic scattering (D I S) experiments, have been performed on the hadrons and nuclear targets, such as the proton, the deuterium, the tritium, the helium 3, etc., [6–46]. To illuminate the consequence of these experiments, the different theoretical models have been introduced by various groups [1–5,47–57]. Recently, the possible use of the unpolarized tritium or helium 3 targets has been proposed for the 11 GeV upgraded beam of the Jefferson laboratory, and     3 the aims are to measure F2n x, Q 2 , using the ratio of the structure functions of helium 3 (F2 H e x, Q 2 )   3 and tritium (F2 H x, Q 2 ) nuclei. Performance of these experiments reduces the systematic errors in both the experimental measurements and the theoretical calculations [57–63]. Because of the importance role of the sea quarks and the gluons at small x (x is Bjorken scaling variable) regions, the numerous experiments and the theoretical works have been concentrated for the investigations of the hadrons structure functions in these area [64,65]. The valence quark exchange (Q E) model, which was originally introduced by Betz et al. [66], was proposed by Hoodbhoy and Jaffe (H J ) to study the quark distributions in the nuclear systems [67–69]. This M. Modarres (B) · M. Rasti Physics Department, University of Tehran, 1439955961, Tehran, Iran E-mail: [email protected] Tel.: +98-21-61118645 Fax: +98-21-8004781 M. M. Yazdanpanah Physics Department, Shahid-Bahonar University, 1439955961, Kerman, Iran

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model was, later on, reformulated and applied to the light nuclei [70] and the nuclear matter [71]. It was also used to derive the spin structure function of the three-nucleon system as well as the proton and the neutron [72]. In the Q E model, the quark structure of nuclei are calculated by using this assumption that the valence quarks are exchanged between the constituents of nuclei, i.e. the nucleons [73]. If one realistically wants to consider the roles of sea quarks (quarks-anti-quarks) and gluons in the Q E model, the formalism becomes very complicated. Despite this fact, the valence Q E model was also used as the initial conditions for the quantum chromodynamics (QC D) evolution equations to calculate the sea-quark and the gluon contributions of the proton S F at the leading order (L O) and the next-to-leading order (N L O) levels [74–77]. In order to include the above degrees of freedom in the Q E model, several effective approaches based on the finite size structure of the quarks have been proposed. Some them are: the constituent quarks C Q [78,79] (and reference therein) and [80–86], the chiral quarks [87–90], the valon [91] and the bag [92], approaches. The C Q idea, which is more appealing to us, was originally introduced by Feynman [54–57] and others [78– 86] and it was applied, by different groups [78,79,93,94], to calculate the S F of hadrons. Recently, the C Q model was also used by us, to determine the bound state parton distributions in the protons Yazdanpanah et al. (Y M R) [95]. In the Y M R a satisfactory improvements and agreements were found between our results and the available experimental data. In the framework of the C Q model, the C Q are assumed to be complex objects, with the point-like (P L) quarks, anti-quarks and neutral gluons as its constituents. Then in this context, again the various ideas have been proposed by various authors. For example, Triani et al. [79] (and the references therein) and Scopetta et al. [93,94] have considered the non relativistic and the algebraic quark models of Isgur and Karl [96–98]) and Bijker et al. [99–101], respectively, in order to calculate the constituent probability distributions of quarks kind u and d i.e. U (z, μ20 ) and D(z, μ20 ), in the protons (see the Eq. (1), below). However, in the Y M R, the Q E model, which is based on the realistic parameters and achieved better results with respect to the Isugar et al. [96–98]) and Bijker et al. [99–101] field theoretical approaches, has been used. In the Y M R, the bound state parton distributions were computed in the protons and they were used to estimate the ratios of neutrons to protons S F. The results showed better agreement in the small x regions with the experimental data [95]. As we remarked above, in the valence Q E, the roles of the sea quarks and the gluons are ignored and only the valence quarks are considered. But, by using the Altarelli C Q model [78], one can calculate the S F of 3 H e and 3 H mirror nuclei in which the contributions of the sea-quarks and the gluons are taken into account and the results are compared with the corresponding data especially, the expected measurements at the 11 GeV upgrade beam of Jefferson laboratory from the unpolarized tritium and helium 3 nuclei targets [57,63]. It is worth to mention that the new experiments have been also proposed in the Jefferson laboratory with the moderate pressure of the 3 H e and 3 H nuclei gas targets [62]. It is also possible to use the Doshitzer, Gribov, Lipatov, Altarelli, Parisi (DG L A P) evolution equations [102–104] to calculate the S F at 3.5 to 4 GeV2 energies in order to compare our results with the available experimental data. Finally, one can also find out, which types of the distributions in the 3 H e and 3 H nuclei are dominant with respect to the free partons distributions i.e., the valence-quarks, the sea-quarks or the gluons. So the paper will be organized as follows: The summary of the C Q model is introduced in the Sect. 2. In the “Appendixes A and B”, the constituent Q E model formulation is given to obtain the constituent quark distributions in the helium 3 and the tritium nuclei and the brief DG L A P Q 2 -evolution of the parton distributions is introduced to obtain the S F at higher energy scales, respectively. The S F of each nucleus in terms of the Q E and C Q model at corresponding scales are presented in the Sect. 3. Finally in the Sect. 4, our results for the different partons distributions, the S F of 3 H e and 3 H nuclei, and their ratio, as well as discussions and conclusions are given.

2 The Constituent Quark Model Let us start with a brief summery of the C Q formalism. In this model the C Q are assumed to be the complex objects and their weak and electromagnetic structure functions are defined in terms of the functions ab (x), that specify the number of P L partons type b inside the constituent type a with a fraction x of its total momentum. These functions are not all independent, but they are restricted by isospin and charge conjugation [78]. The ab (x) are called the structure functions of the constituent quarks [78,93,94]. So, the parton distributions in a nucleon, according to the S F of C Q and the constituent probability distributions of the quark-kind u and d, i.e., U and D, can be expressed as,

The Structure Functions of 3 H e and 3 H Nuclei

1 q(x, μ20 )

=

87

     x 2 x 2 dz 2 2 , μ + D(z, μ0 )Dq ,μ U (z, μ0 )U q . z z 0 z 0

(1)

x

In this equation labels q represents the different P L partons such as the P L valence quarks, u v and dv , the P L sea quarks, qs , and the P L gluons, g, where the μ20 is the momentum scale, at which the C Q model is defined. To extract the various types and the functional forms of the constituent quarks structure functions, it is needed to make the three natural assumptions, namely: the determination of the P L partons by the QC D, the Regge behavior for x → 0 and the duality idea, and finally, the isospin and the charge conjugate invariant. For different kinds of partons, the following definitions of the S F have been proposed [78,93,94]: (i) for the case of valence quarks,  P q v

x 2 ,μ z 0



A−1   x  A + 21 1 − z  , = 1 x  2 (A)

(2)

z

(ii) the related structure function for the sea quarks,  P q s

x 2 ,μ z 0

 =

C

=

G

x z

  x D−1 , 1− z

(3)

  x B−1 . 1− z

(4)

and finally (iii) for the gluons,  P g

x 2 ,μ z 0



x z

The momentum carried by the second moments of the parton distributions are known experimentally at high Q 2 . Their values at the low scale μ20 could be obtained by performing a next-to-leading-order (N L O) evolution downward. These procedure is used to extract the value of the constants A, B, G and the ratio C/D. For example, at the hadronic scale μ20 = 0.34 GeV2 , 53.5 % of the nucleon momentum is carried by the valence quarks, 35.7 % by the gluons and the remaining momentum are belong to the sea quarks. So, in this scale the mentioned parameters take the following values: A = 0.435, B = 0.378, C = 0.05, D = 2.778 and G = 0.135. More information about the calculations and procedures of evaluating these constants can be found in the reference [93,94].



In the Eq. (2) the P (qv ) can be equal to U (u) or D (d). But U d xz , μ20 and Du xz , μ20 are zero, because in the C Q type U , there is no quark valence type d and vice versa. Beside this, as it will be shown, in the next section U (z, μ20 ) and D(z, μ20 ) have different functional forms in the valence Q E model, so, the different forms of u(x, μ20 ) and d(x, μ20 ) are obtained from the Eq. (1) (see the Figs. 1, 2). But in the case of the P L sea quarks (qs (x, μ20 )) and the P L gluons (g(x, μ20 )), as it is presented in the Eqs. (3) and (4), there is no flavor dependent and the single distributions for the sea quarks or the gluons in the nuclear systems is derived. So in this work it is not possible to see the effects of the flavor dependence of sea quarks as well as the generation of gluons in this connection. This will not also be affected when the DG L A P evolution equations are used to scale up the partons distributions.

3 Nucleus Structure Function It is well known that the structure function measures the distribution of quarks as a function of k + (the lightcone momentum of the initial quark) in the target rest frame which is equivalent to boosting the nucleus to an infinite momentum frame. This is usually done in the literature by using an ad hoc prescription for k 0  0 = [(k2 + m 2 ) 21 − 0 ]). It is well known that the resulting S F are not sensitive to as a function of |k|(k this assumption [67–69,109]. So, the valence-quark distributions at each Q 2 = μ2 , can be related to the

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2

(xU(x) , xuv(x))

1.5

1

0.5

0 0.01

0.1

1

x

Fig. 1 The constituent up quarks U (x) (the full curve) and the convolution valence up quarks u v (x) (the dash curve) in the 3 H e for the (m u , 0u ) pairs of (290, 150 MeV) at μ20 = Q 2 = 0.34 GeV2 and b = 0.8 f m. The dots curve are the G RV free valence up quarks distributions at μ20 = Q 2 = 0.34 GeV2 [112] 1

(xD(x) , xdv(x))

0.8

0.6

0.4

0.2

0 0.01

0.1

1

x Fig. 2 The same as the Fig. 1 but for the down quarks and the pairs of (310, 240 MeV)

momentum distributions for each flavor in the nucleon of the nucleus Ai , according to the following equation, ( j = p, n (a = u, d) for proton (up-quark) and neutron (down quark), respectively):  

 z   k+ 1 a 2 a   q j z, μ0 ; Ai = − d k. (5) ρ j k; Ai δ (1 − z) M (1 − z)2 After doing the angular integration, we get, q ja (z, μ20 ; Ai )

2π MT = (1 − z)2

∞

 Ai )k dk, ρ aj (k;

(6)

a kmin

with

a kmin (z) =

z MT 1−z

2 + 0a − m a2 (μ20 )

. MT 2 z1−z + 0a

(7)

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Note that the Eqs. (5) and (6) have been written such that to take the covariant properties and relativistic effects into account [110]. In these equations, the m a (MT ) are the quarks (nucleons) masses and the 0a is the quark binding energies. Moreover, regarding our Gaussian choice for the nucleon wave function in terms of quarks, as illustrated in the “Appendix A”, near x = 1 the above prescription is not working properly because the S F probe quark very far from their mass-shell [67–69,72]. One should note that our approach is different from those works in which only the nuclear effects are considered e.g. Ciofi degli Atti and Liuti [111]. The Eq. (6) takes the following forms for the up and down constituent quarks in the 3 H e and the 3 H targets, respectively (M +M ) (we use the averaged mass for proton and neutron, i.e. MT = M N = p 2 n [73]): 

U

z, μ20 ; Ai



2π MN = (1 − z)2

∞

ρUAi (k)k dk,

(8)

Ai ρD (k)k dk

(9)

u kmin

D



Finally, the target S F F2Ai x, Q distributions as follow [73]: A F2 i



x, Q

+

αs

 2 



 2

z, μ20 ; Ai



∞

in the N L O limit is related to the quark, the anti-quark and the gluon



Q a2 {q ja (x, Q 2 ; Ai ) + q¯ ja (x, Q 2 ; Ai )}

a=u,d,s; j= p,n

{q ja (x,

2π

2π MN = (1 − z)2

d kmin

=x

Q2



Q

2

; Ai ) + q¯ ja (x,

 Q ; Ai ) + 2g(x, Q ; Ai )} , 2

2

(10)

  in which αs Q 2 is the N L O coupling constant, 2

Q β1 ln ln(

1 αs ∼ 2) − , = 2 2 Q Q 4π β0 log( 2 ) β03 ln( 2 )

(11)

where β0 and β1 are the first two universal coefficients of the QC D β functions. So, the form of the S F of the nucleus targets such as the 3 H e and 3 H nuclei take the following forms, respectively:

        2       αs Q 2 αs Q 2 1 9 2 2 3He  2 2 1+ 1+ F2 u v x, Q + 6 dv x, Q 2 x, Q = x 4 3 2π 3 2π                αs Q 2   αs Q 2 1 2 2 2 2 2 1+ + 24 qs x, Q + g x, Q , (12) + 3 3 3 2π 2π

        2       αs Q 2 αs Q 2 1 3 2 2 3H  2 2 1+ F2 x, Q = x u v x, Q + 9 dv x, Q 2 1+ 2 3 2π 3 2π           2       αs Q 2 α Q 1 2 2 2 s 1+ + 24 qs x, Q 2 + g x, Q 2 . (13) + 3 3 3 2π 2π In the above S F the different parton distributions in the 3 H e and 3 H nuclei are calculated by using the convolution Eq. (1) and the Eqs. (2)–(4) and (8)–(9) as: (i) the P L up and down valence quark distributions, u v (x, μ20 ) and dv (x, μ20 ): 

u v x, μ20



1 =

 dz  U z, μ20 U u v z



 x 2 ,μ , z 0

(14)

 x 2 ,μ , z 0

(15)

x



dv x, μ20



1 = x

 dz  D z, μ20 Ddv z



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(ii) the P L sea-quark distributions, qsea (x, μ20 ): 1 qsea (x, μ20 )

=

   dz   U z, μ20 + D z, μ20 qqsea z



 x 2 , μ0 , z

(16)

x

and (iii) the P L gluons distributions, g(x, μ20 ): g



x, μ20



1 =

   dz   U z, μ20 + D z, μ20 qg z



 x 2 ,μ . z 0

(17)

x

So we have found the relation between the parton momentum distributions and the bound nucleon S F as well as the target S F. In our formalism we have two free parameters for each flavor, namely the quarks mass, m a , and the quarks binding energies, 0a , which physically their values should be around 200 MeV with respect to the proton mass [67–69]. On the other hand, as it has been shown in the previous works, references [67–77], and it will be discussed later on, the final results are not very sensitive to these parameters as far as they are chosen in the above range. In order to fix the values of m a and 0a with a = u, d, for each value of b, two pairs (m u , 0u ) and (m d , 0d ) can be founded such that the best fit to the valence quark distribution functions of G RV ’s parton S F would be achieved [112]. The GRV ’s structure functions fit the experimental parton structure functions data over the whole range of x, Q 2 plane very well. As it is shown in the Figs. 1 and 2, our fitted parton distributions for the (m a , 0a ) pairs (Note a = u, d) are (290, 150 MeV) and (310, 240 MeV) at Q 2 = 0.34 GeV2 for b = 0.8 fm (the charge radius for 3 H e(3 H ) is 1.68 (1.56) fm which corresponds to b = 0.837 (0.780) fm. So the b = 0.8 fm is a good option for our calculations). The χ 2 for the up and down quarks fitting parameters are 6.32 and 2.23, respectively. As it was pointed out before, we found that the fitting procedures were not very sensitive to the choice of above parameters (note that present calculations mainely have qualitative nature). Our fitting procedures are not good enough as x → 1, especially, for the nucleus target, because the Fermi motion effect have been ignored by using the leading-order expansion of the nuclear wave function in the Q E model. However the aims of present results are more qualitative nature rather than quantitative, since our models are more realistic, for example, with respect to G RV .

4 Results, Discussion and Conclusion The paper is started by presenting, in the Figs. 1 and 2, the calculated (constituent and P L) up and down valence quark distributions in the 3 H e and the 3 H targets by fitting our C Q distributions to those of G RV . As it was stated in the previous section, for each value of b, the two pairs (m u , 0u ) and (m d , 0d ) are fixed such that, the best fit to the G RV ’s valence u and d quarks distributions functions [112] are  reached.  The G RV ’s partons S F fit the experimental proton S F data very well over the whole range of x, Q 2 plane. So their parton distributions at very low energy scale i.e. μ0 = 0.34 GeV2 should be reliable. In the above Figs. 1 and 2 our fitted up and down constituent distribution functions for b = 0.8 fm (the heavy full curves) with the (m a , 0a ) pairs (note that a = u, d) of (290, 150 MeV) and (310, 240 MeV) at Q 2 = μ20 = 0.34 GeV2 are plotted. The dash curves are the corresponding P L valence distributions calculated from the Eqs. (14) and (15). Unlike the P L quark distributions, the corresponding constituent ones, become closer to those of G RV , but they are not valid as x → 1, especially for the nuclear targets. Because, the effect of Fermi motion has been ignored, by using the leading order expansion of the nuclear wave function (see “Appendix A”). In our future work we hope to omit this approximation and calculate the Fermi motion effect in the framework of Q E formalism and the Faddeev wave functions for the three-nucleon systems (note that in this work still the  q) ). It is also noted that exchange term is approximately calculated due to the leading order expansion of χ ( P, because of the binding and the Fermi motion effect, we expect that for x > 0.4, the Q E model (P L) up and down valence quark distributions behave differently with respect to those have been fitted to produce the free nucleon structure function i.e. the G RV . On the other hand, it is obvious that, since the calculated valence up and down quarks are different in the Q E model, the P L up and down quarks distributions become different after the applications of C Q model, i.e. Eq. (1). But as it will be shown later on this will not affect the sea quarks and gluons, and we find the same P L parton distributions, which are independent of isospin flavor.

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91

In the Figs. 3 and 4, we have plotted the sea quarks and the gluons distributions which have been calculated through the Eq. (1) by using the Q E model constituent up and down quarks distributions. Unlike our several works, (in the recent years), [66–77,106,107], based on the quark exchange picture, between the bound nucleus to explain the experimental data of the D I S of hadrons or nuclei targets, because of complexity of calculations, only the exchange of the valence quarks among the hadron have been considered and the existence and the exchange of the sea quarks and the gluons have been ignored. But now by using C Q model, we have calculated the contributions of the sea quarks and the gluons to the S F of the bound nucleus and the nuclear targets. On the other hand, as we pointed out before there is no flavor dependent in the case of the sea quarks (qs (x, μ20 )) or the gluons (g(x, μ20 )) and, we have single distribution for the sea quarks or the gluons in the 3 H e or 3 H nucleus. One should note that the G RV ’s sea quarks and gluons distributions are flavor dependent. But for comparison, we have plotted their isospin average distributions in the Figs. 3 and 4. So the discrepancies between our result and those G RV should be consider as qualitative nature. In the Figs. 5, 6, 7 and 8 the evolution of the parton distributions i.e. the valence quarks u v (x, μ20 ), dv (x, μ20 ), the sea quarks qs (x, μ20 ) and the neutral gluons g(x, μ20 ) are plotted at the higher energy scale at Q 2 = 4 GeV2 . Since we only access to the experimental data of the ratio of 3 H e to 3 H at Q 2 = 4 GeV2 , so it is needed to

0.2

xq sea(x)

0.15

0.1

0.05

0 0.0001

0.001

0.01

0.1

1

x Fig. 3 The same as the Fig. 1 but for the sea quarks. Note that as explained in the text, in the constituent level there is no sea quarks 0.7 0.6

xg(x)

0.5 0.4 0.3 0.2 0.1 0 0.01

0.1

1

x Fig. 4 The same as the Fig. 3 but for the gluons. Also in the constituent level there is no gluon distributions

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1.2

(xU(x) , xuv(x))

1

0.8

0.6

0.4

0.2

0 0.01

0.1

1

x Fig. 5 The evolved u v (x) valence quark distribution in the 3 H e at Q 2 = 4 GeV2 (heavy full curve). The other curves are the same as the Fig. 1 (just for comparison) 0.5

(xD(x) , xdv(x))

0.4

0.3

0.2

0.1

0 0.01

0.1

1

x Fig. 6 The same as the Fig. 5 but for the evolved dv (x) valence quark

evolve the above parton distributions to the mentioned energy scale in order to calculate the S F and the ratios at this energy scale and compare it with available data. The structure functions of helium 3 and tritium are presented in the Fig. 9. The dash curves show the calculations at Q 2 = 0.34 GeV2 and the full curves are the same S F, but only with the valence quarks, for b = 0.8 fm [73] (the b = 0.8 fm agrees with the root mean square radius of proton). The 2 H data are also given for comparison. Since in the present calculation the sea quarks and the gluons distributions have been considered in the S F of the bound nucleon, we get more realistic S F with respect to our previous work which we have just used the valence S F i.e. reference [73]. Furthermore, the available experimental data are at Q 2 = 4 GeV2 , so we have evolved the P L valence quarks, the sea quarks and the gluons distributions to this energy scale and we have used them to construct the S F of the 3 H e and 3 H nuclei at Q 2 = 4 GeV2 (dash-point-dash curves). Our results are in agreement with the experimental data especially for x < 0.1. We should make this point that here our sea quarks are isospin independent, which in principle affect the low x behaviors of our results. Finally, the ratios of the structure functions of helium 3 to tritium are plotted in the Fig. 10 for b = 0.8 fm (dash curve). The full curve is the same ratios, but only with the valence Q E M up (down) quark distributions U (x, μ20 ) (D(x, μ20 )). The filled circles points are the expected ratios that have been estimated by using the kinematics of the proposed 11 GeV Jefferson laboratory experiment [57,63]. The dotted-curve are the same

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0.8 0.7 0.6

xqsea(x)

0.5 0.4 0.3 0.2 0.1 0 0.001

0.01

x

0.1

1

Fig. 7 The same as the Fig. 5 but for the evolved sea quarks. As one expected the increase of Q 2 leads to observe more sea quarks especially at x < 0.01

1.6 1.4 1.2

xg(x)

1 0.8 0.6 0.4 0.2 0 0.01

0.1

x

1

Fig. 8 The same as the Fig. 7 but for the evolved gluons

ratio but at Q 2 = 4 GeV2 . It is expected that the present ratios be closer to the estimated prediction, because in this ratios the sea quarks and the gluons are considered. It should be point out here that because of the Fermi motion effects for x > 0.8, our result are not close to the estimated prediction. In conclusion, we have calculated the bound state parton distributions in the 3 H e and 3 H nuclei and the helium 3 to tritium structure functions ratios in the framework of the quar k-exchange and the Altar eli constituent quark models. We have treated the u and d quarks, the sea quarks and the gluons as well as the proton and the neutron explicitly in our formalism and we have found satisfactory results compare to the present available data. Our structure functions are more realistic, because we contributed the role of sea quarks and gluon to the structure functions, especially for x < 0.1 which S F pass throw the experimental data. However, because of the large χ 2 in the fitting procedures, the present result should be considered more qualitative nature rather than quantitative. We can improve our calculation by treating Fermi motion effect explicitly in the quark exchange frame work, i.e. calculating the full overlap integral rather than the leading-order expansion we made in the nuclear wave function. One can also test other choice of nuclear wave function with the new nucleon– nucleon potentials [108], and since for the nuclei, it is possible to have the non-vanishing structure function for x larger than one [122,123], we can measure and calculate this behavior by considering the quar k-exchange effect in nuclei with a full realistic nuclear wave function.

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1

3

F2(x)

3

0.1

H

He 3 3

H

He 3

3

0.01 0.01

0.1

He

H

1

x

Fig. 9 The 3 H e and 3 H structure functions for b = 0.8 fm (the dash curves). The full curves are from the reference Modarres and Zolfagharpour [73]. The dash-point-dashes are the evolved structure functions at Q 2 = 4 GeV2 . The 2 H data are from the NMC [30–35] 1.4

3

1.2

3

F2

He

/F2

H

1.3

1.1

1

0.9

0

0.2

0.4

x

0.6

0.8

1

    3 3 Fig. 10 The ratios of structure functions of helium 3, F2 H e x, Q 2 to tritium F2 H x, Q 2 for b = 0.8 f m. The dash curve is from the constituent quark exchange model calculation and the dots curve is the ratios of the evolved structure functions at Q 2 = 4 GeV2 . The full curve are from Modarres and Zolfagharpour [73] and the circle points are from the reference [62,63]

Acknowledgments We would like to thank the Research Council of University of Tehran and Institute for Research and Planning in Higher Education for the grants provided for us.

Appendix A: The Constituent Quark-Exchange Formalism In the constituent Q E model [67–69] it is assumed that the nucleon states are made of three C Q (QQQ), in which each C Q has its own especial quantum numbers, such as the spin, the iso-spin and the color i.e.: |Q = Q†μ |0,

(A1)

and equivalently for the nucleon kind N , [70–77,95], 1 † |N  = N α |0 = √ Nμα1 μ2 μ3 Q†μ1 Q†μ2 Q†μ3 |0, 3!

(A2)

 M S , MT } ({k,  m s , m t , c}) [note that where the indices αi (μi ) designate the nucleon (C Q) states { P, † 1 1 MT (m t ) = + 2 and − 2 for the proton (up-quark) and the neutron (down-quark), respectively]. Q†μ (N α ) denote the creation operators for the C Q (nucleons) with the state index μ (α). We made a convention that a

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 The totally antisymmetric repeated index means a summation over repeated indices and also integration over k. α nucleon wave functions Nμ1 μ2 μ3 are written as,  k1 , k2 , k3 , P),  Nμα1 μ2 μ3 = D(μ1 , μ2 , μ3 ; αi ) × δ(k1 + k2 + k3 − P)φ(

(A3)

 are the “nucleon wave function in terms of the C Q” which is approximated by a where φ(k1 , k2 , k3 , P) Gaussian form (b  nucleons radius)

  4 (3/4) (k12 + k22 + k32 ) b2 P 2 3b 2  = φ(k1 , k2 , k3 , P) exp −b + . (A4) π2 2 6 This wave function is the ordinary symmetric SU (4) quark model wave function which has been supplemented by a Gaussian radial form. The D(μ1 , μ2 , μ3 ; αi ) are the products of four Clebsch − Gor don coefficients j j j Cm11 m2 2 m and the color factor c1 c2 c3 , i.e. 1 1 11 1 1 11 1 1 s s t t Cm2 s 2m s M S Cm2 s2μ m sν m s Cm2 t 2m t MT Cm2 t2μ m tν m t , D(μ1 , μ2 , μ3 ; αi ) = √ c1 c2 c3 √ σ σ αi αi 2 3! s,t=0,1

(A5)

Now, by using the above nucleon creation operators, we can define the nucleus states as, 1 † † † |Ai = 3 = √ χ α1 α2 α3 N α1 N α2 N α3 |0, 3!

(A6)

where χ α1 α2 α3 are the complete antisymmetric nuclear wave functions, which are taken from the Faddeev calculation with the Reid soft core potential [57,63,105–107]. According to Afnan et al. [57,63], the choice of nucleon–nucleon potential does not affect the E MC results. χ α1 α2 α3 could be defined as the center of mass motion of the three nucleons and write the nuclear wave function as,  q)D(α1 , α2 , α3 ; Ai ), χ α1 α2 α3 = χ ( P,

(A7)

where the last term are similar to the one we defined for C-G coefficients in the Eq. (A5), i.e., 1 D(α1 , α2 , α3 ; Ai ) = √ 2



1

S,T =0,1

S1

11 1 1 11 2 2S 2T 2 2 2T C C C . M M M M M M M M M S Si Sα2 Sα3 S Tα1 T Ti Tα2 M Tα3 MT α1

2 2 CM S

(A8)

The momentum distributions of the C Q with the fixed flavor and the nucleon iso-spin projection in the three nucleon system can be written as,  Ai ) = ρμ¯ (k;

Ai = 3|Q†μ¯ Qμ¯ |Ai = 3 Ai = 3|Ai = 3

.

(A9)

The sign bar means no summation on MT , m t and integration over k on the repeated index μ. One who is interested to the explicit calculations of the terms Ai = 3|Ai = 3 and Ai = 3|Q†μ¯ Qμ¯ |Ai = 3, could get more details in the reference [73]. The up and down C Q momentum distributions in the 3 H e and the 3 H nuclei could be calculated by performing all summations on the free repeated indices in the Eq. (A9), for fixed MT = 21 and − 21 , respectively, where MT is the three nucleons system isospin projection [73]:    2 4 9 −1 3He , (A10) ρU (k) = 2A(k) + B(k) + D(k) 1 + I 9 9 8    1 4 2 9 −1 3 , (A11) ρDH e (k) = A(k) + B(k) + C (k) − D(k) 1 + I 9 9 9 8 with



3He

ρD

1 (k)d k = 2



3  ρUH e (k)d k,

(A12)

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where 3 3      3b2 2 3 2 2 27b2 2 3 A(k) = exp − b k , B(k) = exp − b2 k 2 I , 2π 2 8π 2 3 3         27b2 2 12 27b2 2 C (k) = exp − b2 k 2 I , D(k) = exp −3b2 k 2 I , 7π 7 4π 

(A13) (A14)

and ∞ I = 8π

2

∞ 2

x dx 0

1 2

y dy −1

0



 3x 2 d(cosθ )ex p − 2 |χ (x, y, cosθ )|2 . 4b

(A15)

where I is the contribution of the nucleus wave function (χ (x, y, cosθ )) to the Q E momentum distributions [67–69]. Appendix B: DGLAP Evolution Procedure The DG L A P equations which govern the evolution of the quark distribution q(x, Q 2 ) and the gluon distribution g(x, Q 2 ) to the high-energy scale are given by [102–104]:   d αs q x, Q 2 = d log Q 2 2π

1 x

  d αs g x, Q 2 = 2 d log Q 2π

1

         x x dy 2 2 q y, Q Pqq + g y, Q Pqg , y y y

       x x dy  2 2 qi y, Q Pgq + g y, Q Pgg . y y y

(B1)

(B2)

i

x

In the above equations, Pi j ’s are the splitting functions which denote the probability that the j-parton with the momentum fraction y appears as the i-parton with the momentum fraction x. The N L O coupling constant is expressed as 2

Q ln ln β 1 αs ∼ 1

2 2 − 2 , (B3) = 4π β ln Q β 3 ln Q 0

2

0

2

where β0 and β1 are the first two universal coefficients of the QC D β functions. M S is taken to be 0.248 GeV throughout our calculations [113–116].  By defining non-singlet q N S = qi − q j and singlet qs = i (qi + q¯i ) combinations of the quark flavor group, the DG L A P equations expressed as   d αs q N S x, Q 2 = 2 d log Q 2π

1

     x dy q N S y, Q 2 Pqq , y y

(B4)

         x x dy 2 2 qs y, Q Pqq + g y, Q Pqg . y y y

(B5)

x

  d αs qs x, Q 2 = 2 d log Q 2π

1 x

  The Mellin transform of the parton distributions, P x, Q 2 can be write in the N-momentum space as:      M P x, Q 2 = MP n, Q 2 ) =

1 0

  x n−1 P x, Q 2 d x,

(B6)

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  where n is the order of moments and P = qv , q, ¯ g and MP n, Q 2 are the Mellin transform of the parton   distributions, P x, Q 2 . As we mentioned in the calculation of the structure functions of different targets, we assume the SU (3) flavor-symmetric sea quark distributions q¯ = u¯ = d¯ = s¯ = s. Unlike the our previous work [74–77], here the sea quark and the gluon contribution in the static point (hadronic scale) μ20 = 0.34 GeV2 are not zero. By solving simultaneously the Eqs. (B4) and (B5), we lead to the coupled equations for the non-singlet and singlet N L O evolution of the parton distribution in the n-th momentum space, which can be written as,     γ qq /2β0      NS qq 0   αs Q 2 − αs Q 20 αs Q 2 β1 γ0 γ1 NS 2  2 M (n, Q ) = 1+ M N S n, Q 20 − (B7) 2 4π 2β0 αs Q 0 2β0 ⎡        λ− /2β0   S 2 αs Q 20 − αs Q 2 αs Q 2 1 M (n, Q )   P− − P− .γ .P− =⎣ M g (n, Q 2 ) 2β0 4π αs Q 20 ⎛   ⎤  2    2  (λ+ −λ− )/2β0 ⎞ 2 αs Q 0 αs Q αs Q P− .γ .P+ ⎦ ⎠   −⎝ − 4π 4π 2β0 + λ+ − λ− αs Q 20 ⎤  S  M n, Q 20  ⎦ +(+ ←→ −) . (B8) M g n, Q 20 The N L O anomalous-dimension splitting functions, γ s, λs and Ps are defined as in [117–120]. All the above solutions in the n-momentum space can be inverted into the x-space by the inverse Mellin (M−1 ) transformation of the Eq. (B6) [121], 

P x, Q

2



−1

=M

[MP





1 n, Q ] = 2πi 2

c+i∞ 

  x −n MP x, Q 2 dn.

(B9)

c−i∞

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

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