A Novel Feature of Valence Quark Distributions in Hadrons

Examining the evolution of the maximum of valence quark distribution weighted by Bjorken x, $h(x,t)\equiv xq_V(x,t)$, we observe that $h(x,t)$ at the peak should become a one parameter function; $h(x_p,t)=\Phi(x_p(t))$, where $x_p$ is the position of the peak and $t= \log{Q^2}$. This observation is used to derive a new model independent relation which connects the partial derivative of the valence parton distribution functions (PDFs) in $x_p$ to the QCD evolution equation through the $x_p$-derivative of the logarithm of the function $\Phi(x_p(t))$. A numerical analysis of this relation using empirical PDFs results in a observation of the exponential form of the $\Phi(x_p(t)) = h(x_p,t) = Ce^{D x_p(t)}$ for leading to next-to-next leading order approximations of PDFs for the all $Q^2$ range covering four orders in magnitude. The exponent, $D$, of the observed"height-position"correlation function converges with the increase of the order of approximation. This result holds for all PDF sets considered. A similar relation is observed also for pion valence quark distribution, indicating that the obtained relation may be universal for any non-singlet partonic distribution. The observed"height - position"correlation is used also to indicate that no finite number exchanges can describe the analytic behavior of the valence quark distribution at the position of the peak at fixed $Q^2$.


INTRODUCTION
Valence quarks play a unique role in the QCD dynamics of hadrons. They define the baryonic number of the nucleons and represent "effective" fermions with complex interactions among themselves and with the hadronic interior. One important property of valence quarks is that their number is conserved and hadrons can be considered as systems with fixed number of effective (valence) fermions. The continuing progress in experimental extraction of valence quark distributions in a wide range of x and the emerging new possibilities for probing them in semi-inclusive and exclusive deep-inelastic processes create a new motivation for theoretical modeling of their dynamics. This modeling is important since in the case of success one gains a new level of understanding of the QCD dynamics in the hadrons. Even if lattice calculations can reproduce the major characteristics of valence quark distributions they don't necessarily result in a qualitative understanding of the underlying processes. In this respect, observation of new properties and relations in valence quark distributions is significant since it allows one to constrain models aimed at describing the dynamics of QCD interaction.
The possibility of considering nucleons as a system of "effective fermions" (i.e. valence quarks) whose number is conserved, opens a new venue in exploration of the dynamics of the valence quarks from the point of view of universal properties of two component (spin or isospin) fermi systems with conserved number of constituents. This approach is analogous to the recent study of the ultra cold two component fermi atomic systems and atomic nuclei for which, despite 20 orders of magnitude difference in the density, a similar analytic form for the high momentum tail of the momentum distribution was found [1] based on the universal properties of the fermi system. We follow a similar logic in studying QCD structure of hadrons and our focus in the present work is on one of the most distinguishable characteristics of valence quarks which is, their distribution, q V (x, Q 2 ) weighted by momentum fraction, x, exhibits a clear peak [23]. This peak is a hallmark for the bound system of conserved number fermions [24] and is characterized by its height, h(x p ), and the position, x p , both of which evolve with the resolution scale, Q 2 . Since this peak is a product of the magnitude of x and the strength of the valence quark distribution one expects that its position and the height to be Q 2 dependent due to the QCD evolution of the valence quark distribution, q V (x, Q 2 ), whose strength shifts towards smaller x with increasing Q 2 . As a result one expects that both the position of the peak, x p and its height, h(x p , t) = x p (t)q V (x p , t) to be a function of Q 2 (hereafter we use the variable, t = log Q 2 ).

II. "HEIGHT-POSITION" CORRELATION OF THE PEAK OF THE VALENCE QUARK STRUCTURE FUNCTIONS
As Fig.1 shows the height of the peak and its position for valence PDFs in the nucleon diminishes with an in-crease of Q 2 as one expects from the QCD evolution that moves the strength of PDFs towards small x. If now we assume that both the height and the position of the peak of the valence quark structure function evolve due to the evolution of the strong coupling, then these dimensionless quantities can be expressed as: where x n and h n are constants and α s (t) is the strong interaction coupling constant evaluated at t = log Q 2 . Eq.(1) is a single variable function of α s , continuously differentiable with non zero derivate. Thus it is in general an invertible function. This fact allows us to combine Eqs. (1) and (2) representing the height, h(x p , t) as oneparametric function of x p in the form: where Φ is a function of x p variable only. In the following we will explore the implications that Eq (3) may have on partonic distributions of valence quarks.

III. NEW RELATIONS FOR VALENCE PDFS
In general, PDFs depend on the two independent variables of x and t (or Q 2 ). The relation of Eq.(3) indicates that at the peak the x weighted PDFs depend analytically only on one variable, x p , and the t dependence is expressed through the x p 's dependence on t. This situation should result in specific relations for valence PDFs at peak values.
Starting with the relation: we first consider the t derivative using the right hand side of the above equation resulting in: Considering now the middle part of the Eq.(4) the t derivative yields: Comparing Eqs. (5) and (6) one obtains: or in a more compact form: The above equation is rather unique since it allows one to relate the t derivative of the valence quark PDFs, which can be evaluated through the QCD evolution, to the x p derivative of the same distribution.
Using various PDF sets (e.g. [2][3][4]), which have been fitted to high energy data such as deep inelastic scattering, Drell-Yan processes, etc., one can calculate numerically the LHS of Eq.(8) and thus evaluate the correlation function Φ(x p ) as a function of t or Q 2 .
In Fig.2 we present the calculation of the LHS of Eq.(8) using the CT14 PDF set published in Ref. [3] at LO in α s accessed via the LHAPDF library. Here we used the QCD evolution equation to leading order to calculate while dxp dt and are calculated numerically using valence d-and u-quark distributions at LO [3].
As Fig.2 shows the LHS of Eq.(8) is practically constant for the all of Q 2 s, covering four orders of magnitude (1.8 GeV 2 to 3.3 × 10 4 GeV 2 ). This indicates that Fig.3 we present a similar evaluation for the LHS of Eq.(8) but for next-to-next-to leading order (NNLO). As the figure shows, the LHS part again produces an almost constant behavior for the entire range of Q 2 . This suggests again that Figs. (2) and (3) show that the condition of d log Φ(xp) xp = const is approximately independent on the order of approximation in the QCD evolution.
The above observation of d log Φ(xp) xp = const indicates that the correlation function, Φ(x p ) has an exponential form: which is universal with respect to the order of approximation in the QCD evolution equation, with differing  Fig.2 but for next to next to leading order approximation for CT14 [3]. The thick orange line is the D obtained from averaging all the D's for the central value curve, while the shaded orange region shows the region within 1σ of the average. Table 1 contains the PDF propagated errors.
exponents D and overall factors C. The values of C and D for d-and u-valence quarks in LO, NLO and NNLO approximations are given in Table I, which suggests that they converge with the increase of approximation in the QCD evolution equation. Note that the LHAPDF gives PDF values by using spline interpolation between (x, Q 2 ) grid points. This leads to instabilities when calculating PDF derivatives numerically (see [10]), hence the discontinuities and kinks in Fig. 2 and 3.
In Fig. 4 we use the parameters of Table I to compare the function of Eq.(10) with the x p dependence of h(x p , t) for down and up valence quarks in the nucleon in the LO and NNLO approximations for the CT14 param-  eterization. The figure shows that indeed the h(x p , t)x p correlation follows almost ideally an exponential form.
It is worth mentioning that other modern PDF parameterizations using different ansatzs, renormalization and factorization schemes for PDFs give similar results (see Section VI).
Summarizing this chapter, we conclude that we found a new empirical relation for valence PDFs according to which the specific combination of the t and x p derivatives of PDFs results into a constant value for all the range of Q 2 currently being discussed. The relation is: where the constant D depends on the flavor of the valence quark and order of approximation in QCD evolution.

IV. ORIGIN OF THE EXPONENTIAL FORM OF THE HEIGHT-POSITION CORRELATION
Our observation of the exponential form of Eq. (10) in the current work is purely empirical. We used existing PDF sets in the given approximation and estimated the expression in Eq. (8) finding that it results in a constant number, D. It is an interesting problem to understand the origin of the observed exponential form of heightposition correlation. For this we note that while nonperturbative dynamics define the initial shape of the valence PDFs, its change with Q 2 and therefore the heightposition correlation is associated with QCD evolution and the baryonic sum rules that valence quarks must satisfy. Thus the change of the height of the peak of h(x, t) function and its position, x p , is associated with perturbative dynamics.
To understand the origin of the exponential form of the "height-position" correlation (Eq.(10)) one needs to consider the simultaneous solutions of evolution equations for q V (x, t) and h (x p , t) functions at the given approximation together with the condition of the maximum of the function dh(x,t) dx | x=xp(t) = 0. The latter leads to the relation: where h (x p , t) = dh(x,t) dx | x=xp(t) and for the ∂h (xp,t) ∂t term one can derive an evolution equation. For example, in LO approximation: The complexity of above equations (especially in NLO and NNLO approximations) makes it difficult to find an analytic solution in the form of Eq. (10). To understand the origin of such a correlation, currently the above equations are being studied numerically [13] in different approximations, using different ansatzs for PDFs that describe experimental distributions at fixed Q 2 . These studies, which will be presented elsewhere, indicate specific properties of evolution equation whose solutions furnish the correlation function in the form of Eq. (10).

V. THE X DEPENDENCE OF H(X, T ) AT THE VICINITY OF THE PEAK
Using the fact that x p < 1, from Eq. (10) one observes that in the vicinity of x ∼ x p : where C and D are constants defined in Eq.(10) and e is the Euler's number. It can be checked that the above function peaks at x = x p and its peak value corresponds to the terms of the Taylor expansion of Eq.(10) in x p up to O(x 2 p ). It is interesting that even though the constant D is due to dynamics of QCD evolution it defines also the valence PDF in the vicinity of the peak position, x p .
As it follows from Eq.(14) the exponent of the (1 − x) term, xp(t) is defined by the position of the peak x p .
The latter, as we discussed above, changes continuously with t due to QCD evolution. The fact that the exponent of (1−x) term is not a constant and depends on the resolution of the probe indicates a more complex dynamics in the generation of valence PDFs at x ∼ x p . For example, the decrease of x p with t indicates that less momentum fraction is imparted to the interacting valence quarks, which can be due to the increase of the recoil mass of the nucleon, which itself is due to valence quark radiation at large t. Overall this observation indicates that no fixed number of constituent short-range interactions can be responsible for the dynamics of valence quark PDFs at any fixed values of Q 2 and at x ∼ x p . For a fixed number of exchanges, one gets a constant exponent proportional to the number of exchanged particles [5,6]. On the other hand, a smoothly varying exponent can be obtained by considering an effective interacting potential [7] in Weinberg type equations for relativistic bound states [9]. Thus one may expect that valence PDFs at x ∼ x p at fixed Q 2 are generated by mean-field type interactions rather than by a combination of a finite number exchanges between valence quarks.
Note that only moving away from x p towards x → 1 one should expect the mechanism of quark-quark interaction through a hard gluon exchange to become important. Indeed the asymptotics of empirical PDFs indicate that the exponent of (1−x) part of the distribution approaches to a constant value at x ≥ 0.7 − 0.8 (see e.g [3,10]).
In this respect the partonic dynamics for the valence sector can be similar to the one in the nuclear physics with mean field and short-range correlations dominating at different internal momentum regions of the constituents (see e.g. Ref. [11]).

VI. OTHER PDFS AND HADRONS
FIG. 5: (Color online.) The peak position-height relation for the PDF sets NNPDF 30 nnlo [16], ABM12 nnlo [17], CJ15 nlo [4] and JR14 nnlo -FF [18] for the up valence. The shaded regions show the uncertainty at 68% confidence level while the dashed curves are exponential fits for each PDF set. 09 0.10 0.11 0.12 0.13 0.14 0  The observed "height-position" correlation of the peak of the h(x, t) function in the nucleon is a combination of two effects: the dynamics that generate the partonic distribution of valence quarks at given Q 2 and the QCD evolution that shifts the strength of the distribution towards smaller x. The relation we found is robust for all different PDF sets in leading order, which reinforces the expectation that effects is due to QCD evolution. For the next to leading orders the PDF depends on factorization schemes, thus it is interesting to check whether the correlation of Eq.(10) persists in higher order approximations using different schemes. Since all modern PDF parameterizations employ M S scheme [20] we first check the validity of correlation for these sets of PDFs. As can be seen in Fig. 5 the PDF sets NNPDF 30 nnlo [16], ABM12 nnlo [17], CJ15 nlo [4] and JR14 nnlo -FF [18] all showed an exponential relation between the h p and x p for the up valence PDF. In addition, the extracted C and D parameters were all similar to the one obtained from CT14 nlo and nnlo case. This is despite the fact that these sets use different orders of approximation and prescriptions. While the M S scheme is used in the DGLAP evolution equations the renormalization and factorization schemes used to deal with the heavy quark (which ends up impacting the light quark distributions via the momentum sum rule) differ. NNPDF uses the on-shell renormalization scheme for the heavy quarks and FONLL-C for factorization, ABM12 nnlo and JR14 nnlo -FF both use M S for renormalization and FFNS (n f = 3) for factorization, while CJ15 NLO uses the on-shell prescription and SACOT for factorization [19]. They also differ in their initial ansatz, with NNPDF30 using neural networks and starting at Q 2 0 = 1 GeV 2 , while ABM12 nnlo, CJ15 nlo, and JR14 nnlo -FF used simpler functional forms and started at Q 2 0 = 0.5, 1.69 and 0.5 GeV 2 , respectively. In Fig.6 we compare available PDFs which are obtained using both M S and DIS [22] schemes in next to leading order for CTEQ6 parameterization [21]. Here we again observe the exponential form of correlation for both schemes, in which the results for DIS scheme is somewhat closer to that of leading order approximation. However, it is worth noticing that total uncertainty for CTEQ6 parametrization [21] is larger than the uncertainties for modern parameterizations (e.g. Ref. [3]).
From above comparisons one might expect that correlation of type of Eq.(10) should be universal for valence PDFs describing "experimental" distributions, satisfying specific sum rules, such as baryonic number sum rule for nucleons.
It is interesting to explore the possibility of similar correlations also for mesons extending it to the sector of strange and charm quarks. As an initial result we also present in Fig.7 the x p dependence of x weighted valence quark distribution in the π-meson calculated based on the recently obtained PDF parameterization [12]. (Detailed analysis of pion PDFs will be presented elsewhere [13].) As the figure shows the exponential form of Eq.(10) fits the"height-position" correlation reasonably well with the parameters C = 0.202(3) and D = 1.50 (2). This represents a strong indication that the observed correlation (Eqs. (11) and (10)) is universal for any valence quark distribution in the hadron.

VII. CONCLUSION AND OUTLOOK
Concluding, we emphasize that the exponential form of the "height -position" correlation is a result of the specific relation between x p and t derivatives of valence PDFs (Eq.(11)), which results in a constant value D for all range of Q 2 that PDFs are considered. The verification of this relation with PDF parameterizations obtained in LO, NLO and NNLO approximations (e.g. Fig.2 and Fig.3) indicates that it is valid for any order of QCD coupling constant, α s based on M S scheme of fac-torization. Despite large uncertainties our analysis also indicates that Eq.(10) is valid also for the other (DIS) scheme of factorization. Such a universality of the exponential form of the correlation, in our view, is due to the dynamics of QCD evolution which can in principle be studied as a separate analytical problem as discussed in Sec.IV.
It will be interesting to verify the existence of relation (11) for nuclear PDFs as well as for semi-inclusive DIS processes sensitive to the valence quark distributions. Relation (11) can be used in the calculation of valence PDFs using lattice QCD, not just for the proton but for other hadrons whose PDFs are not well constrained by experiment.
Finally, one expects similar effects to be observed also for fragmentation functions, since evolution equations at least in LO have similar splitting functions. In fact, gluon fragmentation functions at small x exhibit features reminiscent to the one discussed in this work (see e.g. Refs. [14,15] and references therein).
Overall, establishing the universality of Eq.(11) will allow to use it to constrain Q 2 evolution of more complex processes in higher order approximations.