QCD at the amplitude level: Fock state interference in heavy quark electroproduction

Quantum chromodynamics (QCD) rigorously predicts the existence of both nonperturbative intrinsic and perturbative extrinsic heavy quark contents of nucleons. In this article we discuss the heavy quark electroproduction on protons induced by the Fock states $|uud+g\rangle$ of three valence quarks in the proton and a nonperturbative gluon, and the $|uud+Q\bar Q\rangle$ non-perturbative state of three valence quarks and a heavy quark-antiquark pair. The first one gives the perturbative contribution when the gluon produces a heavy quark pair, while the second is the intrinsic part, and they produce amplitude interference. We use nonperturbative light-front wavefunctions for these Fock states, which are computed using the color-confining light-front holographic QCD theory. Due to interference of the amplitudes corresponding to the intrinsic and extrinsic heavy quark contribution to the proton a novel $Q$ vs. $\bar Q$ asymmetry emerges in the differential cross section of the electroproduction off a proton $d^2\sigma_{e^-p}/(dx dQ^2)$. Our analysis proposed a novel asymmetry in QCD between intrinsic and extrinsic heavy quark content in nucleon and provides new insights into the physics of heavy quark phenomena in QCD at the amplitude level. This asymmetry is similar to the Brodsky-Gillespie asymmetry discovered before in QED case and confirmed at DESY.


Introduction
The hard scattering model is the main tool for the theoretical analysis of the physical processes that happen in hadron colliders. Its basic ingredients are the parton distribution functions, the hard partonic scattering cross section and in some cases the parton-hadron fragmentation functions. In most cases the collinear approximation is used, in which the parton transverse momenta in the distribution functions is ignored. The theoretical foundations of this model rely on the factorization theorems, which have been proven in specific cases. Nevertheless, although there has been a tremendous amount of experimental work in abstracting the parton distribution functions from data and theoretical work in getting the hard scattering cross section from perturbative QCD, the size of the corrections to the model are not known. All that is known is that these corrections are going to be suppressed by the hard scale that is involved in the parton cross section. One correction comes from the fact that the transverse momenta of the partons in the distribution functions is ignored, but more importantly, there is a multiplication of probabilities (the hard scattering model diagram is not a Feynman diagram), which is certainly an approximation. One way to look at this last point is that there are other intermediate states than the leading parton-parton scattering, that can participate in the process. For example, in J/ψ production, in which the main contribution comes form gluon-gluon scattering, there is the contribution in which two gluons from one hadron interacts with one gluon from the other [1].
From the point of view of obtaining a better knowledge of the properties of QCD, it is therefore interesting to look for reactions in which QCD at the amplitude level is considered. For example, in nuclear shadowing there is interference of Pomeron and Reggeon amplitudes [2], and in single spin asymmetries there is interference of two amplitudes which have different proton spin J z = ±1/2 but couple to the same final-state [3,4]. Notice that in the hard scattering diagram the blob in which the parton comes out of the proton represents a distribution function, but in cases in which amplitudes are considered, it represents a lightfront wavefunction (LFWF) or Fock state, whose square is related to the parton distribution.
One of the rigorous properties of QCD is the existence of intrinsic heavy quarks in the fundamental structure of hadrons, which is part of the Fock state expansion of the proton. In fact, the existence of both nonperturbative intrinsic and perturbative extrinsic heavy quark contents of the nucleon [5], has now been confirmed by many experiments at world-wide facilities. As first noted in [5], the intrinsic contribution comes from the nonperturbative Fock state |uud+QQ . For a review see, e.g., Ref. [6]. An even more surprising and novel feature resulting from intrinsic heavy quarks is the strong asymmetry between the Q(x) andQ(x) distributions in the nucleon eigenstate. The c(x) vs.c(x) asymmetry in the proton has recently been demonstrated by a lattice gauge theory analysis, where the charm quark distribution dominates at large x, and we will use this result in our heavy quark electroproduction analysis, which will obviously produce a corresponding asymmetry in this process. This asymmetry was discovered in Ref. [7] and later was confirmed in Ref. [8]. However, there is an additional asymmetry describing heavy quark content in the proton. This asymmetry arises due to interference of intrinsic and extrinsic heavy quark distributions in the proton. It was discovered before in case of QED. In QED, the interference of the first-Born and second-Born amplitudes [9] leads to an ℓ vs.l asymmetry in the Bethe-Heitler cross section for lepton pairs produced on nuclei γZ → ℓ + ℓ − Z. The lepton asymmetry for electron-positron pairs was measured at DESY [10].
In particular, in the present paper we will study the analogous effects due to the interference of QCD amplitudes for the leading Fock states of the proton producing heavy quarks. These Fock states are shown in Fig. 1: (a) the |uud + g Fock state describing the three valence quarks in the proton and a nonperturbative gluon which produces a heavy quark pair with odd charge conjugation C Q |Q >= −|Q > and (b) the |uud + QQ state describing the five-quark state, which is the bound state of the three valence quarks plus a non-perturbative intrinsic heavy quark-antiquark pair with both even and odd charge conjugation components. The interference contributing to the e − + p → e − + Q +Q + X reaction is given by the imaginary part of the diagram shown in Fig. 2. Our consideration is based on the factorization picture, which is valid at large Q 2 . In comparison with Ref. [8] we consider the interference the Fock states producing intrinsic and extrinsic heavy quark content in the nucleon. It leads to a novel asymmetry in QCD, which is similar to the QED case (as stressed in previous paragraph) and differed from asymmetry induced by intrinsic heavy quark content in the nucleon discussed in Refs. [7] and [8].

Framework for interference of intrinsic and extrinsic heavy quarks in the nucleon
In order to calculate the effects of the interference, we compute the square of the matrix element corresponding to the scattering amplitude e − + p → e − + Q +Q + X. The amplitude is given by the sum of the two terms induced by the two Fock states shown in Fig. 2. Therefore, one gets three contributions: two squares of the amplitudes generated by the individual Fock states and also their interference. Following the ideas of holographic QCD [11]- [14] we proposed [8] the heavy quark and heavy antiquark parton distribution functions (PDFs) Q in (x) andQ in (x) encoding intrinsic  heavy quark and heavy antiquark content in the nucleon. The PDFs Q in (x) andQ in (x) are expressed in terms of the LFWFs ψ in;λ N Q;λ Q ,Lz (x, k ⊥ ) and ψ in;λ N Q;λQ,Lz (x, k ⊥ ) with struck heavy quark Q and heavy antiquarkQ, respectively: (2.1) The LFWFs introduced in Eq. (2.1) correspond to the Fock state |uud + QQ with specific quantum numbers -helicity for the proton λ N =↑, ↓, helicity for the struck heavy quark or heavy antiquark λ Q , λQ = + 1 2 , − 1 2 , and z projection of the angular orbital momentum L z . For the heavy quark PDF Q in (x) the result obtained in LF QCD [5] reads Here N Q in is the normalization constant fixed from data. For example, as pointed out in Ref. [6], in the case of the charm distribution N c in = 6, if there is a 1% intrinsic charm contribution to the proton PDF. Such a choice was motivated by an estimate of the magnitude of the diffractive production of the Λ c baryon in the pp → pΛ c X reaction [5], which is consistent with the MIT bag-model estimate [16] of the probability of finding a five-quark |uud + QQ configuration the nucleon at the order of 1%. In the case of the bottom distribution, N b in = 6 (m c /m b ) 2 [6]. Heavy antiquark [8]. This form of the heavy antiquark PDF was motivated by Ref. [7] and the constraints that the zero moments of the Q in (x) andQ in (x) PDFs should be equal, or that the zero moment of the asymmetry Our prediction for the first moment of the heavy quark asymmetry x c−c = 1 0 dx x Q as (x) = 1/3500 ≃ 0.00029 [8] was in order of magnitude agreement with the prediction of Ref. [7] x c−c = 0.00047 (15). Also our prediction for the fraction of the proton momentum [Q] = 1 0 dx x Q in (x) +Q in (x) = 0.54% carried by charm quark and antiquark was in good agreement with the central value 0.62% of the results recently extracted by the NNPDF Collaboration [15].
To fix the explicit form of the LFWFs ψ in;λ N Q;λ Q ,Lz (x, k ⊥ ) and ψ in;λ N Q;λQ,Lz (x, k ⊥ ) we propose that they are proportional to the scalar functions ϕ Q in (x, k 2 ⊥ ) and ϕQ in (x, k 2 ⊥ ), which are parametrized in terms of the PDFs Q in (x) andQ in (x) of the intrinsic heavy quark and antiquark, respectively [8] Finally, the LFWFs ψ in;λ N Q;λ Q ,Lz (x, k ⊥ ) and ψ in;λ N Q;λQ,Lz (x, k ⊥ ) read The LFWFs ψ λ N λg ,λ X (x, k ⊥ ), describing the Fock state of a gluon (g) and a three-quark spectator X = (uud), where λ g = ±1, and λ X = ± 1 2 are the helicities gluon and three-quark spectator, are listed as [14]: where the functions ϕ (1) (x, k 2 ⊥ ) and ϕ (2) (x, k 2 ⊥ ) are expressed through the gluon PDF functions G ± (x) as [14] ϕ (1) Here G + (x) = G g↑/N ↑ (x) = G g↓/N ↓ (x) and G − (x) = G g↓/N ↑ (x) = G g↑/N ↓ (x) are the helicity-aligned and helicity-anti-aligned gluon distributions, respectively, whose combinations define the gluon unpolarized G(x) = G + (x) + G − (x) and polarized ∆G(x) = G + (x) − G − (x) PDFs. G(x) and ∆G(x) are expressed in terms of derived LFWFs ψ λ N λg ,λ X (x, k ⊥ ) as For G + and G − we use the results that were proposed in Ref. [17]: 8967 is the normalization constant, fixed from the first moment of the gluon PDF [17]: x g = 1 0 dxxG(x) = (10/21)N g . For x g we take the central value of the lattice result x g = 0.427 [18]. These densities obey very important model-independent constraints: (1) at large x the power scaling [17], which is consistent with QCD constraints [19] dictated by matching the signs of the quark and gluon helicities and the even power scaling of gluon PDFs; (2) at small x the gluon asymmetry ratio ∆G/G behaves as ∆G(x)/G(x) → 3x and is consistent with Reggeon exchange arguments [20].
The |uud + g state generates the effective LFWFs ψ ex;λ N Q/Q;λ Q/Q ,Lz (x, k ⊥ ) encoding the extrinsic heavy quark contribution in the proton, which are described by the product of ψ λ N λg ,λ X (x, k ⊥ ), gluon propagator, and the splitting function for the annihilation of the gluon into the heavy quark-antiquark pair where α s is the strong coupling, C F = 4/3 is color summation factor, ξ( , and N Qex is the normalization constant of the extrinsic heavy quark contribution, which is fixed from data. Note that the α s at the one-loop is calculated according the formula α s = α s (µ 2 ) = 4π/(β 0 log(µ 2 /Λ 2 QCD )), where µ = 2m Q is the scale taken at the value of two heavy quark masses, Λ QCD = 0.226 GeV is the QCD scale parameter used in our previous paper [4], β 0 = (11/3)N c − (2/3)N f is the leading coefficient in the α s expansion of the QCD β function. In particular, for the charm case we have N f = 4, µ ≃ 3 GeV, and therefore α s = 0.3.
The extrinsic PDFs Q ex (x) andQ ex (x) are degenerate Q ex (x) =Q ex (x) and are expressed in terms of derived LFWFs as

Electroproduction of heavy quarks and a novel asymmetry in QCD
Next, we specify the matrix element describing the elastic e − +Q/Q → e − +Q/Q scattering. It is given by where α = 1/137.036 is the fine-structure coupling; e Q/Q is electric charge of heavy quark/antiquark; u e (p, s), u Q/Q (p, s) are the spinors of electron and heavy quark (antiquark), which will be taken to be on-shell. Neglecting the masses of electron and heavy quark, the square of the amplitude M(e − + Q/Q → e − + Q/Q) summed over polarizations of the fermions is equal to |M| 2 = 128 (παe Q ) 2 (s 2 + u 2 )/t 2 for both quark and antiquark case. Note that our formalism is correct for any Q 2 , although as an example, we apply it for Q 2 ≫ m 2 in the present paper. Taking into account the effects of finite heavy quark masses m will be done in future work. We use the Mandelstam partonic level variables s, t, u in the proton rest frame, with s = ( is the Bjorken variable, y = (q · P )/(p 1 · P ) is the rapidity, P and m N are the proton momentum and mass. Now we calculate the square of the amplitude M (e − + p → e − + Q +Q + X) induced by the two Fock states |uud + QQ and |uud + g which is averaged over the spins of initial electron and proton and summed over the final fermions polarizations: which is the product of the effective transverse-momentum distribution (TMD) function f (x, k 2 ⊥ ) and |M| 2 /4. The TMD f (x, is split into diagonal contributions of intrinsic (f 1 ) and extrinsic (f 2 ) amplitudes, and their interference (f 3 ): The differential cross section for the electroproduction of heavy quarks is given by Here F Q 2 (x) is the heavy quark structure function, predicted by our approach, which includes contributions of both |uud + QQ and |uud + g Fock states and their interference: where Q full (x) andQ full (x) are the full results for the heavy quark and heavy antiquark PDFs including intrinsic and extrinsic contributions, and their interference: where Combining the interference terms containing in Q full (x) andQ full (x) we derive a novel quantity ∆ Q (x), which we name asymmetry in the differential cross section of electroproduction off a proton: We stress again that this asymmetry emerges due to interference of the amplitudes produced by |uud + QQ and |uud + g Fock states or due the intrinsic-extrinsic heavy quark interference. This novel phenomena is similar to the Brodsky-Gillespie asymmetry discovered before in QED case [9] and confirmed at DESY [10]. For convenience one can define the asymmetry in the normalized form Indeed this asymmetry is induced by the intrinsic-extrinsic heavy quark interference and this phenomena is similar to the Brodsky-Gillespie asymmetry discovered before in QED case [9] and confirmed at DESY [10]. Now we consider numerical applications of our findings. In Fig. 3 we plot the asymmetry x P as (x) as function of x when the extrinsic normalization constant N ex is varied from 0.3 to 0.6 and N c in = 6. In Fig. 4 we plot our predictions for the differential cross section dσ e − p /dQ 2 at y = 1 including all charm quark and antiquark contributions (intrinsic, extrinsic, and their interference) and compare it with data extracted from results of the H1 Collaboration at DESY [21]. Here we use fixed value of N cex = 0.3. For completeness we vary the N c in from 1.5 to 6, which corresponds to the 0.25%−1% of intrinsic charm in the proton. The variation of the N c in is shown by shaded band. One can see that our curve for the cental value of the N c in = 3 fits perfectly the data.

Conclusion
In conclusion, we have confirmed the existence of a novel asymmetry P as (x) in the differential cross section d 2 σ e − p /(dxdQ 2 ) of the heavy quark pair electroproduction in the proton. This asymmetry occurs due to interference of the amplitudes corresponding to the nonperturbative Fock states |uud + QQ and |uud + g in the process of the heavy  Figure 4. Differential cross section dσ e − p /dQ 2 in comparison with data from H1 Collaboration [21].
quark pair electroproduction in the proton. We presented the study of this asymmetry in QCD at the amplitude level. Due to our conjecture that the distributions of heavy quark and antiquark in the |uud + QQ are described by different scalar functions ϕ Q in (x, k 2 ⊥ ) = ϕQ in (x, k 2 ⊥ ) the asymmetry is proportional to the combination of the intrinsic heavy quark and heavy antiquark PDFs: Q in (x) − Q in (x). When Q in (x) andQ in (x) are degenerate, one has P as (x) ≡ 0. The P as (x) asymmetry is a new QCD asymmetry arising due to interference of two Fock states in the proton inducing intrinsic and extrinsic heavy quark content. Together with the Q in (x) −Q in (x) asymmetry considered in Refs. [7,8] it gives important information about heavy quark distributions in the proton. Both asymmetries vanish at the limit m Q → ∞ due to falloff of the normalization constants of the intrinsic and extrinsic contributions of heavy quark/antiquarks in the proton. As an application, we have calculated the asymmetry P as (x), which appear in the heavy quark structure function F Q 2 (x) and differential cross section dσ e − p /dQ 2 and have found reasonable agreement with experimental results from the H1 Collaboration [21].