Higher twist effects in charmed-strange $\nu$DIS diffraction

The non-conservation of charmed-strange current in the neutrino deep inelastic scattering ($\nu$DIS) strongly affects the longitudinal structure function, $F_L$, at small values of Bjorken $x$. The corresponding correction to $F_L$ is a higher twist effect enhanced at small-$x$ by the rapidly growing gluon density factor. As a result, the component of $F_L$ induced by the charmed-strange current prevails over the light-quark component and dominates $F_L=F_L^{cs}+F_L^{ud}$ at $x\lsim 0.01$ and $Q^2\sim m_c^2$. The color dipole analysis clarifies the physics behind the phenomenon and provides a quantitative estimate of the effect.


Abstract
The non-conservation of charmed-strange current in the neutrino deep inelastic scattering (νDIS) strongly affects the longitudinal structure function, F L , at small values of Bjorken x. The corresponding correction to F L is a higher twist effect enhanced at small-x by the rapidly growing gluon density factor. As a result, the component of F L induced by the charmedstrange current prevails over the light-quark component and dominates F L = F cs L + F ud L at x ∼ < 0.01 and Q 2 ∼ m 2 c . The color dipole analysis clarifies the physics behind the phenomenon and provides a quantitative estimate of the effect.

Introduction
Weak currents are not conserved. For the light flavor currents the hypothesis of the partial conservation of the axial-vector current (PCAC) [1] provides quantitative measure of the charged current non-conservation (CCNC) effect. [2]. The non-conservation of the charm and strangeness changing (cs) current is not constrained by PCAC. Here we focus on manifestations of the cs current non-conservation in small-x neutrino DIS. At small x the color dipole (CD) approach to QCD [3,4] proved to be very effective. Within this approach it is natural to quantify the effect of CCNC in terms of the light cone wave functions (LCWF) 1 where j ν =c(k)γ ν (1 − γ 5 )s(p), ∆E = E q − E p − E k and ǫ ν is the four-vector of the so-called longitudinal polarization of the W -boson with the four-momentum q. Notice that The observable which is highly sensitive to the CCNC effects is the longitudinal structure function F L (x, Q 2 ) related, within the CD approach , to the quantum mechanical expectation value of the color dipole cross section, Our finding is that the higher twist correction to F L arising from the cs current non-conservation appears to be enhanced at small x by the BFKL [6] gluon density factor, The color dipole analysis reveals mechanism of enhancement: the ordering of dipole sizes typical of the Double Leading Log Approximation (DLLA) and the multiplication of log's like to higher orders of perturbative QCD. As a result, the component F cs L induced by the charmedstrange current grows rapidly to small-x and dominates F L at Q 2 ∼ < m 2 c [7,8].

CCNC in terms of LCWF
In the CD approach to small-x νDIS [9] the responsibility for the quark current non-conservation takes the light-cone wave function of the quark-antiquark Fock state of the longitudinal (L) electro-weak boson 2 . For Cabibbo-favored transitions the Fock state expansion reads where only ud-and cs-states (both vector and axial-vector) are retained.
In the current conserving eDIS the Fock state expansion of the longitudinal photon contains only S-wave qq states and Ψ vanishes as Q 2 → 0, Here r is the qq-dipole size and z stands for the Sudakov variable of the quark.
In νDIS the CCNC adds to Eq.(6) the S-wave mass term [11,12] and generates the P -wave component of Ψ(z, r), where upper sign is for the axial-vector current, lower -for the vector one and ζ = 2λ -for the 3 High Q 2 : z-symmetric cs-states In the color dipole representation [3,4] the longitudinal structure function F L (x, Q 2 ) in the vacuum exchange dominated region of x ∼ < 0.01 can be represented in a factorized form where g is the weak charge, The light cone density of color dipole states |Ψ| 2 is the incoherent sum of the vector (V ) and the axial-vector (A) terms, The Eq. (6) makes it obvious that for large enough virtualities of the probe, Q 2 ≫ m 2 c , the S-wave components of both F ud L and F cs L in expansion (4) are dominated by the "non-partonic" configurations with z ∼ 1/2 with characteristic dipole sizes [13] In the CD approach the BFKL-log(1/x) evolution [6] of σ(x, r) in Eq.(9) is described by the CD BFKL equation of Ref. [14]. For qualitative estimates it suffices to use the DLLA (also known as DGLAP approximation [15,16]) Then, for small dipoles [17] σ(x, r) ≈ and from Eq.(9) it follows that where G(x, k 2 ) = xg(x, k 2 ) is the gluon structure function and α S ( The rhs of (12) is quite similar to F (e) L of eDIS [15,18] (see [17] for discussion of corrections to DLLA-relationships between the gluon density G and F (e) L ). Two S-wave terms in the expansion (4) that mimics the expansion (5) evaluated within the CD BFKL model of Ref. [19] are shown by dotted curves in Fig. 1. The full scale BFKL evolution of the νN structure function F L (x, Q 2 ) with boundary condition at x 0 = 0.03 is shown in Fig. 2 of Ref. [20].
4 Moderate Q 2 : asymmetric cs-states and P -wave dom- Fig.1). To evaluate it we turn to Eq. (9). For m 2 c ≫ m 2 s in Eq.(10), , where K 1 (x) is the modified Bessel function and one can integrate in (9) over r 2 to see that the z-distribution, dF cs L /dz, develops the parton model peaks at z → 0 and z → 1 [7]. To clarify the issue of relevant dipole sizes we integrate in (9) first over z near the endpoint z = 1.
For r 2 from the region This is the r-distribution for cs-dipoles with c-quark carrying a fraction z ∼ 1 of the W + 's light-cone momentum. Thus, the singularity ∼ r −4 in Eq.(13) together with the factorization relation (9) and σ(r) ∼ r 2 give rise to nested logarithmic integrals over dipole sizes. Indeed, in the Born approximation the gluon density G in Eq. (11) is where .
Notice, that perturbative gluons do not propagate to large distances and µ G in Eq.(15) stands for the inverse Debye screening radius, µ G = 1/R c . The lattice QCD data suggest R c ≈ 0.3 fm [21]. Because R c is small compared to the typical range of strong interactions, the dipole cross section evaluated with the decoupling of soft gluons, k 2 ∼ < µ 2 G , would underestimate the interaction strength for large color dipoles. In Ref. [22,19,23] this missing strength was modeled by a non-perturbative, soft correction σ npt (r) to the dipole cross section σ(r) = σ pt (r) + σ npt (r). Here we concentrate on the perturbative component, σ pt (r), represented by Eqs. (11) and (14).
Then, for the charmed-strange P -wave component of F L with fast c-quark (z → 1) one There is also a contribution to F cs L from the region 0 < r 2 < (m 2 which is, however one L short. Thus, the CD analysis reveals the ordering of dipole sizes x Bj =0.0125 x Bj =0.0175 x Bj =0.008 x Bj =0.02 typical of the DGLAP approximation. The rise of F cs L (x, Q 2 ) towards small x is generated by interactions of the higher Fock states, cs + gluons. The DLLA ordering of Sudakov variables and dipole sizes in the n-gluon state |csg 1 g 2 ...g n results in the density |Φ n+1 | 2 of multi-gluon states in the color dipole space [3] By virtue of (19,20) the csg 1 g 2 ...g n -state interacts like color singlet octet-octet state with the cross section (C A /C F )σ(ρ n ). Then, making an explicite use of Eqs. (11,14) and (13) we arrive at the P -wave component of F L that rises rapidly to small x, In Eq. (22), which is the DGLAP-counterpart of Eq. (3), is the DGLAP expansion parameter with η = C A log(x 0 /x).
Additional contribution to F cs L comes from the P -wave cs-dipoles with "slow" c-quark, z → 0. For low Q 2 ≪ m 2 c this contribution is rather small, If, however, Q 2 is large enough, Q 2 ≫ m 2 c , corresponding distribution of dipole sizes valid for The DLLA summation over the s-channel multi-gluon states, results in [5] F cs L ≈ Therefore, at high Q 2 ≫ m 2 c both kinematical domains z → 1 and z → 0 (Eqs. (22) and (25), respectively) contribute (within the DLLA accuracy) equally to F cs L .
The P-wave component of F ud L is small because of small factor m 2 q /Q 2 , where m q is the constituent u, d-quark mass. Here we deal with constituent quarks in the spirit of Weinberg [27].
This suppression factor, m 2 q /Q 2 , comes from the light-cone wave function Ψ ud ∼ m q (Qr) −1 and is of purely perturbative nature.
In [20] we checked accuracy of the color dipole description of F L (x, Q 2 ) in the nonperturbative domain of low Q 2 making use of Adler's theorem [2], In (26) f π is the pion decay constant, σ π is the on-shell pion-nucleon total cross section.
Invoking the CD factorization, which is valid for soft as well as for hard diffractive interactions, we evaluated first the vacuum exchange contribution to both σ π and F L (x, 0).
The parameter f π in Eq.(26) was evaluated within the CD LCWF technique [28,29]. The approach successfully passed the consistency test: πF ud L (x, 0)/(f 2 π σ π ) ≈ 1 to within 10%. The cross section σ π was found to be in agreement with data. However, the value of f π appeared to be underestimated. It was found that for m q = 150 MeV, commonly used now in CD models successfully tested against DIS data, our F L at Q 2 → 0 undershoots the empirical value of f 2 π σ π /π by about 40% [20], not quite bad for the model evaluation of non-perturbative parameters. One can think of improving accuracy at higher Q 2 ∼ m 2 c which we are interested in.
Notice, that Adler's theorem allows only a slow rise of F ud L (x, 0) to small x, much slower than the rise of F cs L following from our DLLA estimates. The value of the socalled soft pomeron intercept ∆ sof t ≃ 0.08 comes from the Regge parameterization of the total πN cross section [30].
6 Comparison with experimental data.
The structure function F 2 for the νF e and νP b interactions are shown in Fig. 2. From comparison with experimental data [24], [25] and [26] we conclude that the excitation of charm contributes significantly to F 2 at x ∼ < 0.01 and dominates F 2 at x ∼ < 0.001 and Q 2 ∼ < m 2 c . For comparison with data taken at moderately small-x the valence component, F 2 val , of the structure function F 2 should be taken into account. We resort to the parameterization of F 2 val (x, Q 2 ) suggested in [36]. This parameterization gives F 2 val (x, Q 2 ) vanishing as Q 2 → 0 which is not quite satisfactory from the point of view of PCAC. The latter requires Here x = m 2 a /W 2 and σ R π (W ) stands for the secondary reggeon contribution to the total pion-nucleon cross section that diminishes at high cms collision energy as σ R π (W ) ∼ (W 2 ) α R −1 , where α R ≃ 0.5. However, at smallest values of Q 2 ≃ 0.2−0.3 GeV 2 accessible experimentally F 2 val (x, Q 2 ) ≫ F P CAC 2 val (x, 0), remind, the characteristic mass scale in the axial channel is m a ∼ 1 GeV. Therefore, the accuracy of F 2 val (x, Q 2 ) of Ref. [36] is quite sufficient for our purposes. In Fig. 2 the valence contributions to F 2 are shown by dash-dotted curves. The agreement with data is quite reasonable.
One more remark is in order, the perturbative light-cone density of ud states, |Ψ ud | 2 ∼ r −2 , apparently overestimates the role of short distances at low Q 2 (see Ch. 5) and gives the value of F ud L (x, 0) which is smaller than the value dictated by Adler's theorem [20]. This also may lead to underestimation of F 2 in the region of moderately small x ∼ > 0.01 dominated by the light quark current.

Summary
Summarizing, it is shown that at small x and moderate virtualities of the probe, Q 2 ∼ m 2 c , the higher twist corrections brought about by the non-conservation of the charmed-strange current dramatically change the longitudinal structure function, F L . The effect survives the limit Q 2 → 0 and seems to be interesting from a point of view of feasible tests of Adler's theorem [2] and the PCAC hypothesis.

Acknowledgments
V.R. Z. thanks the Dipartimento di Fisica dell'Università della Calabria and the Istituto Nazionale di Fisica Nucleare -gruppo collegato di Cosenza for their warm hospitality while a part of this work was done. The work was supported in part by the Ministero Italiano dell'Istruzione, dell'Università e della Ricerca and by the RFBR grants 07-02-00021 and 09-02-00732.