1. E.V. Bugaev (INR, Moscow) B.V. Mangazeev (Irkutsk State Univ.)
23rd European Cosmic Ray Symposium Moscow, Russia, July, 3 – 7, 2012
Photonuclear interactions at very high energies and vector meson dominance
E.V. Bugaev (INR, Moscow) B.V. Mangazeev (Irkutsk State Univ.)
Let me sit down in order to manage my presentation.

2. Prehistory
The words “photonuclear interactions” in CR physics mean processes of photoabsorbtion by nuclear (or nucleon) target of virtual or real photons, accompanied by the production of hadrons. N γ N γ
E.P.George and P.Trent, Nature, 164, 838 (1949), “Stars in photoemulsions …”
G.Cocconi, V.Cocconi Tongiorgi, Phys. Rev., 84 (1951), 29, “Nuclear Disintegrations Induced by mu-Mesons”.
Not all these processes are important in high energy CR physics, only the so-called diffractive DIS processes (in case of the virtual photons).
Q2 is fixed, x=Q2/s → 0, , “Regge-Gribov limit”.
Now, after 60 years, still there is no theoretical picture for a description of the photonuclear interaction which can be considered as generally accepted.

3. Introduction 1.1 Historical comments
1969 y, J.Sakurai, “Vector-meson dominance and high-energy electron proton inelastic scattering” (VMD)
accumulation of data, advent of QCD
1992 y, В.Badeleck and J.Kwiecinski, “Electroproduction structure function F2
in the low Q2, low x region” (VMD + QCD)
nonlinear gluon dynamics, CGC theory
2003 y, E.Iancu et al, “Saturation and BFKL dynamics in the HERA data at small x” (PQCD)
AdS/CFT correspondence, small-x evolution in the large
coupling limit of QCD like theories
2011 y, R.Brower et al, “Small-x DIS via the Pomeron in AdS” (??)

4. 1.2 Dipole factorization
Coherence length, lc=1/Mx, is small, virtual photon fluctuates into a hadronic system before the collision with the target.
r is the dipole transverse size, z is the fraction of the photon longitudinal momentum.
At high energy where
is the forward scattering amplitude for a dipole with size r and impact parameter b.
From unitarity one has in the black disc limit).
In PQCD theory
i.e., diverges at breaking unitarity.
1.3 Gluon dynamics
Two types of non-linear phenomena at small x:
multiple scattering (the exchange of more than two gluons),
gluon saturation, i.e., non-linear effects taming the rise of the gluon distribution at small x.

5. 1.4 Two types of configurations in DIS
In Regge type model (Forshaw et al, 1999)
in accordance with two Pomeron model by Donnachie and Landshoff , 1998.
Physically, there are two types of configurations in DIS (e.g., Bjorken, 1996).
qq-pairs with large masses M have a small transverse size, r~1/M, and interact perturbatively with the target via gluon exchange. The cross section is proportional to the square of the dipole moment, and is small at
there are configurations where q and do not have large and are aligned along the photon beam direction. From kinematics, the one particle is “slow”, another one is “fast”,
The slow quark has enough time to evolve nonperturbatively (e.g., there is
enough time for a nonperturbative color flux-tube to grow between q and ).
So, on arrival at the target the hadronic system can look like the vector
meson interacting with the target with the large cross section.
It is quite difficult to imagine that these two mechanisms can be studied
with a help of one and the same theoretical model (at least, using QCD, where only perturbation theory methods had been developed so far.

6. 1.5 DIS in saturation models at small Q2
At high energies, PQCD predicts that the small-x gluons in hadron wave function forms a Color Glass Condensate (CGC), and a hard saturation scale, Qs, exists, at which nonlinear gluon recombination effects start to be important. It appears that Qs grows with decreasing of x. In LLA at fixed coupling one obtains:
(Iancu et al, 2002, Albacete et al, 2005).
If Qs >> ΛQCD, the non perturbative effects are cut-off because at
the large, nonperturbative, dipoles (r >1/Q) are deeply in the saturation regime, i.e., for them the dipole scattering amplitude approaches the unitarity bound, and the cross section is large.
Clearly, the qq-pairs with large r have an overlap with hadronic (i.e., nonperturbative) states, but it is assumed that the total probability for hadronic components in photon wave function is correctly given by PQCD (Iancu et al, 2003). So at least for a description of data at small x, small Q2 region,
there is no need for an accounting of the nonperturbative component.

7. 1.6 Main arguments against of this conclusion
Equations of saturation models (e.g., BK evolution equation (Balitsky, 1996, Kovchegov, 1999, )) unitarize the linear BFKL Pomeron theory, so the numerical value of the Pomeron intercept is very important.
it appears that in the leading order the intercept is about 1.5,
too large!
The next-to-leading correction is very large and negative,
too small!
There is need for the resummation of this series.
It is tempting to study the large coupling limit of QCD, in which all higher order perturbative corrections are summed up.
It is possible using AdS/CFT conjecture: there is correspondence
between conformal (gauge) field theory and string (gravity) theory in a
background with extra dimension (“gauge-gravity duality”). When the
coupling is large, g2Nc >>1, then the gauge theory is strongly coupled but
the string (gravity) theory is weakly coupled, i.e., the calculations are
possible. Namely, the boundary values of the fields on the gravity side are
related to the local operators on gauge theory side.

8. It was shown (Kovchegov et al, 2009) using QCD-like CFT (N=4 super Yang-
Mills theory) that the theory with large coupling (i.e., nonperturbative one)
can explain HERA data for F2 rather well. Besides, it appears that in the limit
of high energies the saturation scale becomes independent of x in
contrast with the behavior in PQCD in leading order. The conclusion of
Kovchegov et al is that the nonperturbative model “… could be viewed as
complimentary to the perturbative description based on CGC”.
In the gravity dual theory of DIS the Pomeron is replaced by the “Regge graviton” in AdS space which unites both hard and soft components (Brower et al, 2007), and the intercept is For a good fitting of HERA data at small x one must put j0=1.25 (Brower et al, 2011) which means that the DIS physics at small x is in the crossover region between strong and weak coupling.
The Regge graviton model predicts that the dependence of the effective
intercept εeff of Q2
has a form required by data and in similar with the corresponding prediction
of two-Pomeron model (Donnachie and Landshoff, 2003). There are two
corrections to the conformal approximations: due to an accounting of
confinement (this correction is dominant) and due to the saturation (this
correction is subdominant at Q2~2 GeV2 and x ≥ (Brower et al, 2012) ).
In general, the AdS/CFT models do not contradict and even support the idea
of two independent Pomerons, suggesting for this the simple geometrical
explanation connected with a presents of extra coordinate.

9. Two-component model
From the previous arguments we see no reason to give up the use of the two-component approach (e.g., Bugaev E, Mangazeev B and Shlepin Y, 1999) for a description of DIS at small x. For the perturbative part we use the colour dipole model suggested by Forshaw et al, 1999, where, in the expression for the dipole cross section we keep only the hard Pomeron piece (accounting the dipoles with small transverse size).
For a description of the nonperturbative component, we use the Vector Meson Dominance (VMD) model. Correspondingly, we use, instead of the dipole factorization the more general construction, the double mass dispersion relation for the forward Compton scattering amplitude (Gribov, 1969),
Here, M and M’ are the invariant masses of the incoming and outgoing vector mesons, ρ(s,M2,M’2) is the spectral function. The field-theoretical basis of this formula goes back to Sakurai’s idea of treating the ρ-meson as gauge boson and to hidden-gauge theories, i.e., theories, in which the vector meson is a gauge boson of the hidden local symmetry (HLS).

10. In last several years a lot of papers appeared, in which holographic models of QCD and, moreover, holographic models of hadrons are elaborated. These models are extra-dimensional, essentially nonperturbative (in particular, color confinement is built in), based on AdS/QCD correspondence. Vector meson dominance and hidden local symmetry are natural consequences of these models. Important (for us) predictions of holographic QCD are as follows:
1. There are towers of vector resonances with infinite numbers of particles: ρ, ρ’, ρ’’, ... .
2. The mass spectrum of vector mesons depends on the geometry: M2 ~ n2 (in “hard-wall” models) or M2 ~ n (in “soft-wall” models).
3. There is the current-field identity
4. The pion (as well as the nucleon) electromagnetic formfactor is completely meson dominated,
The last two points show that one can, in some sense, say about the “return of vector dominance”: the “old” vector dominance with the lowest V(1) = ρ is replaced everywhere by a “new”, extended, vector dominance with an infinite tower of vector mesons.

11. It is well known that the consistency of the spectral representation with the approximate Bjorken scaling requires rather unnatural (from point of view of hadron physics) strong cancellations between amplitudes of the diagonal, Vp→ V’p, transitions. It had been shown in our previous works (Bugaev E, Mangazeev B and Shlepin Y, 1999; Bugaev E and Mangazeev B, 2009 ) that such cancellations are not effective. More exactly, there is no motivation for essential cancellations if it is assumed that the vector mesons of VMD models are similar to vector mesons with known properties. On the other hand, if it is assumed that these states are qq-systems with definite mass, rather than vector mesons, the essential cancellations between diagonal and off-diagonal transitions become possible, due to a general feature of quantum field theory that fermion and antifermion couple with opposite sign (the well-known example is the case of two-gluon exchange). Therefore, in some modern versions of VMD there are no vector mesons at all, only qq-pairs , i.e., these models are really not hadronic.
The way out of this situation is reached in the proposed in Bugaev E., Mangazeev B. and Shlepin Y., 1999, two-component model of DIS at small x: interactions of the qq- pair (produced by the virtual photon) with the target nucleon can be described by PQCD if the transverse size of the pair is small and by VMD (in which vector mesons have typical hadronic properties) if it is of the order of typical hadronic size. Nondiagonal transitions are necessarily taken into account.

12. Correspondingly, in this model the structure functions of DIS have two components: the “soft” component described by VMD and the “hard” one described by PQCD. Naturally, VMD is used in its “aligned jet” version (J. Bjorken, 1973), i.e., the configurations are selected, in which the q and anti-q, produced by virtual photon, are aligned along the beam direction and, as consequence, the transverse distance between q and anti-q, on arrival at the target nucleon, is of order of hadronic size.

13. Main results SOFT + HARD SOFT
Figure 1. The total cross section of photoabsorption for the real photon (the dashed line). The solid line is the soft contribution

14. SOFT
SOFT + HARD
Figure 2a. The Q2 dependence of structure function F2 for different values of x.
Figure 2b. The same as left figure. Soft + Hard only.

15. SOFT + HARD
Figure 3. The x dependence of structure function FL for different values of Q2. The experimental points are taken from Aaron et al, (H1 Collaboration).

16. SOFT + HARD
Figure 4. The x dependence of structure function F2 for small values of Q2 in the region of very small x.

17. SOFT
SOFT + HARD X
Figure 5. The x dependence of structure function F2 for small values of Q2 in the region of very small x.

18. Conclusions
We demonstrated that the two-component approach to the theoretical description of diffractive DIS, in which the nonperturbative part is given by the VMD in its modern form (suggested by the AdS/CFT correspondence) is successful in description of HERA data at small x (x<0.08) and Q2≤ 10 GeV2 .
This may be enough for using in CR experiments if one uses muons from the steeply falling cosmic ray muon spectrum.
At Q2>10 GeV2 the perturbative part of F2 becomes dominant (for x~10-4) in our calculations. Note, once more, that we use for the perturbative part the hard Pomeron piece of colour dipole cross section from the work of Forshaw et al, 1999 (without any modification, in the region x>10-6 , where the most of data exists).