{'fraction_non_alphanumeric': 0.04382611158312093, 'fraction_numerical': 0.01373548569810252, 'mean_word_length': 4.09009009009009, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Evidence for excess pions in nuclei

We analize two reactions: the electroproduction at small x and the Drell-Yan production by protons on nuclei. We show that nuclear shadowing in electroproduction is larger if the virtuality of inter- mediate hadrons is properly taken into account. In the Drell-Yan production, initial state interactions of fast partons are included. With this input, the excess pion contribution improves the description of data for both reactions.

1. Following the conventional picture of nuclei, the meson exchange currents result in a certain amount of excess pions in nuclei. This fundamental nuclear property was questioned by some recent experiments. The behaviour of nuclear structure functions at low x [1] and a lack of nuclear cross section enhancement for Drell-Yan (DY) production by protons [2] have led several authors (see, e.g., the Ref. [3]) to the conclusion that there are no excess pions in nuclei. The present analysis of these reactions is based on some new conjectures. In electroproduction, it is shown that virtual hadrons interact only inelastically with targets.
As the result, the nuclear shadowing can be significantly stronger than it has been believed so far. In DY production by protons, as well as in other hadroproduction reactions [4], initial state interactions of fast partons are taken into account. With this input, the data for both reactions are much better described with the excess pion contribution included.
2. In the impulse approximation, the quark (antiquark) distribution in nuclei can be represented as a convolution of nucleon and excess pion quark distributions, where f A N (y) and f A π (y) are the nucleon and excess pion distribution functions in nuclei. The flavour indices and the Q 2 − dependence are implied here and hereafter. Note that we denote by f A π (y) the renormalized excess pion distribution which preserves the total probability conservation [5]. Therefore the functions f A N (y) and f A π (y) are not independent but are constrained by the momentum sum rule One way to find q A (x) is to calculate first f A π (y) using the elementary πN interaction and then to determine f A N (y) from (2), adopting some model for the nucleon momentum distribution. In earlier works of this kind [6] the unrenormalized pion distribution was used for f A π (y), what led to an overestimation of the pionic contribution to q A (x). This was corrected in more recent papers [7], but there is a significant model dependence of calculations, which start from the elementary πN interaction.
Another approach was suggested in Ref. [8]. Since f A N (y) is an invariant function (y is the ratio of two scalar products), it can be calculated in any frame. In the nucleus rest frame, f A N (y) is determined by the energy and momentum distribution of nucleons, which can be measured in independent experiments [9]. In this case the magnitude of the EMC-effect is proportional to (1− < y A N >)/m N = ǫ/m N , where ǫ is the average nucleon separation energy. For heavy nuclei the data give ǫ ≥40 MeV. The A-dependence of ǫ was discussed in Ref. [11].
When f A N (y) is determined, the first moment of the excess pion distribution is fixed by (2) and a detailed form of f A π (y) can be found by using some realistic form of pion momentum distribution, e.g. the one from Ref. [12]. Here we calculate f A π (y) with this method. 3. It was shown [8] that the main part of the EMC-effect can be explained by the binding correction of nuclear structure functions. We here focus on the low-x region and show that the nuclear shadowing and antishadowing can be explained provided (a) the virtuality of intermediate mesons is taken into account while considering their multiple interactions in nuclei and (b) the contribution of excess pions is added. We use the generalized vector meson dominance (GVMD) model, but the above conclusion is valid for some other hadron dominance models, adopting the Glauber theory [10] to describe multiple interactions of virtual hadrons. In the GVMD model, the structure function is given by (see, e.g., the Ref. [13]) where Π(µ 2 ) is the spectral function of mesonic states with the mass µ, σ tot is the total cross section of meson interaction with a target. Total cross sections are connected to forward scattering amplitudes by the optical theorem, In some other models [14], the parton-target amplitudes have been used instead. In the independent particle model, the Glauber result, adopted by many authors, is given by where T ( b) is the optical thickness of the nucleus and b is the impact parameter. For (4) can be rewritten as We now show that (4) and (5) cannot be used for virtual hadrons participating in electroproduction.
In Glauber theory it is assumed that for fast projectiles the overall modification of the wave inside the nucleus is the factor [10, 15] where This equation is also valid for virtual projectiles if their propagation length λ satisfies the condition λ ≫ r A , where r A is the nucleus size. A correction of (6) in the case λ ∼ r A has been considered in Ref. [16] and will be taken into account below. The equation (4) follows from (6) if the elastic scattering amplitudes are expressed in terms of profile functions as [10,15] where k is the projectile three-momentum in the lab. frame. The last equation describes the Fraunhofer diffraction of waves [17], if the diffraction is observed at large distances z from the target. The condition for the Fraunhofer diffraction (and for the validity of (7)) is [17] z ≫ L, L = 2kr 2 t , where L is the healing length of shadow [15] (L characterizes the duration of the elastic scattering process) and r t is the target transverse size. At z ∼ L the Frenel diffraction, which is weaker than the Fraunhover diffraction, takes place instead and at z ≪ L there is no diffraction [17]. But for virtual projectiles the accessible longitudinal distances are only of the order of λ. Therefore the standard derivation of (4) doesn't hold if [18] λ ≪ L.
In that case, a projectile vanishes before the diffraction takes place. Let us obtain the same result using a more formal language.
For stable projectiles, the Glauber result for elastic and inelastic cross sections reads As it follows from this equation, both σ el and σ inel are non-vanishing if Γ( b) = 0. However, this is not the case for virtual projectiles and we now show that σ el vanishes under the condition (9), but σ inel does not. The total interaction probability is given by the imaginary part of the forward projectile-target amplitude f (s, t = 0) ≡< k, P |T |k, P > where k, P are projectile and target momenta and the operatorT describes all projectile-target interactions.
The imaginary part of the amplitude can be represented as the sum of elastic and inelastic channels, Im < k, P |T |k, P >= (2π) 4 2 n δ 4 (k n + P n − k − P ) < k, P |T † |k n , P n >< k n , P n |T |k, P > where |k n , P n > are all possible states of initial particles and |i > are all other states, |i > = |k n , P n >. The intermediate states |k n , P n > and |i > contribute to Im[f (s, t = 0)] if they contain only free particles. In fact, to take the imaginary part of a diagram is equivalent to cut this diagram and to put the cut lines on the mass shell. In the case of virtual hadron-nucleus scattering, P 2 = m 2 A but k 2 = µ 2 , where µ is the hadron mass. It is easy to verify that the first sum from (11) vanishes when P 2 n = m 2 A , k 2 n = µ 2 . A hadron with the momentum k n must be off-mass-shell (the opposite case of an off-mass-shell recoil nucleus is irrelevant for the process under consideration). Due to the uncertainty principle, a hadron with the off-mass-shell momentum k n can be considered as a free particle during the time λ ∼ 1/(k n,0 − ( k 2 n + µ 2 ) 1/2 ). For fast particles, λ is their propagation length. In our case λ is similar for hadrons carrying k n and k. Thus, for virtual projectiles the contribution of the elastic channel, represented by the first sum from (11), is non-vanishing during the time of the order of λ. If L is the duration of the scattering process, then the elastic cross section vanishes in the case (9), in agreement with the above euristic argumentation. In this case the only contribution to Im[f (s, t = 0)] is due to inelastic channel because the states |i > contain new particles, which can be on the mass shell. When (9) is fulfilled, we obtain In electroproduction on nuclei, λ ∼ 1/xm N and L A = 2q 0 r 2 A where q is the virtual photon momentum. In the shadowing region L A ≥ 100λ and we have to use σ A inel instead of σ A tot to calculate F A 2 (x). In most cases, the condition (9) is also valid for separate nucleons and the free nucleon structure function determines σ N inel rather than σ N tot for virtual hadrons. Note that the condition for the Fraunhofer diffraction never holds in electroproduction.
In the independent particle approximation and for A ≫ 1, the Glauber result for the nuclear inelastic cross section is given by In contrast to (5) (13) there is no 1/2-factor, which is present in (5) due to diffraction effects. If λ/r A is finite, for the square well form of nuclear density the equation (13) can be rewritten where ρ is the nuclear density. The exponent inside the exponent simulates the x-dependence of nuclear shadowing due to the finiteness of λ/r A . To calculate nuclear structure functions F A,GV M D 2 (x) at low x, we have used (3) with σ tot equal to σ A inel from (14). Following the data for high energy hadron-nucleon interactions, we assume that for virtual mesons σ N inel ≈ 5/6σ N tot as well as for free mesons. The final results are not sensitive to small variations of σ N inel . We have used the standard set of parameters for low lying mesons. Π(µ 2 ) for the meson continuum was determined from the data for the e + e − -annihilation following the prescription of Ref. [13]. The cross section for the continuum was taken in a form σ N inel = 18mb GeV 2 /µ 2 in order to reproduce the experimental nucleon structure function. For every x-bin, the average Q 2 is the same as in Ref. [1]. At x > 0.175, when the contribution of virtual hadrons becomes negligible, we have calculated F A,EM C 2 (x) using the results of Ref. [8] with ǫ=35 MeV, with the Fermi momentum p F =230 MeV/c and with the flux factors [19] included. Our result for the structure function ratio can be represented as where . The excess pion contribution ∆F A,π 2 (x) was calculated following the preceeding section with ǫ= 35 MeV and with the pion three-momentum distribution from Ref. [12]. In some previous calculations, on-mass-shell pions with E π = p 2 π + m 2 π [8] or static pions with E π = 0 [6] have been used. Here we take the intermediate value E π = p 2 π + m 2 π /2. In this case R deviates by less than 2% from the ratio calculated for the two extreme values of E π . For the number of excess pions per nucleon we obtained n π ≈ 0.07. The elementary structure functions were taken from Ref. [20].
The resulting R is shown by the solid line on Fig.1. The nucleon contribution, represented by the first term from (15), is shown by the dashed line. The earlier GVMD model result [13] for the nuclear shadowing effect is shown by the dotted line. In our method the nuclear shadowing is significantly larger than in previous calculations, because the shadowing is more effective in the inelastic channel. The agreement with the data is restored by the virtual pion contribution. In this case the antishadowing at x ∼ 0.1 is also explained. Though we obtained a good description of the data, we understand that the GVMD model may be too simplified and can hardly provide a correct Q 2 -dependence of nuclear shadowing. But the result of this section is more general then the GVMD model: following our conjecture, the nuclear shadowing will be larger in other hadron dominance models based on Glauber theory. The agreement with data, obtained in some papers [13,14], can be recovered if the excess pion contribution will be included.
4. The DY production by protons probes the antiquark content of a target and for the cross sections of this reaction it was usually accepted The measured R DY [2] is close to 1 (see Fig.2 The factor F describes the projectile parton flux attenuation in nuclei, where σ q abs is the cross section of absorptive interactions of fast initial partons with nucleons. We assume that σ q abs is the same for quarks and antiquarks. In this case the parameter σ q abs can be found from the DY productions by pions, which is not affected by the excess pion contribution. The analysis of Ref. [4] gives σ q abs ≈ 2mb and we used this value to calculate R DY with (17) and (18). In our calculations, the nuclear antiquark density q A (x) includes the excess pion contribution by analogy to (1). This contribution was calculated as described in preceeding sections. We considered only u-quarks and antiquarks with the parametrizations from Ref. [20]. The binding correction for ǫ=35 MeV was introduced as described above. The ratio R DY calculated without the excess pion contribution (the lower dashed line on Fig.2) is significantly below the data points. The full result for R DY calculated with E π = p 2 π + m 2 π /2 is shown by the solid line. The same for E π = 0 is shown by the upper dashed line. As in the case of electroproduction, the pionic contribution improves the agreement with data. The bulk of the pion contribution is fixed by the sum rule (2) but the detailed form of resulting curves depends on the choice of the pion distribution.
We conclude that both electro-and hadroproduction data, as well as the data for K + −A scattering [21], can be explained with the excess pion contribution included.    [13]. The data are from Ref. [1]. Fig.2. The ratio σ A DY /Aσ N DY for the Drell-Yan production by protons calculated without excess pions (lower dashed line) and with two different forms of excess pion distribution (solid and upper dashed lines). The earlier estimation of this ratio with excess pions included (dotted line) and the data points are from Ref. [2].