Vacuum Pole Research Articles

Beauty, charm, and F L at HERA: New data vs. Early predictions

One of the well known effects of the asymptotic freedom is splitting of the leading-$\log$ BFKL pomeron into a series of isolated poles in complex angular momentum plane. Following our earlier works we explore the phenomenological consequences of the emerging BFKL-Regge factorized expansion for the small-$x$ charm ($F_2^c$) and beauty ($F_{2}^{b}$) structure functions of the proton. As we found earlier,the color dipole approach to the BFKL dynamics predicts uniquely decoupling of subleading hard BFKL exchanges from $F_2^c$ at moderately large $Q^{2}$. We predicted precocious BFKL asymptotics of $F_2^c(x,Q^2)$ with intercept of the rightmost BFKL pole $\alpha_{\Pom}(0)-1=\Delta_{\Pom}\approx 0.4$. High-energy open beauty photo- and electro-production probes the vacuum exchange at much smaller distances and detects significant corrections to the BFKL asymptotics coming from the subleading vacuum poles. In view of the accumulation of the experimental data on small-$x$ $F_{2}^{c}$ and $F_{2}^{b}$ we extended our early predictions to the kinematical domain covered by new HERA measurements.Our structure functions obtained in 1999 agree well with the determination of both $F_2^c$ and $F_2^b$ by the H1 published in 2006 but contradict to very recent (2008, preliminary) H1 results on $F_2^b$. We present also comparison of our early predictions for the longitudinal structure function $F_L$ with recent H1 data (2008) taken at very low Bjorken $x$. We comment on the electromagnetic corrections to the Okun-Pomeranchuk theorem.

The BFKL-Regge factorization and F 2 b , F 2 c , F L at HERA: physics implications of nodal properties of the BFKL eigenfunctions

The asymptotic freedom is known to split the leading-log BFKL pomeron into a series of isolated poles in the complex angular momentum plane. One of our earlier findings was that the subleading hard BFKL exchanges decouple from such experimentally important observables as small-x charm F2c, beauty F2b and the longitudinal structure functions of the proton at moderately large Q2. For instance, we predicted precocious BFKL asymptotics of F2c(x, Q2) with intercept of the rightmost BFKL pole aIP(0) − 1 = ΔIP ≈ 0.4. On the other hand, the small-x open beauty photo- and electro-production probes the vacuum exchange for much smaller color dipoles which entails significant subleading vacuum pole corrections to the small-x behavior. In view of the accumulation of the experimental data on small-xF2c and F2b we extend our 1999 predictions to the kinematical domain covered by new HERA measurements. Our parameter-free results agree well with the determination of F2c, FL and published H1 results F2b on but slightly overshoot the very recent (2008, preliminary) H1 results on F2b.

Diffractive beauty photoproduction as a short distance probe of QCD pomeron

High-energy open beauty photoproduction probes the vacuum exchange at distances ∼1/ m b and detects significant corrections to the BFKL asymptotics coming from the subleading vacuum poles. We show that the interplay of leading and subleading vacuum exchanges gives rise to the cross section σ b b ̄ (W) growing much faster than it is prescribed by the exchange of the leading pomeron trajectory with intercept α P (0)−1= Δ P =0.4. Our calculations within the color dipole BFKL model are in agreement with the recent determination of σ b b ̄ (W) by the H1 Collaboration. The comparative analysis of diffractive photoproduction of beauty, charm and light quarks exhibits the hierarchy of pre-asymptotic pomeron intercepts which follows the hierarchy of corresponding hardness scales. We comment on the phenomenon of decoupling of soft and subleading BFKL singularities at the scale of elastic ϒ (1 S )-photoproduction which results in precocious color dipole BFKL asymptotics of the process γp → ϒp .

Global description of0−12+→0−12+reactions utilizing the bare Pomeron

It is shown that the combination of a "bare Pomeron" with intercept ${\stackrel{^}{\ensuremath{\alpha}}}_{P}(0)=0.85$ in conjunction with a reasonable set of secondary Regge trajectories and a canonical absorption prescription is capable of providing a good global fit to practically all ${0}^{\ensuremath{-}}{\frac{1}{2}}^{+}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}{\frac{1}{2}}^{+}$ meson-nucleon scattering data up to lab momenta of 30 GeV/c. The bare Pomeron with intercept lower than 1 has a large real part which greatly facilitates the description of the data. At higher energies, "renormalization" effects can be expected to be important as inelastic diffraction events, and these lead to a renormalized Pomeron intercept very close to or equal to one. The value ${\stackrel{^}{\ensuremath{\alpha}}}_{P}(0)=0.85$ used throughout this intermediate-energy fit is in agreement with current inclusive triple-Regge data and maximum-rapidity-gap distributions. It is also in agreement with certain strong-coupling ABFST (Amati-Bertocchi-Fubini-Stanghellini-Tonin) multiperipheral model calculations. For secondary effects, we have used a family of vector Regge trajectories ($\ensuremath{\rho},\ensuremath{\omega},{K}^{*}$) with a degenerate intercept of about 0.45, and tensor trajectories (${A}_{2},{K}^{**}$) with an intercept of about 0.25. A second vacuum pole emerges with intercept close to 0. The ${p}^{\ensuremath{'}} (f)$ trajectory, not included here, can perhaps be expected to appear in conjunction with the renormalization of the Pomeron. Although no wrong-signature nonsense zeros are included in the parametrization, the $\ensuremath{\rho}\ensuremath{-}{A}_{2}$ and ${K}^{*}\ensuremath{-}{K}^{**}$ pole couplings are nevertheless very nearly exchange degenerate. SU(3) is used to relate most of the other couplings. The (pole + cut) helicity-flip $\ensuremath{\rho}\ensuremath{-}{A}_{2}$ and ${K}^{*}\ensuremath{-}{K}^{**}$ amplitudes also show considerable exchange-degenerate characteristics. We have used a standard absorption prescription to calculate the second-order bare Pomeron $(\stackrel{^}{P})\ensuremath{\bigotimes}$ Reggeon cuts and $\stackrel{^}{P}\ensuremath{\bigotimes}\stackrel{^}{P}$ cuts. An unusual result emerges---the "enhancement" ${\ensuremath{\lambda}}_{i}$ factors for all cuts are less than one. This indicates the presence of higher-order cuts which thus dominate over inelastic intermediate-state production in this approach. The data used in this fit are a representative selection of ${0}^{\ensuremath{-}}{\frac{1}{2}}^{+}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}{\frac{1}{2}}^{+}$ data (including $\ensuremath{\pi}N$ amplitude analysis, hypercharge-exchange differential cross sections and polarizations; ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ and ${K}^{\ifmmode\pm\else\textpm\fi{}}p$ total and differential cross sections, polarizations, and $t=0$ real-to-imaginary ratios; and $\ensuremath{\pi}N$ and $\mathrm{KN}$ charge-exchange differential cross sections and polarizations) up to ${p}_{\mathrm{lab}}=30$ GeV/c and $|t|=1.5$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$.

Measurement of the cross sections for inclusive π− + p → π− + X + reactions at momenta 42 and 51 GeV/ c in the beam fragmentation region

Inclusive cross sections for π − + p → π − + X + reaction have been measured for the interval x = p/ p max = 0.90–0.99 at the momenta 42 and 51 GeV/ c and the values of the momentum transfer squared p ⊥ 2 = 0.04 + 0.26 ( GeV/ c) 2. With the increase of x one can observe some growth of invariant differential cross section at p 2 fixed in the indicated interval. In the Regge pole model such a behaviour may be explained by a predominating contribution of the vacuum pole. Under this assumption we have found that energy dependence of the total cross section for Pomeron interaction with protons at several values of p ⊥ 2.

Some General Consequences of Regge Theory for Pomeranchukon-Pole Couplings

Without reference to specific Regge models, and assuming only the validity of asymptotic Regge-pole expansions, we use unitarity and exact sum rules to draw conclusions concerning the couplings of the vacuum pole ${\ensuremath{\alpha}}_{p}(0)$. If ${\ensuremath{\alpha}}_{p}(0)<1$ we derive an upper bound on the triple-vacuum coupling. Assuming ${\ensuremath{\alpha}}_{p}(0)=1$ we prove either that the ($t=0$) Pomeranchukon-particle-Reggeon coupling vanishes for all Reggeon masses, or the ($t=0$) Pomeranchukon-Reggeon-Reggeon coupling vanishes. We also prove the vanishing of either the Pomeranchukon-Pomeranchukon-Reggeon vertex at zero mass, or the Mueller-like Pomeranchukon-Reggeon-particle-particle vertex, again all Reggeons at zero mass. Our model-independent technique is used to recover the Finkelstein-Kajantie result that the Pomeranchukon-Pomeranchukon-particle coupling vanishes for ${\ensuremath{\alpha}}_{p}(0)=1$; if ${\ensuremath{\alpha}}_{p}(0)<1$ an inequality is obtained for the coupling. The influence of Regge cuts is neglected.

Compton Scattering at High Energies

Using the continuous moment sum rule it is found that the third vacuum pole P is im­ portant for the forward proton Compton scattering, and the residue of the fixed pole at J =0 is also estimated. The third vacuum pole P has zero intercept ap(0)~-0.5. The prediction of the high energy total cross section is consistent with recent experimental results.

Inclusion of Interference Terms in the Amati-Bertocchi-Fubini-Stanghellini-Tonin Multiperipheral Model

An important class of interference terms is included in the ABFST multiperipheral model by adding an extra term to the inhomogeneous term and to the kernel of the integral equation. This additional term corresponds to the contribution to the 2-to-4 cross section from interchanging one or two pairs of particles from two different vertices. It is found that these interference terms change only slightly the position of the output vacuum pole, although they give a non-negligible contribution to the four-particle production cross section.

NegativeP′-pole parameters and vanishing total cross sections in high energy πN scattering

In πN forward scattering the conventionalP′-pole can be simulated by two vacuum poles with α(0)<0 and different signs of the residues. In the regions −0.3≦ αp′(0)≦−0.1 and −0.8≦αp′'(0)≦−0.4 the practically identical fits to the πN scattering data from 8 to 65 GeV/c yield 0.98≦αp(0)<1. The dispersion theoretical sum rule for the scattering lengtha(+) is fulfilled because of the negativeP′ residue; it is used as a finite-energy sum rule withN=8 GeV/c.

Secondary vacuum trajectories and scalar meson exchange

A new second vacuum trajectory passing throughf*(1515) and σ(548), α(t)=− 0.3+t, is proposed, which, together with αp″(t=0)=− 0.5, overcomes the failure of Igi's original superconvergent πN sum rule and avoids the ghost-state of the conventionalP′-trajectory. The negative residue att=0 confirms the repulsive contribution of a vacuum pole component withJ=0 to the generalizeds-reaction potential, as postulated by Chew (1965). TheP- andP′-poles can build the non resonating background.

On the vacuum pole in quantum electrodynamics

The asymptotic behaviour of the ee, e γ and γγ scattering amplitudes is determined in the main logarithmic approximation. The j-plane singularities in the channel with vacuum quantum numbers are investigated near j = 1.

The absorptive regge model and features of πN scattering at high energy

The absorptive Regge cut associated with ϱ exchange is calculated in detail as a function of the ϱ, P and P′ Regge poles in the Fourier-Bessel representation. The ϱ ′chooses nonsense′ at α ϱ = 0 and has no built-in crossover zero. The vacuum pole amplitudes are constrained to satisfy the relation νB (+) A′ (+) = N . Good fits to πN charge exchange and elastic data are obtained for N = −1 and −2. The elastic crossover is well reproduced and is explained as a co-operation between the effect of the cut and having N negative. The effect of diffractive dissociation contributions to the Regge cut is considered. Their inclusion tends to worsen the fits.

On the dynamical equation for the Pomeranchuk pole

It is shown that the recently suggested bootstrap multi-Regge equation for the Pomeranchuk pole does not possess a solution with the vacuum pole α p (0) = 1, as well as a similar Bethe-Salpeter equation.

Asymptotic behaviour of essentially inelastic collisions

The asymptotic value of particle production amplitudes at s → ∞ is investigated for “essentially inelastic” collisions when it is especially large. These collisions are characterized by the production in the c.m. system of two groups of particles at high primary energies, each moving nearly parallel to the momenta p a and p b =− p a of the colliding particles. The momenta of all particles in each group differ considerably by their order of magnitude: p 1 ⪢ p 2 ⪢ … ⪢ p n1 , p n ⪢ p n−1 ⪢ … ⪢ p n1+1 , where the subscripts 1, 2 … n 1 refer to the particles of the first group and n 1+1, n 1+2 … n−1, n to those of the second. The mathematically corresponding limit of a particle production amplitude is determined by the requirement that the squares s ik of the energies of any pair of particles in their c.m. system should increase (or remain large as compared with particle masses squared) as the primary energy increases, s → ∞. It is also necessary that the transverse components k i of all momenta p i of the particles produced and momentum transfers squared t i =( p a− p 1− … − p i ) 2, i=1, 2, …, n should be small. It is shown that in this limit the asymptotic value of the amplitude takes the form of a product of Regge factors Π i=1 n−1 ( s i, i+1 ) jo( ti) γ i (where j 0( t i is a vacuum pole) and can in a sense be compared with the contribution of Feynman-type graphs with n−1 “reggeons”. The quantities γ i=γ i( K′ i, K′ i+1) depending on the projections K′ i and K′ i+1 of the transverse momenta of reggeons before and after the emission of a particle correspond to the vertex parts designating in these graphs the emission of the particle by a reggeon. In a preliminary report on this question 2, 3) the dependence of the quantities γ on the angle between K′ i and K′ i+1 was overlooked. Thus the coefficient of the product of the Regge factors in the asymptotic value of the amplitude depends on the transverse components of the momenta of all the particles produced. As a result the asymptotic value of a particle production amplitude proves to be dependent on the same number of variables as in the general case (i.e. is sensitive to any change of the momenta). The asymptotic behaviour of mixed “essentially inelastic” and “quasi-elastic” collisions is also considered when not particles but n groups of particles are produced and the energy of each particle in the c.m.s. system of a group is low. Collisions of this type are mainly responsible for multiple production at high energies. In conclusion brief consideration is given to the “complicated” types of asymptotic behaviour of “essentially inelastic” processes when the extreme right singularity in the j plane is a branching point or a point of accumulation of singularities etc. It is shown that in this case instead of the factor s i, i+1) j o ( ti) one ought to expect in the asymptotic values of essentially inelastic amplitudes expressions like Σ ν λ ν ( t; ξ i, i+1 )( S i, i+1 ) jv( ti) , where ξ i, i+1 =In( S i, i+1 / m 2) and λ ν ( t, ξ i, i+1 is a function depending on the character of the vth singularity and perfectly analogous to those which determine the corresponding asymptotic values of elastic scattering amplitudes.

ASYMPTOTIC BEHAVIOUR OF π-π SCATTERING AMPLITUDE AND SINGULARITIES IN COMPLEX L-PLANE IN STRIP APPROXIMATION

The contribution of inelastic intermediate states to the singularities in complex l-planc and to the asymptotic behaviour of crossed channel are investigated in the strip approximation. The procedure is to solve the simultaneous equations suggested by Chew, Frautschi and Mandelstam by means of iteration method. The general iteration expressions for the s channel high energy amplitude f(s,t) and the t channel partial wave amplitude ft (t) are derived. It is shown that the divergence difficulty mentioned by Chew, Frautschi and Mandelstam is not present in the iteration results of the above f(s,t). The main results of first iteration are as follows: i) The total cross section falls slowly for not very high energies, but at the end tends to a constant. ii) The diffraction angular distribution will have a long tail. As the energy increases, this long tail extends towards the small angle region. iii) In the t channel a cut will appear along the real axis in complex l-plane; the right ending point of the cut is at l=1 when t=0, and moves to the left as t decreases, with a speed nearly half that of the vacuum pole. Higher order iterations are also investigated; the results are qualitatively analogous to those of first order.

Diffraction Scattering and Singularities in the Angular Momentum Plane

The singularities in the angular momentum plane, which describe the vacuum intermediate state dominating high-energy scattering, are assumed to consist of a single Regge pole and a fixed branch cut. If the energy variable ($\ensuremath{\surd}s$) only enters the equations in the normalized form $w=\frac{s}{2{M}_{1}{M}_{2}}$, where ${M}_{1}$ and ${M}_{2}$ are masses of the particles, the value of $w$ for lower mass interactions (e.g., $\ensuremath{\pi}\ensuremath{-}N$) will be much larger than for high mass interactions (e.g., $N\ensuremath{-}N$), for the same value of the lab energy. In the combined pole and cut model the vacuum pole dominates at lower values of $w$, and the cut becomes dominant as $w\ensuremath{\rightarrow}\ensuremath{\infty}$, so that the $p+p$ scattering between 7 and 20 BeV/c will be mainly described by the single vacuum pole, causing the width of the diffraction peak to shrink, whilst the ${\ensuremath{\pi}}^{\ensuremath{-}}+p$ scattering between 7 and 17 BeV/c will be described by the fixed cut, which leads to no shrinkage. This permits the $p+p$ and ${\ensuremath{\pi}}^{\ensuremath{-}}+p$ data to be combined into one. A fit is obtained to the combined $p+p$ and ${\ensuremath{\pi}}^{\ensuremath{-}}+p$ data for the elastic differential cross section obtained recently by Foley et al. Another consequence of the vacuum-pole-cut model is that, in the energy range under consideration, the total cross section is not yet constant, but has the form ${\ensuremath{\sigma}}_{T}=a+\frac{b}{\mathrm{ln}w}$, which agrees well with ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}+p$ scattering data. This behavior of ${\ensuremath{\sigma}}_{T}$ should also describe the average of the $p+p$ and $\overline{p}+p$ total cross sections. Estimates of $a$ and $b$ using $p+p$, $\overline{p}+p$, and ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}+p$ data agree with those obtained for ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}+p$ alone to within 6%. This strongly supports our method of combining $N\ensuremath{-}N$ and $\ensuremath{\pi}\ensuremath{-}N$ scattering data, and is in good agreement with the pole and cut model. A test of the theory can be obtained from experiments on $K\ensuremath{-}N$ scattering.

Restrictions on Regge pole trajectories

Under the assumption of the univalence of the trajectory of the main vacuum pole in the plane of the complex variable t (square of the 4-momentum transfer) inequalities restricting this trajectory in the physical region of t are obtained.

Non-shrinking Diffraction Scattering and the Nambu-Sugawara-Miyazawa Hypothesis

A partial proof of the Nambu-Sugawara-Miyazawa hypothesis is given. On the basis of unitarity and analyticity it is shown that if alpha 'p (0) = 0 then the vacuum pole trajectory in the Regge-Pomeranchuk hypothesis alpha p(t) = 1. It is also proved that the unitarity and analyticity are not sufficient conditions to bound the total cross sections in the high energy limit. If alpha p(t) = 1 the non-shrinkage diffraction scattering in the high energy limit is obtained. (C.E.S.)

Elasticπ++pScattering from 7 to 17 BeV/c

Differential cross sections obtained for the momentum range 7 to 17 Bev/ c and the 4-momentum-transfer squared range of 0.2 to 0.1 (Bev/c)/sup 2/ failed to show the shrinkage with increasing center-of-mass energy squared that was predicted by Regge-pole theory. Similar results were previously obtained for pi /sup -/-p scattering, while shrinkage was observed in p-p cross sections. These results contradict the dominance of the vacuum pole or three-pole models. (D.C.W.)

Two Vacuum Poles and Pion-Nucleon Scattering

A general expression is given for the pion-nucleon non-charge-exchange scattering amplitude for arbitrary energy and small momentum transfer on the assumption that only the vacuum pole $P$ and the second vacuum pole ${P}^{\ensuremath{'}}$ exist in the upper half $J$ plane. We derive sum rules for non-spin-flip and spin-flip amplitudes and use them, combined with the analysis of the high-energy $\ensuremath{\pi}\ensuremath{-}N$ cross sections in terms of Regge poles, to investigate the behavior of $P$ and ${P}^{\ensuremath{'}}$ trajectories near $t\ensuremath{\approx}0$. For this purpose the importance of a precise measurement of the low-energy partial-wave phase shifts is emphasized. A sum rule for the $S$-wave pion-nucleon non-charge-exchange scattering length can be satisfied with ${\ensuremath{\alpha}}_{{P}^{\ensuremath{'}}}\ensuremath{\approx}0.5$.