Lipatov Pomeron and the Heisenberg chain

TL;DR

This paper discusses an analogy between QCD Pomeron exchange and the Heisenberg chain, suggesting lattice techniques could be applied to high-energy physics phenomena.

Contribution

It introduces a novel analogy that connects QCD Pomeron exchange with the Heisenberg chain, opening new avenues for applying lattice methods.

Findings

Potential application of lattice techniques to high-energy phenomena.

New perspective on Pomeron exchange in QCD.

Bridge between quantum chromodynamics and condensed matter models.

Abstract

It is pointed out that the recently discovered analogy between the QCD description of the Pomeron exchange and the Heisenberg chain may allow to apply lattice techniques to the high energy phenomena.

Equations

$$A(s,t)\sim s^{\alpha (t)},s>>-t,$$

$$\alpha (t)={\alpha }_0+{\alpha }^{\prime }t,$$

$${\sigma }_{tot}=\frac{1}{s}Im{A}_{el}(s,0),$$

$${\sigma }_{tot}\le const.{\log }^2(s),$$

$$F(x,Q^2)=\frac{1}{s}Im{A}_{{\gamma }^{\star }P}(s,t=0)$$

$$s=\frac{Q^2}{x},x=\frac{Q^2}{2M\nu },x\ll 1.$$

$$F(x,Q^2)\le const.{\log }^2(x).$$

$$A_{LLA}\sim s^{1+{\Delta }_{BFKL}(t)},$$

$${\Delta }_{BFKL}(0)=\frac{4{\alpha }_s{N}_c}{\pi }\log (2)$$

$$A(s,t)=is{\int }_{\delta -i\infty }^{\delta +i\infty }\frac{d\omega }{2\pi i}{\left(\frac{s}{M^2}\right)}^{\omega }\tilde{A}(\omega ,t),$$

$${\tilde{A}}_n(\omega ,t)=\int d^2{k}_1\dots d^2{k}_n{d}^2{k}_1^{\prime }\dots d^2{k}_n^{\prime }$$

$$\omega T_n(\omega )=T_n^0(\omega )+{\mathcal{H}}_n{T}_n(\omega ),$$

$$T_n(\omega )=\frac{1}{\omega -{\mathcal{H}}_n}{T}_n^0(\omega ).$$

$$Q_3(z)=\sum _n{f}_n{z}^n,$$

$$(n+1{)}^3{f}_{n+1}=$$

$$\left(n((2n+1)(n+1)+q_2)+i{q}_3\right)f_n$$

$$-(n-1)(n(n+1)+q_2)f_{n-1}.$$

Full text

1 HIGH ENERGY SCATTERING IN THE REGGE LIMIT

Customary lattice methods are considered suitable for the description of the static, or low energy, nonperturbative properties of hadrons. Even though some progress has been made in studying the scattering length, form factors and the deep inelastic structure functions, the finite ultraviolet cut-off $a^{-1}\sim 2GeV$ limits the application of classical lattice techniques mainly to the static phenomena. However, there exists an intriguing mapping, between the description of a high energy scattering and the one dimensional Heisenberg chain, which offers a possibility to use the methods of statistical physics to study the genuine high energy scattering.

The typical high energy scattering is described by the Regge amplitude

[TABLE]

where $s$ and $t$ are the Mandelstam variables and $\alpha(t)$ describes the Regge trajectory exchanged in the $t$ channel. Phenomenologically Regge trajectories are linear

[TABLE]

with the intercept, $\alpha_{0}$, and the slope, $\alpha^{\prime}$, depending on the quantum numbers exchanged in the $t$ channel. The total cross section

[TABLE]

is given in terms of the elastic scattering in the forward direction. The latter is described by the exchange of the Pomeranchuk trajectory with the vacuum quantum numbers. Unitarity restricts the growth of the cross sections. The Froissart bound

[TABLE]

limits the intercept $\alpha_{0}\leq 1$. Regge behaviour occurs also in the deep inelastic scattering at small $x$

[TABLE]

Again unitarity limits the growth of the structure functions at small $x$

[TABLE]

Emergence of the Regge behaviour is qualitatively well understood as the result of the exchange of the ladder diagrams [1]. The quantitative derivation of the Pomeron exchange in QCD was pioneered by Lipatov and is presently actively investigated [2, 3, 4].

2 POMERON IN QCD

Working in the leading logarithmic approximation, $\alpha_{s}\ll 1,\;\;\;\alpha_{s}\log{(s/M^{2})}\sim 1$, Lipatov and others [5] have identified and summed the relevant class of the QCD ladder diagrams in the Regge limit. In addition to the simple ladder diagrams, other diagrams also contribute and the whole class is commonly referred to as the exchange of two reggezied gluons. The result [5] reads

[TABLE]

Therefore QCD reveals the Regge behaviour of the elastic amplitude and the intercept of the Pomeranchuk trajectory is known. However the amplitude (9) violates unitarity, hence a lot of subsequent work has been devoted to find unitarity corrections to Eq.(9). In particular the scheme termed generalized leading logarithmic approximation (GLLA) was proposed [6, 7] which identifies the non-leading contributions which should restore unitarity. In the GLLA the Mellin transform $\tilde{A}(\omega,t)$ of the scattering amplitude

[TABLE]

is given as a sum of contributions $\tilde{A}_{n}$ from the exchange of $n$ reggeons (reggeized gluons) in the $t$ channel. For fixed $n$ they are given by the evolution operator $T_{n}(\omega)$

[TABLE]

Where $\Phi_{A(B)}$ are the wave functions of the hadron $A(B)$ in the reggeon basis. The evolution operator conserves the number of reggeons and satisfies the following Lippmann-Schwinger equation

[TABLE]

which can be solved symbolically as

[TABLE]

Comparing Eqs.(12) and (10) we see that the largest eigenvalue of the Hamiltonian of $n$-reggeons determines the high energy behaviour of the scattering amplitude. Moreover in the large $N_{c}$ limit only planar diagrams give leading contribution, and consequently ${\cal{H}}_{n}$ describes the system of $n$ degrees of freedom which possesses only the nearest neighbour interactions. At that point the analogy with statistical systems emerges. It becomes even more appealing after transforming to the impact parameter representation, $\vec{k_{i}}\rightarrow\vec{b}_{i}$, and after replacing transverse variables by the complex coordinates, $\vec{b}_{i}=(x_{i},y_{i})\rightarrow z_{i}=x_{i}+iy_{i},\overline{z}_{i}=x_{i}-iy_{i}$. It was shown [8, 3] that the resulting Hamiltonian describes the linear chain of the quantum spins with the nearest-neighbours interactions in a complete analogy to the Heisenberg chain of ordinary spins. There are two complications, however, which make this problem nontrivial and challenging. First, the spin operators in question are infinitely dimensional. In fact they generate the $s=0$ unitary representation of the $SL(2,C)$ group. Second, the nearest neighbours interaction is nonlocal in the $z,\overline{z}$ space. In spite of this complications, Faddeev and Korchemsky [3], using the generalized Bethe ansatz have confirmed Lipatov’s hypothesis [9], that the n-reggeon problem is holomorphically separable and in principle, exactly solvable. The $n=2$ hamiltonian was diagonalized explicitly and BFKL slope, Eq.(9) was reproduced.

3 THE ODERON CASE

The $n=3$ case, which corresponds to phenomenologically important oderon problem, is solved only partially. Analytical results are available for integer values $(h=n)$ of the Casimir operator of two spins, $\hat{q}_{2}=(s_{i}+s_{j})^{2}=-h(h-1)$. Also the asymptotic expansion of the maximal eigenvalue of ${\cal{H}}_{3}$ in $1/h$ was derived. All these results are based on the Bethe ansatz approach. We have studied the solution $Q_{3}(z)$ of the corresponding Yang-Baxter equation by analogy to the standard theory of Fuchsian equations of the second order. Even though the Yang-Baxter equation for $Q_{3}(z)$ is of the third order, the elements of the general theory apply. In particular, we have found that the case considered corresponds to the degenerate situation with the characteristic exponent $r=0$. Hence, the equation has one regular and two irregular solutions around $z=0$. This is independent of the eigenvalues of the two Casimir operators $\hat{q}_{2}$ and $\hat{q}_{3}$ which parametrize the problem. The coefficients $f_{n}$ of the power expansion

[TABLE]

are determined by the following recursion relation

[TABLE]

This recursion is equivalent to the recursion derived in Ref.[3] where the expansion in terms of the Legendre polynomials was studied. In particular, for integer $h$ $Q_{3}(z)$ is a finite polynomial in $z$ only for the discrete values of $q_{3}$. The quantization condition of $q_{3}$ which follows from Eq.(14) is the same as one derived in Ref.[4] The straightforward power expansion (13) may, however, prove more suitable for the study of the analytic structure of the solution and in consequence may allow the analytic continuation for arbitrary $h$.

4 LATTICE FORMULATION

In closing we would like to comment on the feasibility of applying lattice Monte Carlo methods to solve this problem numerically.

Contrary to the standard applications, physically interesting questions arise already for the finite number of exchanged reggeons, i.e. for finite length of the Heisenberg chain, say $n=3$. However the volume of the transverse parameter space is infinite, hence the normal difficulties associated with this infinite volume limit would arise. In fact the hamiltonian ${\cal{H}}_{n}$ has the global conformal invariance, therefore the critical slowing down is expected to occur. So, the challenge is how to simulate effectively a conformal theory on the lattice?

The nearest neighbours interaction is nonlocal which also complicates practical applications. Since, however, the system is critical, one is bound to employ the nonlocal (cluster) algorithms. Therefore the nonlocality of ${\cal{H}}_{n}$ may not be the main problem.

The choice of the representation may help in designing a practical approach. The ”$z$” representation is the one possibility. However one may try to employ the $SL(2,C)$ symmetry to simplify the problem. Note that one of the analytical solutions of the Pomeron $n=2$ case uses solely the group theoretical information on the spectrum of the unitary representations of the $SL(2,C)$ group.

Finally, even though the problem is certainly far from trivial, we should be aware that the $n=2$ case was solved analytically, hence the spectrum of the excitations of elementary interaction is known. Combining this knowledge with the recent progress in simulating critical systems may lead to a realistic proposal for the numerical approach.

J. W. thanks G. Korchemsky, L. Lipatov and M. A. Nowak for the discussion.