New Unitary Relations between the QCD sum rules

New unitary symmetry relations are obtained between the QCD sum rules for ΣΛ transitions. Also for QCD SR’s with baryon distribution amplitudes the unitary relations are constructed, proved in a simple quark model and found to be consistent with the existing calculations.


Introduction
QCD sum rules (QCD SR's) introduced by [1,2] have become one of the most important tools while studying baryon or meson couplings and the corresponding form factors with photons, mesons or baryons.At the same time, the complexity of the formulas used in calculations has increased in obviously non-linear manner.So the control of one's own and other authors calculations becomes a non-trivial problem.Recently, we have proposed simple relations between the quantities involving Σ hyperons and those involving Λ hyperons [3].They are generalized easily to any Σ-like and Λ -like baryons containing heavy quarks.Our formulas seem to be useful and for example they were inserted into the codes by Lai Wang and Frank Lee while studying magnetic moments of the octet baryons [4].
It is interesting that also the transition Σ − Λ quantities could be obtained from the corresponding quantities of the Σ or Λ baryons.We want here to propose one more relation to control the complex calculations of the transition Σ − Λ quantities which has the merit that it does not involve other baryon quantities, neither of Σ nor of Λ baryons.
The formula is rather trivial for nonrelativistic quark model or simple unitary symmetry model.However, it shows us the way to overcome many difficulties in the framework of QCD sum rules, as we shall see later on.In the very tedious QCD sum rule calculations it proves to be extremely useful.Its applicability was demonstrated in [3] on the example of sum rules for magnetic moments of octet baryons.In this report we are interested with the Σ − Λ transitions.In [3] they were related to the Σ and Λ quantities as (we write in terms of correlators [1,2]) where d ↔ s means interchange of the flavors d and s.The main point of our report is that it can be formulated in some "auto-sufficient" way.Our new relation between the corresponding correlation functions reads as a e-mail: ozpineci@metu.edu.trb e-mail: zamir@depni.sinp.msu.ru

Master formulas for the octet
As the main ingredient of the polarization operator is the matrix element of the T-production of the interpolating currents, we now put our attention to it.The corresponding formulas are often named as "master formulas" [4].We begin with the master formula for the Σ 0 [4], [7]: .. where V could be meson or photon.It is vacuum state for the mass sum rule (SR).S is the quark propagator [1], [2].We write for our purpose just the first line, as for us it is important that in every line the quark flavors index is of the form Now, we construct Λ−quantity using the relation from [3] .. and we retain for a moment only these terms to show explicitly the group-theoretical structure.In fact, every three lines as to the quark flavors are of the form in agreement with [3].
Transition magnetic moment "master formula" could be read from [3] as One can see that every two lines, as to the quark flavors, are of the form Let us give an example.In [5] a QCD SR for the magnetic transition moment was written, which reads in terms of various condensates a q = −(2π) 2 qq , b = g 2 s G 2 and so on [5], [6] (for a moment we maintain a u , a d different, the same for other condensates and quark masses m u , m d ): e u a u m d − e d a d m u ) − 2e s a s (m u − m d )− a s (e u m d − e d m u ) − e s (a u m d − a d m u ) + 2(e d a d m u − e u a u m d )+ 2m s (e d a d − e u a u ) + m s (e u a d − e d a u ) + e s (a d m u − a u m d )]+ L 4/9 36 (2κ − ξ)] × [a s (e u (2κ u − ξ u )a u − e d (2κ d − ξ d )a d ) + e s (2κ s − ξ s )a s (a u − a d )] − M 4 4L 28/27 [(e u a u χ u − e d a d χ d )m s + e s a s χ s (m u − m d )]+ 1 18 [(e u a u (2κ u − ξ u ) − e d a d (2κ d − ξ d )m s + e s a s (2κ s − ξ s )(m u − m d )]M 2 − M 2 6 [(e u a u κ u − e d a d κ d )m s + e s a s κ s (m u − m d )]× ×[ln( M 2 Λ 2 ) − 1 − γ EM ]] = β Σ β Λ √ 3μ Σ 0 Λ e − m2 /M 2 (1 + A Σ 0 Λ M 2 ) + ...One can see that each term in this formula satisfies our new relation ΣΛ | (u↔s) + ΣΛ | (d↔s) = ΣΛ .00045-p.3 QCD@Work 2014 Tr[γ 5 CS be u Cγ 5 S c f s ]] + ... New relations (we restrict ourselves with the first terms) yield