Vsemayr A. Shakhbazyan

Vsemayr A. Shakhbazyan

Renormalization Group,

Asymptotic Freedom,

AndMore...

QueensCollege, CUNY, Flushing, NY11367

Abstract

A new approach to Quantum Field Theory (QFT) is reviewed. The generalized Renormalization Group (RG) method is developed. It is proven that any investigation in the framework of the QFT must be done in strict accordance with the rules of Mathematical Physics (MP). The negligence of these rules creates meaningless artifacts – the ‘ghost’ states. It is shown that the appearance of them is the result of distortions of analytical structure of the QFT vertices. In the framework of Quantum Electrodynamics (QED), it is explained how the Landau ‘ghost’ appears when using the usual RG method. It is shown also how to remove it in this presented generalized RG.

Relying on the MP reasoning, the non-perturbative asymptotically free solutions were found for the effective interactions in QED and conventional Symmetric Pion-Nucleon Theory (SPNT) correspondingly.

The direct connection between SPNT and Quantum Chromodynamics (QCD) is stated.

Some applications of these solutions are presented. In particular, the double logarithmic asymptotics are calculated when using the above non-perturbative solution for the effective interaction in the combined model of SPNT/QCD. Perspective and valuable ‘Induced Identities’ are stated. They are results of the analytical continuation of the differential RG Lee equations for the QFT vertices to the ‘mass-shell’ region.

They permit solvingunlimited number of QFT vertices.

Preface

This is a completely new theory. Preliminary remarks are presented in the Introductionto provide the reader some taste of the product. The first and second sections are the introductory ones; just for fast reading and browsing readers, mostly interesting is ‘what is this about’. The first section mostly informs about the new connection between the SPNT and QCD. The second section is for the people who are eagerly interested ‘how to get rid of the‘ghosts’. They are provided by the key references in the subject.

The whole story is given in the third section. It must be really enjoyable for the comprehending reading. Here are givenfirst thefirst a, b, c’s of the RG method. Then it is explained in details what is wrong in the usual wayof operating with RG. It is explained how the ‘ghost’ state appears as a result of the rude distortion of the analytical structure of the QFT vertices, and the right way is given for their removal in this generalized approach. The challenging issue of the asymptotic freedom in QFT is mentioned. Then comes the backbone of the section: finding the general way to the asymptotic freedom in QFT. Selectedwas a special ‘boundary condition’ which provided the non-perturbative asymptotically free solution for the effective interactions both in QED and SPNT. After this exciting achievement, the connection between the SPNT and QCD was found. This section is provided with detailed references and footnotes. The author tried not to miss any of the nuances in this very complicated and confusing subject.

Of course, after such efforts in the fundamentals of QFT the applications of the obtained results are very welcomed. Indeed, since the density of the non-perturbative (i.e. singular) states is infinite, the possibility of the enormous number of reasonable solutions is very high. Therefore, the field of applications is very wide. It is worth mentioning that every non-perturbative solution has the perturbative tail, and the asymptotically free non-perturbative solution has also asymptotically free perturbative solution. This circumstance opens the possibility of improvement of the regular, perturbative solutions literally in every model of the QFT. The double-logarithmic asymptotics obtained when using the non-perturbative solution for the SPNT/QCD effective interaction is very important, for example, in calculations for deep inelastic scattering and others.

Very valuable are the ‘InducedIdentities’, which are obtained from the RG differential Lee equations by the analytical continuation from the group manifold of the normalization parameters to their ‘mass-shell’values, i.e. to the physical theory. It opens unlimited number of physical processes which can be investigated by those identities.

ACKNOWLEDGMENTS. I am deeply grateful to all my colleagues and friendsfor many lengthy discussions about this work.

I present my gratitude to the staff and friends in the Mathematical Department of the QueensCollege, CUNY, and in theCollege of Professional Studiesof the Saint JohnsUniversity .Contacts with them helped very much for the final perfection of this work.

I sincerely appreciate the help ofDr. Ann Wintergerst, who carefully proofread the text.

Introduction

(Preliminary Remarks)

I

Sometimes scientists make mistakes – accidentally or being involved with an attractive idea; sometimes it happens with the best scientists.

Let us use some simple analogy from Mathematical Physics. When you are trying to solve the problem of eigenvalues of the oscillating string, you must impose the correct boundary conditions on the ends of that string. When you keep both ends of the stringfixed, you receive one set of eigenvalues; if you prefer to keep one end of the string fixed, and the other end freely moving, you must impose on the loose end some rules for its movement. But you cannot omit the condition about the behavior of the loose end and not say anything about that movement! In that case the problem won’t be correctly stated.

There are also “natural boundary conditions” for the behavior in infinity (see any appropriate problem in Quantum Mechanics).

Unfortunately, mistakes of the above type happen even in the modern Quantum Field Theory (QFT). When using the method of the Renormalization Group (RG) and looking for the asymptotics of vertices in any model of QFT over the quantity k2, which is much more than m2 (k2 » m2, k2 is the square of the momentum, m is the mass), you must dismiss all the quantities, which are small in comparison with the big k2, right? The point is, however, that in the renormalizable theories all big terms enter as the logarithms of k2: ln (k2/m2). In the “machinery” of the RG these logarithmic termsare usually taken split in to the ln (k2/λ2) and ln (m2/λ2), where λ2 < 0 is the square of the normalization momentum, and usually the “mass terms” are dismissed from the beginning.

But if k2 » m2, it does not mean that ln (k2/λ2) is always much more, than ln (m2/λ2). Every high school student knows that the logarithm of 0 (formally) is equal to infinity:

ln0 = - ∞. So, when you are looking for asymptotics over “big logarithms”, you are analyzing the quantities with big (but finite!) k2, and you are trying to dismiss “small” terms of mass-logarithms. In reality, however, you are dismissing infinitely big terms in comparison with finite (although big) terms. Every educated mathematician knows that such wrong doing destroys the analytical structure of the expression you are analyzing.

Unfortunately, many theoreticians who used (and are using) the RG-method, did not pay attention to this obvious fact. On the language of QFT, they were (and are) neglecting the severe infrared singularities.

In order to correct the situation in QFT, you must express the quantities containing the mass terms through the quantities of the massless theories and then expand the first ones to the series over the degrees of 1/[ ln (m2/λ2)]. Since these logarithmic terms are big (remember that ln0 = - ∞), the expansion over the degrees of 1/(lny) makes sense (here m2/λ2 ≡ y).

So, I expressed the generalized Gell-Mann-Low function

Ψq (y, (gλ)2) (which contains the mass-term y, see above; gλ is any coupling constant) through the Callan-Symanzik’s β-function, which is designed for the massless theories. Both the Gell-Mann-Low and Callan-Symanzik functions are the infinitesimal operators of the Lee group theory used in the RG-method. After very long and tedious work, I was able to find the first several terms of the expansion of the Ψ-function over the degrees in 1/[(lny)]n. The coefficients of that expansion do not contain dependence on the mass terms. So, you can either completely dismiss the dependence on y, or you can take into account terms of (order of) O(1/(lny)), or terms of O(1/(lny)2), or terms of O(1/(lny)3), etc. It depends how many corrections for inclusions of mass-terms you want to take into consideration.

I did this work for Quantum Electrodynamics (QED) and Symmetric Pion-Nucleon Interaction Theory (SPNIT). Italso can be done for any conventional QFT model. The point is that you can embed a massless theory, such as QCD, into the massive theory of SPNIT. You can do that when searching for the correct solution for the high energy asymptotics.In this way I found the non-perturbative asymptotically free solutions for the above mentioned models. Moreover, you can return to the physical theory taking k2/λ2 → - k2/m2. It was torturous and tiresome work, but I have done it because my model-independent approach really opens the way for correct searching inall the branches of QFT, including such models as string theory, superstring theory, and many others that could appear in the future. It is amazing that even the archaic SPNIT – the old traditional theory of interactions of the neutral and charged π-mesons with nucleons – can be saved and connected to the modern QCD!

Beginning with Stückelberg and his disciples, Gell-Mann and Low and their followers, Bogoljubov and his disciples, Landau and his disciples – all of these brilliant scientists made the above-mentioned mistake! As a result, there appeared in QFT such artifacts as “ghost states” – the meaningless entries that tortured the QFT more than one half century. I removed them and comprehensively explained how they appear and whythey cannot exist in the correct approach.

It makes sense to describe how this mess happened. It was in the middle of 1950swhen some leading scientists – (in view of the horrendous difficulty of

Schwinger-Dyson‘s (SD) system of QFT equations) – were trying to find a more simple way to extract some new information, not connected with the perturbation theory. L.D. Landau (LDL) with disciples cut the SD equations, limiting the vertex parts by the “ladder approximation”. A concept of ‘leading logarithms’ was created in this way. The LDL’s approach was used to calculate the Z3 constant in QED. It appeared equal to zero!

N. N. Bogoljubov (NNB) with disciples created the new axiomatic in QFT relayed on distributions; the powerful R-operation was created for removal of singularities from QFT models (Hepp and Zimmerman developed NNB’s workfurther). When trying to introduce the Lee group parameter (the normalization parameter λ), N.N. Bogoljubov and D.V. Shirkov (DVS) dismissed, unfortunately, the mass terms (see above). Actually, the term ‘ghost state’ was coined because of the work of NNB and DVS and their coauthors. The easiness to manipulate the Lee-group differential equations helped in that, I guess, very much. After all, combined with the results of zero values of renormalized coupling constants in all conventional models of QFT, obtained by LDL and his disciples, the concept of the ‘Landau ghost’ was created.

All that intense activity gave a big push to the development of the QFT. Many brilliant scientists participated in those efforts. But, unfortunately, it was impossible to escape from the ‘Landau ghosts’; it remained ‘immortally’ alive in the ‘old’ QFT. You can take any text-book in QFT and convince yourself that nobody takes into account the crucial role of the dismissed mass-terms (see, for example, the excellent Quantum Field Theory of Itzikson and Zuber).

Besides the distortion of the analytical structure of QFTs vertices, the other misleading happened in all that amazing development. Implicitly the subtle question was avoided: the RG-equations – the Lee differential group equations – are, in reality, the equations of evolutions of QFT vertices outside the mass-shell of the QFT-models under consideration! Such a transformation was inevitable because the essence of the RG-method was in introducing the continuous Lee-group parameter (just in attempting to escape theSchwinger-Dyson(SD) equations difficulties!). But the poison of that misleading is becoming quite obvious: RG-Lee-Equations are just the rules that must be imposed on the high-momentum behavior of the QFT-vertices, but it is quite not enough yet! Without the rules imposed on the infrared ends – on the ‘the physical surfaces’ of the vertices – the MP boundary problem is not stated at all!

That’s the point!!!

Therefore, it appears that QFT-vertices became to contain the functional arbitrariness when being prepared for the RG investigation.

So, the bottom line is: you must remove that functional arbitrariness when you return to the physical theory. Because of that process, the RG-equations transform to identities, and the ‘ghost states’ completely disappear from the conventional models of the QFT. This transition is equivalent to imposing a “natural boundary condition” to the asymptotical behavior of the effective interaction. As always in Mathematical Physics, something new appears: the non-perturbative asymptotically free solution pops up.

It is not accidental that only I came to these statements. In 1960 I was the first scientist who stated in the framework of the RG that the Lee evolutionary equations do not represent the connections on the ‘mass-shell’. Moreover, I was the first one who created the multi-parameter renormalization group. In confirmation of that the reference to my article is presented; it was published in the Russian Journal of Experimental and Theoretical Physics(JETP), vol.39, #2(8), p.484 (1960) (in Russian).

II

Sometimes, it is very instructive to explain a new statement in a simple and transparent way. Let us repeat the first application section in the ‘Applications’ section– considering the removal of the Landau ‘ghost state’.

We investigate the behavior of the “ghost state” in QED. For the effective interaction in RG we have

ξ (x, α) = ξ (x/t, α′) (1)

α′ = ξ-1(x/t, ξ (x, α)) (2)

In the circle of convergence 3π/α for ξ (x, α) the “main log” series converges to the

ξ (x, α) = α/[1 – (α/3π)lnx] . (3)

(the famous ‘Landau ghost’).

In the circle of convergence for the r.h.s. of (1), the “main log” series converges to

ξ (x/t, α′) = α′/[1 – (α′/3π)ln(x/t)] . (4)

Substituting (4) into the r.h.s. of (1) and solving for α′, we find that

α′ = ξ (x, α)/[1 + (ξ (x, α)/3π)ln(x/t)] ≡ ξ-1(x/t, ξ (x, α)). (5)

It is easy to check the correctness of all these results by just checking the back substitution to obtain identity.

Thus, we have the equation:

ξ (x, α) = ξ(x/t, ξ-1(x/t, ξ (x, α)) . (6)

Following the way of obtaining Ovsjannikov’s equation (the earliest version of the Callan - Symanzik equation, published in 1956), let’s differentiate the eq. (6) over t and let t = 1. Then, we obtain the following equation:

∂ξ(x/t, ξ-1(x/t, ξ (x, α))/∂x

+ [∂ξ(x/t=1, ξ-1(x/t=1, ξ (x, α))/∂ ξ-1][∂ξ-1(x/t=1, ξ (x, α))/ ∂x] = 0, (7)

the “generalized” Ovsjannikov’s equation.

Now, let’s put in (5) t = 1. Then,

ξ-1(x, ξ (x, α)) = ξ (x, α)/[1 + (ξ (x, α)/3π)ln(x)]. (8)

Substituting into (3) ξ-1 instead of α, we have that

ξ (x, ξ-1) = ξ-1/[1 – (ξ-1/3π)lnx]. (9)

After calculating, we have

∂ξ (x, ξ-1)/∂ ξ-1 = [1 + (ξ (x, α)/3π) lnx]2. (10)

Next,

[∂ξ (x, ξ-1)/∂ ξ-1][ ∂ξ-1(x, ξ (x, α))/ ∂x]

= ∂ξ (x, α)/∂ x – ξ2(x, α) (1/3πx). (11)

Substituting (10) and (11) into the eq.( 7), we have that

∂ξ (x, ξ-1)/ ∂x + ∂ξ (x, α)/∂x – ξ2(x, α)(1/3πx) = 0. (12)

Since

ξ (x, α) = ξ-1(x, ξ (x, α)), (13)

we have that

2∂ξ (x, α)/ ∂x – ξ2(x, α)(1/3πx) = 0, (14)

and solving for ξ (x, α), we now obtain that

ξ (x, α) = α/[1 – (α/2*3π)lnx]. (15)

We obtained a surprising result: the “ghost pole” has shifted to the bigger value of the lnx:

lnx = 2*3π/ α. (16)

If we take as the starting point the value (15), then repeating the above consideration, we will find

ξ (x, α) = α/[1 – (α/4*3π)lnx]. (17)

Repeating n times, we obtain:

ξ (x, α) = α/[1 – (α/2n*3π)lnx]. (18)

Actually, we’ve obtained the series of the “ghost pole” expressions:

ξ1(x, α), ξ2(x, α), …, ξn(x, α), … (19)

with

ξi(x, α) = α/[1 – (α/2i*3π) lnx]. (20)

If n → ∞, then this functional series tends to the following limit:

ξn(x, α) → α. (21)

Thus, the limiting point is just the coupling constant of the QED. We again obtained the result presented in the basic text.

It is instructive to evaluate the radius of the limiting convergence circle:

R∞ = limn→∞(2n*3π/α) = ∞. (22)

This means that the coupling constant α, as the limiting value of the effective charge of QED, exists for all the values of lnx, i.e. the “ghost pole” completely disappeared! (No need to plunge into the works of Johnson, Baker, &Villey, or Adler’s ambiguous ‘loop wise’ summation).

So, acting ‘in my way’, i.e. step-by-step changing the boundary condition, we can get rid off the ‘ghost state’. Amazing, isn’t it? Think over the advantage of the correct approach! Just educated and responsible approach!

Tolerance toward the misleading in QFT is dangerous. The point is that such a fundamental branch of science as QFT enters into everything, for example, also in astrophysics. It is very well known how many consequences such interplay can create – stranglets, black holes, and many others. Now, if the asymptotic freedom can be realized in the non-perturbative way, then we can expect a lot of surprises.

It is amazing to see how the peculiarities of the non-perturbative solution can change the behaviors of the vertices in QFTor how the “perturbative tale” of that solutionappears. Maybe, the most amazing thing is that instead of solving SD equations it is possibleto handle it with much easier“induced identities”: in this way you can work out problems in Regge-pole theory, or in the deep-inelastic scattering problems; actually, you can investigate the peculiarities in any inclusive (or semi-inclusive) processes in QFT.

The presented approach has the advantage to try anything you want to consider. It is natural because you don’t have the wrong ‘ghost’ states in QFT (the main psychological obstacle for ‘unfortunate’ QFT specialists trying to solve the problems and because of that obstacle bouncing here and there to find the right way for solutions!).

This first sectionis devoted to the preliminary and introductory understanding of this new approach in QFT.

1. A Possible Connection of the Pseudo-Scalar Symmetric Pion-Nucleon Strong Interaction Theory with QCD

In the one–loop approximation of Quantum Chromodynamics aconnection of πNNSI and QCD β–functions is found. The non–perturbative asymptotically free solutions are found both for πNNSI and QCD, provided the correct transition of the massive πNNSI theory to its massless asymptotic case is taking place. As a result, the connection between observable quantities of πNNSI and non-observable ones of QCD is stated.