§:

(1/2) β′(αλ) – (2/ αλ) β (αλ) = 0. (7.33)

Solving this equation we obtain

β (αλ) = C αλ4, (7.34)

where C is an arbitrary constant.

Now, the equation (7.33) can be rewritten in the form when instead of αλ the effective interaction is taken:

β′(r∞) – (4/ r∞) β (r∞) = 0. (7.35)

Its §:(1/2) β - Queens College, cuny, Flushing, ny 11367 solution will be as follows:

β (r∞) = β (αλ) (r∞/ αλ)4. (7.36)

Next, using (7.36) in the equation for the effective interaction

∂r∞(x, αλ)/∂x = (1/x)β(r∞(x, αλ)), (7.37)

we obtain

r∞3(x, αλ) = αλ3/[1 –(3β (αλ)/ αλ)lnx]. (7.38)

For the positive value of the cubic §:(1/2) β - Queens College, cuny, Flushing, ny 11367 root, we have

r∞(x, αλ) = αλ/[1 –(3β(αλ)/ αλ)lnx]1/3. (7. 39)

Substituting instead of β(αλ) the solution from (7.34), we have

r∞(x, αλ) = αλ/[1 –(3Cαλ3)lnx]1/3. (7.40)

Let’s show that for physical α, C < 0. Since

β(αλ) = limy→0 Φ1(y, αλ), β2(αλ) = limy→0 Φ2(y, αλ), (7.41)

in the limit λ2 = – m2 we §:(1/2) β - Queens College, cuny, Flushing, ny 11367 have this condition:

[д2q∞(1, α)/дα2)β (α) – 2y(д2q∞(y, α)/д αдy)|y=1

+ (1/2)(дq∞(1, α)/дα)β′(α)] = 0. (7.42)

Calculating by means of (7.6) all derivatives and substituting them into (7.42), we obtain that

4α/3π + (1/2) β′(α) = 0. (7.43)

From (7.34) we have β′ (α) = 4Cα3 and

C §:(1/2) β - Queens College, cuny, Flushing, ny 11367 = – 2/3πα2. (7.44)

Thus, we finally have the physical asymptotically free effective interaction

r∞(–k2/m2,α)=α/[1+(2α/π)ln(– k2/m2)]1/3 , (7.45)

→ (π/2)1/3α2/3/ln1/3(– k2/m2),

when k2 → – ∞.

If we substitute the numerical value of α = 1/137, we will obtain approximately

r∞(– k2/m2, α) ≈ 0.04374[1/ln §:(1/2) β - Queens College, cuny, Flushing, ny 113671/3(– k2/m2)] (7.46)

Let’s now consider the symmetric pseudo-scalar hadron theory. In this case

q∞(y, αh) = αh/ [1 + (5/4π) αhlny] (7.47)

and

(1/2)β′(αh)–(2/αh)β(αh)=0, (7.48)

we have

β′ (αh) =(4/αh) β (αh) (7.49)

or

β′ (r∞(x, αh)) =[4/ r §:(1/2) β - Queens College, cuny, Flushing, ny 11367∞(x, αh)] [β (r∞(x, αh))]. (7.50)

The solution of this equation gives us

β(r∞(x, αh)) = β (αh)[r∞(x, αh)/ αh]4. (7.51)

Now, the Lee equation for the genuine effective asymptotic interaction in this §:(1/2) β - Queens College, cuny, Flushing, ny 11367 case of QFT is of the form:

∂r∞(x, αh)/∂x = (1/x)β(r∞(x, αh)) (7.52)

or, substituting (7.51) into (7.52), we obtain:

∂r∞(x, αh)/∂x = (1/x) β (αh) [r∞(x, αh)/ αh]4. (7.52a)

The solution §:(1/2) β - Queens College, cuny, Flushing, ny 11367 of this equation is:

r∞3(x, αh) = αh3/[1 – (3β(αh)/ αh)lnx]. (7.53)

The positive root of this equation gives us

r∞ (x, αh) = αh/[1 – (3β(αh)/αh)lnx]1/3 (7.54)

or

r∞ (x, αh) = αh/[1 – (3Cαh3)lnx]1/3 , (7.55)

where §:(1/2) β - Queens College, cuny, Flushing, ny 11367 C = –(5/2π αh).

Finally, we have that

r∞(x,αh)=αh/[1+(15/2παh)lnx]1/3 (7.56)

≈ (2π)1/3[αh5/3/(15)1/3]/ln1/3x

Calculating the constant coefficients,

we have (when taken αh = 15):

r∞ (– k2/m2, αh) ≈ 68.2606 / ln1/3( – k2/m2). (7.57)

Let’s §:(1/2) β - Queens College, cuny, Flushing, ny 11367 investigate the most interesting for now, the option of quantum chromodynamics (QCD) for hadron physics.

In the simplest case

βQCD(αs)=–{[11–(2/3)f]/4π}αs2, (7.58)

where αs is the quark-gluon coupling constant and f §:(1/2) β - Queens College, cuny, Flushing, ny 11367 is the number of quarks. Equating the expressions for β (αh) and βQCD(αs), we obtain the relation between αh and αs:

αh2 = {[11– (2/3)f]/30}αs2 (7.59)

8. Conclusion

This lengthy and cumbersome way was really worth §:(1/2) β - Queens College, cuny, Flushing, ny 11367 trying. We proved that the right understanding of the nature of the renormalization group gives us the possibility to get rid of many painful misunderstandings in quantum field theory. On the other §:(1/2) β - Queens College, cuny, Flushing, ny 11367 хэнд, the correct use of the RG-method gives us great power for investigating the key points in the background of QFT.

First of all, the understanding that the introduction into §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the vertices of QFT, a dependence on the normalization momentum (in order to be able to introduce a continuous group parameter to use the theory of the Lee groups), leads to the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 inclusion into expressions of QFT a functional arbitrariness. This is a challenge to correctly eliminate this arbitrariness when we return to the real physical theory.

In order to handle this situation, the normalization function §:(1/2) β - Queens College, cuny, Flushing, ny 11367 q for the effective interaction rq was introduced. This q-function bears all the information about the terms type of ln(λ2/m2), artificially introduced into the vertices of QFT. We must §:(1/2) β - Queens College, cuny, Flushing, ny 11367 be very careful when dismissing terms of the type ln(λ2/m2) in comparison with the terms ln(p2/m2), when p2 is much bigger than m2 (p2 >> m2). The danger is §:(1/2) β - Queens College, cuny, Flushing, ny 11367 that you can throw out the severe infrared singularities; because of that, the analytical structure of the QFT expressions can be severely distorted. We have shown that careful handling of this situation removes §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the well-known past “ghost states” in the conventional field theories. It appears that RG-equations do not provide any regular method for summing up the perturbative terms in QFT. The correct §:(1/2) β - Queens College, cuny, Flushing, ny 11367 return to the physical theory proves that the Lee-group equations must transfer into identities (that’s the reason why we name the q-function a “translator”). We have shown §:(1/2) β - Queens College, cuny, Flushing, ny 11367 how the “ghost state” is being created because of the wrong actions in the perturbation theory.

Next, the relation between the normalization function q and Callan-Symanzik’s function β was investigated. Since §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the β-function governs the high-momentum behavior of the effective interaction r (do not mix it up with the auxiliary value rq, which plays a temporary role, but is necessary for the definition §:(1/2) β - Queens College, cuny, Flushing, ny 11367 of the q-function), these relations (actually, there can be an infinite number of such relations; we used only the second order relation) give us the right tool to go to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the massless limit in QFT. It was found that the physical coupling constant always exists as the solution for the high energy effective interaction r. Then, it was shown that if the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 mass terms go to zero as or faster than 1/(ln3m), when m tends to zero (m→0), (of course, m is dimensionless – mass being measured in some scale), there exists the asymptotically free solution §:(1/2) β - Queens College, cuny, Flushing, ny 11367 for the effective interaction, which tends to zero as 1/[ln1/3(p2)], when p2 → ∞!

This result takes place for all types of conventional QFTs.

Investigated first was QED, then symmetric pseudo–scalar §:(1/2) β - Queens College, cuny, Flushing, ny 11367 pion–nucleon strong interaction theory, and its connection to the QCD. The impressive connection between the pion–nucleon coupling constant αh and the quark–gluon coupling constant αs was found. Because of that, the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 connection between the strong interaction coupling constant and the characteristic mass–scale of QCD becomes evident:

αh2={[11–(2/3)f]/30}αs2, αs2=4b2ln2(Λ/μ)]-1, (8.1)

b = (11 – (2/3)f)/4π.

Since the numerical value of αh is well §:(1/2) β - Queens College, cuny, Flushing, ny 11367-known, the scale ratio in αs appearing as a result of dimensional transmutation can be found.

Actually, the issue of asymptotic freedom mentioned in the subsection 6 of this part 3 in this review §:(1/2) β - Queens College, cuny, Flushing, ny 11367, hints that the asymptotic freedom might be the mandatory property of any type of QFT.

The last remark: the results (7.45), (7.56), (7.58), and (7.59) cannot be obtained in the perturbation theory. As was mentioned §:(1/2) β - Queens College, cuny, Flushing, ny 11367 by Dyson, the convergence radius of the perturbative expansion in QFT, in general, equals zero[17].

Appendix I

Let’s rewrite it from section 7:

r∞(x, αλ) = αλ + Σ∞n=1βn(α) Ln. (7.21)

It is easy to obtain that

∂ r §:(1/2) β - Queens College, cuny, Flushing, ny 11367∞/∂L|L=0 = β; ∂2 r∞/∂L2|L=0 = 2 β2; ∂3 r∞/∂L3|L=0 = 6 β3; … (AI.1)

…; ∂n r∞/∂Ln|L=0 = n! βn; …

Next, since

∂ r∞/∂L = β (r∞), (AI.2)

then

∂2 r∞/∂L2 = β′ (r∞)∂ r∞/∂L = β′ (r∞) β (r∞) (AI §:(1/2) β - Queens College, cuny, Flushing, ny 11367.3)

Subsequently using all the definitions in (AI.1) and all the derivatives of (AI.2), we obtain that

βn (r∞) = (1/n) β (r∞) βn-1 (r∞), (AI.4)

and

βn (αλ) = (1/n) β (αλ) βn-1 (αλ). (AI.4a)

Appendix II

Let’s accept that §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the general integral

rq(x, y, αλ) = q (y/x, r (x, y, αλ)), (4.1)

has continuous derivatives of the second order over its arguments.

Then, differentiating eq.(4.1) over x two times and setting x = 1, we obtain §:(1/2) β - Queens College, cuny, Flushing, ny 11367:

Ψq,2(y, αλ) = д2q∞(y, αλ)/дαλ2 Φ12(y, αλ) (AII.1)

– 2y (д 2q∞(y, αλ)/ д αλ∂y) Φ1 (y, αλ)

+ (дq∞(y, αλ)/дαλ) Φ2 (y, αλ)

+ 2y (д q∞(y, αλ)/ ∂y)

+ y2(д 2q∞(y, αλ)/ ∂y2),

where

Φ1(y, αλ) = (дr(x §:(1/2) β - Queens College, cuny, Flushing, ny 11367, y, αλ) /дx)|x=1, (7.25)

Φ2(y, αλ) = (д2r(x, y, αλ) /дx2)|x=1, (7.26)

Ψq,2(y, αλ) = ∂2rq (x, y, αλ)/∂x2|x=1. (7.26a)

Now, using the expansion

rq,∞(x,y,αλ)=αλ+Σ∞n=1φn(αλ)lnn(x/y), (7.20a)

we §:(1/2) β - Queens College, cuny, Flushing, ny 11367 receive the following expansions over the degrees of lny:

Ψq,2(y, αλ)=Σ∞n=1(–1)nnφn(αλ)[lnn-1y + (n – 1) lnn-2y], (AII.2)

дq∞(y,αλ)/дαλ=Σ∞n=0(–1)nφn′(αλ)lnny, (AII.3)

φ0(αλ)=αλ, (AII.3a)

– y д §:(1/2) β - Queens College, cuny, Flushing, ny 11367 q∞(y, αλ)/∂y =Σ∞n=1(–1)n-1nφn(αλ)lnn-1y, (AII.4)

– y д 2q∞(y, αλ)/дαλ∂y = Σ∞n=1(–1)nnφn′(αλ)lnn-1y, (AII.5)

y2д2q∞(y, αλ)/∂y2=Σ∞n=1(–1)nnφn(αλ)[–lnn-1y+ (n – 1)lnn §:(1/2) β - Queens College, cuny, Flushing, ny 11367-2y]. (AII.6)

Comparing the formulas (AII.3) – (AII.6), we come to the relation

Ψq,2(y, αλ) = 2y д q∞(y, αλ)/∂y + y2 д2q∞(y, αλ)/∂y2. (AII.7)

Using it in (AII.1), we §:(1/2) β - Queens College, cuny, Flushing, ny 11367 obtain that

д2q∞(y, αλ)/дαλ2 Φ12(y, αλ) – 2y (д 2q∞(y, αλ)/ д αλ∂y) Φ1 (y, αλ)

+ (дq∞(y, αλ)/дαλ) Φ2 (y, αλ) = 0, (AII.8)

or using the designations used when y → 0, we have

limy→0[(д2q∞(y, αλ)/дαλ2) β2(αλ) – 2y §:(1/2) β - Queens College, cuny, Flushing, ny 11367(д 2q∞(y, αλ)/ д αλ∂y)β(αλ)

+(дq∞(y,αλ)/дαλ) β2(αλ)] = 0. (AII.9)

Now using the recurrent formula (see Appendix I)

β2(αλ)=(1/2)β′(αλ)β(αλ), (AII.10)

and putting it into (AII.9), we finally have

β(αλ)limy→0[д2q∞(y, αλ)/дαλ2)β (αλ) – 2y(д2q∞(y §:(1/2) β - Queens College, cuny, Flushing, ny 11367, αλ)/д αλдy)

+ (1/2)(дq∞(y, αλ)/дαλ)β′(αλ)] = 0. (7.29)

References

1. F.Dyson, Phys. Rev. 75, 1736 (1949).

2. L.D. Landau, A.A. Abrikosov, I.M.Khalatnikov,

Doklad. Akad. Nauk USSR (in Russian), 95, 497, 773,1177(1954)

3. L.D. Landau, I.Ya. Pomeranchuk,

Doklad. Akad §:(1/2) β - Queens College, cuny, Flushing, ny 11367. Nauk USSR (in Russian), 102, 489 (1955);

I. Ya. Pomeranchuk, Doklad. Akad. Nauk USSR (in Russian),

103, 1005 (1955).

* In the middle of the 1950s there was a huge hue and cry concerned with that statement. Actually, it §:(1/2) β - Queens College, cuny, Flushing, ny 11367 was saying that the physical charge (the renormalized electrical charge) becomes zero. The followers of that statement were relying on the asymptotical series for the renormalization constant Z3, containing the cut-off §:(1/2) β - Queens College, cuny, Flushing, ny 11367 momentum Λ(→ ∞). The disciples of L. D. Landau мейд a lot of efforts to justify that bizarre statement (the most famous one of them was the ‘two-limiting technique’ of §:(1/2) β - Queens College, cuny, Flushing, ny 11367 Pomeranchuk). However, if one considers the ‘leading logarithms’ series in Z3, as a double series over both the coupling constant (α/3π) and the ‘cutting logarithm’ ln(Λ2/m2), then it appears that the series over the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 ln(Λ2/m2) can converge only inside the convergence circle with the radius of convergence equal to (3π/α). Therefore, the approximate expression for

Z3 ≈ α/ [1+ (α/3π) ln (Λ2/m2)] (*)

has sense only inside of that circle. Outside §:(1/2) β - Queens College, cuny, Flushing, ny 11367 of it the series over the degrees of ln (Λ2/m2) cannot be summed up to the expression (*); it simply diverges! In other words, it is impossible to tend the cut-off momentum to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 infinity (in the denominator of (*)) to obtain the zero value for Z3. This simple remark shows how dangerous conclusions are relayed on the asymptotical series, every term of which tends to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 infinity. You can cut-off the horrendous Schwinger-Dyson system of equations (the infinite system of integro-differential equations, obtained after the removal of all divergences and renormalization of the charge §:(1/2) β - Queens College, cuny, Flushing, ny 11367 and mass of the theory in question) in the ‘ladder approximation’, using the most refined technique and being after all deadly wrong! There exists a widespread saying that every simple result can §:(1/2) β - Queens College, cuny, Flushing, ny 11367 be obtained in a simple way. The above reasoning, declining the ‘zero-charge’ difficulty (contemptuously named by some Western theoreticians as the ‘Moscow zero’) confirms that simple truth.

4. K. Johnson, M. Baker, R. Willey §:(1/2) β - Queens College, cuny, Flushing, ny 11367, Phys. Rev. 136, B111 (1964);

ibid., 163, 1699 (1967); M. Baker, K. Johnson, Phys. Rev.

D3, 2516 (1971); D3, 2541 (1971).

5. S.Adler, Phys. Rev. D5, 3021 (1972).

6. M. Gell-Mann, F. Low, Phys. Rev. 95, 1300 (1954).

7. E. C. Stueckelberg, A. Peterman, Helv §:(1/2) β - Queens College, cuny, Flushing, ny 11367. Phys. Acta,

26, 499 (1953);

N. N. Bogoljubov, D. V. Shirkov,

Doklad. Akad. Nauk USSR (in Russian),

103, 203, 392 (1955);

ZhETPh (in Russian), 30, 77 (1956);

D. V. Shirkov,

Doklad. Akad. Nauk USSR (in Russian),

105, 972 (1955);

V. A. Shakhbazyan, ZhETPh (in Russian), 37, 1789 (1959);

ZhETPh (in §:(1/2) β - Queens College, cuny, Flushing, ny 11367 Russian), 39, 484 (1960).

8. E. C. Stueckelberg, D. Rivier,

Helv. Phys. Acta, 22, 215 (1949);

E. C. Stueckelberg, J. Green, E. C.

Helv. Phys. Acta, 24, 153 (1951);

N. N. Bogoljubov, Doklad. Akad. Nauk USSR (in Russian),

81, 757, 1015 (1951);

Izvest. Acad. Nauk §:(1/2) β - Queens College, cuny, Flushing, ny 11367 USSR, ser. fiz. (in Russian),

19, 237, (1955);

see also [16], chap. 4.

9. N. N. Bogoljubov, Doklad. Akad. Nauk USSR (in Russian),

^ 82, 217 (1952);

N. N. Bogoljubov, O. C. Parasjuk,

Doklad. Akad. Nauk USSR (in Russian),

100, 25, 429 (1955);

Izvest. Acad §:(1/2) β - Queens College, cuny, Flushing, ny 11367. Nauk USSR, ser. mat., (in Russian),

20, 585, (1956);

Acta Math, 97, 227 (1958);

K. Hepp, Comm. Math. Phys., 2, 301, (1966);

W. Zimmerman, 1970 Brandies Lectures;

Lectures on Elementary Particles and Quantum Field Theory,

eds. S. Deser, M. Grisaru, H. Pendelton

(MIT §:(1/2) β - Queens College, cuny, Flushing, ny 11367 Press, Cambridge, 1971);

Ann. of Phys.(N.Y.), 77, 536 (1973).

10. D. Gross, F. Wilczeck, Phys. Rev. Lett., 30, 1343 (1973);

H. Politzer, Phys. Rev. Lett., 30, 1346 (1973).

11. P.G.Hoel, S.C.Port, C.J.Stone,

Introduction to Probability Theory;

Houghton Mifflin Company §:(1/2) β - Queens College, cuny, Flushing, ny 11367, Boston;

a) Chapter 4,

b) Chapter 8.

12. L. V. Ovsjannikov, Doklad. Akad. Nauk USSR (in Russian),

109, 1112 (1956).

13. C. G. Callan, Phys. Rev. D2, 1541 (1970);

Symanzik, Comm. Math. Phys., 18, 227 (1970).

# In the early days of the RG method §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the normalization condition was in the form of the formula (4.3) in the text of this section. It was done just simply to restore the ‘leading logarithm’ series in the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 framework of the RG. However, it was wrong since it created the ‘ghost’ states in the conventional models of the QFT. Therefore, the arbitrary normalization function, the q-function, is the crucial §:(1/2) β - Queens College, cuny, Flushing, ny 11367 starting point on the way to the correct theory.

14. N. N. Bogoljubov, D. V. Shirkov, Introduction to the Theory

of Quantized Fields, Moscow, 1984 (in Russian); New York, Interscience Publishers, 1959 &1980.

15.C. Itzykson, J.-B. Zuber, Quantum §:(1/2) β - Queens College, cuny, Flushing, ny 11367 Field Theory,

1980, McGrow-Hill, Inc.

§ Actually, this equation is a result of some general reasoning, which could be named as the quantization of the boundary conditions. However, this inspiring §:(1/2) β - Queens College, cuny, Flushing, ny 11367 issue is outside of the scope of this work.

16. G. M. Fichtenholts. Course of the Differential and Integral

Calculus, Moscow, 1948.

17. F.Dyson, Phys. Rev. 85, 631 (1952).

Now, let’s go to applications.

4. Applications

I

The other way for removing §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the “ghost state”

Let’s investigate the behavior of the “ghost state” in QED.

ξ (x, α) = ξ (x/t, α′) (1)

α′ = ξ-1(x/t, ξ (x, α)) (2)

In the circle of convergence 3π/α the “main log” series converges to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the

ξ (x, α) = α/[1 – (α/3π)lnx] . (3)

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 §:(1/2) β - Queens College, cuny, Flushing, ny 11367 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 by the back substitution to obtain identity §:(1/2) β - Queens College, cuny, Flushing, ny 11367.

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), let’s differentiate the eq.(6) over §:(1/2) β - Queens College, cuny, Flushing, ny 11367 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, ξ-1(x/t, ξ (x, α))/∂ ξ-1][∂ξ-1(x/t, ξ (x, α))/ ∂x] = 0, (7)

the “generalized” Ovsjannikov equation.

Now §:(1/2) β - Queens College, cuny, Flushing, ny 11367, 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 §:(1/2) β - Queens College, cuny, Flushing, ny 11367, 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 §:(1/2) β - Queens College, cuny, Flushing, ny 11367, 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 §:(1/2) β - Queens College, cuny, Flushing, ny 11367 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 §:(1/2) β - Queens College, cuny, Flushing, ny 11367 → ∞, 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. Thus, we again obtained the result presented in §:(1/2) β - Queens College, cuny, Flushing, ny 11367 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 §:(1/2) β - Queens College, cuny, Flushing, ny 11367 effective charge of QED, exists for all the values of lnx, i.e. the “ghost pole” completely disappeared!

What happens with the Callan-Symanzik β-function

β(α) = ∂ξ (x, α)/∂x|x=1? (23)

Here we also have the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 series of values: {(α2/2n*3π)}, and

(α2/2n*3π) = 0, β∞(α) = 0. (24)

Thus, the “limiting point” of the βi’s is the fixed point of the RG for the effective charge of QED.

This result is valid for all conventional models §:(1/2) β - Queens College, cuny, Flushing, ny 11367 of QFT.

We see that the set of invariant charges with the “ghost poles” can have a sequence which tends to the coupling constant. However, such a behavior is not unique §:(1/2) β - Queens College, cuny, Flushing, ny 11367. The type of behavior depends on the order of the generalization of Ovsjannikov’s equation you are using◊.

◊ This simple derivation, repeated in the Introduction, is instructive not only from a scientific point of §:(1/2) β - Queens College, cuny, Flushing, ny 11367 view but also in other respects. According to the opinion of the famous American physicist S.A. Goudsmith, who first discovered (together with Uhlenbeck) the spin of the electron, the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 scientific system in Russia is the old-fashioned one of the ‘leading professors’ system. Unfortunately, the members of the ‘scientific schools’ of academicians L. D. Landau and N.N. Bogoljubov, were §:(1/2) β - Queens College, cuny, Flushing, ny 11367 more interested in the rightness of the ‘schools’ statements than in the genuine scientific truth. Even until now the disciples of the L.D. Landau believe in the ‘zero-charge’ difficulty in QFT. The §:(1/2) β - Queens College, cuny, Flushing, ny 11367 followers of N.N. Bogoljubov, who always opposed that meaningless statement, did not find time for finding the above presented simple solution. Actually, the reason for this ridiculous situation is the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 obsolete relations in the ‘leading professors’ system. Too bad the influence of the powerful ‘schools’ prevents the development of science.

^ II

The perturbative “tail” of the

non-perturbative solution in QED

The non-perturbative §:(1/2) β - Queens College, cuny, Flushing, ny 11367 solution for the QED invariant charge is of the form:

r (-k2/m2, α) = α/[1 + (2α/π)ln(-k2/m2)]1/3, (1)

or

r (x, α) = α/[1 + (2α/π)lnx]1/3 . (2)

In the case when x ~ 1 and, therefore, lnx ~ 0, we can use the perturbation theory §:(1/2) β - Queens College, cuny, Flushing, ny 11367 and calculate the β-function.

β(α) = ∂r(x, α)/ ∂x|x=1 = - 2α2/3π. (3)

Together with the conventional P.T., we obtain this result:

r(x, α) = α – (α2/3π)lnx. (4)

Thus, the “perturbative tail” of the non-perturbative solution has the correct §:(1/2) β - Queens College, cuny, Flushing, ny 11367 behavior.

In the past, there were many intuitive hints that the perturbative contribution must give the right contribution (the negative one in the second order). However, until the appearance of the QCD, all §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the efforts in that direction were in vain. Thus, our consideration provided, at last, that long awaited result for all traditional models of QFT without reference to the Yang-Mills §:(1/2) β - Queens College, cuny, Flushing, ny 11367 Lagrangian. Some remark of warning is in order. The non-perturbative solution, found in this work, is singular. Therefore, the possible contribution of the “tail” depends on the type of singularity of §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the n-p.t. solution. If it is so strong, that there will be no derivatives of it, then, of course, there will be no “tail”.

^ III

The effective interaction in QCD in

the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 “twice-logarithmic” Approximation

Let’s consider a contribution to the QCD “running constant” of the “ladder diagram” series contribution, which give the biggest values in any QFT approximation scheme. Moreover, we §:(1/2) β - Queens College, cuny, Flushing, ny 11367 will try to take into consideration the “twice-logarithmic” part of it.

First, here are some designations. The momentum variable is x = Q2/2pq; N is the number of colors; b = (11/3)N – (2/3)nF §:(1/2) β - Queens College, cuny, Flushing, ny 11367, nF is the number of aromas. The block of the above-mentioned diagrams is designated ash(x, αs(Q2)).

Thus, we are looking for the asymptotics of h over x, trying to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 improve, according to our advanced RG machinery, its asymptotical behavior over x.

We will take as input the now widely known expression

h0(x, αs(Q2)) ≈ [ξy]1/2, (1)

where

ξ = lnln(Q2/Λ2), (1a)

y = (8N/b)ln(1/x §:(1/2) β - Queens College, cuny, Flushing, ny 11367). (1b)

After applying the machinery of the basic text, we will find the following result:

h(x, αs(Q2)) = (2)

where

κ = x/ζ, ζ is the RG dimensionless shifting momentum variable,

or

h(Q2/2pq, αs §:(1/2) β - Queens College, cuny, Flushing, ny 11367(Q2)) = (3)

This consideration is valid for the region of κ = x/ζ ≈ 0.01.

Now, we can find the β-function for this asymptotic behavior:

β = ∂h(x, αs(Q2))/∂x|x=κ=

= [1 – 2ln(1/κ)]{h0(κ, αs(Q2))}-2ln(1/κ), (4)

or

β(αs §:(1/2) β - Queens College, cuny, Flushing, ny 11367(Q2)) = [1 – 2ln(1/κ)]{exp[lnln(Q2/Λ2) ln(2pq/Q2)]1/2}-2ln(1/κ). (5)

We know that β-function is the infinitesimal operator which generates the asymptotic perturbative evolution of the vertex under consideration. However, we also know §:(1/2) β - Queens College, cuny, Flushing, ny 11367 that there exists the non-perturbative asymptotically free solution for the hadron physics effective interaction (see the formula (7.56) in the fourth paper of the basic text). In the region §:(1/2) β - Queens College, cuny, Flushing, ny 11367 x ~ 1 (lnx ~ 0) we can calculate the β-function for this solution (the explanations of it see a little later):

βdr(αh) = - 5/2π. (6)

Thus, since

αh2 = {[11 – (2/3)f]/30} αs2(Q2) (7)

(see (7.59) in the fourth paper of the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 basic text), we have

βdr({[11 – (2/3)f]/30}αs2(Q2)) = – 5/2π. (8)

Thus, the evolutionary part of the non-perturbative solution’s β-function is the negative constant – 5/2 π.

Now, we can explain the meaning of this phenomenon. This “perturbative §:(1/2) β - Queens College, cuny, Flushing, ny 11367 tail” that accompanies the non-perturbative solution also improves the genuine perturbative part (derived from the summation of the appropriate diagrams) of the asymptotical effective coupling (since it gives the negative §:(1/2) β - Queens College, cuny, Flushing, ny 11367 contribution). Because the evolutionary equations are linear, this “drizzle” contribution just adds to the perturbative part, so the solution (3) transfers to the new one:

(3) → hdr+p.t.( Q2/2pq, αs(Q2)) = = (9)

Why is the name §:(1/2) β - Queens College, cuny, Flushing, ny 11367 of this “tail” the “drizzle” part of the evolution equation?

Imagine that you have thrown a stone into the water. What happens? There appears a splash, waves, sprays, and … the drizzle §:(1/2) β - Queens College, cuny, Flushing, ny 11367, – a cloud of tiny droplets of water, surrounding the region of the event. After a while, the stone goes to the bottom, waves are quieted step-by-step, big water droplets §:(1/2) β - Queens College, cuny, Flushing, ny 11367 drop onto the surface of the water and disappear, but the drizzle remains and it remains a rather long time, much more than the time of the sudden consequences of the thrown stone event §:(1/2) β - Queens College, cuny, Flushing, ny 11367.

This “drizzle” is the by-product of the non-perturbative solution.

In mathematical terms, we can imagine the non-perturbative solution for the effective coupling as follows. It has §:(1/2) β - Queens College, cuny, Flushing, ny 11367 two parts: the first part is the singular one; the second one can be imagined as the famous Cantor set of diffusely distributed points without closed neighborhoods, the set without the interior §:(1/2) β - Queens College, cuny, Flushing, ny 11367 part. We can introduce this picture, indeed, because for each value of Q2, h0 is in the degree of the constant number – 5/2π, provided that Q2/2pq is, actually, a constant ~ 0.01, and Q2 is the big §:(1/2) β - Queens College, cuny, Flushing, ny 11367 running asymptotical variable, and we cannot imagine any kind of expansion of h0 over it.

This “educated guess” may be useful for understanding what is going on with the “tail” part of §:(1/2) β - Queens College, cuny, Flushing, ny 11367 this non-perturbative solution, although it is not, of course, strictly stated.

Finally, the “asymptotically free” effective coupling in this case is of the form:

r(x, αs(Q2)) = (10)

here §:(1/2) β - Queens College, cuny, Flushing, ny 11367 A = (2π/15)1/3[{[11 – (2/3)f]/30}αs(Q2)]5/3, x = Q2/2pq. This expression shows a surprising picture. It is “twice-logarithmical” superficially since the 2pq/Q2 is actually a constant. The only structure giving (slow) rise to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 the logarithmical grow, comes from the lnln(Q2/Λ2). The transition to the physical theory corresponds to the condition ζ = κ. Then, from x = Q2/2pq and κ = x/ζ follows κ2 = Q2/2pq, and the physical effective §:(1/2) β - Queens College, cuny, Flushing, ny 11367 interaction takes the form:

r(Q2/2pq, αs(Q2)) = A[ln1/3(Q2/2pq)]- 1 +

+ {exp[(lnln(Q2/Λ2))(8N/b)ln(2pq/Q2)]1/2}ρ , (11)

where

ρ = [Q2/2pq]1/2 – (5/2π) – 2ln[2pq/Q2]1/2 . (12)

◘ Actually, these expressions can §:(1/2) β - Queens College, cuny, Flushing, ny 11367 contribute to the QCD-corrections in the processes of the deep inelastic proton-nucleon scattering (also in proton-nucleus scattering), to e+e- - annihilation, and in the any exclusive, inclusive (or §:(1/2) β - Queens College, cuny, Flushing, ny 11367 semi-inclusive) process in any model of the past, present, or future QFT.

^ IV

General approach to the notion of the convergence radius

To determine the convergence radius of the power series, there are §:(1/2) β - Queens College, cuny, Flushing, ny 11367 two possibilities.

1) Definition by Cauchy-Hadamard:

r = (1)

2) According to D’Alamber’s criterion:

r = (2)

Let’s equalize both variants. Then,

. (3)

Now, let’s take instead of a’s β’s, the coefficients of the asymptotic §:(1/2) β - Queens College, cuny, Flushing, ny 11367 expansion of the effective interaction in our machinery. We have:

, (4)

or according to the recurrence conditions,

(βn′ )-1| = . (5)

In QED we have:

βn′, (6)

or

βn′(x)dx. (7)

The result of calculating gives §:(1/2) β - Queens College, cuny, Flushing, ny 11367 us the following:

. (8)

According to Cauchy-Hadamard, the radius of convergence is

. (9)

Now, let’s use the Gell-Mann-Low equation:

. (10)

For our non-perturbative asymptotic solution we can use the result:

. (11)

Then, for the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 asymptotical value of the convergence radius (which corresponds to the k2→∞), we will have

R∞ = R(α) , (12)

where R(α) is given by eq.(11), and r∞ is the asymptotical effective interaction.

Thus, the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 evolutionary part diminishes (or increases) the value of the convergence radius depending on the sign of the β- function. The interplay of the singular and the evolutionary parts can provide some results, which, maybe, can §:(1/2) β - Queens College, cuny, Flushing, ny 11367 improve the convergence properties of the QFT models.

□ This general consideration opens the way for investigating any of the convergence properties in QFT.

^ V

The “induced identities (IIs)”

In the basic §:(1/2) β - Queens College, cuny, Flushing, ny 11367 text we showed that when returning to the physical theory, the Lee differential equations of RG transform (in perturbation theory of QED) to identities.

Let’s see what happens in the general case §:(1/2) β - Queens College, cuny, Flushing, ny 11367. For simplicity, consider the 4-vertex function in QED. The Lee differential equation in our formalism looks like this:

, (1)

where

, (2)

or

, (3)

where

{si} are the fixed variables (5 of the 6 in all, one being the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 asymptotical one).

In the physical theory, in the asymptotic -we have for our non-perturbative solution, the following expression:

. (4)

This is an identity. We name these identities as the “induced identities”. They are induced §:(1/2) β - Queens College, cuny, Flushing, ny 11367 by the renormalization invariance. The peculiarity of this phenomenon is that it follows from the RG equations, which are off-mass-shell relations, and when the return to the physical theory is performed §:(1/2) β - Queens College, cuny, Flushing, ny 11367 correctly, we obtain these “induced identities”. They exist for all QFT models, for any imaginable RG Lee equation in the theory. These identities may include both non-perturbative and evolutionary parts §:(1/2) β - Queens College, cuny, Flushing, ny 11367.

It is very instructive to investigate this huge consequence of the RG-invariance. Let’s hope that this point will attract the attention of the scientific world. Although RG cannot provide any dynamic §:(1/2) β - Queens College, cuny, Flushing, ny 11367 information besides the conventional ones which follow from the field equations, the solutions of these identities will shed additional light on the structure of QFT.

☻So, no need to torture §:(1/2) β - Queens College, cuny, Flushing, ny 11367 oneself in attempting to solve the SD eqs. Just try skillfully to manipulate the IIs.

***************************************************************

6. Conclusions

I tried hard to find the non-perturbative asymptotically free solutions in the QFT. Let’s speculate §:(1/2) β - Queens College, cuny, Flushing, ny 11367 a little about this issue.

We understood that every non-perturbative solution may have a perturbative ‘tail’, which repeats the properties of the non-perturbative one; it is asymptotically free if the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 ‘predecessor’ was asymptotically free.

Can we suggest that there is a rather deep connection between these sides of the same property? Actually, we know that the material world consists of the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 sources – protons, electrons, etc. Also, it is well-known that the self-interaction of a source is singular – actually, the non-perturbative one. So, this interplay of the non-perturbative effective interaction with its perturbative §:(1/2) β - Queens College, cuny, Flushing, ny 11367 tail reflects, in reality, the common situation with the self-interaction of the source and its interaction with the outside world. May we suggest that this statement deserves some §:(1/2) β - Queens College, cuny, Flushing, ny 11367 recollections upon the issue?

***

This lengthy tale deserves, I guess, the attention of readers who are skillful and experienced in Quantum Field Theory. In short, results are the following:

- ^ The “immortal” Landau ‘ghosts §:(1/2) β - Queens College, cuny, Flushing, ny 11367’ are removed from the all traditional (the ‘old’) QFT models.
- The possibility of non-perturbative asymptotically free solutions is established, and one of them is actually found and investigated.
- The unique §:(1/2) β - Queens College, cuny, Flushing, ny 11367 and correct way of reasoning in the method of the Renormalization Group is established. It is developed in strict accordance with the requirements of principles of Mathematical Physics.
- ^ The instructive examples of connections of the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 old models with QCD are developed.
- The “Induced Identities”, following from the Lee differential RG-equations, can be obtained for unbound number of processes in elementary particles physics. They can be §:(1/2) β - Queens College, cuny, Flushing, ny 11367 very helpful for solutions of numerous vertices in QFT and, in support for solutions of horrendously difficult Dyson-Schwinger QFT equations.

When a person has enough knowledge and ability to §:(1/2) β - Queens College, cuny, Flushing, ny 11367 investigate the fundamentals of the QFT, he becomes amazed by the brilliance and perfection of the key investigators who have built the huge edifice of the Quantum Field Theory. Even their mistakes and misleadings are §:(1/2) β - Queens College, cuny, Flushing, ny 11367 very instructive since they permit penetration into the subtleties of their mental laboratories, so to speak. Besides, many of the newest developments in QFT, such as string, superstring, or supersymmetric theories, were §:(1/2) β - Queens College, cuny, Flushing, ny 11367, at least in part, responses to the difficulties of the conventional theories.

But the challenge to return to the enigmatical puzzles of the conventional QFT remains in order. The §:(1/2) β - Queens College, cuny, Flushing, ny 11367 infinite many possibilities of the non-perturbative solutions require that the responsible investigators would always be ready either to help God’s outlines or to encounter the devil’s intrigues. So, the §:(1/2) β - Queens College, cuny, Flushing, ny 11367 obligation of the QFT theoreticians is to try time and time again to find the most perspective solutions in this Generalized Quantum Field Theory.

Let’s follow this call and

GOD BLESS US §:(1/2) β - Queens College, cuny, Flushing, ny 11367 ALL!

Vsemayr Shakhbazyan

Ph.D. in Mathematics and Physics,

Senior Scholar in Theoretical Physics.

New York, 08/10/2009.