научная статья по теме RENORMDYNAMICS, MULTIPARTICLE PRODUCTION, NEGATIVE BINOMIAL DISTRIBUTION, AND RIEMANN ZETA FUNCTION Физика

Текст научной статьи на тему «RENORMDYNAMICS, MULTIPARTICLE PRODUCTION, NEGATIVE BINOMIAL DISTRIBUTION, AND RIEMANN ZETA FUNCTION»

ЯДЕРНАЯ ФИЗИКА, 2013, том 76, № 9, с. 1228-1239

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

RENORMDYNAMICS, MULTIPARTICLE PRODUCTION, NEGATIVE BINOMIAL DISTRIBUTION, AND RIEMANN ZETA FUNCTION

©2013 N. V. Makhaldiani*

Joint Institute for Nuclear Research, Dubna, Russia Received July 17,2012

After short introduction, we consider different aspects of the renormdynamics. Then scaling functions of the multiparticle production processes and corresponding stochastic dynamics are considered. Nonperturba-tive quasi-particle dynamics is considered on the base of the toy QCD—O(N)—sigma model. Last section concerns to the NBD—Riemann zeta function connection.

DOI: 10.7868/S0044002713080217

1. INTRODUCTION

In the Universe, matter has mainly two geometric structures, homogeneous [1] and hierarchical [2].

The homogeneous structures are naturally described by real numbers with an infinite number of digits in the fractional part and usual Archimedean metrics. The hierarchical structures are described with p-adic numbers with an infinite number of digits in the integer part and non-Archimedean metrics [3].

A discrete, finite, regularized, version of the homogenous structures are homogeneous lattices with constant steps and distance rising as arithmetic progression. The discrete version of the hierarchical structures is hierarchical lattice-tree with scale rising in geometric progression.

2. RENORMDYNAMICS

In Quantum Field Theory (QFT) existence of a given theory means, that we can control its behavior at some scales (short or large distances) by renormal-ization theory [4, 5].

If the theory exists, than we want to solve it, which means to determine what happens on other (large or short) scales. This is the problem (and content) of Renormdynamics (RD). RD unifies different renorm-groups in one society.

The result of the RD, the solution of its discrete or continual motion equations, is the effective QFT on a given scale (different from the initial one).

E-mail: mnv@jinr.ru

2.1. General Method of Solution of the RD Equation

The expression of the ft function can be obtained in the following way. The canonical dimensions of the bare fields and constants in the d-dimensional space—time are

[m] = 1, [A] =

d- 2

[Ф]

d- 1

(1)

= d - [A] - 2[Ф] =

4- d

d = 4 - 2e, [ab] = [g2b] = 2e,

ab = ß2eZa,

0 = dab/dt = d(ß Za)/dt = 2e ( £ d(Za) da\ da ^ \ da dt J dt

= ß(a,e) =

—eZa

d(Za) da

= -ea + ß(a),

where

m = a^zl)t

d-4

ß(a, e) = —-—a + ß(a)

(2)

is d-dimensional f3 function and Z\ is the residue of the first pole in e expansion

Z(a, e) = 1 + Zie"1 + ... + Zrae~n + ... (3)

Since Z does not depend explicitly on fx, the ft function is the same in all MS-like schemes, i.e. within the class of renormalization schemes which differ by the shift of the parameter f .

The higher residue of the pole expansion can be defined from (1),

0 = eZa +

д(Za) da da dt

(4)

aZi aZ2

= e a-\---1--T- +

+

£

2

(aZi)' , (aZ2)'

+ ..J (—£a + ß(a)) =

+

= ß - a(aZ{)' + aZi + ~(aZ2 - a(aZ2)' +

£

+ ß(aZi)') + ... ^ ß(a) = a

2 dZn+i Q, N d(aZn)

a —:— = p(a)-

da

da

2dZ1 da

n> 1,

This condition gives the restriction on the charge, correspondingly, mass, size, ..., of the cumulative quasi-particles p — Fluktons.

We have a dual symmetry with respect to the change a± ^ aT, or a ^ Z .If a = a- is not small, we can expend with respect to a+ = ab/a = Z = 1 — — a.

2.3. Discrete Renormdynamic Interpretation If we consider ab as effective coupling constant at

, ,, , , . , , , ,, . the nth RD iteration, ab = an, and take a = an+\, we

where the last equation includes also the previous .,,, n+1

will have

one, if we define, Z0 = 1 Than we find

Zn = BnZo, (5)

È—faf)= f ¿z

" a2 da / da '

B' = Id4l{af> = Idz>-

Z = (1 — B/£)-1 Zo = £(£ — B)-lZ0.

These formal considerations are correct in the nonstandard (non-Archimedean) analysis [6]. There is another way of finding Z,

Z (a) =

ab

da

2e '

-£a + ß(a)

= t.

(6)

For any multiplicative renormalized quantity A, Ab = Z (£, a) A,

A = -Y(£,a)A, y = Z/Z, (7)

/ a

/ f i Y(£,a)

A = exp — / da-—- .

1 i -ea + ¡3{a)

for A = a, y = £ — (3(a)//a.

2.2. Dual Symmetry

Let us take for renormalization constant in critical dimension

^ a± =

0 < Z(a, 0) = 1 - a < 1, 0 < a < 1, ab = a(1 — a) 1 ± VI - 4ab

(8)

2

0 <ab<-.

The solutions for the observable values of the coupling constant a± has the property

a+a- = ab, (9)

which is similar with the Dirac's quantization

aeam = 7 = (47r)2a6 = eg, eb = (10)

4 ab

n2 1 64-/T2 ~ 4

n 12.

an = an+1(1 — an+l), 1

0 < an < an-|-i < , an

(11)

2.4. Solution of the RD Equation RD equation, for the coupling constant a a = (3\a + (32a2 + ... can be reparametrized,

a(t) = f (A(t)) = A + f2A2 + ... +

+ fnAn + ... = £ fnAn,

(12)

n>1

A = biA + b2A2 + ... = bnAn,

ni

(13)

à = Af'(A) = (biA + b2A2 + ...) x

x (1 + 2f2A + ... + nfnAn-i + ...) =

= ßi(A + f2A2 + ... + fnAn + ...) + + ß2(A2 + 2f2A3 + ...) + ... + + ßn (An + nf2An+i + ...) + ... =

= ßiA + (ß2 + ßi f2)A2 + (ß3 + 2ß2f2 + + ßif3)A3 + ... + (ßn + (n — 1)ßn-if2 + ... +

+ ßifn)An + ... =

— ^ ^ A bnin2fn2 &n,ni +n2-i — nnin2>i mi...mk>0

= Y, " Anßmfr f f (n,m,mi,..,mk),

nm> i

m!

f (n,m,mi, ...,mk) =

,mi +2m2+... +kmk "m,mi +m2 +•••+mk

Ön

mi!...mk !

bi = ßi, (14)

b2 = ß2 + f2ßi — 2f2bi = ß2 — f2ßi,

£

2

£

£

a

b3 = $3 + 2/2/2 + /3/1 - 2/262 - 3/361 =

= & +2/ - /3)Pi, 64 = $4 + 3/2/33 + /2 $2 + 2/302 - 3/4bi -

- 3/3b2 - 2/263, ... bn = 3n + ... + 3l/n - 2/2bn-1 - ... - n/nbl, ...

So, by reparametrization, beyond the critical dimension ($1 = 0) we can change any coefficient but $1.

We can fix any higher coefficient with zero value, if we take

f2-ßl>

f — if2 f — ^ /3~ 2ft1 +hl (n-l)/3i ' '

In this case we have simple scale dynamics,

A = (p/ßo)-2eAo = e-2er Ao,

g = f (A(t )).

A = ß2A2 + ß3 A3,

e.g. 64 = 0 when

f ßi ßä , ,2 h-j2+jh + f2,

f2 remains arbitrary and we may put f2 = 0. We can solve (17) as implicit function,

uß3/ß2 e-u = ceß2

1 , fta

U=A + W

2.5. Renormdynamic Functions (RDF)

We will call RDF functions gn = fn(t), which are solutions of the RD motion equations

gn = ßn(g), 1 < n < N.

(20)

In the simplest case of one coupling constant, the function g = f (t), is constant g = gc when ft(gc) = 0, or is invertible (monotone). Indeed,

g = f '(t) = f '(f-i(g)) = ß(g).

(21)

(15)

(16)

We will consider in applications the case when only one of higher coefficient is nonzero.

In the critical dimension of space—time, $1 = = 0, and we can change by reparametrization any coefficient but $2 and $3. From the relations (14), in the critical dimenshion ($1 = 0), we find that we can define the minimal form of the RD equation

(17)

Each monotone interval ends by ultraviolet and infrared fixed points and describes corresponding phase of the system.

Based on real experiments and computer simulations, quantum gauge theory in four dimensions is believed to have a mass gap. This is one of the most fundamental facts that makes the Universe the way it is. In the lattice (gauge) theory approach to the RD (see, e.g., [9]), recently running coupling constant dynamics were calculated for SU(2) Yang-Mills model [10]. The result is in agreement with perturbative calculations at small scales; at an intermediate scale the coupling constant reaches its maximum then decrease. So, at the maximum, we may have nontrivial zero of the ft function, which corresponds to the conformal invariance of the glu-odynamics at this point. Beyond this point we have another phase, strong coupling phase with decreasing coupling constant similar to the abelian theory.

If we approximate the form of the curve near maximum as

a(t)= ac - 6\t — tc\r

(18) for the ß function we obtain

a =

ß(a, t) = sign(tc — t)bn (

ac a

6

(22)

. (23)

(19)

then, as in the noncritical case, explicit solution for a will be given by reparametrization representation (13).

If we know somehow the coefficients ftn, e.g. for first several exact and for other asymptotic values [7] than we can construct reparametrization function (13) and find the dynamics of the running coupling constant. This is similar to the action-angular canonical transformation of the analytic mechanics (see e.gv [8]).

Of course this is not usual ft function, function of a only. It depends also on t. For t > tc we have perturbative phase. For n > 1, ft(ac,t) = 0. Explicit dependence on time variable in one coupling case indicates on implicit two coupling cases.

In the case of the two coupling constants,

gi = fti(gi ,g2), (J2 = ft2(gi,g2), (24)

we can reformulate RD as

gi = g, g2 = f2(t) = т,

dgi dg_ = . fti (g,r)

d92 dr~9 P[9'T) ft2(g,T)'

(25)

n-1

2.6. Ultraviolet (Infrared) Fixed Point for QCD (QED)

Perturbation theory results for QCD (QED) give negative (positive) (3 function, in one-loop approximation

a = ß2a

QCD :ß2 = ( y ~ Y

(26)

QED ß2 = -■

ß(a, £) = —£a + ß(a),

(27)

has stable ultraviolet (infrared) fixed point for negative (positive) value of £,

£ = (3(a)/a. (28)

The fundamental quark and gluon degrees of freedom are the relevant ones at high temperatures and/or densities. Since these degrees of freedom are confined in the low temperature and density regime there must be a quark and/or gluon (de)confinement phase transition.

It is difficult to describe the phase transition because there is not known a local parameter which can be linked to confinement. We consider the fractal dimension of the hadronic/quark—gluon space as an order parameter of the (de)confinement phase transition. It has a value less than 3 in the Abelian, hadronic, phase, and more than 3, in non-Abelian, quark—gluon, phase.

2.7. Two TeV Scale Unification of the Standard Model Coupling Constants

According to the LEP and Tevatron data, the standard model (SM) coupling constants at the Z-boson mass scale take the values (see, e.g., [11])

a\(mz ) = 0.017, ai(mz )-1 = 58.8, (29) a2(mz) = 0.034, a2(mz )-1 = 29.4, a3(mz ) = 0.118, a3(mz )-1 =8.47.

Our aim is to

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