ИНСТИТУТ ТЕОРЕТИЧЕСКОЙ ФИЗИКИ

Transcription

1 АКАДЕМИЯ НАУК УКРАИНСКОЙ ССР к S к S ИНСТИТУТ ТЕОРЕТИЧЕСКОЙ ФИЗИКИ ITP E December, 1979 V.A.Miransky DYNAMIC MASS GENERATION AND RENQRMALI2ATICHS IN QUANTUM FIELD THEORIES

2 В.А.Миранский 1ТР Е Динамический механизм генерации масс частиц и перенормировки в квантовой теории поля Показано, что динамический механизм генерации масс частиц может вызвать нарушение мультипликативных перенормировочных соотношений и привести к расходимостям нового типа в массивной фазе. Для устранения этих расходимостей необходимо, чтобы затравочные константы связи принимали вполне определенные значения. Обсуждаются фазовые диаграммы калибровочных теорий поля. T.A.Miransky Препринт Шститута теоретической физики АН УССР Киев 1979 Dynamic Mass Generation and RenormaliBationa in Quantum Field Theories It is shown that the dynamic maae generation can destroy the multiplicative renormalization relations and lead to new type divergences in the massive phase. To remove these divergences the values of the bare coupling constants must be fixed. The phase diagrams of gauge theories are discussed. Preprint of the Institute for Theoretical Physics Academy of Sciences of the Ukrainian SSR Her 1979

3 Academy of Sciences of the Ukrainian SSR Institute for Theoretical Physics Preprint ITP E V.A.Miransky DYNAMIC MASS GENERATION AND EENORMALIZATIONS IB QUANTOM FIELD THEORIES Kiev

4 УДК В.Л.Миранский Динамический механизм генерации масс частиц и перенормировки в квантовой теории поля Показано, что динамический механизм генерации масс частиц может вызвать нарушение мультипликативных перенормировочных соотношений и привести к расходимостям нового типа в массивной фазе. Для устранения этих расходимостей необходимо, чтобы затравочные константы связи принимали вполне определенные значения. Обсуждаются фазовые диаграммы калибровочных теорий поля. It is shown that the dynamic mass generation can destroy the multiplicative renormalisaticm relations and lead to new type divergences in the massive phase. To remove these divergences the values of the bare coupling constants must be fixed. The phase diagrams of gauge theories are discussed. Dynamic Mass Generation and Renormalizations in Quantum Field Theories V.A.Miransky 1980 Институт теоретической физики АН УССР

5 Divergences in field theories with seal invariant lagrangians (for example,massless electrodynamics and chromodynamics)lead to a breakdown of scale invariance.in the normal phase of these theories where all bare and physical masses are equal to zero the breakdown is determined by the multiplicative renormalization relations [л] for Green functions r w (here {p]=(p it p ij4.,.p n ) f /\ piaye the role of a cut-off parameter, Л is the coupling constant) It ie commonly believed that the dynamic mass generation doesn't destroy these multiplicative relations. The consequence is that the physical mass ев are renormalization group invariant and the form of the massive phase Green functions <3oeea*t depend on (Л, л}< Is this scale invariance breakdown the шовь general? Sine* the renormalization relations (1) are proved only for the noiv mal phase of the theory, the answer to this questinn le not.

6 obvious. In this paper we indicate the dynamic mass generation mechanism which destroys the multiplicative relations (1). In the massive phase divergences of a new type occur. To remove these divergences the value of the bare coupling constant must be fixed. This value coincides with the critical point separating the maesless and the massive phases of theories. A number of dynamic phenomena in quantum field theories have analogues in quantum mechanics (for example, the tunneling process). So, it seems useful to examine the scale invariance breakdown for the massless Dirac equation with the Coulomb potential - dfy (t= *%;) For the values d.< i (^Jc - /J'/) the scale invariance of this equation is manifested in the absence of hydrogen-like stationary levels. In this case there is only a continuous spectrum with energy t^o and F^O (Dirac sea). However, when ci>i the situation is drastically changed. For these supercritical values of ck the fall into the Coulomb centre (collapse) takes place [2]. in this case it is necessary to complete a definition of the problem by introducing a cutoff Л~ fio at small distances '. The consequence is that the scale invariance becomes broken. For simplicity, a cutoff of the form Vfl)= 'ft &{T ' Zo) * t e«-v] (3) will be used. Following Refs. ^2,j7 * we shall look for the solutions corresponding to the Breit-Wigner (BW) resonance levels (J/n <(j which define an outgoing positron wave. The physical content of this solution is as follows: the lifetime of a resonance T^ defines the time of vacuum rearrangement under which the electron-positron pairs are created spontaneously from the vacuum; The mathematical reason for this is connected with the fact that the Dirac operator for the strong Coulomb field (A>1 has nonzero defect indices, [2].

7 -5- the positrons go to infinity and the electrons are coupled to the Coulomb centre. As a result, the supercritical charge of the centre is shielded. In the region Z T'Zc the solution of the Dirac equation with the potential (3) is expressed through the Whittaker functions [1] Щ, &0(Р- -idtsq^fitt j JC = -?iet ), and in the region 1 Xo through the elementary functions. By equating the logarithmic derivatives of these two solutions at ^- %Q, we get the energy spectrum for the /2 8W levels * - ёл (sap - iaaf) At small distances ( JC«J ) the Whittaker function where J/, CL are some numerical constants. In the limit f\~*oo, o( У 1 this function has an infinite number of seroes - the typical evidence of a collapse (the absence of the ground state) fzj. This agrees with the fact that the 6nergy <f ' becomes infinite in this limit. (jj However, there is another limit: A~* OO, *canst'. in this limit the value is fixed, the wave function (я) and all BW resonances with fl^ disappear ( -+Q,ft&C ).

8 -6- We can regard this limit as some renormalization procedure (compare with (2)): However, as can eae.ily verified, relations (1) for the function (5) are not valid. Moreover, in the limit A~*OO, с г ccmt the form of this function is drastically changed: oscillations disappear (see (?)). The physical reason for this is clear: removing the cutoff and preserving the energy to be finite we get rid of the collapse and, as a consequence, get rid of its manifestation, oscillations. Thus, the violation of the relations (1) is determined by the very essense of the fall into the centre phenomenon. We emphasize that the quantum-mechanical divergence considered above is not connected riththe loop divergences of field theories. In the massless QED the problem of the Coulomb centre is substituted by the positronium problem and B'.V resonances by tachyons. In Ref. [$] wi examined (in the ladder approximation) the Bethe-Salpeter (BS) equation for the positronium and showed that tachyon levels appear only for the supercritical values of the coupling cho><*c - Щ/ Tne mass spectrum of tachyons has the form - - /f /vn r j о - /\ expl ffi-r^j, «--#-; (9) /z - i г where /\ is the Lorentz-invariant cutoff. The occurence of tachyons implies the instability of normal phase vacuum. The vacuum rearrangement is determined by 'Just as in the Coulomb centre problem, the dynamics at the snail distances is primarily responsible for this tachyon solution. Therefore the coupling do nust be identified with the bare coupling coiistant.

9 -7- tachyon quantum numbers. In our case the BS wave function of the tachyon solution пав the form /"5/ where and г pectively (P' L =-/?/ ), *- (Ъ&р Xf'fy are the relative and the total momentum, res and the scalar functions (11) where ^ = /л*л /, /J ~ // - К, г is a hypergeometrie function. Since Eq. (10) iirdlies the tachyons have the chiral charge (Js = - 2«Therefore the vacuum rearrangement must lead to the chiral symmetry breakdown (fermion mass generation). Why is the normal phase vacuum rearrangement for massless QED so different from the vacuum rearrangement for the supercritical Coulomb centre? Let us imagine a test positron with a supercritical charge do ><^c Li^e the Coulomb centre, it creates an ev'-pair from the vacuum and binds the created electron thus forming a tachyon. But now the fate of the remaining positron is quite different: since its charge is also supercritical, it, in turn, creates an fv-pair, and eo on. As a result, the tachyon condensate with a chiral charge Qs arises. The physical reason for vacuum stabilization accompanied by the occurence of a fermion mass is clean as is known from the problem of the Coulomb centre, the critical value o<r grows with increasing fermion mass [2]. In massless QED the electron ac-

10 -вqulree a mass but the bare coupling constant remains unchanged. As a result, the vacuum becomes stable. The value of the physical fermion mass /71 can be determined by requiring that the tachyon mode disappears and the pseudobcalar Goldetone appears '. As it turns out to be Using asymptotic formulas for the hypergeometric functions we find that for large < d (small distances) Xs.T ~ ( r /f*j Cds(fiFT& fa t const). This asymptotic form corresponds to the collapse (сотр. with (5)).However, in the limit A~+co t //ff*tf)kootbb coupling cons tant is fixed, the function ^W/y = Щ (13) and all tachyons with /7Ъ-Z disappear. In Ref. [*?] we identified the critical value Ыс with the Gell-Mann-Low fixed point for the baxe coupling cko As it is seen, this value is the boundary of the collapse region f or ca o Just as the Coulomb wave functions the wave functions (12) don f t satisfy the multiplicative relations (1). The formal cause of this violation is connected here with the violation of the relation <А./{ /2& л where jj is the renormalization constant of the photon propagator. Indeed, despite the fact that in the lad-

11 -9- der approximation the constant ^ = 1, the charge renormalization (13) takes place for this solution. Thus, for supercritical do >'-*< there ie an additional (purely dynamic) renormalization of the charge. It is useful to compare this dynamic mass generation mechanism with the known Johnson-Jiak.er-Willey(JBW) mechanismz"67. In the approximation with a bare vertex and a free photon propagator [_6 f 7] the ultraviolet asymptotic behaviour of the solution of the Schwinger-Lyson equation for the fermion mass function/^/^cpropagator J - Aa/i»j- А//У) ) ie when the coupling c{ 0 ^o( c. This is the JBW solution. However, when the constant o(o becomes supercritical, the asymptotic behaviour is changed in a drastic way The bare fermion mass [7] f/ ). (16) Therefore for the JBW solution the bare mass is equal to zero only after the cutoff is removed. Another situation occurs for the solution (16) with the characteristic oscillations: in this case the condition (17) with zero bare maae determines the physical mass spectrum of the form (9) For the JBW solution the mass parameter /# doeen't depend on the cutoff ( $j\ ~C ) and the mass renormalizetion (18) takes place. This leads to an explicit rather than a spontaneous violation of ^.-symmetry [i]. In the examined approximation the charge renormalization for this solution is absent C^r/ )

12 -10- So in this саэе the renormalization group Q -function /i/ equals zero, which leads to the power asymptotic (15) for For the solution (16) the mass parameter /77. depends on A ^f О ) but Щ _ Ш J. <&# фд -0. The mass renormalizatlon is absent, but the additive charge renormalization (13) takes place. As a consequence, the multiplicative relations (1) are destroyed and the violation of the fc -symmetry (in the above approximation when tf s -anomalies are absent) is spontaneous. We would like to indicate a soluble model, the massless Thirring model, where fermion mass generation of the considered type occurs. In the normal phase of this model the charge renormalization is absent ( p -function equals zero) which leads to the scale covariant (with anomalous dimensions) Green fmictions [&]. However, as it was recently shown by Me Coy and Wu/"9/, in a regularized (by a lattice cutoff) version of this model the massive phase exists for the.supercritical coupling constant $t<$c - - Jl /l, When cutoff is removed and the fermion mass remains finite (О 4. /77 < & )i the coupling а Ьесотев fixed: q= 9 C = - fyz. The critical value 9c coincides with the boundary of the collapse region established by Coleman /io/. The phase diagram of the Thirring model is as follows. Since the (3-function equals zero for the subcritical values of the coupling constant%9> ~ J $k t these values form the line of the fixed points. The massive phase C^V- "^ ) bas only one fixed point * & ~ 9c ^ ~ J fe % separating the massless and the massive phases. In the approximation considered above the phase diagram of massless QED is very similar: fi>~ О for all &о/э(с» and the fixed point 0<c separates massless and massive phases '. In connection between phase diagrams of 4 dimension gauge theories and some ^-dimension models was discussed in Refs [Л1/using a different approach.

13 -11- the full theory the diagram that we suggest takes the following form: when <X a <?o( c -^ i, the vacuum polarization effects renonnalize the charge and lead to the Gaussian infrared stable fixed point o(=0 /i2/. The consequence is that when cutoff is removed, the free field theory corresponds to all values &o <"*^c (Landau-Pomeranchuk-Fradkin zero-charge situation [л] ). When c< 0 becomes larger than oi c t the additional charge renormalization arises. This renormalization works in a direction opposite to that of polarization effects: whence for the former сх/ч decreases with «increasing A (see Eq, 13 ), for the latter O(/\ increases as A increases (/j< l)fol. Go one can expect that in QED th non-gaussian fixed point ck c exists and the nontrivial '. ocal ( /\= со ) field theory corresponds to this value of ck o. Another situation occurs in quantum chromodynamics (QCD). As was shown in Refs. /"13»1^7» * ne Bathe-Salpeter equation of a pure Yang-Mills theory has the colourless tachyon solution i.e. the critical 9c x О here. A similar situation takes place Ln QPD with a small enough number of quark flavours f\b] The vacuum rearrangement corresponding to this tachyon can result in the gluons acquiring a mass. The colourless nature of ;he tachyon gaurantees the equality of the тавв for all gluons the role of gluon mass generation for the quark confinement is.iscussed in Ref. /15.7). The results of the present paper lead naturally to the folowing questions. Does an additional dynamic divergence correeond to the occurence of this colourless tachyon? Or is its t>ccu ence (as well as the colour tachyon pole in tha gluon propagatoi etermined by ordinary divergences? These questions will be disuse ed elsewhere. I am grateful to P.I.Fomin, V.P.Gusynin^ Tu.A.Sitenko for seful discussions.

14 -12- SEFERENCES 1. Bocoliubov N.N., Shirkov D.V. Introduction to the Theory of Quantized Fields, Moscow, Nauka, 1973, Zeldovich Ya.B., Popov V.S. Electronic Structure of Superheavy Atoms, Uspekhi fiz. nauk, 1971, 105, N 3, ; Rafelski J,, Fulcher L.P., Klein A. Fernions and Bosons Interacting with Arbitrarily Strong External Fields, Fhys. Rep., 1978, J58C, N 5, 229->D1. ; Fomin P.I., Miransky V.A. On the Dynamical Vacuum Rearrangement and the Problem of Fermion b'ass Generation, Phys.Lett., 1976, 64B, N 2, Bateman H., Erdelyi A. Higher Transcendental Functions,v1, New York, McGraw-Hill Book Company, 1955, Fomin P.I., Gusynin V.P., Miransky V.A. Vacuum Instability of Massless Electrodynamics and the Gell-:/.ann-Low Eigenvalue Condition for the Bare Coupling Constant, Phys.Lett., 1978, 8» K 1» ; 6. Johnson E. f Baker M., r; illey R. Self-Energy of the Electron, j Phys.Rev., 1964, 136B. N 4, j 7. Pagels H. Departures from Chiral Symmetry, Phys.Eep., 197^» 16C, К 4, 219-3'ii2; Fukuda R., Kugo T. Schwinger-Dyson Equation for Massless Vector Theory and the Absence of a Fermion Pole, Nucl., Phys., 1976, 117B. H 1, Klaiber B. The Thirring Model, in: Lecture in Theoretical i Physics, Boulder Lectures, bew York, Gordon and 2reach, 1968 I ! 9. Me CoyB.M., V/u T.T. Dynamic lhass Generation and the Thirring Model., Phys.Lett., 1979i 8^, N 1, lo.coleman S. Quantum Sine-Gordon Equation as the Massive Thir- ring Kodel, 1975, 11, N 8, ba.gdal A.A. Phase Transitions in Gauge and Spin Lattice Systems, Journ.of Exp.and Theor. Phys., 1975, 2., N 4, ; Tolyakov A.il. Quark Confinement and Topology of Gauge Theories, Nucl.Phys., 1977, 120, К 3, 429-^58;

15 -13- Kadanoff L.P. The Application of Renoxmalisation Group Techniques to Qaark and Strings, 1977, 49., H 2, , 12. Wilson E.G., Eogut J. The Renormalization Group and the -Expansion, Phys.Bep., 1974, 12C, H 2, * Fukuda R. Tachyon Bound State in Tang-lulls Theory and Instability of the Vacuum, Phys.Lett., 1978, 222» H 1 i * Gusynin V.P., Miransky V.A. On the Vacuum Bearrangement in Maesless Chromodynamica, Phys.Lett., 1978, 2 B» N 5, * Wilson E.G. Quantum Chromodynamice on a Lattice, In: New Developments in Quantum Field Theory and Statistical Mechanics, Carge'se Lectures, New York, Plenum, 1977, 't Hooft G.Qn the Phase Transition Towards Permanent Quark Confinement, Hucl.Phys., 1978, 138B. 1-35; Hack G. Colour Screening and Quark Confinement, Phys.Lett., 1978, 2 St N 2, Received December 7, 1979 Киранский Владимир Адольфович Динамический механизм генерации масс частиц и перенормировки в квантовой теории поля Редактор А.А.Храброва Техн. редактор О.В.Угрюмова Зак. 19 Формат 60x90/16 Уч.-изд. 0,5 д. Подписано к печати 29. XIIЛ979г. Тираж 295 Цена 4коп. Офсетная лаборатория Института теоретической физики АН УССР

16 4 коп. I с. " I- > ~ J Препринты Института теоретической физики АН УССР рассылаются научлым организациям и отдельным ученым на оснопе взаимного обмена. Наш адрес: Киев-130 ИТФ АН УССР Информационный отдел The preprints of the Institute for Theoretical Physics are distributed to scientific institutions and individual scientists on the mutual exchange basis. Our address: Information Department Institute for Theoretical Physics , Kiev-130, USSR