Fortschritte der Physik / Progress of Physics: Volume 32, Number 11 [Reprint 2022 ed.] 9783112656266

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 • 1984 Number 11 Board of Editors F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief

F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakhakov Calculations in External Fields in Quantum Chromodynamics. Technical Review

585-622

AKADEMIE-VERLAG • BERLIN ISSN 0015 - 8208

Fortschr. Phys., Berlin 32 (1984) 11, 585-622

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copics are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the authors name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawing should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawing and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp, ...): black underlined Greek letters: red underlined Boldface Greek letters: red interlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c C, k K, o 0, p P, s S, u U, v V, w W, x X, y Y, z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, bj, M i l , Mij, Wn^ Please differentiate between following symbols: a, a ; a , a , oo; a, d; c, C, c ; e, I, S, e, k, K, x, x; x, X, x, X;l,l;o,0 a, 0; p, g; u, U, U. v, v, V, one can perform integration over the fermion degrees of freedom considering the vector field configuration Alla(x) as a given external field. Then

(P2-Wo

= Tr c / d*x{xI (P 2 - m 2 )!^- 1 '

02(x); (

/

™6-2M

594

V. A. Novikov, M. A. Shifman et al.: Calculations in External Fields

As a result we get g1 However what if we want to make the next step and find which higher-order terms are actually denoted by dots in eq. (1.16)? To make further step we use the already known trick — shift of variable P by a vector q in the quantity

where F(G) is arbitrary function of the field G. Concentrating ourselves on the coefficient of q2 we get the following relation T r

{ ( P T ^

=

¿

i ' ^

7

F(

-G> -

T r

{

(

P

^

• (1.18)

(The first term in the right-hand side of eq. (1.18) has appeared from the regulator contribution —Tr {(P2 — m2)jl0 F(G)} which must be added to eq. (1.17) in order to make the whole expression convergent). Differentiating eq. (1.18) with respect to the quark mass one readily gets the trace of any power Tr {(P2 — m2)~n F(G)}. The final result for I 0 has the form jgi /o

=

3-2«.«*.««

r J

( d X

*

TTc

I

gi

f ^ "

~

2m*

(1.19) [ A S

« *

m

=

J

-

^

f

2

l n

^

^

-

*

Let us consider now the next commutator { ( P ^ W =

~~ %

Tr

{(P 2 - m 2 f

^

=

S

+

T

r

{(pi -

W 2 )5

•

The second term in the last equality can be omitted since it induces only the terms of an order higher than Gi. Indeed, the matrix element (x) iy^x),

y(0) iyiW(0)})

where xp is the heavy quark field. First of all we rewrite it identically as follows tf _ x 384ji4

¿'5

+ xPtfn, re"

' '

'

where we have used also eq. (2.21). Having checked that the result (2.32) is transversal, as it should be, we contract indices and v and go to the momentum space (the formulae collected in Appendix D will help us). Finally, Hfipiq) I light quarks, i~ = ¿ j ? 2 In Q2 -

± {g*G*),

(2.33)

which exactly coincides with the answer obtained first in ref. [1]. The whole computation takes no more than 10 minutes. I t is. worth considering one more pedagogical example, only indirectly connected with the subject of the present review — the triangle anomaly in divergence of the axial current. Here the advantages of the proposed computational scheme are demonstrated from another point of view. If we regularize the axial current, following Schwinger, by virtue of e splitting, then =

+ ®) W s ( e x P f

dZp

dy)j f(x —

= y)(x + e) {—igA(x + e) ys — ystg^(x

e)j

— e) + igy^y^G^O)}

y{x — e),

where the third term has emerged from differentiation of the exponential and, as usual, A„ = 1/2 This first step is more or less standard. (Notice, however, that we have used expansion (2.7) having substituted A^{y) = 1/2 yeGeit(0)). The further derivation is essentially shortened as compared to the normal route by use of expansion (2.31) for quark propagator in the Foek-Sehwinger gauge. Only the second term of the expansion is operative. Namely, W

= -V = - J

3

Tr

[-21VU0)

0?,(0) Operator

¡DO]2

Coefficient

C-q'^g2

e)

Operator In

q2/p2

Coefficient

YylDGJY C-q-tgf

b)

Fig. 7. Operators (DO)2 and yiy(DG) y> in the correlation function of vector currents. Diagram a gives an infrared logarithm reflecting mixing of the operators

Let us draw the reader's attention to one curious fact. In the quark propagator (2.34) it is impossible to separate the structure singlet with respect to Lorentz and colour indices, Strictly speaking, this structure can be separated with zero coefficient as it is seen from eq. (2.31) with y — 0. At the same time,»in gluon propagator it is explicitly present (see the last term in eq. (2.35). Not accidentally. Analytical properties of the propagators are such that 8(x, 0) does not admit singularities of the type In x 2 while D ^ x , 0) does admit it (here the bar denotes the Lorentz and colour averaging). One can easily prove the latter assertion following the line of reasoning of ref. [6a], In this work a related problem was discussed: analytical properties of quark two-point functions in self-dual fields. 2.5.

Quark Operators

Effects due to quark condensate are essential almost in all applications. They play the dominant role, for instance, in the sum rules for q mesons, D mesons and, especially, baryons. As a rule, calculation of the corresponding coefficients is much simpler technically than for gluon operators. Almost always, after drawing the relevant graph, we immediately read off the answer. Still, even here there exist some useful devices which will be demonstrated below in a few examples.

612

V.

A . NOVIKOV, M . A . SHIFMAN

et al.: Calculations in External Fields

As in other training exercises consider the correlation function (2.16) of vector currents, jp = y>yMy), where ip is the massless quark field. Let one of the quark line be soft. Then, cutting it (fig. 7), we get ^ ( ^ » / ^ ¿ V l i W i - f O ) ) ) yA*> °) yMO) + m

y,m

x) y„v(®)}, -

(2.37)

where the cut lines are represented by the Heisenberg operators ip(x) and y>(0). For large q, to the leading order in q*1, the field ip(x) can be substituted by y>(0). Accounting for the fact t h a t (ipj{0) yijp{0)) = 1/3 df 1/4 dap we arrive at the following expression = ~

(vw) Tr { y M

+ vM~9)

yA,

(2-38)

which vanishes if the quark mass is neglected. The zero result in the chiral limit is obvious beforehand since the structure ytp is chirally non-invariant and appears in the correlation function (2.37) only being multiplied by the quark mass m. Thus, at least this general property is fulfilled and there is no error. Feeling complete satisfaction from the internal selfconsistency we now put the problem of determining the coefficient in front of myiip. For this problem the approximation ip(x) = y>(0) in eq. (2.37) is not sufficient, and it is necessary to account for the next term of expansion in (2.9) V(x)

= y(0) + xeDM 0)

bearing in mind t h a t DQyi will eventually become &ip = —imyj. I t is no surprise t h a t xeDe%p(0) induces a new term in IT^(q), namely, - /

d*x{f(0) xeDeyllS{x, 0) yyy>(0) + y(0) yvS(0, x) ypfiM°)>

•

Observing that {yjD Mp) = j

B-y>) =

« m ( w )

we reduce this extra term to 1 ~ _

d _

Tr ( - y ^ ( i ) yvye + yrS(-q)

yhye).

(2.39)

Combining now eqs. (2.38) and (2.39) we obtain = ( ^ f ( 2 This result was exploited many times in ref. [1], One more example referring to the operator of dimension 6,

.

4

0

)

appearing in eq. (2.16) already in the chiral limit. Correction to n ^ q ) associated with this operator is of order q~l. The corresponding diagram is depicted in fig. 7 b. The consistent procedure requires, generally speaking, expanding ip{x) up to terms 0(xs), so

613

Fortschr. Phys. 32 (1984) 11

t h a t in Il^iq) we deal with three different structures, -> —2 f eigx dlxxp{0) {y^S>( 0)}}.

(2.44)

The Fock-Schwinger gauge condition (2.2) is nonlocal in the momentum space, and one should work very hardly in order to write Green function D a satisfying this condition. Even having overcome all difficulties one would certainly get an expression useless from the practical point of view. However, there is absolutely no need in this work. Indeed, let us assume t h a t the Fock-Schwinger condition is imposed on external field

614

V . A . NOVIKOV, M. A . SHIFMAN

et al.: Calculations in External Fields

Substituting eq. (2.43) into standard QCD Lagrangian we get L = - J (G^)„t - j

(Z>„ ex V) s + J

(D«V)

(2.46) D^a,"

=

+ gfabc(A/)ext

a/.

This Lagrangian describes dynamics of field a* in the background field (AMa)ext and still possesses gauge freedom; namely it is invariant under the transformations ext — 17-1(3:) {A,(x)) ext

ext

< = U~\x)

a„(x)

a^x) U(x) + j

V~\x)

d^x),

with arbitrary unitary matrix U(x). It is evident that the first of these transformations does not violate condition (2.45): x ^ A ^ x ) ) ^ t automatically vanishes provided X/lA^x))^t = 0. To extract the maximal profit from this gauge freedom we fix the gauge of a^ as follows. We add to Lagrangian (2.46) the term 4- W V ) 2

2

(2.4V)

\

l\ I \ I \ Jc. •it

\

\ \ ¡Ac

Jt a)

- H

I I I I I I I I I Jt

Jt b)

0s

K g . 8. Operator in the two-point function of gluon currents. Crosses mark soft lines annihilated by the vacuum (a); b — symbolic picture of the same two-point function in the external field. Gluon propagator (—•=•«»•-) and vertices (circles) are taken in the external field

and, certainly, the ghosts according to standard rules. This is the so called background gauge extensively exploited by DE W I T T [24], 'T HOOFT [ 1 8 ] and in many recent works, both purely theoretical, and of more pragmatic character [ 2 5 — 2 7 ] , Gluon propagator, eq. ( 2 . 3 5 ) , is given just in this gauge. The exercise we concentrate on in this Section is the calculation of the Gs correction in the two-point function of scalar gluonic currents, IS =

G%{0)

2>.a,(0)}> G%{0)j

(2.50)

where subscripts (0, 2) refer to dimension of gluon operators and subscript (ext) is omitted. In particular, (0) marks the free propagator ffvô^ab • One can easily convince oneself that after separating Lorentz-singlet structure the first term in eq. (2.50) vanishes. The second term, in turn, reduces to 16ash'f

d4xeiqxG*v(0)

= 16ocshJ

• 4 this expression is singular — it has a pole — the corresponding residue, however, is proportional to (p 2 ) n ~ 2 . As was already noted, any polynomial in p2 is inessential and can be omitted. The non-singular part of eq. (D.4) evidently contains In p2. Separating the nonsingular part and returning to the Minkowski space we have

/

' d^r (x2)n

jp* =

ii Dn 2i~2niT2 2 2 A ' (p )«- In ( - p 2 ) . r(n— 1) J »

(D.5)

This expression generates a series of other useful relations, for instance (xtr „,, ^ J dW^ -

(xVr

in22i~3n~2ln\ (2n + I - 1)! (ra + I -

2 2 n+l 2 2)! (£?)" (—P ) ~ In (— p ),

(D.6)

where f, rj are auxiliary vectors, £2 = ^ = 0,

Sp = np = 0,

|ij*0.

We encounter with transformation (D.6) in the problem of mesons with spin n. Its derivation is given in ref. [23],

Fortschr. P h y s . 32 (1984) 11

621

The key observation on which the derivation is based is the following

I

"-l~(2

n ) \ J

a X e

'

while integral a

XC

{x2^n+l

is already known.

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[12] K . G. CHETYBKIN, P h y s . L e t t . 126B, 371 (1983). [13] F . DAVID, Nucl. Phys. B 234, 237 (1984); M. SOLDATE^ P r e p r i n t SLAC-PUB-3054, S t a n f o r d , 1983; Ann. P h y s . (N. Y.) in press. [ 1 4 ] A . V A I N S H T E I N , V . Z A K H A B O V , V . N O V I K O V , M . S H I F M A N , Uspechi Fiz. N a u k 1 3 6 , 5 5 3 ( 1 9 8 2 ) . [ 1 5 ] A . V A I N S H T E I N , V . ZAKHABOV, V . NOVIKOV, M . SHIFMAN, Y a d . F i z . 3 9 , 9 ( 1 9 8 3 ) .

[16] S . N . N I K O L A E V , A . V . R A D Y S H K I N , P r e p r i n t J I N R P2-82-914, D u b n a , 1982. [17] N . ANDREI, D . J . GROSS, P h y s . R e v . D 18, 4 6 8 (1978).

Phys. Rev. D 14, 3 4 3 2 (1976). Nucl. P h y s . B 1 5 9 , 4 2 9 ( 1 9 7 9 ) ; V . N O V I K O V , M . S H I F M A N , A . V A I N S H T E I N , V . Z A K H A B O V , Nucl. P h y s . B 2 2 3 , 4 4 5 ( 1 9 8 3 ) .

[18] G. T'HOOFT,

[19] M . T . GRISARU, W . SIEGEL, H . ROCEK,

4

Fortschr. Phys., Bd. 32, H. 11

622

V . A . NOVIKOV, M . A . SHIFMAN

et al. : Calculations in External Fields

[ 2 0 ] V . F A T E E V , A . SCHWABZ, Y U . T Y U P K I N ,

Preprint Lebedev Physical Institute

NO. 1 5 5 ,

Mos-

cow (1976);

C. CBONSTROM, Phys. Lett. 90B, 267 (1980); M. S. DUBOVIKOV, A. V. SMILGA, Nucl. Phys. B 185, 109 (1981). [21] S. NIKOLAEV, A. RADYTJSHKIN, Nucl. Phys. B 218, 305 (1981). [22] M. SHIFMAN, Nucl. Phys. B 173, 13 (1980). [ 2 3 ] M. SHIFMAN, Y a d . F i z . 3 6 , 1 2 9 0 (1982).

[24] B. S. DE WITT, Phys. Rev. 162, 1195, 1239 (1967). [25] J . HONEBKAMP, Nucl. Phys. B 36, 130 (1971); B 48, 269 (1972); R. KALLOSH, Nucl. Phys. B 78, 293 (1974). [26] G. 'T HOOFT, Nucl. Phys. B 62, 444 (1973). [27] M. T. G B I S A E U et al., Phys. Rev. D 1 2 , 3203 (1975); L. F. ABBOTT, Nucl. Phys. B 185, 189 (1981). [ 2 8 ] A . V . SMILGA, M . A . SHIFMAN, Y a d . F i z . 3 7 , 1 6 1 3 ( 1 9 8 3 ) .

[29] S. NARISON, E. DE RAFAEL, Phys. Lett. 103B, 57 (1981). [30] M. B. VOLOSHIN, Yad. Fiz. 29, 1368 (1979); Nucl. Phys. B 154, 365 (1979); Preprint ITEP-21, Moscow 1980; J . S . B E L L , R . A . BERTLMANN, Nucl. Phys. M B 1 7 7 , 2 1 8 ( 1 9 8 1 ) ; R . A . BERTLMANN, Nucl. Phys. B 2 0 4 , 3 8 7 ( 1 9 8 2 ) . [31] E. V. SHURYAK, Nucl. Phys. B 198, 83 (1982); V . L . CHERNYAK,

A . R . ZHITNITSKY,

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(1983).

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The journal "Portschritte der Physik/Progress of Physics" is devoted t o t h e theoretical and experimental s t u d y of t h e fundamental constituents of m a t t e r and their interactions, so to elementary particle physics, classical and q u a n t u m field theory, theory of graviation, thermodynamics and statistics, nuclear physics, laser physics, and plasma physics. The articles should have in general review character. Manuscript should be addressed to t h e Editor-in-Chief of the journal Prof. Dr. F , Kaschluhn Sektion Physik der Humboldt-Universität zu Berlin D D R - 1 0 8 6 Berlin, P S F 1297 or to editors or to members of the Advisory Board.

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Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. F . Kaschluhn, Prof. A. Lösche, Prof. Dr. R . Rompe. Verlag: Akademie-Verlag, D D R - 1 0 8 6 Berlin, Leipziger Str. 3—4; Fernruf: 2236221 und 2 2 3 6 2 2 9 ; Telex-Nr.: 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 1 0 4 0 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki". D D R - 7 4 0 0 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 H e f t e eines Jahres bilden einen Band. Bezugspreis je Band 216,— M zuzüglich Versandspesen. Preis je H e f t 1 8 , - M. Bestellnummer dieses Heftes: 1027/32/11. Urheberrecht: Alle Rechte vorbehalten, insbesondere die der Übersetzung. Kein Teil dieser Zeitschrift darf in irgendeiner Form — durch Photokopie, Mikrofilm oder irgendein anderes Verfahren — ohne schriftliche Genehmigung des Verlages reproduziert werden. All rights reserved (including those of translations into foreign languages). No part of this issue m a y berreproduced in any form, b y photoprint, microfilm or any other means, without written permission f r o m t h e publishers. © 1984 b y Akademie-Verlag Berlin. P r i n t e d in the German Democratic Republic. AN (EDV) 57618

Contents of the following issues:

R. C. EDGAR: Being Discrete about Yang and Mills: Basic Techniques of Euclidean Lattice Gauge Theory H. V. KLAPDOR: Beta Decay Far from Stability and Its Role in Nuclear Physics and Astrophysics L. L. DERAAD, J r . : Source Theory Treatment of the Casimir Effect: A Review E. W. MIELKE: On Pseudoparticle Solutions in the Poincaré Gauge Theory of Gravity P . GAIGG, M. SCHWEDA, 0 . PIQUET, K . SIBOLD: T h e One-Loop E f f e c t i v e Action of t h e Super-

symmetric CP(N-L) Model A. W. HENDRY, C. B. LICHTENBERG: Properties of Hadrons in the Quark Model

Progress of Physics is indexed in Current Contents/Physical, Chemical & Earth Sciences.