0 6 h e A M H e H H bl M MHCTMTYT RAB PHbiX

MCCnBAOB3HMM

AY6Ha

I lf I~ -1'7 E2 - 12249

D.I.Kazakov, O.V.Tarasov, A.A.Vladimirov

ON THE CALCULATION

OF CRITICAL EXPONENTS

BY THE METHODS

OF QUANTUM FIELD THEORY

1979

E2 - 12249

D.I.Kazakov, O.V.Tarasov, A.A.VIadimirov

ON THE (:AL(:ULATION

Of' CRITICAL EXPONENTS

BY THE METHODS

OF QUANTUM FIELD THEORY

Submitted to )J(3T4>

Ka3aKOB .£].11., H ap. E2 - 12249

0 BbltrHCII€HHH KpHTI.PI€CKHX HHlleKCOB MeTO.QBMH KBBHTOBOii

reopuu nonR

¢lyHKUHSI fenn-MaHHa-Jloy H 8HOM81IbHbl€ pa3MepHOCTH KBBHTOBOITOIIeBOfi

MO.ll€1IH c/J 4 Bbi't!HCII€Hbl B 'lJeTbipexneTII€BOM llpH61IH)K€HHH B rflopM81IH3Me

paaMepaoH peaopMuposKu. Ha ux ocaoae aaiineHbi xo:::}(P4luuueHTbi f -paano­)K€HHR lliiR KpHTH't{€CKHX HH.Q€KCOB llO CTelleHH f 4 BKIIID't{HT€1IhHO. Dng 06pa­

QOTK)I PSI.UOB f _ -p831IOW.:€HHR HCI101Ib30B8H M€T0ll CYMMRpOBBHHR, BKIII01.{810lllHi1

MOllHIPHUHpOBBHHOe rrpeo6pa30B8HHe EopenSI H KOH!flopMHOe OT06p8)K9HHe. no­nyqeHHbie 3H8't{€HHSI KpKTH'tJ€CKHX HH.Q€KCOB HBXO/lSITCSI B XOp011I€M COI'IIBCHH

c aKcnepuMeHTOM H c peaynhraTaMu apyrwx Teoperuqecxux nonxonos.

Pa6ora Bbmonaeaa s Jla6opaTopHH TeopeTuqecKoH <PuauKu 011.HH.

npenpKHT 05'be.llKHeHHOrO HHCTKTyTa H.llepHbiX KCCnellOB8HHH • .Qy5Ha 1979

Kazakov D,I., et al, E2 - 12249

On the Calculation of Critical Exponents by the Methods of Quantum Field Theory

The Gell-Mann-Low function a rx:l the anomalous dimensions ' . . (411)2 ~ 2

of the quantum field model f 1 ~---g(¢2) are calculated in four-nt 4!

loop approximation using dimensioml renormalization scheme. Pro­ceeding from them the coefficients of E -expansions for critical ex­ponents are found up to E

4 • These expansions are treated by a summation method based on a modified Borel transform and a conformal rna.pping. The obtained \!alues of critical exponents are in good agreement with the experimental data and the results of other theoretical approaches.

The investigation has been performed at the Laboratory of Theoretical Physics, JINR,

Preprint of the Joint Institute for Nuclear Research. Dubna 1979

1. Introduction

Deep analogies between statistical physics and quantum field theory/1/ can be efficiently used to obtain quantitative predictions on the behaviour of statistical systems in the neighbourhood of phase transition/21. The main part in this ap­proach is assigned to the renormalization group methods/)/ and

E - expansion/4/. Calculating the ordinary Feynman diagrams of quantum 9'4 model in 4-2£ dimensions and solving the renormalization group equations, we can express the critical exponents of phase transitions in the form of the series in £ The physical (three-dimensional) case corresponds to the value of

E. .. 1/2. Much progress in this direction has been achieved by the

authors of papers/5,G/. They succeeded in calculating the contri­butions of all three-loop and some four-loop diagrams.However,in view of the asymptotic character of the obtained series in £ further progress requires not only taking into account the diag­rams of higher orders but also using the methods of "improvement" and summation of these series. The realization of this program is the aim of the present paper.

ln recent years several authors/7,B/ have developed simple and efficient methods of computing the contributions of Feynman diagrams to the renormalization group functions. Application of this technique enabled us to compute .analytically the contribu­tions of all relevant diagrams and to complete four-loop calcula­tions in the <p4 model.

3

This gives us an opportunity to construct the series in E up to E~ for all critical exponents. F0r these series we apply the method of summation we have developed earlier/9/ including the modified Borel transform and conformal mapping of the in­

tegrand.

The values of critical exponents obtained by this method are in good agreement with the experimental data as well as with the results of other theoretical approaches/101.

2. Renormalization Group Functions of the <pq Model in Four­Loop Approximation

The present section is a pure field-theoretical part of our paper. It contains the calculation of renormalization group func­tions (anomalous dimensions and Gell-Mann-Low function) of the

9P4 model in the framework of dimensional renormalization.

We consider Q(~) -symmetrical model of n -component sca­lar field with the Lagrangian

'r' ~ a. Q. 2. Q. Gl.. {6,-1 I. a. a..)2. (1)

~ = 2 '7>rcp'rCf- ;e. 'P cp - 4T ~s ~~ Cf . a.= 4.,2, ..• ,11-.

To calculate the Feynman diagrams we use dimensional regulariza­tion and the procedure of minimal subtractions or, in other words, the dimensional renormalization scheme/11 1. Namely, from each di­vergent integral we subtract only singular terms of its Laurent expansion in E. • where e =(4-d.)/2 and d is the dimension of space-time. In terms of the renormalization constants this means that they are expanded into the series in the inverse powers of

€, ~

z ( t ,~) = i + L )I={

Cv(~) f:..V

Renormalized Green ~unctions /; are obtained from zed ones through the following limiting procedure

-4

(2)

the regular:i.-

• I

f;(iPLr 1, m 2

, ~) = Um Zr(t,9)r{JpJ I m: .9s ,f-)' (3) £.~0

where

m! = Zmff, ~)m2 9a == {r'")j Zit {fJ)Z:2.t tJ). (4)

Here lPJ are the momentum arguments of the Green function~ , · !" is the renormalization parameter and Zm , Z 4 and Z2. are the renormalization constants of the mass, four-vertex f.; and inverse propagator ])-J. • !;_ , respectively • Z m and 9 S allow the expansions analogous to (2):

Zm ( ~ ,~) = i +! ~",(~) , V"'i C

9e. == (f''")~ ( ~ + f a~(~))

(5)

(6)

11•.1 c. The functions avl9) , ~vl9) and Cv {9) (their mass-independen-ce was proven in/121) can be unambiguously calculated in pertur­bation theory by requiring that the limit e- 0 in (3) exists. The quantities at ( ~) , /, 1. { 9) and Ct ( #) are related to the functions entering into the differential renormalization group

equation

[/"'~• +["I~~~ + 0 mi9Jm~0m•-ifr13~J;{Irl,l';m:J) (7)

by the following relations:

j->(9) 5 ~lr•kE~ =fi~~ -J.)Q.i(9), (e)

r m r ~) = ') tn m: I :z. == '0 en.,. ma ,~&

Y' (a):ofnZrJ 2 =-ale~{~). Dr o 1tn.f''- ma,3" <I'D3

~ ~~ g! CJ)' (9)

(10)

5

Tabl~ 1. The contribution of a diagram to an appropriate renor­

malization constant is eiven by a product of expres­

sions from the second and third colwnns in the case of

~2 and ~~ and from the second and fourth columns in

the case of Zq. 2 •

~=~;~;~;;~=====~~============~===============~~==============~==

Diagram

>CY

E7

x:x:x

~ -60-

~ <:Z H !51

Singular contribution

.!.. E.

i .YE. i

-"? 1..-E. 2E2.

I. - E./2

"6Et i E!>

1..-E.

2Ei" !- 9/lf~

6Ez.

2 ';l3)

E.

~-E.-E2

3e 3

. 2 i-E-E

3E 3

J.-3E+4E2. 6€3

~-2e. +e t 3f!l

Combinatoric a! factor for z2 and Z

~ /1.+8 0 2 9 d

.!.. n.+2. a2 6 .3 a

Combinatorical factor for z,2

j_ lt+2 0 2. ~ d

-~n..+6n.+2002 _.!_~ 0 2 t l ~ 4 2.1 (J " 9 d

_3

~tL+-22 0

2

21- d

_.!. n2+ J.On +1.6 a~

"' 21- a

_.!_ ~ 02 2. 3 (J

~ n.\.1Jn.2+24n. ,...yiJ cl' 1.1. (n.t-2. )3 o 5 s S! a ! 21 ~

5 3/t 2+22n. +56 3 2 i! ~ ..!_ n..2+ LDtt+16 0 3 2 21- <I

5n.+Z2 3 -~

2 3 31L +22n.+5"6 3

2 !t ~ 1. n..2+20tt+60 0 3 2 gL 4

t

6 IL +ZOn+6o 03 H d

2. ..s. 3/L +2211. +56 3

'I 8! 9

6

.!. t..M2.)2 a 3

it 9 <J

.!_ Utt-2.)2 a 3 6 g- <l

.!. (1'1.+2.)(11+!.) 03 it 2f <I

(M2.)(n+t.) 0 3

27- 4 2

.!_ Ot+2.) 0 3 lt s J

.------------------~--~------ ------

~ ..,.. ~

~ "' ~ ~ "" ~ "'(I

~0\)1...-1 ~O"'J\n- ~! ; ~Ot)l~ ~ ~ .s:!«l +C\1 ('I ..,('j + '-' .5 ~. ~ 5 -t[ ~ -..j ao '-.)I ~l\0 '-"~.-.

I """~ «< I I I I

~~ ~ ~ .... ......_ + ~ ~ ~

..:f" ~ ~ ~ ~ Covv ~ ~ ~ ~ ~~ ~ ~~.., ~ ~ 0 In ~ ~ ~ ~ ~ ~ ~ ~ ~ .q- co ... ~ ... ... ... + "' ~ t ...... ~ ... c s:: l:t Z! ..... + ~ ~ ~ 0 C"( \,I)

f'IC ... s:: ~ <;;)N) +r(\ (()N) oQH\ I'll """" "' ...> C'( ~"'" N ~ ~ ... ~ +-t i-"'t--f.-..1 ......... C'\1"""' ,.,~ +c-& ~c-c 1'1-c.a ... C'4 oJ co<~"' t'll

... <"Q ('I <:'t If) oq oq ..,J:; ~ ... c r:: ~ I:! ~ ~· + <;;;) CN N,.. ao .,. ~

oq + ~ ... ....... + .... ~ ...... ~ ... "s:: lti .s::::. ... ... "'" ..,... Z! .., ... "k I() '0 ..,...!::! ~ ~ .t:: ~

I ...., 1 c-~ ~r~ ..._.1-'t' lfll~ ..,1._.. ..,,~ oft'~ ..,..!<'1 1'()

1 ..,..! OQ -.i I 1 I I

N N uJ foti\JJJ <'<I C>1 IU ::!!!" ~ u.) VJ ... "'t

I() +on ~ + VJ \JJ + I It) uJ + HI 1.\J ""' ~ I ~ ..., w '!". \ "' w w"' tJJ """ IIJ Ul ~~~ OJ w lU 1.1) w ~ w 10 ""'

;;]' coo to w ~ <.'\1 C>l c-1 lU f ct _.,._ 1'11 Ol \0 I 0..0 1 C... , ..... ~<.'\11, I I

I ~rl) I -..! I ....t ~ ~ ~ ..,.. ..... I I I I I I I I I

~~$~ rxv~1 7

>c<Q>< i-2E+E.2. 3 3n..3+24n.2-rAOn.+B6 ~ - i (ft.+~)3 ~ l.t - :3 £"~ 1-li ' 2./.j3 3 .It 2f

~ _ 1- S'E. ?13 e.2-H€.!+61l3)i _ 2 n.3 +J..J.jn 2+::;.6hd52 ~'f -1:. (.lt+2)2(n+!) 8 it

1. 2 f."' 2. 21(3 2 H

~ 5"-i0t-f2+6£

3 '3 11 n 2+-=t6n. +1..5"6

9lt 2.

- _1_ (__ltd· 2) 9 It 2lf E 't -2 2.1f3 J.t -9-

JZr ~'f(5) 2.

-.3 2n. t-5"5"n +:U6 't

€ 243 ~

-$ _ ~£3)- "t>(-rts)-i1C~t))E. 2

-6 5"/t +- 62tt +- H-6 8 1 2£2. !H3

~ - 1..- ! E.+ f e 2+- ~ £3

-b 1-n.2t-1-2n+!..61( slt g E. 't 2lf3

0 2. ~ ~· 3 2.

- !..- £- £- f+2}l3)£ _ _!It 1'"14tr +'1{>1t+1S2 9

't 4f..'t 2 2.1(3 I

<!> 2 '3 3 l 2.

-J.-2t-2E. t-6f -6)/3)~ _J.. n +iAn +!On +141( 't _ i. (!t+-2) "L(n+~)3 t-b£'f ! 21f3 9 A H

Q .!-'3 E.+ff. 2. ,!, 2 2.

- -~ It +i!.tz +96fl +1.2g 'f - i. (h.t-2.) {It+-!,) 8 't 1..2 f!. "f 2.1(3 ~ 4 .H

4 i it£.+ 121-t.. 2. 5172.+-321'!. +-41( - .1... (lt+-2.) 2. 3- 'f - - ~ 12. - ~It 1..6 E. 3 H 6 9

4 1.-4£ +5'E. 2t- 6€3-i21'f5)E

5 3 i.in

2+?-6n +i5'6 ~.It - .! (.n+7.){ S'rt+22)~ ~

- -- 24~ .lj 3i 1.2£" 2.

2 3 '3 3n.3+2Jin 2+AOn+i"!6 9 ~ 1

<1 !.-E.- E - E +21l'3)c '3 -.!. (I'L+2)Cn-t6n+2P)lf

- .4 E. .It Jt g $1 3 243

iO 'li 2 sn 2+ 3211 +-41.( l.f ~

~ ?- ;tE- It£ _ i Ut+1) ~.It

- -4,8£_3 - .H 8 6 T

<tl 2. !> 3 3 2. _ 1. (/t+?.) ( 11 \6nt2o)

8 ~

i-3E. +E + 5£ -6)l3)~ _ _l n. +!On t-"f211+iGO glt - 2 2'13 " .8i 1..2 E. Jt

3){3)- 3.(}l'5)+t1f't)) E. 2.

<fr7 _

3 S"n +6'21'1 t-1.1-6

81t _ 1 Clt-tll(S"/'1+22)

3/.t

- 2E:2. 243 2 Bi

~ 2 3 - .!. 111

2+ =121'1 + i64 g " .... i (11+2 ){5"11+72) 8 ~ i-3E. +t +S"f - !U.~t 2 243 . "' u.

<1 .1- j £ + t f 4+ t E ~ _

3 1..tn 2-tr6n-fiS6

9'1 _! (lf-tZ)(Sn +22\s.lf

- g £'t 21{3 2 8i.

~ !. -3 E. +e 2 + ?f ~6.,{3)€.~ 3 2. '2.

_ 3

1t +H11 +%tt+1.5"2 8 ~t - J. (lt+2.){M +'/1t20)~4 - 2 Ai

:1.2 e. Jt 243

~ 2. 3 1 - 6 ;.,1. + -::J-,2, t-.16/.t 8" _ (n-t-'2.~(5n-t22) S,. _ i- 6 e.+ 19 e -30t +2lt~{3)f

2lt t.l.j 2Ji3 ~i .

4_ 2 :3 =1-112. + T2 n + :1.6'1 glt _ (ll+2)(5n+-2~~ 9 .~t t-6e +J.9€. -30E -6 - 2'-!e'~ 243 u

Hence, to find the Gell-Mann-Low function J3(9) and w-· nalous dimensions o(9) WC have to knoW the coefficientl3 Rc :11£ ir

the renormalization constant expansj_ons 1T' the inverc ,- ;-y,,~ ,. of

E. To perform tY~~=~ ~orreGpc.~'.:lJ :·:g ~: .. :-.:..":.· :~· L f:

proxi.mn.! l ?t; in thP t..-··:~-;.c;y ( 1) ~He uae \/ ·'l?. ··

papers/ 1 'b/. 'rlle f'·•mC~ j O!l<O j3 ( ._; '1 (, ( tj ': bli -· r:, ( ~: _ :);> ..;J.i' '-& Q/ v...,

Puted- ,_,,---.oY·r' lng t~ ,-•. , fl;) ('J",' As r'rr Y' (!l'i 0 t ' -_ +r . ...,_,,__.~.I.,L.>.L' J<..l' "i"'\"-'•j Ve '-' oma./.--. ,~··· .. ~~·

be more con?ement to us: the equaHty rmq)=tz.(J_-':./jJ>-· low.1ng :from Ztn=-Zif2-Z2 i where '(,_,z{~j ::''11!-n.Z'f ·-· and Zcpl is the renormalization constant of the two-vvint u:cs:

function with <p 2 insertion, i.e,,< <t>lx)<.jl(o)Jdy o/2.(:1 J) . 'Ine contributions of various diagrams to z2 (for propagate:-;, 7" and Zlj> 2 (for four-vertex) are given in Table 1. F-or tr,~> ::'t-·

normalization group functions we obtain the followin,::: exn.an~- -· ons:

~2 3 r2(9) =- {lt+2) _ 9 {11+2)(11+8)

36 16·2r + 5"g 4

(ht2)_

64'· .Si 2

·(-n+Hn+iOO) +0(9 5),

am<~)= 0 502. 3 .st..(n+2) - -<J (n+2) + @ (ht-2)(5n + 31-)_ 6 36 ...,,.,

1"£..

- ~lf{h+2)[-n\t-5'f$n r 3!.060 +.lt&~t3)(.3n\iOn+6 64·243 '

+2~&~l4)(5n+22)J +0(8 5 )

(11

( 12)

O.t,(~)=-l(n+B)+_t(5nt-2Z)- 93 [3sn'--- ;n) 6 9 16·27

1 8-Y 3 " + 91f2n. r 2992 + 96 'ft3) ( 517.+22) r +--Lr- [;n -~ .'_;.{•:'Jr,~., _j g. 24~

+2062ltrt +4991.2 + 24~{3)(63n 2t-:t64tt+2332- +?

· ( 5n2. + 62h + it-6) + 2fl>0[(5)(2n\)s-n +1.S6 )j + ::-:•' J'"),

10

~<~) =- ~( r~,c~)-2f:t(~))=

= 91 (tt t- 8) - 9

3( 3ttHlt) + glt [33 1l t- 922tt. t-2 960 +

6 6 16·2.1

+ 9€.";£3)(5"1l.t-22)J- L [- !itt3

t- 6320tt2+A045"6tt"" (14 )

32·243

H96~1:i& + 961l3)(Bh1-t.:U.;tt +2332)- 2U VJt)(5nt+62h tH6)t

+ H2o":)<5H2rt2 -r5S"tt.-rl&6) J + oc~'"). In particular, for n. =1 we have

( 15) Y ( a2. 0 ~ G5 It 02 ~).:. _g_ __ <I + -Q

12. i6 i92 <I )

Om(~)= 1- ~9\ ~f'- ~9"{~~9 ~ ~(3)+2}'1~)), ( 16)

o J.t c ~) = - i a + 3 9 ~.- ~ 3 { ~~ + 6 "?{3)) + 9 It ( o/ + < 17)

t-39~[3)-9~(1() +60*(5") ),

~(f)== 1-~2.- ~=1~3 +- g~t(~~5 + b~/3))-9~(3;:9+ (18)

+ 3 9l { 3) - 91 ("' ) +- 60 ~ ( 5") ) • The obtained exp~sions of the renormalization group func-

tions can be used in quantum field theory (in four-dimensional space-time) for the investigation of ultraviolet behaviour of the Green functions. They can also be utilized in different at­tempts to go beyond the perturbation theory by continuating its results into the region 9 '?-- 1 of coupling constant, as it was made for example, in papers/9,l4/, In the present paper we use the obtained results to determine the critical exponents of phase transitions in the framework of field-theoretic approach to critical phenomena based on the e -expansion.

3, Critical Exponents and e -Expansion

The renormalization group method being transferred from

II

quantum field theory to statistical mechanics has been Yery sue­cessfull for the description of critical behaviGur of various systems in the neighbourhood of second order phase transition. Scaling invariance-taking place -in critical phenomena finds its natural interpretation in terms of renormalization group: Scal­ing behaviour near the critical point caused by the appearance of long-range order can be described in terms of Euclidean quan­tum field theory possessing an infra-red stable fixed point. The fixed point go of renormalization group equation

P\~· ~ ( ;~, ~) = ~( 9 ( r.: ' ~)) ( 19)

is det-ermined by the equation foCgo)= 0 and is called j_nfra-red stable one, if t:l...'{D0 )>0. In this case effective charge

- .t. j.J (f 2 lJ {P7f'-,.J 9) tends to !Jo when the momentwn argument p tends to zero, i.e.,on large distances~ In the presence o£ infra-red stable fixed point the dimensionless Green functions obey the scaling laws for small [>2 131

[ 2 (p2f (rl~o) (2o) R p ..... o

with the powers equal to the value of anomalous dimensions at

~= 9o. A systematic approach to the description of critical pheno­

mena in terms of Juantum field theory of 9'~ model has been de­veloped in papers 15• 161. There have been found direct relations between the anomalous dimension {20) and critical exponents, characterizing scaling behaviour of statistical quantities in the neighbourhood of critical temperature lrc . Thus, for example. at T- Tc asymptotics or correlation function r ( x) for I X J + oo is defined by the exponent ?

r< x) 1iJ:-o 1~ 1 cl-2+ 7 Correlation length '!f for i=T-T<:- 0 obeys scaling law,

12

(21 )

the following

I r -v Wll

,__ i { 1 + con.st· t + . ·~~ t~o • (22)

where W characterizes the deviation from scaling. All other and Y /lf:,f critical exponents can be expressed in terms of ~ r determining the power beha-For example, critical exponent

viour of sucseptibili ty X ,

X rv { -(J i + Cortft• tuJ~···J ±-+o

(23

is related with ? and \1 b_y the equation

r = ( 2 - 1) v. (24)

Baaed on the analogy between grand partition function and quantum field generating functional, we can establish the follow­ing identities between statistical and field-theoretical quanti­ties:/lG/ The quantum field ip io identified with the order

parameter, the temperature difference lr-lrc with the square -of the bare mass m ~ , the correlation function r ( x) with the propagator < {_f( x) <.p(o)) , the aucsepti bili ty )(. with the

quantity ]) ( 0) , where :D ( p2.): J d.:x: eiPX('<pcx)<f-lo))is the Fourier transform of the propagator, and the inverse correlation length T-! corresponds to the physical mass n1ph which defines the

position of the pole of l>(p2 ): b- 1(p2:-m;J.)= 0 . To find the critical exponents on the basis of ~~ theory we have to inves­tigate quantitatively the power behaviour of mph and JDlo) for m~- 0 and also the behaviour of :D ( p2) in the limit p2~ 0

at tn~=O • The renormalization group equations reduce this analysis to finding of anomalous dimensions ~ttl ( ~) and r2 {~) at the infra-red stable fixed point 3= ~0.

It can easily be seen from eq.(14), that fo -function vanishes at ~ =0. Thus for a£ =4 an infra-red stable fixed point fJ =0 and all anomalous dimensions "( (So) also vanish. This means that in the limit p2~ 0 interaction disappears, i.e., the <p4 theory in four dimensions behaves like a free

theory ln the infra-red region.

13

The idea of E -expansion/4/ is the following: We start from the a{ =4 case being a zero approximation and then cons­truct a perturbation theory in powers of 2 E = 4-d. To pass then to the physically interesting case d =3 we must put 2E =1. Indeed, performing renormalizations in the cp" model at E :j:. 0

one should use the ~ -function

_foE(9)=-e~ + f-'Cg), (25)

where ~ ( 9 ) is the four-dimensional Gell-Mann-Low function. The function .fot:. ( S) vanishes at some point j'o (E) and has a positive derivative at this point. For £ small infra-red stable fixed point 9o(E.) is of an order of E , that enables us, reexpanding t( g0 CE)) in power series in E , to express all critical exponents in terms of these new expansions, the so-cal­led E -expansions.

We present now the explicit relations between the critical exponents and anomalous dimensions r(go)• The corresponding formulas have been obtained in/161. However, we think it to be expedient to reproduce here the derivation of these relations in the framework of dimensional regularization scheme used through­out this paper.

Taking the Fourier transform of (21) and comparing it with the asymptotic expression of the propagator

D(pz)-- (p2fi+r2c~o) p2-90 •

(26)

we get

1 = 2 t2 (So). (27)

The other exponent V is connected with the physical mass tnph· The latter is the renormalization invariant and obeys the

renormalization group equation without anomalous dimensions,

(f"•~, + ~< (9)_!} + (m(3)m~~,) m:h ( ,',f'',3)= D. (2el

14

Accordi~£ tc the theorerr· of homogeneous functioDa,

::.-, 2?J ' ~ - 2' ~~-~ _:::_., + r.; -. ) r;: · = ,'r, Ph ( 9 J

1 1 ~.M"' 'Om• r>,, ., . T:-Ier.~ excl~din£ 'v,2,~, t t'rorr. (281 with the help of (29), we have

,,.,.,

'· ( '"'t Q') ~ A

.t-o~~g)ro.a - i-~r,~a)] tn~h = o. : 2--; !11'! -·-.,-~ '6 "1:-

(30)

'.is.i.ne:; the ptandard r.wttwds of analysis of such equation.:/31 , we

come tc the conclusioL that

So far as

Hence

f'1

2 mPh

and

,_- i

~ =

1 I 2. \

,_2 l m ; :!.- ;r (II \ •u-'SI-0 ul1'\ 00)

3 are fixed we have

2. m 2. .J_ ....., me. = ~

:i i

2(i- Om (~ 0) m r-- ..j...

l'h t-i> 0 I.

and, comparing (33) with (22), we find

1 v =

2(1-o,;,(So)) Taking into account the second term in tl;le expanslon fective charge near the fixed point,

(31 i

(32)

03)

(34)

of the ef-

- ( oz ' 2 p.,' ( a \ -D ~, 0) = ~() ~ con.sf· (~) J- aoJ+ ... 0:)/ ;j r d ,... .

we can find the correction- to the solution (Ji) nh.J.ch determiaes the exponent U)

I

W =: 2~£J8oJ · (Jo'

Let us also derive equation (24). For ])Co) the followi~z

15

renormalization group equation is true:

[ tl--o~2. + ~t(S)~~ + Orn(~)tn~~t+ o:~.(3)JD(o)= 0. 07

}

Proceeding in the same way as for eq.(28), we come to

J) (O} ~o l- (2( So)

(m2)- ~-o,..tUo)

and therefore

r = !- d'2.(9o)

i-om(9o) = (2-7)v.

{38)

(39)

Now we are in a position to write down the E. -expansions up to c If for qo (E) and critical exponents 1 , )I and W on the basis of eqlil. {27), (34) and (36) and the functions ( 2 ( 9) , r m ( $) and f.> (g) calculated in four-loop approximation. It

should be noticed that the coefficients of E -expansions are independent of the renormalization procedure. Our results which are given below are in agreement with those of paper/

6/ where

the corresponding calculations have been performed up to € 3 for

9J~>,V and W and up to E~ for f . In order to obtain the standard form of E. -expansion we choose the expansion parame-

ter to be 2 E. •

a (E.)== ~ (2f.) + 9(3nHlf>czoz + 3(Z03 l~>~n'\.uonz~ do h+& (n+s)3 &lnt-~)sr

+t160tt +lj5lf4- %"2'(3)(n+&)<Snt22)j + (20lt I-5n..5-l 16(11+8)1 (40)

- 261011.-t- 55M n3 + 52=1-b'ltf-~ o093121'1. + 529 H2 + i920}lS)(h+-!)1•

'(2h\5511+Uf.)-2'-S1Ut)(tt+&')3(5ht22.)- %fl3)(-6311 11-

-lf22n:?.+lfH2tl+39lj32ttt-~Z512)J + 0(£s), 2. (41)

}J = (1'1+2.) (2 o2.[1 + __!!__ (-n'l+Sl.n 1" 21-2) +Q..U_ . ( 2.{h +! )2 it(h+ 1:. )2 16(tt+!) 'f

. (-s~"'- z30n 3 + H2~ n1 + "192Dn + J,M.ltlf- 3.!VtV3){hf8)(S"n+Z2 )J} 1'

+ 0( £5') }

l6

J 2. 3 .2- = 2- !!:!:3 C2.£.)-~ 1-131'1 t-.Y1()(2.f.~- (nn)tu) [- ~n3

+-452n1-t

).1 HT! 2(n+S)'3 & I ht-8)!1'

+ 2.6=t2n + .5312- .96~l3)(n+!){5h+:22'J- lM+z) tzn"[- 3~s- 39~h"-t-32 (h+l!.)'f

3 2 2. -+12900n + %H5'2n + 21996~1'1 + 35H,2.0-+- !2.AO~(S)l11+2>~·

· (znz+5S"t\ +i~')- 2!8)l.it){h+.!.)3 {5•H22)-i6~(3)(11+!)·

·( 3n•- i94n·\ iit&n1

+" 9.lfi2n +-iH.l!.~ )] ~ o (t: s), (42)

W == C2t')- 3 ( 3 ntilf)(2f.'>2

+ (2f.)3

[ 33t/+ 538n 2+ (n+A )t ~fn.t!)~~

+lf2!!n +9S"'S f"96Vl)(nt-&)lS'I'1-t22)J- (_2.£.':/' 6[-5n 5

i-

!6(n+&)

+ ilt&l> n * + it66i6 n3 + 4i952An2 + nsoo&On + 2.599552 +

+32){'!.)(11+!.)(.H9n3+ 1.6Hn 2

t5r4An+ 116HJ+ ' 43 )

+ i92Dzt5){11+~)2 {2n 2 t55'11 t .!2.6) -2U}Ut){11+&)3{511+22)]+

+ O(es)

4. Summation of the e -Expansion Series

It is well known that perturbation theory series in the coupling constant ~ are asymptotical. In recent years the tech­nique of estimating the high order coefficients has been develop­ed/171. The series of £ -expansion, arising when solving the equation f.'t. ( 8oCE))= 0 , are also asymptotical. As it was shown in paper/181 asymptotic estimates of the coefficients of perturbation series in ~ lead to the following estimates of the coefficients of high orders of € -expansion -l {ZE.) = 2,t-2E.)kfk:

k

-f ,....._ k: I k kl. lc k~oa · a.. C. , (44)

where f stands for $o , 7 , -lfv ,or W and parameters a. and b are respectively

17

r~ for go,

3 b= 3+ ~ for ?., ' a.=-

IL+-A ' 4+1! for '!)I'

(45)

2 for 5+11 w

2

From (45) it follows that E -expansion series have zero radius of convergence~ Therefore, the direct substitution of

E = !f2 into {41 )-(43) cannot lead to any reliable conclusi­on about the values of critical exponents at physical point

d = 3, To come to the value E.= l.f2 we use the method for surnrnation of the asymptotical series we have developed earli­er/9/, It takes into account the exact coefficients of lower

orders (40)-(43) and, on the other hand, the information obtain­ed from asymptotic estimates (44),(45}. It is based on a modi­fied Borel transform

~ 00 - ~ b+3'12 + < u .. ) = ot :x. e 2fq_ (-x-) s ( x )

2E.Gt 2EO.. 0

(46)

Then

~ J.:· Bl:x.) = L.f.-X) B,..

k

·h Bk= a..ltrthb+f)

(47) .......... c. k., ... \(.~12. •

The series (47) for B (x) possesses the unit circle of conver­gence. The function 8(x.) , as it follows from (44), is free of singularities in the integration region [ 0, OQ) and has a square root branch point/19/ at ~ =-1. In order to perform the analytical continuation of Blx) beyond the unit circle, we use conformal mappine; X~ W with

Wlx.)=~-1 YHx + 1

(46)

It maps the integration region [o,oo) into the interior of

the unit circle while the cut (- oo,-1] is mapped into its boun­dary. Inside this circle the series in ~ , obtained by reex­pansion of the function &(:x.(~)), is convergent, The coef­ficient at W 'll is determined by the coefficients tk of the initial E -expansion with k ~ N , Therefore, if we trun-

II

cate the series in W at N's term

Jl X >. ~ l .>.) k

B !x) ~ (_¥.!) ~ B~o; w , k

(4'))

we obtain for f[2E.) an approximate expression .f,.,(2f.) corres­pondinG to N -th order of pertubtation theory.

Of great importance here is the specific value of the pa­rameter A introduced in (49). It determines the power of asym­ptotic behaviour of -f11 {2t) for large E.

fJI (2E.) >.

~ (2.~) £-.e>O

(50)

For the solvable models with the known asymptotic behaviour in the coupling constant J7 , we find that the best convergence of

the sequence of approximants f11(~) t.a the true function ffl} is achieved only for the choice of ~ , which is consistent with asymptotics of -f ( g) t"or g-. oo /9/, As far as the behaviour of the critical exponents for large E is unknown, we fix ~

just from the requirement of steepest convergence of our appro­ximation procedure.

Consider the set of quantities

6N = i- .f"{i)

.fN-i (i)

(51)

characterizing the relative variation of t,., when taking into account the next term of perturbation theory. If we guess the true asymptotic behaviour of j. {2 E.) , the relative errors ll,; should decrease very fast. Numerical analysis shows the exis­tence of a sharp minimum of li::J N I at the definite values of

). , which appear to be different for each critical exponent, We have checked this method on solvable models and obtained very nopeful results/9/, In our case an application of this me­

thod leads to the following values of A :

19

1 ,2 ... 1 ,4 for go,

>-= / 2 -!- 3 for ? ' (52) 1,0f1,3 for ijy,

0,1 ... 0,9 for w

To define the value of ). we used also the other method pro­posed in paper/201, It is based on the fact that the asymptotics of f~o~ (2f.) for large E. is defined by the asymptotics of the coefficients B~'A) for k ~ oo , which was estimated numerical­ly. The values of A obtained by this method lie within the bounds (52),

Now using the £ -expansions (41)-(43) and the values of .>. (52) we can calculate the critical exponents at the. point 2e =1 according to equations (46) and (49) for )o/ = 4. The

results of calculations are summarized in Table 2. The value of errors are taken to be equal to ! fl..tt•f'l (1 ). We present also the value of 9o ('2.f .. ) for E = i/2 :

tt = 1 It = 2 It= 3

9o(i) = 0,488 :!: 0,006 0,435 ! 0,006 0,392 :!: 0,006

It is interesting to observe the variations of the values of critical exponents with the number of terms of perturbation theory taken into account. Below we write down (in the case /1. =1) the values of the exponents obtained both by direct substitution of 2 E. =1 into (41), (42) and by using summation methods desc­ribed in this section (respectively h PTii), V PTC:I.) and

(N W ?Nii), VN(i} ).

}/ 1 2 3 4 PT

fH li) - 0,0185 0,0372 0,0289

2" (:1.) - 0,0320 0,0332 0,0333 v Tr(i) 0,6 0,645 0,595 0,731 II 0,626 0,625 0,628 VH Ci) 0,620

As one can see from this table, an application of special

20

I Table 2, Comparison of our results (the fiTst column) with the

calculations in cplf model in 3 dimensions (the second column), with high temperature expansion in 3 -dimensi­onal Ising ( tt =1 ) and Heisenberg ( n. =3) models (the third column) and with the experiment (the fourth co­lumn). The numbers given in the second, third and fourth columns are taken from paper 1101,

••===~==7===~=====~==============================================

'l v w

~ Jl

u.J

~ J)

w

0,0333;t0,0001 0,628 ;t0,002

0,781!:, 0,015

0,0352;t0,0001 0,666 ;t0,004 0,777:!: 0,015

0,0354;t0,0001 0,100 ;t0,001

n = 1

0,0315!:,0,0025 0,6300!:,0,0008

0,782:!: 0,010

n = 2

0,0335;t0,0025 o,6693;t0,0010 0,778 ;t0,008

n = 3

0,0340±_0,0025 0,7054;t0,0011

0,119 ;t0,001 0,119;t0,006

0,638+0,002 -0,008

e,043;t0,014 0,715+0,025

-0,015

0,016;t0,014 0,625;t0,005

0,6'75;t0,001

=================================================================

methods of summation drastically improves the approximating pro­

perties of perturbation theory.

The summation technique developed in this paper has been also used to find the values of the critical exponents at 2E =2 which corresponds/21 1 (for /1. =1) to the two-dimensional Ising model allowing an exact solution. For the exponents ? and ~ we obtain the results: 7~(2) =0,18 and V"t£2)=0,92 to be compa­red with the exact values :l./'1 and 1, re~pectively.

Concluding this section we would like to note that the cor­rections ~€1 by themselves, i,e,,without an application of spe­cial summation methods, do not improve the agreement with the va­lues of critical exponents known from the literature, while the

21

method p~oposed above improves this agreement drastically, It is an additional argument in favour of its efficiency and gives a direct confirmation of applicability of quantum field theory ap­proach based on c -expansion to the evaluation of critical exponents.

A c k n o w 1 e d g e m e n t s

We are grateful to K.G.Chetyrkin, A.A.Migdal, D.V.Shirkov and V.A.Zagrebnov for useful discussions.

R e f e r e n c e s

1, A.M.Polyakov. JETF 22• 1026, 1968. A.A.Migdal. JETF 55, 1964, 1968.

2. K.Wilson, J.Kogut. Phys.Reports ~. 75, 1974. 3. N.N.Bogoliubov, D.V.Shirkov. Introduction to the theory of

quantized fields, Moscow, 1976. D.V.Shirkov. Nucl.Phys. B62, 194, 1973.

4. K.Wilson, M.E.Fisher. Phys.Rev.Lett. £§, 240, 1972. 5. E.Brezin, J.C.Le Guillou, J.Zinn-Justin, B.G.Nickel. Phya.

Lett.~. 227, 1973. 6. E.Brezin, J.C.Le Guillou, J.Zinn-Justin. Phys.Rev. Q2, 1121,

1974. 1. A.A.Vladimirov. Preprint JINR E2-12248, Dubna, 1979. 8. K.G.Chetyrkin, F,V.Tkachev. Preprint INR ll -110, Moscow,

1979. 9. D.I.Kazakov, O.V.Tarasov, D.V.Shirkov. TMF, l2• 15, 1979.

10. J.C.Le Guillou, J.Zinn-Justin. Phys.Rev.Lett. ~. 95, 1977. 11. G.•t Hoeft. Nucl.Phys. B61, 455, 1973. 12. J.C.Collins. Nucl.Phys. ~. 341, 1974. 13. J.C.Collins, A.J.Macfarlane. Phys.Rev. £12, 1201, 1974. 14. V,S.Popov, V.L.Eletsky, A.V.Turbiner. JETF 74, 445, 1978.

V.L.Eletsky, V.S.Popov. Phys.Lett. ~. 411, 1978. 15. K.Wilson. Phys.Rev.Lett. £§, 548, 1972. 16. E.Brezin, J.C.Le Guillou, J.Zinn-Justin. Phys.Rev. ~. 434,

2428, 1973.

22

17. L.N.Lipatov. JETF 72, 411, 1977. E.B.Bogomolny. Phys.Lett. ~. 193, 1977.

18. E.Brezin, J.C.Le Guillou, J,Zinn-Justin. Phys.Rev. Q!2, 1544, 1977.

19. C.A.Truesdall. Ann.Math.46, 114, 1945. 20. O.V.Tarasov. Preprint JINR P2-11879, Dubna, 1978; to appear

in Lett.Math.Phys. 21. J.Hubbard. Phys.Lett. 22!• 365, 1972.

Received by Publishing Department on February 16, 1979

23