IL NUOVO CIMENTO VOL. 16 A, N. 4 21 Agosto 1973

Self-Consistent Approach to the Fine-Structure Constant.

F. EI~GLERT

Faeult~ des Sciences, Universit~ Libre de Bruxelles - Bruxelles

(rieevuto il 31 Ottobre 1972; manoscritto revisionato ricevuto il 9 Febbraio 1973)

S u m m a r y . - - Massless quantum electrodynamics is formulated as a field-theoretical (~ bootstrap ,~. The resulting equations are interpreted as self-consistent equations for the physical electric charge following Adler's conjecture. The group-theoretical properties of these equations are investigated. I t is shown that they are not invariant under the eonformal group 04~ in conventional gauges, but that in other gauges a generalized (( bootstrap )) can be formulated which admits 04,2 as its iuvariance group. The relevance of these group properties to the determination of ~ and to massive quantum clectrodynamics is suggested.

1 . - I n t r o d u c t i o n .

The poss ib i l i ty of d e t e r m i n i n g the f ine-s t ruc ture c o n s t a n t a = e~-/4~ f r o m

q u : m t u m elect rodym~mies nlone h:~s r e c e n t l y been rev ived b y ADLEIr (1). I n

t he p re sen t work we sh~ll i n t e r p r e t Adle r ' s t h e o r y in t e rms of se l f -geuer~t ing

in te rue t ions (2) t h e r e b y m~king t he connec t ion wi th the so-called ((conform~l

b o o t s t r a p ~ (~.4). The :tim of this progr:~m is to obt~fin se l f -consis tent equut ions

for a ,~ud to use these eqm~tions to g:fin some ins igh t in to the possible group-

theore t ic~ l s ignif icance of the phys ic~l electr ic chal'~'e.

(1) S. L. ADLER: Phys. I~ev. D, 5, 3021 (1972). (2) F. ENGLERT and C. DE DOMINIClS: NUOVO Cimento, 53A, 1021 (1968). (3) A. ~/[. POLYAKOV: J E T P Lctt., 12, 538 (1970). (4) A. A. MIGDAL: Phys. Let&, 37B, 98, 386 (1971).

557

5 5 8 F. ~ N G L ] ~ T

We first briefly summarize the work of BAKE~, Jom~so~ ~, VVTYIm~Y (5) and ADLER (~). I f quan tum electrodynamics admits a Gel l -Mann-Low l imit (~), t hen the photon propaggtor renormal ized by a subt rac t ion procedure at q 2 = _ ~ts exists when m-+ 0 (7) where m is the physical e lectron mass and simply becomes

(1.1) ~)~,(q~') = (g~, - - Vq~, q,/q~) q-2.

Here ~/is a gauge pa rame te r and the coupling cons tant is ~0, the solution of the Gel l -Mann-Low eigenvalue equat ion. When (1.1) is used to eva lua te the re- normalized vacuum polar izat ion tensor

(1.2) ~v(q 2) = (glAy- ql~ qy/q2) q2ff~R(q2) = (gI~Y- ql~ qy/q2)q2[~(q2) __ ~7~(-- ~2)],

one would obtain (5) if m were s t r ic t ly equal to zero at the outset (massless quan tum electrodynamics)

q2 (1.3) ~z'(q ~) = FZ(~o) log _ ~2,

~nd the consistency between (1.1) and (1.3) would require

(1.4) F~(~o) = o .

The Federbusch-aohnson theorem (s) now implies (~) t h a t it is sufficient to solve the equat ion

(~.5) F(~o) = 0 ,

where F(r is the coefficient of the single logar i thm appear ing in the cont r ibut ion to 7g,, f rom the single-fermion loop. Moreover this theorem also implies (~) t ha t F(~o) vanishes with an essential s ingular i ty in ~,.

ADLER (~) now shows tha t if m is a rb i t r~ry (and in par t icu la r a rb i t ra r i ly

small), the eigenvalue equat ion

(~.6) F (~ ) = 0 ,

where ~ is the physical coupling constant , is consistent with the Gel l -Mann-

Low limit. Assuming convergence of a <~ loopwise summat ion procedure ~, he

(5) i . JOHNSON, M. BAK1~R and R. WILL, r : :Phys. Rev., 136, B 1111 (1964); K. JOHNSON, R. WILI.E~ and M. BAKER: Phys. Rev., 163, 1699 (1967); M. B~tK~ and K. 5omvsozv: Phys. Rev., 183, 1292 (1969); Phys. ]~ev. D, 3, 2516, 2541 (1971). (s) M. GELL-MANN and F. E. Low: Phys. ]~ev., 95, 1300 (1954). (7) By the limit m->0 we mean the limit m/2-->O where ;t~ is an arbitrary space- like momentum at which a subtraction is made. (s) P . G . F~D~RBUSH and K. JOH~CS0~: Phys. ]~ev., 120, 1296 (1960).

SELF-CONSISTENT APFROACtI TO THE FINE-STRUCTURE CONSTANT 5 5 ~

indeed proves t ha t (1.6) guarantees finiteness of massive quan tum electro- dynamics for large momenta . Thus if (1.6) is correct , ~ may be in te rpre ted as

tha t value of the renormal ized coupling constant for which massless quan tum elcetrodynamics exists.

The eigenvalue equat ion (1.6) is thus ent i re ly ch~r~cterized by the one-loop approximat ion to ~,~ in massive quan tum electrodynamics. In Sect. 2 we shall write the coupled Sehwinger-Dyson equ:~tions for the corresponding renormalized quanti t ies and argue why these equations are of :~ <~ boots t rap ~> type for asymp- tot ic momenta or equivalent ly for massless quan tum electrodynamics.

Section 3 contains a pre l iminary discussion of the electron propagator and of the ver tex funct ion obta ined by imposing covariancc umler the conformal group 04. z. I t is shown tha t despite the f:,ct t ha t the COilform~fl electron prop-

agator is the correct one and tha t the conform:~l ve r t ex funct ion is consistent bo th with the Ward iden t i ty aml with g;~uge covariance, this cannot be the correct solution of the <(bootstrap >> equations. Addit ional form factors for the ver tex funct ion must appear because of the nonconformai n.~ture of the photon propaga tor (1.1).

In Sect. 4, the group-theoret ical propert ies of the solution of the <~ boots t rap )) equat ions ~re studied. This is done by introducing ~r conformal photon prop- agator with a g~uge depending on an e• point (9,1o). The ver tex funct ion is then obt:~ined by a l imit ing process in which the nonlocal gauge is projected out. A generalized <(bootstrap >> depending on the externa l point is formula ted and its invariance group O~. 2 exhibi ted.

We summarize and discuss in Sect. 5 the content of the self-consistent equa- tions and thei r gener,.lization to massive qm~ntum electrodynamics.

2. - Se l f -cons is tent formula t ion o f mass l e s s q u a n t m n e lee trodynamics .

If vacuum polarizat ion loops are omit ted , the asympto t ic form of the re- normalized electron propagator in massive quan tum electrodynamics is deter- mined through the C:~ll'~n-Symanzik equation (1:)to be (12) (the bare mass is zero)

(t: )(-p~l ~`~, (2.1) S-I(P)=/I(~)C: ~ . , :r \ - ~ V / (Y'P) ,

(.' z 2 = c 1 ~ , 1 \"~/ ' (2.2)

(a) R . A . ABDELLATIF: Quantum electrodynamics with no photon sell-energy insertions, University of Washington dissertation (1970). (lo) S. L. ADLER: preprint NAL-THY-58. (11) C.L. CALLAN: Phys. Rev. D, 2, 1541 (1970); K. SYMANZ:K: Commun. ~fath. Phys., 18, 227 (1970). (1~) S. L. ADLER and W. A. BARDEEN: Phys. Rev. 19, 4, 3045 (1971).

560 F. ~NGL~RT

/~ is ,~ photon mass provid ing an inf ra- red cut-off and A the u l t rav io le t cut-off

m o m e n t u m ; y(a) depends l inear ly on the gauge p a r a m e t e r ~ and can be var ied

f rom - - c~ to + c~. One should notice t h a t despi te the fac t t h a t Z~ is not the renorm,dizat ion cons tan t of the comple te theory , the set of graphs considered

does in fac t sat isfy the W a r d iden t i ty and therefore Z., = Z,. The renormal ized ve r t ex funct ion / ~ in the absence of v a c u u m polar iza t ion loops t hen obeys the

Schwinger-Dyson equat ion with an inhomogeneous t e r m propor t iona l to Z2 as g iven b y (2.2). Therefore if y(~) ~ 0 we m a y wri te in the l imi t ,/1 -+ c~ a

homogeneous in tegra l equa t ion (Fig. 1)

(2.3) F~, = f T',,SF~,SF, D ,, + i r reducible

/rr/cZuc/bte

Fig. l. - Graphical representation of eq. (2.3). O = / ' ~ (renormMized vertex func- tion), ~ = S (renormalized electron propagator), ~ = D ~ photon propagator).

p rov ided each in tegra l on the r igh t -hand side converges.

p r o p a g a t o r is in configurat ion space

(2.4) D,,(x) ---- g,,x-" -- �89 --2xox, x-4).

I n (2.3) the pho ton

The coupling cons tan t e has been included in the definit ion of T' and ~](go, x l - ' ) . _ _

--2x~x~x -~) is the space- t ime expression of the longi tudina l gauge t e rm. I n

all t h a t follows x ~ will mean (x2-- is) ~/2. For m o m e n t a large compared to m we shall search for sc~de inva r i an t solutions

of (2.3); m will therefore drop f rom the equ'~tion and we m a y in te rp re t (2.3) in this l imi t as an equat ion for i , for all finite moment.~ when m -> 0.

I n the following Sections we sh~dl argue t h a t each in tegra l on the r ight-

hand side can be defined in near ly all gauges ~nd t h a t the singul~rities occurring

for exceptiom~l gauges are simple poles. Therefore / ' , as defined f rom (2.3)

can be cont inued ana ly t i ca l ly even if Z~ diverges and (2.3) m a y in fact be used

SELF-CONSISTENT APPROACU TO TIlE FINE-STRUCTVRE CONSTANT 5~1

in an :~rbitrary gauge (13). We also note tha t the constant C~ which appears

in (2.1) aml (2.2) and which depends on the (, photon mass )) drops from (2.3)

bec~use of the Ward ident i ty (see (2.5) below).

Equat ion (2.3) is :m equation for ~t self-~enentting inter~ction. ] t must

be coupled with :~n eqm~tion for the eleelron propagator . This can be done th rough the Ward iden~ i ly

(2.5) (,

~x~' x , , = - - - [ 6 ( x ~ - - x l ) - - ~ ( x ~ - - x ~ ) ] S - ~ ( x , - - x 2 ) .

Equ~tion (2.5) implies :~ un i ta r i ty eqmttion for S ~nd the (( bootstr'~p ~ equation

tha t one e~a~ write for d S ( p ) / d p ~ (~) ~ts in the (~ bootstr,~p ~ approach to the phase t ransi t ion problem in stat ist ical mechanics (1~).

I n order to determine a one has to complete (2.3) and (2.5) by a consistency

condit ion on the photon propagator . This is equivalent to (1.6) and one muy write

Yg#v(Xl-- 3~2) ~ 0 f o r x, : / : x ~ .

This eqm~tion however c:mnot be expressed un,~mbiguously in terms of I ' ,

S and D,~ because it leads in general to divergent integrals (~5). A simple way out is to follow MACK aml SY~IA~z1~: (15) and to write

( 2 . 6 ) ( X l - - X 2 ) : T T F I v ( f l - - X 2 ) : O , X 1 ~ : X 2 .

Equ~tion (2.6) c~m now be expressed in terms of gr,~phs contuining / ' , S and

D~ ~md the graphical exp~nsion is shown in Fig. 2. Note tha t eq. (2.6) is gauge inw~rfimt while (2.3) and (2.5) :~re only gauge covarian~.

(13) Tile detailed mechanism by which one may reach by analytic continuation the region Z~ = ~ will be discussed in a separate paper (14) in a specific model. It will be shown in this model that even the boundary point (Z 2 finite) is correctly obtained by analytic continuation. Thus one may conjecture that (2.3) is correct not only in nearly all gauges but in fact in all gauges. If in a particular gauge, conditionally convergent integrals do occur, the only consistent procedure for their evaluation si then to consider these integrals as limits of convergent integrals in nearby gauges. (14) F. ENOL]~RT, J- M. F~/~I~E and P. NICOL~TOPOULOS: Vertex Bootstrap /or the F.i~e Structure Coastant prcprint TH. 1671 - ~!ERN (M~y 1973). (~5) G. MACK and K. S~')IANZIK: DESY 72/19, Iiamburg (1972). The proof of these authors is done in the context of a (~ conformal bootstrap ~) but does not depend on that assumption provided the integrals convcrgc. (16) M. D'ExAMO, L. PARIS1 and L. PETITI: Lett. Nuo~,o Ci.mento, 2, 878 (1971).

37 - II .\'~lot~o ( ' imeMo A.

562 F. ENGLERT

The sys tem (2.3), (2.5), (2.6) en t i re ly character izes (~:) the one-loop approxi - ma t ion of q u a n t u m e lec t rodynamics when m--~ 0 in the Gel l -Mann-Low and

Adler limits. They can also be viewed, as seen f rom the discussion in the In t ro -

duct ion, as describing comple te ly massless q u a n t u m eIectrodynnmics.

a b

c d

Fig. 2. - Graphical representation of cq. (2.6). ~ ,~ = (x~--x '~)S(x 1 - x2), b , d b d

I I = irreducible Bethe-Salpeter kernel K(x, , , % , x~, x~), [ - - - 7 - 7 = a C a c

= ( x ~ - x~) K(x~ , x~, x~, x~).

The fac t t h a t these equat ions are of the (, boo t s t r ap )) t y p e points towards

a group- theore t ica l content which we shall now invest igate .

3. - Conformal e lectron propagator and vertex funct ion .

Homogeneous in tegra l equat ions of the t ype (2.3) a d m i t solutions which are covar ian t under a conformal group (in the presen t e~se 04. 2) if the prop- agators a re conformal ly eovar ian t . This p r o p e r t y has been p roven for the

scalar <(bootstrap ,) by POLYAKOV (3) and the proof can be s t r a igh t fo rward ly

general ized for par t ic les wi th spin (4). The presen t p rob lem is different be-

cause as discussed below, there is no choice of ~ which makes the pho ton prop-

aga to r (2.4) conformal ly eovar iant . Howeve r it will be convenient first to

forget this fact and wri te down the behav ior of the solutions of (2.3) and (2.5)

which would be obta ined under the (false) a s sumpt ion of eonformal symmet ry .

(17) In ref. (2) it was argued that ill a theory of self-generating interactions the self- energies are zero. The proof however rests on the assumption that the graphs used in the self-energy equation are convergent integrals. This is no~ the case in the <( conformal bootstrap )> and this fact motivates the use of derivative equations (see ref. (1~)) such as (2.6). Amusingly enough, in the present problem where the only elementary particle is the photon, the statement is indeed correct for this particle but in an apparently highly nontrivial way!

SELF-CONSISTENT APPROACH TO TIIE F I N E - S T R U C T U R E CONSTANT ~

The f ield-theoret ic consequences of confot mul s y m m e t r y are wel lknown (~.,.lS).

Under a special inf ini tes imal confornml t r ans fo rma t ion defined by the vector

(3.1) ~ # ' - 2 x ' ( c . x ) - - d ' x ~

scalar, vector and spinor fields t r ans fo rm ~s follows:

(3.2)

~cf (x ) = 2 l z ( c . x ) c f ( x ) ,

~A~ , ( x ) - - 2 l , ( c . x ) A ~ , ( x ) - - 2(x ~ % - - c ~ x ~ , ) A ~ ( x )

~ , ( x ) = 2 1 / c . x ) ~ ( x ) - � 8 9 ,

~ ( x ) - ' . ' ~ . / c . x ) ~ ( x ) + �89

where Is, l s and l v are respec t ive ly the dimensions of % A~ and ~ in uni ts

of length. I f the v a c u u m is i nva r i an t under the eonformal group, the var i~t ion

of a n y Poincar6 inw~ri~nt fo rm fac tor cont r ibu t ion to an n-poin t funct ion under (3.1) mus t be equal to the cor responding v~ria t ion of tile n-point funct ion ob-

t a ined th rough (3.2), a t least for noncoinciding space- t ime points: the ana ly t i c behav iour for coincident a rgumen t s is de t e rmined by causali ty. This procedure

gives uniquely the fo rm of all conformal ly cow~riant p ropaga tors and 3-point funct ions; higher-order Green 's funct ions ~rc de te rmined up to a rb i t r~ry func- t ions of ha rmon ic rat ios (3).

The eonformal e lectron propagar is

(3.3) S . ~ - i = c ( ~ , . x ) , ~ x ~ ,

where e is :~n a r b i t r a r y cons tan t , x ~ . ~ , = X l - - X 2 t t and (y.x)~2 = ~xl~.~. This is indeed the correct answer , as seen f rom (2.1) with the identif icat ion

(3.4) 1 ~ - ( 7 - 3 ) / 2 .

The dimension of the fe rmion field depends l inear ly on the gauge p a r a m e t e r ~.

There are two traeeless eonformal fo rm f~ctors for the th ree-po in t funct ions

G~, = (.Q[TJ~,(x3)yJ(Xl)f2(x2)[a} where J , is the e lec t romagnet ic current (9.~8). The

conformal ve r t ex funct ions h~ve the same form prov ided l F is replaced by

- - l~. - - 4 (this ensures t h a t the dressing by ex te rna l electron propaga tors gives

(is) G. MACK and A. SALAd: A n n . o / P h y s . , 53, 174 (1969). (19) I am indebted to J. M. ]~R~RE and P. NICOLETOPOtYLOS for pointing out to me that Migdal's form factors (ref. (~)) are not linearly independent. In this context see also ref. (0).

5 6 4 v. ~ G L ~ a T

the correct Green's functions). These are

(3.5)

(3.6)

"~'A A. (.4) , . x2 ) ~, l--1 .. , I--1 --21F--8--~ ' = ~ . A / ~ (X3, X l , ~`4(~) '~13)X13 Yt t (7 X)32X32 X12

~B B (B) . . . . I+1 3 [ " ~'32"~ : ~ .r~, I~.~, ~,,, , ~ ) : ~ , ( x ~ j ~ ) ~ , , [~og~)(~ .~)~2x ,~ ~ 1o ~,

2A, 2~ are a rb i t r a ry constants , 1 is the dimension of j , . These are odd under charge conjug,~tion; ano ther form f~etor (C) with the same behaviour under charge conjugat ion can be wr i t t en down but we shall disregard i t because it is

not traeeless ~nd therefore can be decoupled f rom A and B in the (, boots t rap ~) eqm~tions. In Appendix A the re la t ion of A, B and the conformal form factors

obta ined by MIGDAL (4) is given; a procedure t h a t yields d i rec t ly A, B (and C)

is p resented in Append ix B. I t is seen t ha t A .~nd B are consis tent with (3.3) and the Ward iden t i t y

(2.5) if and only if

(3.7) 1 -- - - 3 ,

aS expected (20). (Under this condit ion C is pure t ransverse; see Appendix B.) I t is also seen tha t the corresponding expressions for G, h'~ve a space-t ime

dependence on the gauge only through a12 as required Fina l ly we remark t ha t if (3.3), (3.5), (3.6) and (2.4) are inser ted into the

r ight-hand side of eq. (2.3) power count ing would indicate t h a t all integrals converge in the ordinary sense for --o5 < 1 F < - - ~ (for a detai led descr ipt ion of convergence cri ter ia ~nd power count ing procedures, the reader is refer red to MACK and TODOROV (22)) and power count ing is indeed the correct cr i ter ion as has been shown by SPEER (23). In this range 7 is negat ive and therefore Z2(A = c~) -- 0 ; this shows the consistency of the present procedure.

Moreover, outside this range, the integrals can be ana ly t ica l ly cont inued for near ly all values of l F because the only singularit ies of the integr~fls are poles

arising from F-functions at l~ = - - ~ § n where n is any integer; thus one may

enlarge the t~bootstrap ~ equat ions even for gauges which make Z2 diverge. However , the ve r tex funct ion cannot be s imply :r l inear combinat ion of A

~nd B because the photon propagator (3.4) is not conformal ly eovar iant ; using

(3.1) and (3.2) one indeed shows t ha t the only eonformal p ropaga tor for '~ vector

field is

[ 2x12.~x12,~ r2z, (3.8) <.QITA~,(x~)A~(x2)]~) : ~t[g,,~--- x~2 - ) ' ~2 �9

(~0) K. C. WILSON: Phys. Rev. D, 3, 1818 (1971). (21) K. C. JOIINSON and B. ZUMINO: Phys. l~ev. Lett., 3, 351 (1959). (22) G. MACK ~md I. T. TODOROV" Trieste preprint IC/71/139 (1971). (23) E. SPEER: Journ. Math. Phys., 9, 1404 (1968).

S E L F - C O N S I S T ] ~ N T A P P R O A C I [ 2 '0 ' F i l e F I N E - S T R U C T U R ] ~ C O N S T A N T 5 6 5

For l, = - - 1 we see tha t only the g~uge t e rm of the photon propagator is

conformally covariant (see eq. (2.4)). We shall now discuss this point and

the related depar ture of F , from conformed covariance (~).

4. - Group theory of the self-consistent equations. The generalized {{ bootstrap . .

The (( dynamical ~) conformed symmet ry of the ~ boots t rap )) equations is

violated by the photon i)ropao'ator (2.4). This might appear surprising, be-

c~use in the present formulat ion only the photon ~ppears :~s e lementary and it

is well known tha t M~xwell's equations :tre conform~lly eovariant (~5). The

origin of the trouble arises in the quant izat ion procedure; the gauge-invari~nt

free action fF~,~P'~d4x, where F is the electromagnetic field tensor, is in-

deed eonformally inw~riant but this L:~grangian cannot be quantized in terms

of the electromagnetic potent ia l bec:~use of the wmishing of the conjugate

m o m e n t u m to A0. In fact the photon-free propagator should be the inverse

of the coefficient at AvA~ in the action, which in momen tum space is g , ~ q 2 - - q , q~. This is indeed a conformal object but does not ~dmit :~n inverse.

I n ordin:~ry formul~tions of qu~ntum electrodym~mics, one overcomes this

difficulty by adding" a g~uge te rm to the L~gr~ngian and this destroys in general

the explicit conformal invarianee of the qm~ntized theory.

To restore eonformal inv~rianee one may t r y 1o construct :~ conformal

gauge-dependent L~grangian. Indeed ABI)ELLATIF (+) and ADLER (~0) have ob-

ta ined the following couformally cow~riant propagator (.o6):

2 2 ~ ~ log Xlo ~- (4.1) D~v(;~l.. ;.%~ ; x2,) : g/,,' .r12 + ~Xl# log x,,1 ~x2~ "

- - _ _ ~ ):~12,1~ 12.VX12) + 2 ~ , log x12 ~ h)g x,,2 - - 2 log x,1 log ~1~,2 - - ~ (gm.J~12 ) . , -a , C ;)/'2 v ~')/ ' l / t ~,/'., v ~'~

(24) A formal solution involving only conformal form factors can however be constructed. This is because one can show that the departure of conformal

invariance due to the photon propagator in integrals of the type /. ~

is ~-.,q-4cil,(q."c"+ c#q ~) ~ . Therefore if one chooses linear combinations of form factors A, B, C such that the longitudinal equal-time contribution drops out. then these linear combinations will be self-reproducing in (2.3). Such a solution is however of no physical interest because it would imply e - -0 by the Ward identity. (2s) See for instance T. ]~ULTON, R. ROIItCLICH and L. WITTEN: Rev. Mod. Phys., 34, 442 (1962). (2G) This photon propagator is equation 110 of ref. (10) where x 1, and x 2, are made to coincide in x, and where we have added a conformal gauge term (3.9). See also ref. (9).

566 F. rNGLrRT

o r

(4.2) . . 8 [ l o g x 1 2 ~ l o g x a l ] - - D, dx~2 , Xla , X2a ) ---- gt,~x~ q- ~ 2 ~x,, J

8 ~Xl/~ ~X2v J ~Xitt ~X2y

Here G is an a rb i t r a ry externa l point to be varied in a conformal t rans fo rmat ion ; its cont r ibut ion to every g~uge-invariant quan t i t y vanishes because it appears only in the gauge terms. A direct check of the conformal invariance of (4.1)

by use of (3.1) and (3.2) is s t ra ightforward. We ma y formal ly in te rpre t (4.1) by adding an addi t ional scalar field Cf(Xa)

of dimension zero in the photon propagator and write

(4.3) c D~,jx~2 ; Xla , X2a ) : < ~ r [TA~,(xOAJx2)cf(x~)Iag~> ;

eonformM invar iance of the vacuum 1~} in (4.3) now gives (4.1). Taking the

l imit xo--> c~, one has

(4.4) l im <Y2~ITA~,(x~)A~(x2)q~(G)[/2~> = (Y21TA~,(xl)AJx 2) IT2>. xa--->co

The r ight-hand side is the previous photon propaga tor (4.4) and one sees t h a t in this l imit conformal invariance is lost.

Quantum electrodynamics can be rewr i t ten consis tent ly in this conformal gauge; this amounts to in t roducing the field ~(x~) in every Green's funct ion except in gauge invar ian t ones (such as (~lTJ~,(x~)J~(x~)l~Qc}), where it would drop out automat ica l ly . Le t us first s tudy the implications of (4.3) and (4.4) on the group-theoret ical proper t ies of the (( boots t rap ~> of Sect. 2.

I f one inserts in the r ight-hand side of (2.3) a l inear combinat ion of A and B,

the resul t will not look eonformally covariant . However if we replace the

photon propagator by (4.3) we obtain a eonformal object having the trans-

fo rmat ion proper t ies of

(4.5) F.(x~; x l , x2; xo) = <gqYJ.(x~)~ ' (x , )C. ' (x~)q~(xo) lg~ ,

where y / i s a fictitious spin-�89 field of dimension l• 4. Inser t ing back (4.5) into the (~ boots t rap ~) now yields an object with the s,~me t ransformat ion prop- erties. Thus if we assume tha t a solution of the ((bootstrap ~> equat ions does indeed exist and tha t inversion of limits is permissible, we obtain from (4.4)

(4.6) F~(x3; xl, x:) = lim (,,[2~lTJt,(x3)~f'(xl)(p'(x2)qg(x~)l~c}. Xa-->co

SELF-CONSISTENT APPROACII TO TIlE F I N E - S T R U C T U R E CONSTANT 5 ~ 7

3Ioreover, the Ward iden t i ty (2.5) implies

(4.7) ~'"F,(x~;Xx,X~) 0,

where the pr ime symbol means tha t the der iva t ive is t aken at noncoincident ex te rna l points.

Consider inste~d of (4.7) the equat ion

(4.8) ~'~'<[2 ~ 1TJ~,(x~) y)'(x~) (p'(~,~) c~(x )I~Q~> = O .

This equat ion is conformMly invari,~nt. To prove this we consider only the t ransformat ion of x3 under (3.1) bec,~use x~, x~ and xa are untouched by the de- r iva t ive operat ion. This gives the following var ia t ion of (4.8):

(4.9) [--2(c.x) ~ - - 2 ( x ~ c ' - - c~x )] ~ ~.0 I l J , ( x ) ... }~i::'

+ [-- 6(c "x) 6~ - - 2(x ~ c, - - c ~ ;%)] ~"~<.Q~ lTJ~(z) ... I ~ } +

~- [-- 6c z - - 2(c ~ - - 4c~)]<~ ~ ITJ~(x) ... [~> = - - 8(c "x) ~ " < ~ ]T,~(x) ... ]~> .

Equa t ion (4.9) implies t ha t the lef t -hand side of (4.8) is ~ conformal object with t r ans format ion propert ies

(4.1o) <t2~ ~'(x~)~'(x~)~(xo)1~99 ,

where qS(x3) is a scalar field of dimension - - 4; this proves the s ta tement . Now (4.8) implies in general (4.7) ~nd the converse is also t rue because if

(4.8) is not satisfied then the divergence for x,~ finite is of the form (4.10) and this conformal form f~ctor will have a nonzero l imit when Xa--~ C~ because ~(x,) has dimension zero. Thus provided the convergence assumptions used here are correct, the vertex J u n c t i o n / ~ ( ~ a ; x~, x2) is the l imi t w h e n x 4 ---> c~ o / c o n / o r m a I

/ o r m / a c t o r s sa t i s /y ing (4.8). In this l imit conformM covariance of the ver tex funct ion ma y be kep t :~s in the form factor A, B or it m ay be lost us will be shown below.

We shall not derive the most general solution of (4.5) and (4.8). We simply r emark tha t the solution of (4.5) depends on a rb i t r a ry harmonic ratios (,_,7) of

x12x3~/x~x23 and x~.x32/x2,~x~3 and tha.t (4.8) implies a res t r ic t ion on these ~r- b i t r a ry functions. However we wish to show by a physically re levant example

how the hidden group-theoret ical propert ies of Fu(x3; x~, x2) may be brought to l ight by using (4.6) and (4.8). One cun easily obtain a conformMform factor saris-

(27) The general form of this solution has already been given (with lp-- -- ~ but that restriction is unimportant) in the work of ABDELLATIE (ref. (9)).

5 6 8 y. ENGLERT

fying (4.8) b y mul t ip ly ing A on the left b y (7" ~)1 log (X~/Xla)~ on the r ight by (F.~)~ log (X=~/X~a)~ ~md b y raising the power of x~= b y two units . This is ~n

immediu te consequence of eq. (B.5) of Append ix B. Le t us therefore consider the following conformal fo rm fac tor ob ta ined by adding to t h a t one 2B:

( 4 . n ) x : , xo) =

2 1 2 - - - - ~D(~?'~)llOg~1a(~2"X)I2(~2"X)~]t~(~]'X)31(~?'X)12(~?'~)2110g X~----X2~ (X~aX~) --~ X~-~ ~-~ .

This gives in the l imi t Xa---> CO a new permiss ible fo rm fac tor D which is not

conformal ly covar ian t :

(4.12) r (D)z l im ,,# t, ix=; x~, x2; x,,) -4 -2~-s = - - 2,(7"x)23y,(V'x)3~(x~x32) x12 = Xa--->oo

~ D ~ ~ 4 --4 1~ 2l F 5

I n the canonical l imi t (l~ = 2 - - ~ - ) , (4.12) reduces to the convent ional F e y n m a n

d iagram drawn in Fig. 3. I t s conspicuous absence in the canonical l imi t of the

conformal fo rm factors A and B indicates the re levance of the present extension

of conformal s y m m e t r y .

x 3 ~ A

(~,.x)~3 x ~ . xh2 x3-2 ~ / X

X go-~ X122 X

Fig. 3. -- Canonical limit of the form factor (4.12).

Thus we h,~ve seen t h a t the ( (boots t rap ~) equat ions are not conformal ly

covar ian t bu t the electron p ropaga to r is and bo th the ve r t ex func t ion and the

photon p ropaga to r are the l imi t of eonformal objects when an addi t iona l field

is p ro jec ted out. These group- theoret ica l p roper t ies ~rise in fact f rom the following fe~tures :

i t i s poss ible to ]ormula te a ( (con]ormal bootstrap ~ d e p e n d i n g on an ex terna l

p o i n t xa and this general ized ~ bootstr,~p ,) yields the ~ boo t s t r ap ~) of Sect. 2

when ~(x,) is p ro jec ted out. These s tu tements are immed ia t e consequences of a formula t ion of massless

q u a n t u m e lec t rodynamics in the eonformal gauges (4.3). Indeed as s ta ted b y ABDELL~_TIF (9) the t e rms in (4.2) which depend on xa are less s ingular and the

renormal iza t ion p rog ram can thus be t aken over in these gauges. Moreover the

S E L F - C O N S I S T E N T A P P R O A C H TO T H E F I N E - S T R U C T U R E CONSTANT 569

only traceless conformal fo rm fac tor for

is

( y ' X ) 1 2 ' X12 2 [ k"--i

and this does not depend on x~ (9). Thus to formuh~te a conformM <( boo t s t r ap ~

it suffices to reph~ce in (2.3), (2.5) and (2.6) F~,(xs; x,, x2) and D,, as given b y (2.4) b y their general izat ion F,(x3; x,, x~; x,) and D~, while keeping S(x,2).

Polyakov ' s t heo rem implies the conformal covariance of these general ized

boo t s t r ap equations. I f we now push x~ to inf ini ty we are, bec~use of (4.4), led

again to the boo t s t r ap of Sect. 2 and conformal covar iance is lost for the photon

p ropaga to r and in genera l for the ve r t ex funct ion bu t not for the electron

p ropaga to r which is un touched b y the l imi t ing process. A consequence of this

nice p r o p e r t y is t h a t one recovers the ADLER-:BARDEEN- (,2) fo rm of S (eq. (2.1))

d i rect ly f rom the group- theore t ica l solution of the <~ boots t rap . ,)

We close this Section by r emark ing t ha t the previous discussion concerning the convergence of the <~ boo t s t r ap ~> integrals app ly for sui table choices of ge-

neral ized F~.~ as for ins tance to (4.11). This leads us to suspect t h a t the con- vergence proper t ies of the ((bootstrap)) equat ion are not spoiled by the new fo rm factors which have to be introduced.

5. - S u m m a r y .

Let us wri te F , - - ~ ~ F ~ where i runs over M1 permiss ible fo rm factors i

sat isfying 0 .5) and (4.8). (The ~, m a y include coefficients of t e rms in a series expans ion of ha rmon ic ratios.)

Equa t ion (2.3) would in principle de t e rmine c~ = ]dl~) for an a rb i t r a rygauge p a r a m e t e r ~]. The W~trd iden t i ty (2.5) would t hen al low us to re la te e to l~ (2s). Thus c ~ = h~(e). I n s e r t i n g this in (2.6) would then provide an eigenvalue equa t ion for ~.

(2s) A linear relation between ~l and lu is provided from gauge covarianee (see ref. (20) or (12)). We have not been able to show by a direct proof that the (( bootstrap ~> equa- tions are consistent with this relation. I t is however not unreasonable to expect such consistency because the right-hand side of (2.3) closely duplicates the conventional field-theoretical equations from which this relation may be derived. We shall therefore not consider the possibility that this may yield an additional equation for e. We also mention that because of departure from straightforward conformal invariance, the Ward identity need not be verified diagram by diagram. We should in fact not expect such a simplifictttion to hold in the present theory because it does not occur in more conventional theories either. I t is therefore not a trivial problem to approxi- mate the solution of the equations by cutting off the series.

570 r. ~NGLERT

One ma y hope tha t this p rogram can even tua l ly be simplified for the fol- lowing reason. The depar tu re f rom eonformal sy m m et ry of the boots t rap equat ions of Sect. 2 has led us to generalize these equat ions in Sect. 4 in such ,~ way tha~t conformal symmet ry is re ins ta ted. This was done at the cost of in t roducing an a rb i t r a ry ex te rna l point which is s t rongly reminiscent of a pro- jee t iv i ty (see in this context ref. (~o)). I f such a projee t iv i ty has itself a group-theoret ical content , then the possibility exists t h a t ~ could be obta ined more d i rec t ly f rom group theory.

Final ly we remark t ha t eq. (2.3) is not res t r ic ted to massless quan tum elec-

t rodynamics . In fact if ~ is indeed de termined by the self-consistency of massless quan tum electrodynamies, t hen it follows from Adler 's work (~) t h a t the form

of expression (2.2) for Z2 is correct in the physical massive q u an tu m electro- dynamics. Thus eq. (2.3) would still be t rue in massive quan tum electrody- namics and it follows from the discussion of Sect. 4 tha t the electron mass could then be a t t r ibu ted to a spontaneous break-down of conformal invarianee, the physical charge being still given by the unbroken solution.

We are grea t ly indebted to R. ]3ROUT for m a n y discussions on all aspects of this work. We thank P. MANNHEIM for in teres t ing comments on charge

conjugation propert ies of conformal form factors. We wish especially to t h an k S. ADLE~ for sending us so p rompt ly his recent work (ref. (~o)).

A P P E N D I X A

Relation of form factors A, B with Migdal's form factors.

(A.1)

(A.2)

(A.3)

1V[igdal's form factors for the ve r t ex funct ion are

--4 4 --2~F--7 A~ = ~(y'x)137,~(?'x)32 Tr ((7"xl~2(7"xh2)x~3X~aX~ ,

,'r fl --4 --4 -2~F-7 B 1 = 17~, (y .x)uy#(y 'x)327 ~ Tr (y (y'x)12y (7"x)n)xlax32x~2 ,

2.8 )'0 - 4 --4 --2/p-7 C1 = ~a~,~(y'x)lay#(r'x)a:(~o Tr (a ( ~ ? ' X ) 1 2 O" (y'X)12)X13X32X12 �9

A tedious bu t s t ra ightforward ea.leulation yields (for 1 = - - 3 )

(A.4)

(A.5)

(A.6)

A I = A

B t = 4 B ,

C1 = 3 2 B - - 8 A .

~ELF-CONSISTENT APPROACH TO THE FINE-STRUCTURE CONSTANT 571

A P P E N D I X B

Some useful relations for the construction of conformal form factors.

The fol lowing re la t ions follow d i r ec t ly f r o m (3.1) and a n t i c o m m u t a t i o n rela- t ions of D i r ac m a t r i c e s :

(B.1)

(B.2)

(B.3)

(B.4)

~X12 ~-- [C" (X 1 -~- ,5:2) ] X12 ,

~(r'x),. = It. (x~ +x~)] (y.x),2--�89 e), (v'xh] (7"x,~) + ~(y" x),~[(y-c), (r 'xh],

{[(r'c), (~,.x)] r . - r . [(~'c), (r 'x) ]} = - 2 (x%~- x.c~)~,

(B.5) 3(y.~)~ = - -2 (c . x ) , (y'O)~--�89 -k ~ (y.~)~ [(~.c), (~.x)].

A p p l y i n g these re la t ions to A a n d B a n d c o m p a r i n g wi th (3.2), one imme- d i a t e ly checks t he c o n f o r m a l cova r i ance of these f o r m factors .

We see t h a t t he fo l lowing f o r m fac to r is also eonfo rmal ly covar i an t (and odd u n d e r charge con juga t ion ) :

(B.6) [lo '~ xa~'l+x x "l~X--2lF--9--I

One sees t h a t C is d ivergence less even a t equal t i m e and the re fo re would no t give a n y c o n t r i b u t i o n to t h e W a r d i d e n t i t y w i t h o u t be ing incons i s t en t wi th i t ; a d i rec t eva lua t ion of i ts Fou r i e r t r a n s f o r m at zero p h o t o n m o m e n t u m shows t h a t C(0 ; p, p) = 0.

�9 R I A S S U l q T O (*)

Si forma l'elettrodinamica quantistica come un (< bootstrap >> di teoria dei campi. Si interpretano le equazioni che ne risultano come equazioni autocoerenti per la carica elettrica fisica, seguendo la congettura di Adler. Si esaminano 18 propriets di queste equazioni nella teoria dei gruppi. Si mostra che esse non sono invarianti rispetto al gruppo conforme 04.2 helle gauge convenzionali, ma che in altre gauge si pus formulare un <~ bootstrap >> generalizzato che ammette 04~ come suo gruppo di invarianza. Si sug- gerisce quanto siano importanti queste proprieth di gruppo nella determinazione di e nella elettrodinamica quantistica con massa.

(*) Traduzione a cura della Redazione.

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