Self-consistent approach to the fine-structure constant

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

    Self-Consistent Approach to the Fine-Structure Constant.


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

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

    Summary . - - 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. - In t roduct ion .

    The possibi l i ty of determin ing the f ine-structure constant a = e~-/4~ f rom qu:mtum electrodym~mies nlone h:~s recent ly been revived by ADLEIr (1). In the present work we sh~ll interpret Adler 's theory in terms of self-geuer~ting interuetions (2) thereby m~king the connect ion with the so-called ((conform~l

    boots t rap ~ (~.4). The :tim of this progr:~m is to obt~fin self-consistent equutions for a ,~ud to use these eqm~tions to g:fin some insight into the possible group- theoretic~l significance of the physic~l electric 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 ETP Lctt., 12, 538 (1970). (4) A. A. MIGDAL: Phys. Let&, 37B, 98, 386 (1971).


  • 558 F. ~NGL]~T

    We first briefly summarize the work of BAKE~, Jom~so~ ~, VVTYIm~Y (5) and ADLER (~). I f quantum electrodynamics admits a Gell-Mann-Low limit (~), then the photon propaggtor renormalized by a subtraction procedure at q2=_ ~ts exists when m-+ 0 (7) where m is the physical electron mass and simply becomes

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

    Here ~/is a gauge parameter and the coupling constant is ~0, the solution of the Gell-Mann-Low eigenvalue equation. When (1.1) is used to evaluate the re- normalized vacuum polarization 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 strict ly equal to zero at the outset (massless quantum 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 (~) that it is sufficient to solve the equation

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

    where F(r is the coefficient of the single logarithm appearing in the contribution to 7g,, from the single-fermion loop. Moreover this theorem also implies (~) that F(~o) vanishes with an essential singularity in ~,.

    ADLER (~) now shows that if m is arbitr~ry (and in part icular arbitrar i ly small), the eigenvalue equation

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

    where ~ is the physical coupling constant, is consistent with the Gell-Mann- Low limit. Assuming convergence of a 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).


    indeed proves that (1.6) guarantees finiteness of massive quantum electro- dynamics for large momenta. Thus if (1.6) is correct, ~ may be interpreted as that value of the renormalized coupling constant for which massless quantum elcetrodynamics exists.

    The eigenvalue equation (1.6) is thus entirely ch~r~cterized by the one-loop approximation to ~,~ in massive quantum electrodynamics. In Sect. 2 we shall write the coupled Sehwinger-Dyson equ:~tions for the corresponding renormalized quantities and argue why these equations are of :~ type for asymp- totic momenta or equivalently for massless quantum electrodynamics.

    Section 3 contains a prel iminary discussion of the electron propagator and of the vertex function obtained by imposing covariancc umler the conformal group 04. z. I t is shown that despite the f:,ct that the COilform~fl electron prop- agator is the correct one and that the conform:~l vertex function is consistent both with the Ward identity aml with g;~uge covariance, this cannot be the correct solution of the > equations. Additional form factors for the vertex function must appear because of the nonconformai n.~ture of the photon propagator (1.1).

    In Sect. 4, the group-theoretical properties of the solution of the depending on the external point is formulated and its invariance group O~. 2 exhibited.

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

    2. - Self-consistent formulat ion of massless quantmn eleetrodynamics.

    If vacuum polarization loops are omitted, the asymptotic form of the re- normalized electron propagator in massive quantum 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), (.'

    z2=c1 ~, 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 providing an infra-red cut-off and A the ultraviolet cut-off momentum; y(a) depends l inearly on the gauge parameter ~ and can be varied from - - c~ to + c~. One should notice that despite the fact that Z~ is not the renorm,dization constant of the complete theory, the set of graphs considered does in fact satisfy the Ward ident ity and therefore Z., = Z,. The renormalized vertex function /~ in the absence of vacuum polarization loops then obeys the Schwinger-Dyson equation with an inhomogeneous term proport ional to Z2 as given by (2.2). Therefore if y(~) ~ 0 we may write in the l imit ,/1 -+ c~ a homogeneous integral equation (Fig. 1)

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


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

    provided each integral on the r ight-hand side converges. propagator is in configuration space

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

    In (2.3) the photon

    The coupling constant e has been included in the definition of T' and ~](go, x l - ' ) . _ _ --2x~x~x -~) is the space-t ime expression of the longitudinal gauge term. In

    all that follows x ~ will mean (x2--is) ~/2. For momenta large compared to m we shall search for sc~de invar iant solutions

    of (2.3); m will therefore drop from the equ'~tion and we may interpret (2.3) in this l imit as an equation for i , for all finite moment.~ when m -> 0.

    In the following Sections we sh~dl argue that each integral on the right- hand side can be defined in nearly all gauges ~nd that the singul~rities occurring for exceptiom~l gauges are simple poles. Therefore / ' , as defined from (2.3) can be continued analyt ical ly even if Z~ diverges and (2.3) may in fact be used


    in an :~rbitrary gauge (13). We also note that 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 identity (see (2.5) below).

    Equation (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 through the Ward iden~ ily

    (2.5) (,

    ~x~' x , , = - - - [6 (x~- -x l ) - -~(x~- -x~) ]S -~(x , - -x2) .

    Equ~tion (2.5) implies :~ unitarity eqmttion for S ~nd the (( bootstr'~p ~ equation

    that one e~a~ write for dS(p) /dp ~ (~) ~ts in the (~ bootstr,~p ~ approach to the phase transition problem in statistical mechanics (1~).

    In order to determine a one has to complete (2.3) and (2.5) by a consistency condition on the photon propagator. This is equivalent to (1.6) and one muy write

    Yg#v(Xl-- 3~2) ~ 0 fo 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 - -X2) :TTF Iv ( f l - -X2) : 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 that 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 - I I .\'~lot~o ('imeMo A.

  • 562 F. ENGLERT

    The system (2.3), (2.5), (2.6) entirely characterizes (~:) the one-loop approxi- mation of quantum electrodynamics when m--~ 0 in the Gel l -Mann-Low and Adler limits. They can also be viewed, as seen from the discussion in the Intro- duction, as describing completely massless quantum 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 fact that these equations are of the (, bootstrap )) type points towards a group-theoretical content which we shall now investigate.

    3. - Conformal electron propagator and vertex funct ion.

    Homogeneous integral equations of the type (2.3) admit solutions which are covariant under a conformal group (in the present e~se 04. 2) if the prop- agators are conformal ly eovariant. This property has been proven for the scalar


    The field-theoretic consequences of confot mul symmetry are wel lknown (~.,.lS). Under a special infinitesimal confornml t ransformat ion defined by the vector

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

    scalar, vector and spinor fields transform ~s follows:


    ~cf(x) = 2 lz (c .x )c f (x ) ,

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

    ~ , (x ) = 21 /c .x )~(x) - 89 ,

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

    where Is, l s and l v are respectively the dimensions of % A~ and ~ in units of length. I f the vacuum is invar iant under the eonformal group, the vari~tion of any Poincar6 inw~ri~nt form factor contr ibution to an n-point function under (3.1) must be equal to the corresponding v~riation of tile n-point function ob- tained through (3.2), at least for noncoinciding space-t ime points: the analyt ic behaviour for coincident arguments is determined by causality. This procedure gives uniquely the form of all conformally cow~riant propagators and 3-point functions; higher-order Green's functions ~rc determined up to arbitr~ry func- t ions of harmonic ratios (3).

    The eonformal electron propagar is

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

    where e is :~n arb i t rary constant, x~.~,= Xl - -X2t t and (y.x)~2 = ~xl~.~. This is indeed the correct answer, as seen from (2.1) with the identification

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

    The dimension of the fermion field depends l inearly on the gauge parameter ~. There are two traeeless eonformal form f~ctors for the three-point functions

    G~, = (.Q[TJ~,(x3)yJ(Xl)f2(x2)[a} where J , is the electromagnetic current (9.~8). The conformal vertex functions h~ve the same form provided l F is replaced by - - l~. - - 4 (this ensures that the dressing by external electron propagators gives

    (is) G. MACK and A. SALAd: Ann. o /Phys . , 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).

  • 564 v. ~GL~aT

    the correct Green's functions). These are



    "~'A A. (.4) , . x2 ) ~, l--1 .. , I--1 --21F--8--~ ' = ~.A/~ (X3, X l , ~`4(~)'~13)X13 Ytt(7 X)32X32 X12

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

    2A, 2~ are arbitrary constants, 1 is the dimension of j , . These are odd under charge conjug,~tion; another form f~etor (C) with the same behaviour under charge conjugation can be written down but we shall disregard it because it is not traeeless ~nd therefore can be decoupled from A and B in the (, bootstrap ~) eqm~tions. In Appendix A the relation of A, B and the conformal form factors obtained by MIGDAL (4) is given; a procedure that yields directly A, B (and C) is presented in Appendix B.

    I t is seen that A .~nd B are consistent with (3.3) and the Ward ident i ty (2.5) if and only if

    (3.7) 1 -- - -3 ,

    aS expected (20). (Under this condition C is pure transverse; see Appendix B.) I t is also seen that the corresponding expressions for G, h'~ve a space-time dependence on the gauge only through a12 as required

    Final ly we remark that if (3.3), (3.5), (3.6) and (2.4) are inserted into the right-hand side of eq. (2.3) power counting would indicate that all integrals converge in the ordinary sense for --o5 < 1F


    For l, =- -1 we see that only the g~uge term of the photon propagator is

    conformally covariant (see eq. (2.4)). We shall now discuss this point and the related departure of F, from conformed covariance (~).

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

    The (( dynamical ~) conformed symmetry of the ~ bootstrap )) equations is violated by the photon i)ropao'ator (2.4). This might appear surprising, be- c~use in the present formulation only the photon ~ppears :~s elementary and it i...


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