Fortschritte der Physik / Progress of Physics: Volume 32, Number 9 [Reprint 2022 ed.] 9783112656228

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 1984 Number 9

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 . topuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: E . d ' E m i l i o , M. M x n t c h e v

Physical Charged Sectors in Quantum Electrodynamics I. Infra-Red Asymptotics II. The Charge Operator

I P

473-501 503-523

AKADEMIE-VERLAG • BERLIN

ISSN 0015-8208

Fortschr. Phys., Berlin 32 (1984) 9, 473-523

EVP 10,- 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 copies 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-spaces 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 drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings 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 (vektors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He p, ...), elementary mathematical functions like Be, Im, sin, cos, exp, ...): black underlined Greek letters: red underlined Boldface Greek letters: red interlined twice Upringth 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 8, u V, 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, 6«, Mi J, Mfr Please differentiate between following symbols: a, a; oc, a , oo; a, d\ c, C, c ; e, I, g, e, k, K, x, x, X, x, X; Z, 1; o, 0, a, 0; p, Q-, U, V; U. % v, V; /» ?y]+ = * = g^y"*',

&=

g^y*8'-

The Fourier transform and its inverse are defined according to kv) = / d'xe^fix), where dtp =

f(x) = /

dtpe^f(p),

dip/(2n)i.

Gk(Rn) is the space of ¿-times countinuously differentiable complex functions on R". R") is the Schwartz test function space on Rn, , iyt).

(1.9b)

Postulating for (1.9) equal-time canonical Poisson Brackets (PB's) (which we do not explicitely write down for the sake of conciseness), one can check that {j0(x),

= 0,

(1.10)

{Ux),B{y)\\^_v« = Q,

(1.11)

Fortschr. Phys. 32 (1984) 9

{?'o(*)> V(y)}\*°=y° =

477

- y) w(y),

(1.12a)

{?'(«). y { y ) \ = -igS(x - y) f{y),

(1.12b)

{?•(*). b(y)U*°=v° = 0.

(1.12c)

[B(z), A^y)) = 8,D(x - y),

(1.13)

{B(x), y>{y)} = igD(x — y) y>(y),

(1.14a)

{B(x), y(y)} = -igD(x

- y) y{y),

(1.14b)

{B{x), My)} = 0,

(1.14c)

where the Pauli-Jordan commutator function m

= ^

(1.15)

6(2«) d(z*)

has the well known properties • -D(z) = 0,

(1.16a)

D{0,z) = 0,

(1.16b)

D(0, z) = »{«)

(1.16c)

(the dot in (1.16c) means time derivative). Equations (1.10—12) are immediate consequencesof the equal-timecanonicalPB's for(1.9). As for (1.13), by imposing the requirement of translation invariance and Lorentz covariance, one has {B(x), A,(y)) = F,(x - y) where, because of (1.8), F^x — y) satisfies •

- y) = 0.

(1.17)

Moreover, by repeated use of equal-time canonical PB's and employing (1.4a), one gets JV(0, a - y ) =

5(® - y)

(1.18)

F„(0, x - y ) = {B(x), ¿M(y)}|*._f. = {f(x), A,ly))U=f

- &{Fi„(«), A^y)}U.=„.

= grt 8'd(x -

y). (1.19)

Equation (1.17) with the initial conditions (1.18, 19) has a unique solution given by the right hand side of (1.13). The PB's (1.14) can be verified in analogous way. The next step of our discussion in this section is the study of global and local gauge symmetry properties of classical spinor electrodynamics. As usual the global (£7(1)) gauge transformations are defined by f(x) i-> y)(x) eigl,

tp(x) A^x)

A„(x),

B{x) h> B(x).

A € R1

(1.20) (1.21)

478

E. D'EMILIO, M. MINTCHEV, Physical Charged Sectors in QED. I.

For the local gauge transformations we adopt the following definition: y>{x)

•^/¿(x)

ip(x) i-> ip{x) ei(x),

(1.22)

Ap(x) -f- dpA.{x),

(1.23) (1.24)

B(x)y+B[x), where the real function A £ G2(R4) satisfies: EM(x) = 0,

(1.25)

lim A{x) = 0, |a|-wo

>| = (%2 + ae2)1'2.

(1.26)

As easily seen, the Lagrangian density (1.1) is invariant under both the above sets of transformations even without imposing (1.26)4). As shown below, this condition is essential in proving the existence of locally gauge invariant, but globally noninvariant fields. Moreover it implies that the intersection of the sets of global and local gauge transformations contains only the identity. Remark. Examples of (real) functions A £ G2( R4) obeying (1.25, 26) are provided by ^ i^T

cos

-1-

siML

,

a > 0,

(1.27)

where P x and P 2 are (real) polynomials. Let us define the current JliA(x)

= A(x)^llB(x),

^ „ = 5,,

—

(

1

.

2

8

)

which, owing to (1.8) and (1.25), is conserved, i.e. a«j/(®) = 0 .

(1.29)

Equations (1.7) and (1.29) imply that the charges Q = f d3xj0(x),

(1.30)

QA = j d3xJ0A(x),

(1.31)

are constants of motion. Moreover a straightforward calculation, which makes use of (1.10—14) shows that Q and QA generate respectively the global and the local gauge transformations (1.20, 21) and (1.22—24). At this stage we are prepared to construct locally gauge invariant classical fields carrying nontrivial global charge. Let us consider ¥(x; /) - e^/^Vs-vMKsO y,(x),

(1.32a)

T(x; f) = y{x) e ^ P V ^ s - » )

(1.32b)

/„€-$"(R 4 ),

(1.33a)

where

4)

(i = 0, - . . , 3

Usually (1.22—24), where A(x) is supposed to satisfy only (1.25), are called gauge transformations of the second kind and obviously contain the set of global (first kind) gauge transformations (1.20, 21).

Fortschr. Phys. 32 (1984) 9

479

satisfies dj"(z)=d(z),

(1.33b)

/ »

(1.33c)

= U»)>

and whose Fourier transform admits the representation U P )

= */

d

i q

F(

P

, q)

R8),

F ( p , q ) e # " (

+ o.

(c+

.

. c ± 6 R1.

(1.34) (1.35)

Equations (1.33 b, c) lead respectively to the normalization condition (c + + c_) / diqF(p,

q) = 1,

V P € M4

(1.36a)

and F(p,q)=F{-p,q),

V 2 € M*.

(1.36b)

Remark. To our knowledge, fields of the type (1.32) have been introduced in electrodynamics firstly by DIRAC [34]. In the context of the gauge A^ = 0 he chooses f^ in the following way: /„(*) = 0 ,

U(z) =

(1-37)

Such a choice is convenient in performing locally gauge invariant quantization, but is not suited for the investigation of the I R problem we are interested in. Let us check, for example, the local gauge invariance of W: W(x; f) i-> ei°Si'vf»(x-»*AHv)+i>i'A(vy\ =

-iA(x)+igf

e

d'yAWSff^x-tfip^.

(1.38a) f)

,

. f)#

(1.38b)

In obtaining (1.38b) we have used (1.33b) and the fact that the surface term, arising in the integration by parts of the second term in the exponent in (1.38 a), vanishes owing to (1.26) and (1.34). In the same way one can show that If is locally gauge invariant. On the other hand W and W transform under global gauge transformations according to V(x-, f)

e-W^x;

The gauge properties of pact form:

/),

W[x-, f)

e^lT{x;

/).

(1.39)

and T , verified above, can be expressed in the following com-

{QA, W(x; /)} = {Q*. T(x; /)} = 0 ,

(1.40)

{Q,V(x-,f))

= igT(x-,l),

(1.41a)

{Q,nx;f))

= -igT(x;f).

(1.41b)

The P B ' s (1.40, 41) can be checked independently by employing (1.10—14) and (1.28, 30, 31). Remark. I t is useful to notice that eqs. (1.13, 14a, 14b) imply {B{x), W{x- /)) = {B(x), T(x; /)} = 0.

(1.42)

480

E.

d'Emilio,

M.

The equation of motion for (i* - m)

Physical Charged Sectors in

M i n t c h e v ,

QED. I.

is

/) = 9V^"{x;

/) V(x; /),

(1.43)

where A"(x; f) = / d*y[gt"d(x - y) - 8"f'{z - y)] Av(y).

(1.44)

I t follows from (1.32 a) and the first equation in (1.6). Analogously one can write down the equation of motion for T . In using the fields (1.32) for the study of the I R problem, we will need a distinguished subfamily of the set of functions {/^J satisfying (1.33—35). This subfamily is given by (1.34) with F{p, q) = (271)* d(n — q)

(1.45a)

V P € M1,

w £ M4 being arbitrary. Because of (1.36a), eq. (1.45a) implies 2c + = (l + c),

2c_ = (1 — c),

(1.45 b)

c € R1.

Eqs. (1.34, 45) lead to _

»

I" 1 + c 2 [mp + is

1 — c "I wp — it J'

(1.46 a)

or in coordinate representation, + 00

W*)

=

n

" j f ^[t1 + — oo

^

- (! -

The fields obtained from (1.32, 44) by fixing

c

)

( - {x),

exp ig y J doc(( 1 + c) 6(ot) — (1 — c) 0(—«))»M,,(a; — aw) W(x\ n, c) + oo

= f(x) exp —iff

Y

J

¿«[(1

+

c)

6(a)

-

(1

-

C) 0 ( - « ) ]

n'A^x

-

ocn)

— 00

+ 00

A^x-, n, c) - A ^ x ) d p J — Z^gA^xpy^ip + dmijhp.

(2.3)

483

Fortschr. Phys. 32 (1984) 9

Furthermore dimensional regularization [41] of Ultraviolet (UV) divergences is understood where needed. The subtraction scheme will be specified later. The fact that Zx =

(2.4)

is a well known consequence of the invariance of (2.3) under (1.20, 21). In this framework we will show more precisely that : a) the operators

+ 00

*P(x\ n) = t„ : exp iff J 1/2

fdc : ip{x), dot e({, the vacuum, such that the Green functions are expressed in terms of -)f as vacuum expectation values of the corresponding time-ordered products. Consider furthermore the subspace I V = {0 € D( : B'(x) 0 = 0},

(2.6)

where B~ is the annihilation operator corresponding to B (the splitting of B in creation and annihilation parts is possible due to (1.8)). One can show in perturbation theory that all the elements of the family {D/, (•, -) { : f 6 R1} are isomorphic as inner product spaces. Let \D', (•, •)} be an arbitrary representative of this family. Then the quantum version of (1.4a) which follows from (2.3), and (2.6) imply that the Maxwell equation is weakly satisfied on D', i.e., .

(0,

-

*> = 0

\/0,X€D',

or, equivalently, it holds in strong form on the factor space D'/D", where D" — {0 £ D':

(0, 0) = 0}.

The physical state space can be defined without making use of any Hilbert space structure on D (see Ref. [44]), according to ^phys =

D'/D",

(2.7)

where the bar stands for the completion with respect to the Hilbert topology induced

by (•, •) on D'/D".

Finally from the quantum counterparts of (1.42) one has

. [B~{x), 9%; »)]_ = [B~(x), T{y, w)]_ = 0

(2.8a)

and consequently

[£-(*), e(y)l = [B~(x), ê(y)\. = 0.

(2.8 b)

484

E . D'EMILIO, M . MINTOHEV,

Physical Charged Sectors in

QED. I .

Therefore the fields {!?(«; n), P(®;»), «(«), «(«)}

(2.9)

leave D' invariant and give rise to globally charged states belonging to D'. Furthermore, these fields are nonlocal with respect to F^,, owing to the convolution in the exponent. This is the reason why they evade the theorem of Ref's. [6] and [7]. The next sections and Ref. [27] are devoted to the study of the charged sectors defined by means of (2.9). 2 B.

Feynman Rules

The following graphical notation is adopted. a) Propagators: = iSc{p) = i(p — m + ie)~l, "

"

*

= -¿M2

(2.10)

~ (1 - r1) kek,\ Ec(k; p)

= -iE%(k; p, f )

(2.11)

where the distribution Ee(k; ft) admits [26, 39, 45] the representations E°(k->fl) = 1 (fc2 + ie)-> ke JL In [(-fc 2 - &)/|A] = w - lim f(fc2 - A2 + ie)~2 + in? In (A2//«2) d{k)} = w - lim ^ d

~ ^ ( P + ¿e)"2 L a W ( - A 2 - ¿e)]"4

tf_>o+ do 1 (1 + 0)

(2.12a) (2.12b) (2.12c)

(the derivative in (2.12a) as well as the limits in (2.12b, c) are in weak sense). The form (2.11) of the photon propagator is quite unusual in the literature (see however Ref. [26]). It deserves therefore some comments. 1. The occurence of the mass scale ¡x can be better understood by noting that the distribution (k2 + ie)~2 (which one would be naively tempted to write in (2.11) instead of Ee{k\ fi)) is defined only on those (test) functions vanishing at k = 0. As well known, (fc2 + is)~2 can be extended to (test) functions not vanishing at k — 0. A set of causal extentions satisfying Lorentz invariance and the normalization condition (dictated by (1.3, 4a) with ¡7 = 0): (fc2)2 Ec(k; /¿) = 1

(2.13)

is given by the one-parameter family represented in (2.12a—c). 2. Concerning (2.12 a), an integration by parts is understood together with the" prescription of neglecting the surface term at k = oo. This certainly is a correct procedure when Ec{k\ ¡x) acts on test functions of sufficiently fast decrease at infinity. It has to be handled with care when one computes UV divergent integrals. 3. The parameter X appearing in (2.12b) is typical of the representation and has nothing to do with the "photon mass" used, e.g., in Ref. [2] as IR regulator. From this representation one can argue that when a certain Feynman graph has formal IR power counting zero, the usual diverging logarithms of the "photon mass" are replaced by In n (see Ref. [26]).

485

Fortschr. Phys. 32 (1984) 9

4. The connection of (2.11) with the usual form of the photon propagator in the gauge £ = 1 becomes manifest on taking into account the identity k*Ec(k; fi) = (k2 + ie)-1.

(2.14)

We emphasize, however, that (2.14) holds only when the left hand side acts on functions regular at k = 0. It is well known that this is not always the case, as, e. g., in the calculation of the matrix elements of the scattering operator. We will discuss this point in full detail in Section 3 B. Here we only anticipate that avoiding (2.14) and using (2.11) enable us to give an IR finite representation of the S matrix (see also Ref. [26]). b)

Vertices:

The first thing to be realized is that, in expanding Green functions involving W(x; n) and T(x; n), apart from the usual vertex

- — ig7ll,

(2.15)

there appear some extra vertices originated by the expansion of the exponents in (2.5). The peculiarity that distinguishes these extra vertices is that they are external, i.e., they are initial or final points of fermion lines. The Feynman rule for them is

*VL = -A/*' = (-,>. ...

(21.6)

As shown in (2.16), the extra vertices are denoted by empty circles. In what follows all the graphs containing at least one vertex of type (2.16) will be referred to as extra graphs. The simplest example where extra graphs occur is the one-loop contribution to the propagator (T¥(x; %) *P{y; w2)). They are pictured in Fig. 1. Fig. 2 represents all the graphs contributing to the one-loop approximation of the three-point function (T^(x; ^(0) T(y\ n2)). It is evident that no extra graphs appear in the expansion of the Green functions relative to and F ^ only. Two comments about the vertices (2.16) are in order, i) The analytic properties of the extra graphs are the same of the usual graphs with the same external lines, occurring in a given order of perturbation theory. Graphs (b)—(d) have for example a cut along the real positive axis of the complex p2-plane, with threshold at p2 = m2, much the same as graph (a). This is so because the analytic properties are determined only by the propagators (2.10, 11), as long as the vertices (2.16) satisfy the condition (1.33 c).

Pig. 1. The graphs contributing to G(1)(i>,

n2; e, I)

486

Physical Charged Sectors in

E . D'EMILIO, M . MIÏTTCHEV,

I.

QED.

A Fig. 2. The graphs contributing to G^(1)(2>,

k; e, |)

ii) The superficial degree of divergence of a single One Particle Irreducible (OPI) Feynman graph is D =

(2.17)

4 - - f - b - c ,

where b(f) is the number of external boson (fermion) lines and c is the number of empty circle vertices. Therefore graph (d) is UV convergent, while the UV divergence of the (amputated) graphs (b, c) contributes only to the wave function renormalization constant. Contribution to the mass renormalization is given only by graph (a). Finally we fix the notation for the Green functions : •••; x2f,

ylt...;

x

= (T 2f-i;

Vj

-

qi

• • -, P f , ii, • • -, 97-1 ;h,

=

F^fa)...),

qf+h+

(2.18) klt...)

n

i f \ h, •••; h, • ••)

-

+ kx+

•••),

(2.19)

...;k1,...)

Pi ; • • • ; Pi, Pf ; 21 ; • • • ; 37-1, 37-1 ; 37 ; h, • • • ; K • • •).

(2.20)

Fortschr. Phys. 32 (1984) 9

487

where in the right hand side of (2.20) qf = E P i + E h t=l i

+ E h - e\i i t=l

•

(2.21)

Moreover, any time we refer to unrenormalized Green functions, the dependence on the parameter of dimensional regularization e and on the gauge parameter £ will be explicitely written down. 3.

Green Functions

3 A.

Electron Self-Energy Function

One might consider the propagator %) !P(0; n2)). However, for our purposes (see eq. (3.12) below), it is sufficient to consider the case % = w2 = n. Let us start by examining the one-loop self-energy contribution 3f {1)(p, n; e, £) relative to the unrenormalized propagator G{p, n ; s , f ) = G(p, n ; n ; e , £).

(3.1)

The italic letter is used to distinguish the sum of graphs (a)—(d) from the contribution of the first graph alone, usually denoted in the literature by E ilf (P> e> £)• One starts from -i

n-e, f) = -g*

j

d^JcE^k-,

p, ()

>

X S —k, one obtains ac - i - 9>™{p, n;e,

/

d^kye8%p

f) = -g\? -

-

m) j

k) y a E e ( k ; p) &

C

T

dt_2ckE c(k;

T fi) t

1

mile" -4- lein" ^

+

1 +

ra 2jfe 2T

tfi

— J .

(3.5)

Remarks: a) The dependence on f can be eliminated by a partial renormalization, as it is evident from the first term in the r.h.s. of (3.5), in-which the bare structure p — mis factorized.

488

E . D'EMILIO, M . MINTCHEV,

Physical Charged Sectors in Q E D .

I.

b) After the above partial renormalization, n; e) coincides with the unrenormalized one-loop self-energy relative to (Tf(x) f(0)) in the gauge n^A* — 0. As eq. (1.51) suggests, this property remains true also for the Green functions (2.18), provided that % = w2 = ••• = »2/, but it is lost if the last condition is violated. Note that the study of the fields (1.49) forces one to consider the general caseTO;^ »,• (see e.g. eq. (3.22b) below, where nx ^ n2). Before pursuing the calculation of ¿?a)(p, w; e, f) any further, it is useful to make a comparison between the IR behaviour ^ ~ m of (3.5) and that of £a)(p; e, £). To this purpose it is convenient to consider the respective imaginary parts and discuss the possibility of imposing on-shell normalization conditions. Since -1*

I

"

[«-'-*+m]=(¿?

«-+3» (t

- ' + ' » •

where y is Euler's constant and A is the mass scale of dimensional regularization, the first line of (3.5) gives no contribution to Im £f{1) and one gets the ^-independent result I m Sfm{p,

» ; e , £)|,_ 0 = 2JI2 / djcy^p

- ]t + m) ya(2n) 6(p° - Jfl)

[

«eZ.®

m'he

"1

r

= j 0(P°) e(p° - »•)

+ (5) ^ - 1 + ¿(/J)] - 2m + 2*r

- 1) jar

(ntf J B(fi)}, (3.6)

where (npf A({i) = 2 ^ + ( l - ^ ) ( l - i l n l ± | ) ] ,

(3.8)

'

(3-9)

Eq. (3.6) shows that Im

- m2),

m; e, |)| e = 0 =

(3.10 a)

much the same behaviour of graph (a) alone, which gives Im Z*\p,

e, l ) U o = j

6(p°) B(p» -

~ %>2 — m2),

m2)

-

l)

(l +

- 4m]

p

(3.11)

A completely new situation arises if one chooses (compare also with [13]) n„ = j v

(3.12)

Accordingly, from (3.7) /? = 0 and Im ^ (1) (î>, p; s, l)|«=o =

m

) Im d2a)(p, m) + m Im d5il](p, m),

(3.13a)

Fortschr. Phys. 32 (1984) 9

489

with Im d2w(p, m) = j dip0) 6(p2 - m2) [l - (m2/?2)2],

(3.13b)

Im dsa)(p, m) = j

(3.13 c)

6(p°) 6(pi - m2) [1 - m 2 /p 2 ] 2 .

Therefore Im

p; e, i)|,_ 0

^(j»* —»•)»),

(3.10b)

i.e., the choice (3.12) leads to a smoother IB. behaviour, allowing for on-shell normalization conditions. We note that (3.11) and (3.13) have the same UV behaviour, in agreement with the comment subsequent to (2.17). The final result for the unrenormalized self-energy reads p;e, i) = dma) +

f)+

f)] (jr - m)

{1)

— d2 (p, m) {p — m) + dsa)(p, m) m. w

(3.14)

{1)

Here dm is the mass renormalization, Z2 (fixed e.g. according to the minimal subtraction scheme) is contributed by graph (a) alone and f (1) 6), which has equal contribution from (b) and (c), is normalized so that the full sum of graphs is normalized on shell, i.e.7), y(P,P)\?=m = 0,

(3.15a)

= 0,

(3.15b)

where S?(p, p) is the renormalized self-energy function. Finally d»w(p, m) = -

~

m%

IV%) [(1 + m2/p2) In (1 - *e - p*/m*) + l],

dsa)(p, m) = - j (1 - m 2 /p 2 ) 2 [In (1 - ie — p^/m?) + p2/m2].

(3.16a) (3.16b)

The renormalized propagator &(p) = G(p, p) = i{d2{p, m) [p — md5{p, m)] + ie}-1

(3.17a)

can be given also the parametrization nP) = jP — Tib \ ," f " • v C

+ mb(p, m)].

(3.17b)

For the structure functions a(p, m) and b(p, m), eqs. (3.16) imply a(p, m) = 1 + — [(1 - m 2 /p 2 ) 2 In (1 - p 2 /m 2 - ie) - (1 - m 2 /p 2 )| + &(oc2), n 4 (3.18a) b(p, m) = 1 + — 4- [ - ( 1 - m2/p2) p2/m2] + @{'(k; n), as can be read off from (1.51). On the other hand, the behaviour p% ~ m2 is controlled by the classical currents (oc pM), responsible for the emission of soft photons. This amounts to an effective IR vertex p,Tv,,(k;n). The choice (3.12) is the only one, which due to (3.4a), annihilates the contribution of the classical soft currents, responsible for the perturbative logarithms of (0.1a). The above explanation is furtherly reinforced by the nonperturbative analysis [46] of the mass-shell behaviour of the field e(x), carried out in the Bloch-Nordsieck approximation [1]. It will be useful for the discussion in the next subsection to report the IR asymptotic contribution of graphs (a) + (b) alone: (a) + (b)|„^p ~ i ^ Sc(p) [am'D^p) + Z™(e, f ) + 1 ^(s, £) + 0(ln(l - is — p2/m2)) (a) + (b)|„=p - i - J 8"{p) ôm™S°(p) + 71 3 B.

(3.20a) I) + j Cll)(s, |)J.

(3.20b)

The One-Loop Vertex Green Function

The one-loop approximation to the unrenormalized Green function G„W(IÎ, ni;n2;l;e,è)

= iSc(q) A™(p, n,-q, w2; e, f ) iS°(p)

(3.21)

with q = p — I, is represented in Fig. 2 by graphs (A)—(H). Graphs (A)—(D) are One Particle Reducible (OPR) self-energy like graphs. Their rôle will be discussed later. Let us turn our attention to the OPI graphs corresponding to (E)—(H). Their contribution is

Fortschr. Phys. 32 (1984) 9

491

given by

n = -«V2 / d^2ekE%(k; fji, i) X y, _

(jr

_

m)

^ . J ^ ( p _ ft) y ^ f l _

k)

r

- < # - » )

nfi-i

K k ) l

(3.22 a)

= - ¿ w r ' - l ) / di-2tkEc(k; fx) - v f i t - w . M) »V.. { [ y X

- ») ¿ g j

S

^

r

£ — ]k + m D(i, g )

where the fermion denominators are

D(lc, r) = k2 —

+ r 2 — m2 + is.

Remarks: a) The dependence on £ is factorized (compare with remark a) of the previous subsection). b) The integrand of the second integral in (3.22b) is a product of distributions with coinciding singularities at k = 0. I t is well known, that such products are in general illdefined and a fortiori, not associative. Therefore it is not surprising that, if k2 in the numerator firstly multiplies Ec{k\ fi), one obtains the illdefined expression

/ djc[{k2 + is) (%£) (n2k) D(k, p) D(k, q)]-1. On the contrary, when k 2 is attached to the fermion part one has / djmjt,

fi) {^[(tijk)

(n2k) D(k, p) D(k, g)]" 1 }

(3.23)

which is well defined. The question arises how to incorporate the rule (3.23) in the Lagrangian. This can be done in the Hertz formulation of QED by replacing in (1.55a) gA" f y ^ -> gC{n

w ) .

(3.24)

where the free Hertz potential C^