A ν-dimensional analysis of the axial-vector current divergence

Nuclear Physics B64 (1973) 244-252 North-Holland Pubhshmg Company

A v-DIMENSIONAL ANALYSIS OF THE AXIAL-VECTOR CURRENT DIVERGENCE Horaclo Oscar GIROTTI* Departamento de F{sica, Facultad de Ctenctas Exactas, Umverstdad Naclonal de La Plata, Calle 115, esq 49, La Plata {C C 67), Argentina Received 4 June 1973 (Revised 23 July 1973) Abstract In a formal generalization of some aspects of quantum field theory to a space of contmuous dimensmnallty, we analyze the behawor of the axial-vector current divergence as an analytic functmn of the number of dimensions of the space (v) It is found that (0 talS~(x)~o is an analytic functmn of v with simple poles at v = 6, 8, etc The analytlclty of (0~1~)o at v = 4 results from the cancellatmn of the (v-4) behavmr of '~')'g,'gs} with the singularity (v-4) -1 of the product of the Dlrac field operators taken at the same spacetlme point For v 4:4 we have local and non-local contributions to (~]5) o

1. Introduction So far the dimensional renormahzatlon method proposed by Bolhnl and Glamblagl [1,2] and independently by 't Hooft and Veltman [3] has been used at the level fo perturbation theory. What this means ts that only the Integrals correspondmg to closed loops of the Feynman graphs associated with the process under analysis are regularized by analytical extension in the number of dimensions of the space (u) whale external wave functions, the Dlrac algebra for gamma mamces and so on are kept m their usual four-dimensional form Only very recently Akyeampong and Delbourgo [4] and Bollini and Glamblagi [5] have generahzed the algebra of the Dlract gamma matrices for a space of continuous dimensionallty. They have also carried out an apphcatlon of the above mentioned generalization to the VVA (vector-vector-axial-vector) mangle graph. Following the line of thinking of the authors of references [4] and [5] we try, m the present paper, to reformulate the well-known problem of the so-called anomalous axial-vector current divergence in massless electrodynamtcs [6,7] in a way [8] which, m some sense, implies pushing a little further an the direction of extending formally quantum field theory to a space with a continuous number of dimensions The analysis of the results thus obtmned is our main purpose in the present work. * Member of Consejo Naclonal de Investlgaclones Clentfflcas y Te'cnlcasde Argentina

H 0 Girottt, The axial-vector current divergence

245

Far from being rigorous, the correctness of the extensions to a v-dunenslonal space of the Dlrac algebra for gamma matrices, the two-point electron Green function and the equal time antlcommutator of the spin one-half field operators that we are presently proposing, can only be justified in the hght of the final results The idea is as follows We start by considering the Lagranglan density* d~(X)

:

1

/ao-

-~(~"~(x))~ .~(x)+e~(x)~, ~ ( x ) A " ( x )

(1)

as describing massless splnor electrodynamlcs in a v-dimensional space (v 1s In general a complex number). The 7 u matrices appearing m the Lagranglan density are square matrices of dimension d(u) verifying the following algebra [2]

7u7 p + "Tp"/u = 2gup

(2)

The electromagnetic field-strength tensor is defined by

Fuo = ~oAu - 3uAo • Next, we look for the change in the Lagranglan density 6J2(x) induced by the infinitesimal axml-vector gauge transformation @(x) ~ ( l + l A ( x ) 7 5 ) @ ( x ) ,

~(x)

~ f(x)(l+tA(x)'Y5),

(3)

A~,( x) ~ A~,(x) , where A(x) is the infinitesimal gauge function and ")'5 is also a square matrix object of dimension d(v) which is required to satisfy** Tr (75")'ulTu2) = 0 ,

(4a)

Tr (757ulTu 2 . .Tu2n+l ) = 0 ,

(4b)

for n a positive integer and*** {7~,')'5} = 7~,75 + 3'57u :P 0

(4c)

An easy calculatmn shows that 6 2 2 ( x ) can be written

* All repeated greek ln&ces are to be summed from 0 to u - 1 Our metric is goo= g l l = - g 2 z =+1 andguluZ = 0 1 f u l e:u2 ** The u-dimensional extension of the 3'5 matrix is by deflmtlon non-hermltean 3"+ = -3"03'53"0 *** In the language of Bolhnl and Glamblagl (see ref. [5]) { 3"u,3"s} is an evanescent anUcommutator since it behaves as (v-4) as v-* 4

H 0 GzrottL The axtal-vectorcurrent divergence

246

~(x) = (~ a. ~(v~,,~,5). ,(x)-½ ~Oc) { "r,,,v5 } a. 6(x)

+te~(x)(Tu,75}t~(x)AtL(x))A(x) - (71-~(x)[7#,~s ]" ~(x))0~'A(x), which Implies that

/~(x) -

8.8

~(0.A(x))

-

l ~(x)I.r~..rsl.

C~(x)

(5)

is the current density connected with the gauge transformation (3) whose divergence is given by

Ou]5(x )_ 8.ff _ te~(x){Tu,75}. $(x)AU(x )

8A(x)

+ ½(OU~(x)){Tu,75}" ~(x) - ~ ~(x) ( "),u,75 ) -

0U~(x).

(6)

As we can see, the axial-vector current divergence is expressed as a sum of local products of field operators hawng as common factor the evanescent antlcommutator (Tu,'y5 }. It wall be shown that the ( v - 4 ) behavior of (7u,75) cancels out with the singularity ( u - 4 ) - t , characteristic of the local product of field operators, yielding a non-vanishing result for Ou/5u(x) at v = 4. Now we assume, and these are in fact our fundamental assumptions, that for a space of continuous dxmensionahty one can still write*

Gba(X',X")

= (01T($b(X')~a(X"))[O)

(7)

for the two-point electron Green function and

( ~b(X'),f a(X") )X,o=X,o,= (70)ba~(V-l)(x'--X ")

(8)

for the equal time antlcommutator of the Dlrac field operators. Here, ~Ob(X) and (')'0)b a are some specific representanons m a v-dimensional space of the electron field operator and the 3,0 matrix, respectwely. Working in the approximation in which the electromagnetic field is handled as an external field, we use eqs. (7) and (8) to write the vacuum expectation value (V.E.V.) of the right-hand side ofeq. (6) m the following final form

(OulSu(X))o = - r e

Tr ({7u,75

+½ Tr[{Tu,75}

1)m

}G(x,x))AU(x)

(~G(x',x")-

X --~ X XH-~ X

* (v- 1) dimensional vectors are in bold type

o~G(x',x"))l.

(9)

H 0 G1rottl, The axtal-vector current divergence

247

Thus, our purpose of reformulating the problem of the axial-vector current divergence in massless splnor electrodynamlcs through the extensive use of the v-dimensional regulanzation scheme has been completed. To explore the vahdlty of eq. (9), and therefore of all assumptions we have made so far, we focus our attentmn on the analytic properties of (O"]~(x)) 0 a s a c o m p l e x function of v. These properties and their implications are presented m sect 3. Sect 2 contains a compllatmn of algebraic relations concerning gamma matrices m a v-dimensional space which are useful for the subsequent calculations The concluslons are included an sect. 4.

2. Algebraic relations for gamma matrices defined in a v-dimensional space

It is important to note that we are not assuming any particular form for 75 (besides eqs (4)) This remark is in order because in ref [3] a particular extensxon of 3'5 has been proposed for a v-dimensional space. Needless to say, all relations quoted in this section are verified by 3`5 as defined m ref. [3] but one can not dxscard the possibility of having other extensions of 75 to a space of continuous dtmenslonallty From eqs. (2) and (4a) it follows that (a) Tr (3`57,1"ru23,,33`,4) is completely antlsymmemc in all indices. (b) Tr ({3`. ,75 }3'. 13`"23`" 3 ) = 0 .

(10)

Using eq (2) we find an exphcit expression for Tr((3'.,3`5}3`.13,.23`.33,.aT.s) which reads Tr ({7~z,75}')'.17.23`.33`.43`.5)= 2gu. 1 Tr (753`.23`.33`.4")'~s) 2g.u 2 Tr (3`53`#13`/.t33`.43`.5) -I- 2g.. 3 Tr ("~53`.13`.23`.43`.5) --

2 g . . 4 Tr (')'53`.13`.23`.33`.5) "t- 2g.. s Tr (')'53`.13`t/,23`.33`bt4)

(11)

When this last equation is combined with (a) one obtains (c) Tr((3,.,75}7.13,u23,u33,.43`.s)is antlsymmetnc in the indices/l 1 . tl 2, #3. ~t4, /a5 . (d) gU"l Tr({7.,3'5} [-I 7.,) = ( - 1 ) / + 1 2 ( v - 4 ) Tr(3`5 1-[ t

3`.t)

(12)

14=1

for i, 1 = 1, 2, 3, 4, 5. Working along slmdar lines one can show that Tr

( {3`.,75}[ 7I )7 . , k~lk"3k"sk"7~O, /=1

(13)

H 0 Gtrottt. The axial-vectorcurrent divergence

248

7

t=l

f

kUlkUakUSka l kUlkV3kU7k ~ kUlkUSkU7k~ l > kla3k~lSktiT k~

( 1) Tr(757u27g47#67gT) Tr(75TuzTu4Tu6T#5) (-1) Tr(757~27#47#67#3)

2(v 4) fd~kk4f(k2 ) V

(14)

Tr(757u27v47v67vl)

and Tr (('¥~x,~/5}

!i]lT~x,)fdVkf(k2)k~Xlk#3k#Sk~X7kU9k#X

_ 2(v-4)v Tr (757u27uaTu67u8)fdVkk6f(k2).

(15)

This is the complete set of algebraic relations involving the Dirac gamma matrices which will be used in sect. 3.

3. Analytic behavior of (Ou]uS)0 in the v-complex plane From eq. (9) it follows that the analytic properties of (au]~) 0 as a function ofv are mainly related to those of the two-point electron Green function which, in the external field approxnnatlon, can be written

G(xt, x-)= --~aF(X-X l . . . . . ). +. ¼le f dUYSF(X" -Y) 7" A(y)SF(Y-X') +

~e2f d~Y 1 fd~y2Sv(x"-y2)7" A (y2)SV(Y2-Yl) 9"" A(Y 1)SF(y 1-x')

+ O(e3) •

(16)

Here, SF(X"-x' ) is to be understood as the v-dimensional Fourier transform of the free massless electron propagator, that is to say [2]

SF(X,,_x, ) = (27r)~dd 2i ( vp ~7" p e-tP • (x"-x') -

-

H 0 Gtrottt, The axtal-vectorcurrentdtvergence

249

When this last expression is inserted back in eq. (16) and use is made ofeqs (4a) and (10) one easily establishes that (ouIs) 0 does not contain the orders e 0 and e 1 in the coupling constant, while eq. (10) alone is enough to determine that only the second term of the right hand side of eq. (9) makes a non-vanishing contribution of order e 2 to (Oul5)0 which, after some straightforward calculations, is found to read

(Ouj 5 (x)) (ez)=

4

( - (2rr) 1 ) v (3v 2 ) 2 V e 20/ d ) , )0d z z f d U q l f d v q 2

×Ie 2t(ql+q2) XA~l(2ql)A,2(2q2)(ql+q2)U4(ql X g ~ S Tr

{yu,3,5}[ [ 7ul ,=1

dVk

q2)U3

[k2 +-D] 3 '

(17)

where D is a function of the mvarlants q~, q~, q l ' q2 and of the Feynman variables y and z A " ( q l ) is the v-dimensional Fourier transform of the electromagnetic field A " ( Y l ) . After using eq. (12) and the result

i f dvk [k2+D]--k23 -tzr~v4 P(2-72,,l"'~D~V-2 in eq. (17) It turns out, from the resulting expression, that the V.E.V. of the axialvector current divergence behaves, as a function of v, in the following form

(O,j5(x))ge2)~ (_-~v) o 1 r(_-~v) 9 1

(18)

This result says that (au/5(x)) 0 is an analytic function of v with simple poles at v = 6, 8, etc. In fact, for integer v, v = 4 is the space of highest even dlmenslonallty for which (3"]5)0 is finite. On the other hand, eq. (18) confirms our previous assertion that the finite and non-vanishing value acquired by (Oujs) 0 at v = 4 is a consequence of the cancellation of a ( v - 4 ) factor, intrinsically contained in the evanescent antlcommutator, with a pole ( v - 4 ) -1, essentially originating in the local product of the Dlrac field operators. It is also worth mentioning that in this approach (0u15)0 s different from zero even for complex values of the regularizing parameter (v), where the theory is supposed to be convergent. This seems to be a characteristic peculiar of all analytic regularlzatlon methods [10] and it contrasts with the result obtained for the same problem using gauge-lnvanant cut-off procedures (see ref. [7]). Algebraic rearrangements let us recast eq. (17) into the more familiar form

u.s (e2)(--1)urr~Ve2( v) (3 lu(X)) 0 = l P 3- ~ Tr('YS~/ulTu27u3~fu4) (21r)v

2

x fdvyl fdVyzFU3Ua(Vl)'Kz(yl,Y2,X,v)FUXUZ(y2),

(19)

H 0 Gwotti, The axial-vector current dlvergence

250 where

K2(Yl,Y2,X,V ) -

1f

1

(2~r) ~ o

1

f d zfd q, fdvq2

o

X k~v-2(q21,q2, ql.q2;y,z ) e '[q' (yl-x)+q2 ()'2-x)]

(20a)

and

K(q2, q2, ql q2,Y,Z) = q 2 z ( 1 - z ) + q 2 y z ( 1 - y z )

+ 2q 1 q 2 I I +yz(1-z)]

(20b)

[ ~z,Ju'0 5 \(e2) conThe particular form of the kernel K 2 indicates that for arbitrary v, ,v tains two different kind of terms. One of them is purely local and reproduces the value found by Adler [6] for (3ujS~)O, the other one as non-local and can be separated from the ennre result replacmg an eq. (19) the kernel K2(Y 1,Y2,X,V) by

n'(2) n (2~); X

d y f dzzfd~qlfd'q2 0

([lnlf(q2pq~,ql q2,y,z)] n etql (Yl--X) e lq2 (Y2

x)

This xs a consequence of the non-local character acquired by the theory for v 4 : 4 The remaining contributions to (au]5)0 coming from orders h~gher than e 2 of the power expansxon can be calculated in a similar manner. We wall show that their analytical structure ts such that all of them behave as O ( v - 4 ) for v ~ 4 From power counting arguments it is easy to see that orders higher than e 4 only involve m o m e n t u m integrals which are convergent at v = 4. Therefore, they wall appear multaphed by the factor v 4 contained an the evanescent anticommutator. The contributions to ( /3u,5~ ,u,0 of order e 3 and e 4 do contain superficially divergent parts at v = 4, the c o m p u t a n o n of which can be carried out with the help of eqs. (13), (14) and (15) and yields*

(27r) v

--}-

+1

P 3--~

Tr('),53¢,i2T~14T.6T.s)

×f d~Ylf d'Y2f d~Y3K3(Yl,yZ,Y3,X,V)[(A'4(yz)A~':(y3)

(21)

-A':(Y2)AU4(y3))FUS'6(yl)-(Ala6(yl)AU4(y2)-AU4(Yl)Ata6(y2))FtZS'2(y

* The lack of vector gauge mvanance of expressions (21) and (22) must not be surprising since they do not represent the enme contributions to (atal~(x))o of order e3 and e4

3

H O. Gtrottt, The axial-vectorcurrentdtvergence l(--1)vTr~v e 4

v

251

v

× f dVYl f drY2 fdVY3 fdVY4 K4(Yl,Y 2,Y3,Y4,X,U)[(AU2(y4)AU4(Y3) -AU4(Y4)AU2(y3))(AU6(y2)AUS(yl )-AUS(Y2)AU6(yl)) ] ,

(22)

where the kernels K 3 and K 4 are such that for v ~ 4

K3()'l,)'2,)'3,x,v~4 ) ~ (5(v)(), 1 x)6(V)(Y2-X)6(V)(y 3 x) + O ( v - 4 ) ,

(23)

K4()'I,Y2,Y3,Y4,X.U~'4) ~ 6(V)(yl x)6(V)(Y2-X)~(V)(Y3-X)6~V)( Y4 x) + O(v-4).

(24)

The combined use of eqs. (21) (24) together with property (a) of sect 2 tells us that (Ou15)~e3)and (~15)(0e4) also behave as O ( v - 4 ) when v --' 4 Thus, we conclude that only (Olal5)(e2)IS sl~gmficatlve in the vicinity of v = 4. We want to end the present section remarking the slmphclty, elegance and power of the v-dimensional regularlzatlon method as compared with the non-covarlant eseparation device.

4. Conclusions It is not difficult to convince oneself that the change in the Lagranglan density (1) induced by the set of transformations (3) cannot be compensated by the introduction of another gauge field (say Bu) which under the axial U(1) group undergoes the transformation

Bu-~B u +g lOuA(x) with g a new couphng constant. Furthermore, one can in fact show that in this last case the axial-vector current divergence is still given by eq. (9) with the obvious exception that the two-point electron Green function must now be calculated taking into account the presence of both gauge fields. With the present scheme of regularizatlon, the non-safe [1 l] character of the U ( I ) ® U(1) chlral group can be read off directly from the corresponding Lagrangian density The analytical structure in the v-complex plane of the V.E V. of the axial-vector current divergence says that (Ou]5) 0 is an analytic function of v with simple poles at v = 6, 8, etc. The finite and non-vanishing value acquired by (~u]5) 0 at v = 4 is a consequence of the cancellation of a ( v - 4 ) factor, intrinsically contained in the evanescent antlcommutator, with a pole ( v - 4 ) - 1 , essentially originating m the local product of the Dlrac field operators

H 0 Gtrottt, The axtal-vector current dwergence

252

On the other hand, and since (Ou/5,(x)) 0 is different from zero even for values of the regularlzat~on parameter in the region of convergence of the theory, one can not attribute the failure in the axial-vector Ward Identity, naively derived, to the mathematically dlegal step of translating the integration variable an a hnearly &vergent integral. It is also worth of mentioning that for arbitrary v, (Ou]5,) 0 contains both local and non-local contributions. The slmphclty, elegance and power of the v-dimensional regularlzatlon method contrasts with the more famdlar and non-covarlant e-separation device. It is a pleasure to thank Professors C.G. Bollinl and J.J. Glamblagl for continuous help and encouragement.

References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

C G Bolhnl and J.J Glamblagl, Phys Letters 40B (1972) 189 C G Bolllmand J J. Glamblagl, Nuovo Cxmento 12B (1972) 20 G "t Hooft and M Veltman, Nucl Phys B44 (1972) 189 D A Akyeampong and R Delbourgo, Imperial College, preprmt IC/72122 C G. Bolllm and J J. Glamblagl, Umversldad de La Plata, preprmt (1973) S. Adler, Phys Rev 177 11969) 2426 S Adler, Perturbation theory anomahes, 1970 Brandeis Summer Institute in theoretical physics, vol l B Zumlno, Proc. of the Topical Conf m weak mterachons (Geneva 1969) p 361 D.A Akyeampong and R Delbourgo, Imperial College, preprlnt IC/72/32 H Fanchlottl, C.A Garcfa Canal, H O Glrottl and H Vucetlch, Nuovo Clmento Letters 3 I1972) 9 H Georgl and S Glashow, Phys Rev D6 (1972) 429