Dimensional regularization and finite calculations in gauge theories

Nuclear Physics B84 (1975) 132-140. North-Holland Publishing Company

DIMENSIONAL

REGULARIZATION

AND FINITE CALCULATIONS

IN GAUGE THEORIES W.J. MARCIANO

Department of Phystcs, New York Umverszty, New York, NY 10003 Received 29 July 1974

Abstract: The leading finite corrections to a lowest order relationship between renormalized coupling constants in an SU(2) × U(1) gauge model of weak and electromagnetic interactions are presented. The calculations were carried out in two different gauge mvafiant dimensional regularxzation schemes. A brief outline and comparison of the technical aspects of these methods is given. For our particular calculation we find that both yield the same finite results as expected.

1. Introduction The advent of gauge theories which unite the weak and electromagnetic interactions has gwen us the o p p o r t u m t y to perform various finite calculations. Gauge mvariance along wxth renormalizability implies that certain ratios o f renormalized quantities in these theories are finite and computable in perturbation theory. A clear framework witlun which corrections to certain lowest order relationships may be stud~ed can be found in ref. [ 1]. There the finiteness of certain relationships was exphcitly demonstrated and a method for carrying out the finite corrections was given. The particular calculation we present here is the leading correctxon to the lowest order relationship g 2 = e2/(1 - R ) + O ( a 2 ) ,

(1)

where gw is an infrared finite, weak coupling constant, e is the physxcal charge o f the electron and R xs the ratio o f the renormalized masses M 2 / M 2, often called cos20w in the literature. All of these quantities appear as parameters m the original SU(2) X U(1) Weinberg-Salam model which we investigated [2]. Besides serving as an example of the calculability of higher order corrections to a natural relationship, this computation may be used to express radiative corrections to decay rates in terms o f the fine structure constant ct and the various masses of the theory rather than the weak coupling constant. Since the degree of divergence of individual F e y n m a n graphs is very high in the

tv.J. Marctano, Dimensional regulartzatton

133

umtary gauge which we used, amplitudes must be regularized m a manner that preserves the gauge invanance of the theory. The old Pauh-Villars method nawely applied, rmght break the gauge mvarlance and yield incorrect finite results. For this reason, calculations in gauge theories are being performed via the new method of dimensional regularizatlon which explicitly preserves the gauge invarlance of the theory [3, 4]. In an attempt to deal with spinors m this formalism, various modifications of this technique have been presented. The exposition of this calculation presents us with an opportunity to compare two particular methods and to comment on some of the technical aspects of their use. Our plan is the following: In sect. 2 a brief outhne of the original method of dv mensional regularizatlon is given from a user's point of view. A rewsed form of th/s technique prescribed by Bardeen to overcome spinor difficulties is also exammed [5]. In sect. 3 we exhibit the results of our calculation and comment on its use m expressions previously obtained. Since both forms of dimensional regulanzatlon are valid for this calculation, we are able to compare them and show that they do indeed yield the same finite results. Finally, we close w~th some comments and conclusions concerning this calculation and the various techniques employed.

2. Dimensional regularization We present here only a rough outline illustrating how one actually uses the technique of dimensional regularization in carrying out a calculation. For a rigorous treatment of this subject we must refer the reader to the literature [ 3 - 5 ] . The procedure as outlined in the original proposals o f dimensional regularization is very straightforward and easily applicable. All internal momentum variables over which we are to integrate in evaluating Feynman diagrams are taken to have n rather than 4 components, while external momenta are left as four vectors. On the one loop level, Feynman integrals now have the form

f( dnk F(p, k ) .

(2)

27r) n

Power counting assures us that for small enough n the integral is ultraviolet convergent. All algebraic manipulations performed on the mtegrand are to be carried out in n dimensional space-time using the metric g00=l,

g~/=-6q,

i,]=l,2...n-1,

g0i=0,

(3)

which implies g U = n. Translation of integration variables, Feynman parametnzatlon and symmetric integration are all allowed. The n-dimensional one-loop integral Is finally evaluated by using the formula [1]

134

W.J.Marclano,Dtmenstonalregularizatton

( dnq (q2)r _i(-1) r-m cr_m+½n r(r+½n)P(m-r-~n) J('~)n (q2_C)----'--~ (16rrZ)~n F(~n)P(m) '

(4)

after which all Feynman parametrizatlon integrations must be performed. All originally divergent integrals exhibit simple poles at n = 4. The fimte contributmn of any Feynman graph is just the constant term one gets after a Laurent expansion about n=4. Some graphs contain spinor propagators internally, so we must generalize the algebra of the Dirac matrices to n dimensmnal space time and f(n) dimensmnal spin space. The n 7-matrices satisfy the equation 7u7 v + 7VTU = 2gUz' ,

(5)

where guy is the metric of (3). When manipulating 7-matrices reside the integral, eq. (5) must be adhered to; otherwise the gauge invariance will be lost and incorrect finite results will be found. So for example, 7UTu = n not 4 m this method. We will not exhibit the form o f f ( n ) , the 7-matrices nor a generalized 75- Instead we shall assume the following properties: Our 3'5 anticommutes with the other 7-matrices and the trace rules are Tr(odd number of 7's) = 0 , Tr(I) = f ( n ) = 4 + G ( n - 4 ) + O ( n - 4 ) 2 + . . .

(6)

The G will not appear in final physical results and may be set equal to zero. We shall verify this fact for our calculation in sect. 3. This method of handling spinors runs into trouble when an eum~ which has no natural extension outside of n = 4 appears. To overcome this problem, Bardeen has suggested a revision of this method [5]. Observing that most divergent integrals are regularized even if we only extend the internal meson part to n dimensions, he suggests leaving the 7-matrices in four dimensions. In this approach, spmor propagators have four not n components. If_k represents the first four components of k and w all the rest, then the integral of (2) breaks up into two parts f d4k F l ( k '

- dn-46o

(7)

The F l ( k , p) comes from the spinor propagators and vertices; while F2(w , k, p) contains the meson parts extended to n dimensions. The d n - 4 w integral can be evaluated by slightly revising formula (4) 0.e. do not include the Wick rotation). We are then left with a d4_k integration. The integrand contains the untouched spinor parts and a factor which looks hke an analytic continuation in the exponents of the meson propagators. The factors in the denominator can be combined by using a generalised version of Feynman parametrizatlon

W.J. Marctano, Dtmenstonal regulartzatton 1 _ I'(r+m) f l arb m P ( r ) I ' ( m ) J

135

xr_l(l_x)m_ 1 dx { a x + b ( 1 - x ) } r+m

(8)

and the d4k integration can then be carried out. As before, the fimte part Is obtained from a Laurent expansion about n = 4. The two methods yield different finite resuits for individual graphs; but when graphs are summed both give the same result. Pure spinor loops must be handled by some alternate regularlzatlon prescription m the Bardeen scheme * Although the infrared divergences associated with this calculation were regulated by gwing the photon a small mass ?~, we could have instead regulated them with dimensional regularizatlon also [7]. Integrands that carry an infrared divergence are convergent for n > 4. However, in this paper dimensional regularlzation will be restricted to ultraviolet dwergences alone.

3. Finite corrections In the Weinberg-Salam model the relationship

g2o = e2o/(l - R o )

(9)

is natural. Only two of the three parameters go, eo and R o need be independently renormahzed. Therefore, the counterterms for both sldes of this equation have the same divergent parts [8]. So, to lowest order this same relationship holds for the renormalized parameters and higher order corrections to it are finite and calculable. We define the renormalized coupling constant g~v~w to be the coefficient of M o = -(i/x/2)exa~Tx ((1-75)/2)Vv~ found from summing all Feynman diagrams which can contribute to the decay I4/~ £ + ~ , with the three particles on mass shell. Or in terms of the bare couphng constant of the original Lagrangian 1

1

1

~ Z 2v~ r ZW ~ /Z 1Qv~W' g~v~W =g o Z 2~

(lO)

where £ stands for either lepton, the electron or the muon. The g~v~w that we have defined contains infrared and lepton mass divergences; so in ref. [9] we found it convenient to define a new renormalized coupling constant gw independent of m~ and the infinitesimal photon mass ?,. We found:

* Even in the first method we presented, we have not carefully addressed ourselves to the problem of pure spmor loops for other possible generalizations of "ts. For a recent examination of this problem see ref. [6].

W.J Marciano, D,mensional regularizatlon

136

M,~ +m~ - -

Mw-m

_

In

MW1 m~

/

J

Imp/1

+Sln M W - l n 2 MW +O • m~ m~ M~ J.J

(11)

In that paper we computed, among other things, the total decay rate of the W boson into £ + ~ + 3' including both soft and hard photons. Expressed In terms ofg W it was: gw ~ [ 1 -m212 2 i m2_~2W3 I 1 -I- c t ( 77 _ . ~7r2 + O ( m~_~))3 P~ =4-~n MW 1 + -M~r j ~ i~

(12)

After we evaluate the O(c~2) corrections of eq. (1), we will be able to rewrite this expression for P~ in terms of the electron charge e and the masses m e, rnu, Mw, M z and Me of this particular model. If we merely replaced g2w by e2/(1 -R), the decay rate would no longer be accurate up to order a 2. We begin by exhibiting the relationships between the renormalized and bare parameters found from the Feynman diagrams of figs. 1-5. Using some of the notation and results of ref. [ 1], we have [ go [13/ 1 lnMw)+A gw =go 1 - 16rr 2-/--f~L--~_4 +13' +

e--e

M~

I1 o

+O(g

e-----L-° 2 /~¢---~1 + lnMw) } O(e~)] 161r2 / 3 \ n - 4 ½3' + +B + ,

Mz2 - 1--'R 8~2

(13)

(14)

(15)

where 3' is the Euler constant which comes from our expansion of the I~function and A, B and C are real finite expressions given below. Eq. (9) constrains these renormalizations such that

2 { ' e2° 1 - R

( 6M2

6M2~]

(16)

Combining these relationships we find the finite corrections to eq. (1): g2W=l---SR- 1-~-~

I---~-B+( l - R )

2

'

(17)

W.J.Marciano,Dimensionalregularization

137

where we have replaced e2[41rby the fine structure constant a. Using (17) the decay rate given in (12) may be rewritten as Mw (1

m2

m2] 2 -

-

2n ~12

m2

0

1-R

(1-R) 2

"

In a similar way, calculations like the radiative corrections to muon decay may be rewritten in terms of e rather than gw [ 10]. We now give the expressions obtained for A, B and C: A=~

4 G - 5 4 R + 3 8 - - ~ - 25 + R1 20 - 2 r 2 + 6 r -

+ (r 3 - 9r2 + 9 r - 6 ) l n ( r ) +

60R + 6 - ~ - + 2 R 2

(-72R-158+~___0_ 2R 215 R_~) Qo(R)

lnR (19)

-(r3-~r2+16r-18)Qo(1) + (-24-3-~+~3)So(R) + (48R+96)To(R)] , B =-]G- ~ - ~ ln(memu]M2 ) ,

(20)

C = ~ [ ( - 1 6 R + 2 8 - _~) G - 96R 2 - ~-~R + ~ + (-~r3R 2 +

3r2R + 32R

- 9r - 48 + 76 R

+ 121 3R 7 R2

R12 + ( R _ 1)r2 1 2/~3

)

In R

+ ( ½ r - ½m 2 - 3 + 3R)r21n(0

-

66 _~6 +__1_1) 96R + 112 - - ~ - +R2 2R 3

(Qo(R) _

_ (½r3_4r2+14r_24)Qo (1)+(~r3R2 where

Re(R2Po(R)))

(21)

4r2R+14r_2_~__) Qo(~rr)]

138

I41..1.Marcmno, Dtmenstonal regularization

v.¢

~,y,Z

',V'

V

I

,,

•

Z

!

~ I

"l

v~,

£

y:'

I

I t

I

I

I

lJ F

~,

Z

s

I

I

I

i

I

I

I

h- W

I

I

i

Fig 2.

Fig. 1.

I /

i

q~,¥, Z,

ii

!

e,

.e,V p

.°

~w

~w Fig 3.

Figs. 1, 2 and 3. Diagrams contributing to I~v~W coupling renormahzation.

R-

2 2 - MCv/Mz,

2 2 r =M;/MCv

1 Qo(R) = f

R dx

o

1

'

Rx2-x+l

So(R) = f

1

Po(R)= f

o

~ In(x) l+x

o

'

1

dx

1

x2-x+R

,

To(R):

f o

dxllnRX2-X+lx 1-x

The expression for A was obtained from those graphs illustrated in figs. 1-3. The two regularization schemes gave different fimte results for individual diagrams contalmng both internal bosons and fermions. Only figs. 1 and 2 contain graphs of this kind. If we sum all of the differences that the two methods gave for the graphs in fig. 1, we find that the Bardeen method gave an extra (~ + ~R)Mog3/16zr 2. However, the diagrams of fig. 2 evaluated a la Bardeen gave an extra -(¼ + [R)Mog3o/16~r2; these contributions exactly cancel. This exphcitly shows that both gauge invanant regularization prescriptions do give the same finite result. The expression for B was obtained from those graphs illustrated in fig. 4. Notme that B contains lepton mass singularities, Le. parts that dwerge as m~ -+ 0. These arise because with the conventional definition, e/e o contains lepton mass singularities. So, total decay rates expressed in terms o f g w are free of In m~ singular terms; but when expressed m terms of e such terms do appear.

N.J. Marctano, Dimensional regularization

e

eve

e

139

e

Fig. 4. Dmgrams contributing to charge renormahzatlon. ¢

~,y,Z W _ - - . p

, .

", .

.

.

.

Z

W

~ o _

~s

"

Z

.

q~,W

~,y,W,Z W

,"

W

Z

~

J

Z

_A__.O._L ~e,~u

e,~

e) tl, ~e, ~l.t

w z

:'-',, z w

Fig. 5. Diagrams contributing to the W and Z meson two-point functions. The expression for C was obtained from those meson self-energy graphs Illustrated in fig. 5. The Po(R) term in C m a y have an imaginary part which should be dropped. Other imaginary parts have already been discarded in arriving at the results of (21). These imaginary contribuUons arme because of the mstabdlty of the W and Z bosons. Finally, we note that A, B and C all contain G terms which come from pure spinor loops. However, when they are inserted into eqs. (17) or (18) the terms revolving G cancel and no G dependence is left as claimed m sect. 2.

4. Conclusion

The leading finite corrections to a lowest order natural relationship between masses and coupling constants have been presented. Even on the one-loop level, such

140

I¢.j. Marciano, Dtmenszonal regulartzatzon

a calculation proved to be long because of the numerous particles involved. The performance of this computation provided us with a test for some of the aspects of dimensional regutarization. We have exphcitly demonstrated that the physical results are independent of the G which came from expanding f ( n ) the generahzed dimension of spin space. The equivalence of the results gwen by two different dimensional regularization methods was also checked by this calculation. In total, the technique o f dimensional regularization proved to be a very straightforward, powerful and easily apphcable method for performing calculations. I wish to thank Professor A..Sirhn for many illuminating conversations.

References [1] C.G. Bollim, J.J. Glamblagi and A. Strhn, Nuovo Clmento 16A (1973) 423. [2] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264; A. Salam, Elementary particle physics, ed. N. Svartholm (Almqulst and WikseUs, Stockholm, 1968). [3] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [4] C.G. Bollim and J.J. Giamblagi, Nuovo Cimento 12B (1972) 20. [5] W. Bardeen, Proc. of the 16th Int. Conf. on high energy physics, Batavia, vol, 2 (1972) 295. [6] R. Delbourgo and V.B. Prasad, Impe.nal College, London, preprint ICTP/73/27, D.A. Akyeampong and R. Delbourgo, Nuovo Cimento 17A (1973) 578; 18A (1973) 94. [7] R. Gastmans and R. Meuldermans, Leuven preprint (1973). [8] H. Georgl and A. Pals, Phys. Rev. D10 (1974) 355. [9] W.J. Maxciano and A. Strlin, Phys. Rev. D8 (1973) 3612. [10] T.W. Appelquist, J.R. Primack and H.R. Qumn, Phys. Rev. D7 (1973) 2998; D.A. Ross, Nucl. Phys. B51 (1973) 116.