QCD non-leading corrections to weak decays as an application of regularization by dimensional reduction

N~clear Physics B187 (1981) 461-513 © North-Holland Publishing Company

QCD NON-LEADING CORRECTIONS TO WEAK DECAYS AS AN APPLICATION O F R E G U L A R I Z A T I O N B Y DIMENSIONAL REDUCTION G. ALTARELLI Istituto di Fisica dell' Universitd di Roma, and Istituto Nazionale di Fisica Nucleate, Roma, Italia

G. CURCI Scuola Normale Superiore, Pisa, and Istituto Nazionale di Fisica Nucleate, Pisa, Italia

G. MARTINELLI CERN, Geneva, Switzerland

S. PETRARCA Istituto di Fisica dell' Universit& di Roma, and Istituto Nazionale di Fisica Nucleate, Roma, Italia

Received 23 February 1981 We compute in QCD with massless quarks the two-loop anomalous dimensions of the four-fermion operators relevant to weak non-leptonicamplitudes and discuss the physical implications for strange and charm particle decays. Our main goal was to derive the complete order a corrections to the inclusive decay width of a charm (or heavier) quark. We find that the correction terms in the effective hamiltonian show the same enhancement-suppression pattern as the leading terms. Also, for a free c quark, the non-leptonic decay width is further increased (and the semileptonic branching ratio further depleted) with respect to the leading approximation. Implications for D (especially D +) decays are discussed. The calculation was completely done by a recently proposed variant of dimensional regularization called dimensional reduction. Much of this paper is devoted to exposing and resolving (within two loops) the field theoretic subtleties associated with this method which was particularly suited to this computation.

1. Introduction It is k n o w n that g l u o n effects, as o b t a i n e d from the s h o r t - d i s t a n c e o p e r a t o r e x p a n s i o n [1] a n d Q C D in the l e a d i n g l o g a r i t h m i c a p p r o x i m a t i o n (LLA) are i m p o r t a n t [2, 3] for w e a k n o n - l e p t o n i c a m p l i t u d e s . I n particular, these effects w o r k in f a v o u r of the A T = 1 rule in s t r a n g e particle decays. T h e p r e d i c t e d a m p l i t u d e ratio ( A 1 / E / A 3 / 2 ) L L A ' ~ - - 3 - - 4 is, however, n o t sufficient to r e p r o d u c e the o b s e r v e d v a l u e ( ~ 2 0 ) . T h e r e m a i n i n g e n h a n c e m e n t is p r e s u m a b l y d u e to l o w - e n e r g y effects in the m a t r i x elemertts [4] (including those of " p e n g u i n " o p e r a t o r s [5, 6]). 461

462

G. Altarelli et al. / Corrections to weak decays

For heavy quark decay (especially for charm) a substantial increase in the non-leptonic width is obtained, which leads to a prediction [7] for the (quark) semileptonic branching ratio B sL, which is considerably smaller than the free field value. For charm, the prediction in the L L A is typically B sL -~ 13-16% as compared with the free field value of - 2 0 % . Until recently, the results for a charm (c) quark were directly taken as relevant for charm particles, the light constituent quarks being taken as inert spectators. After the experimental finding* of a quite different lifetime for D Oand D + by D E L C O , also confirmed by M A R K II and by observations in emulsions, one is now led to a picture of charm particle decays based on both c quark decay and W exchange between constituent quarks (in the s- or t-channel) with real gluon emission [9]. However, the c quark decay prediction should remain essentially valid for D ÷ (provided the spectator is really inert [10]) because, in D ÷, the annihilation process can only occur at the Cabibbo suppressed level. Since a value of B sL for D + close to 20% is being currently reported [8] it is important to verify whether or not the L L A is supported by a study of the next to leading corrections. In order to investigate these matters we computed the first non-leading Q C D corrections to the effective weak non-leptonic hamiltonian (a s u m m a r y of our results has already been published elsewhere [11]). The main ingredients for this calculation are the two-loop anomalous dimensions in the massless theory of the four-fermion operators of dimension six in the short-distance expansion for the product of two weak currents. For peculiar technical reasons, related to the chiral structure of weak interactions and the definition of the Levi-Civita tensor e,vo~ in less than four dimensions, we were led to use in the calculation a variant of dimensional regularization called dimensional reduction. Regularization by dimensional reduction was recently proposed by Siegel [12] and has been further studied by other authors [13, 14]. It consists in continuing to n < 4 only coordinates and m o m e n t a , while all other tensors are kept in four dimensions. The familiar fourdimensional Dirac algebra is then essentially maintained and a great technical advantage is obtained. This simplicity is however paid for by the appearance of a n u m b e r of field theoretic subtleties. Some of them have already been studied, for example [14] the correct prescriptions for reproducing y5 and other anomalies. In this p a p e r we study, in detail, several intriguing aspects of this regularization method and finally explicitly show that within two-loop accuracy all potential difficulties can be kept under control. All G r e e n functions of interest to us are demonstrated to be renormalizable and to satisfy the correct renormalization group equations in four dimensions. In particular, we illustrate the procedure to be followed in order to extract two-loop anomalous dimensions in the minimal subtraction scheme. The resulting non-leading corrections to the effective hamiltonian are found to be of normal size, so that the corresponding perturbative expansion appears well* For an up-to-date review of the experimental situation, see ref. [8].

G. AItarelli et al. / Corrections to weak decays

463

behaved to the order explored. Moreover, the new corrective terms are found to possess the same pattern of enhancement and suppression as the leading terms. Thus all the significant implications obtained from the leading-order Q C D hamiltonian are considerably reinforced by the present computation. We then proceeded to compute the complete expression to order a for the nonleptonic width of a c quark or any other heavy quark. The present result supersedes an incomplete calculation of the same quantity in another paper [15]. When taken together with the known corrections [16] to the semileptonic width our result leads to an expression for B sL up to and including terms of order a. In this case, also, we found a confirmation of the predictions in the LLA: the nonleptonic width is further increased and B sL is further decreased with respect to the leading corrections. It is, therefore, concluded that a semileptonic branching ratio of D ÷ close to 20% would definitely indicate a significant difference between the non-leptonic width of the meson and that expected for a free quark. This violation of the parton model approach to D ÷ decay, if ever observed, should be attributed presumably to the destructive interference between diagrams differing by the exchange of the two a antiquarks present in the final state, namely the spectator and the one originating from the c quark decay, as suggested in ref. [10]. The paper is organized as follows. In sect. 2 we set up the general framework. In sect. 3 we introduce dimensional reduction and explain its particular fitness to this case. Examples of the computation of diagrams in dimensional reduction are discussed in sect. 4, where all important features are exposed. In sect. 5 we use the wisdom so obtained in order to derive general rules for computing anomalous dimensions and physical quantities in two loops via dimensional reduction and minimal subtraction. Sect. 6 is devoted to a description of the computation of the non-leading corrections to the weak non-leptonic effective hamiltonian. In sect. 7 we discuss the structure of the resulting hamiltonian and its physical implications for strange particle decays. In sect. 8 we consider the heavy quark decay. Finally sect. 9 contains a summary of our results and the physical conclusions.

2. General framework Let us consider a non-leptonic weak process induced by charged currents. In the lowest order in the weak coupling the transition matrix element is given by the time-ordered product of the two weak charged currents folded with the W propagator: Hvi = g2

I d4xDw(xZ'Mzw)(FIT[J~" (x), J~ (0)]lI),

(2.1)

where Mw, gw, and D w are the W boson mass, coupling and propagator, respectively. For flavour-changing amplitudes the leading contributions in the limit

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G. Altarelli et al. / Corrections to weak decays

Mw-~Oo arise from the four-fermion operators of dimension six in the shortdistance operator expansion for the T-product [2, 6, 17]. In the particular example of charm-changing processes the relevant terms are of the form HaC=~ =~/}GF{C+(t, a ) ( F l O + ( O ) [ I ) + C - ( t , a ) ( F [ O _ ( 0 ) ] I ) + ' ' .} FI

(2.2)

where t

In ( M 2 / / x 2 ) .

=

(2.3)

H e r e / z is a reference mass scale, a = a(/z) being the renormalized Q C D coupling* at the scale/x, and 1

-t

-

t

-

-t

t

O± =~[(s C)L(Ud )L ±(UC)L(S d )L] /N±I\ , = / 2 N - - ) (g C)L(t/d )L ± ~A ( y t A c ) L ( u t A d ' ) L .

(2.4)

The shorthand notations for left-handed (right-handed) currents, (t~lq2)L,a

=

I~l'Ytz (1 ~

"Y5)q2

,

(2.5) (t~l tAq2)L.a = q13/ ~ (1 ~ v5)tAq2 , were used here; s' and d' are the Cabibbo-like quark mixtures coupled to the c and u quarks respectively, in the standard electroweak theory [19, 20]; l A are the SU(N)co~our matrices in the quark (fundamental) representation, with the normalization Tr (tAt B) = I(~AB .

(2.6)

The second equality in eq. (2.4) is obtained through Fierz rearrangement of (tic)(Yd'), according to the identity , ,_ tA q2)L , . (qlq2)L(l~3q4)L = 1/V (qlq4)L(q3q2)L + 2 ~, ,_ tqx tA q4)Ltq3 A

(2.7)

We also recall the additional relation A~(ql

N a- 1

1

- t Aq2)L(q3-tAq4)L = ( - - 2 - ~ - ] ( /g l l q 4 ) L ( g l a q 2 ) L - - N A

,

A A q4)L(q3 t q2)L.

(2.8)

In eq. (2.2) the dots stand for non-leading terms from operators of higher dimension and also for "penguin" operators [5, 21, 22] of dimension six, which are present in some channels for non-degenerate quark masses. * For a review of perturbative QCD, see, for example, ref. [18].

G. Altarelli et al. / Corrections to weak decays

465

w

Fig. 1. Penguin diagrams in lowest order.

These operators are of the form

A

(qltAq2)L E (qftAqf)L+R, f

(2.9)

where the sums over A and f refer to colour and flavour, respectively. They can mix with O± at order a through the diagram in fig. 1 (penguin diagram) for non-degenerate quark masses, i.e. when the G I M cancellation [23] is evaded. However, for charm-changing amplitudes penguin operators can only arise at the Cabibbo suppressed level and in fact are quite small even at that level, as a consequence of existing bounds on the b quark couplings. Thus penguin contributions can be completely neglected in the total non-leptonic width of charm and we shall not deal with these terms in the present paper. Actually for definiteness we shall from now on drop the primes from s' and d' in eq. (2.4) and explicitly deal with the dominant transition c-~sud. Of course, the situation with respect to penguin diagrams and in general to mass effects is different for strange particle decays. We shall c o m m e n t on that in due course, our main target being here the case of charm decays. In studying c decays, the coefficients C± can reasonably be taken from the massless quark theory. The present statement would be obvious in the absence of b and t quarks, because in most of the range b e t w e e n / z ~ mc and Mw the neglect of all quark masses would certainly be allowed. In the presence of heavier flavours we can still maintain the previous statement with some words of justification. The first point is that all mixings with operators such as (/~C)L(tib)L are irrelevant for the same arguments as for penguin terms. The remaining effects amount to some distortions at virtual m o m e n t a near the b and t thresholds, in the transition regions between one value and the other of f, the n u m b e r of excited flavours, which appears in the relevant formulae of the massless theory. This is directly apparent from a look at the diagrams involved in a calculation of C± at the level of first subleading corrections. These diagrams (fig. 2) only show b and t internal lines in the gluon vacuum polarization. Thus a reasonable approximation is to b r e a k the range of virtual m o m e n t a from mc to M w into mc-rnb, rob-rot, m t - M w and apply the massless theory result in each interval with f = 4, 5, and 6, respectively. Alternatively we can take some effective value of f between 4 and 6 and simply forget about the existence

466

G. Altarelli et al. / Corrections to weak decays

~

i

"~-

. . . . . . . . . . .

f

J

Fig. 2. The 28 independent two-loop diagrams for the anomalous dimension of the four-fermion operators of dimension six. Replicas differing by up-down, left-right reflections of diagrams are not shown. "Penguin" like diagrams are absent in the massless theory. They are irrelevant for transition involving four different flavours as in c ~ sdu. of b a n d t q u a r k s in o u r p r o b l e m . This is t h e a t t i t u d e w e shall m a i n l y t a k e in the following. In o r d e r to c o m p u t e C±(t, ~) in the massless t h e o r y at s u b l e a d i n g accuracy, we c o n s i d e r t h e r e n o r m a l i z a t i o n g r o u p e q u a t i o n for C±: -O--+/3(ot) _O~+'~±(a)}C±(t, a ) = O

3t

Oa

(2.10)

w h e r e / 3 ( a ) is d e f i n e d b y / 3 ( ~ ) = - Oa -

Oln/x

2 =-ba2(l +b'a +...),

(2.11)

467

G. Altarelli et al. / Corrections to weak decays

with b and [24] b' given by b-

11N-2/ 12rr '

b'-

1

[~N2_~Nf+f/N],

(2.12)

(4~)2~

and where 3~±(a) are determined in terms of y±(a) and y1(a), i.e. the anomalous dimensions of the operators O± and of the weak current, respectively, ~±(a) = 2 y 1 ( a ) - y ± ( a ) .

(2.13)

The solution of eq. (2.10) has the familiar form ~(t)

.~

/3(a') d a '

,

(2.14)

where a (t) is the running coupling at the scale Mw[as =- as(O) is the running coupling at the scale/, ] and C±[a(t)]=C±[O,a(t)].

(2.15)

It is well known [18] that beyond leading order the form of ~±(a) depends on the exact definition of the renormalized operators O± and the renormalized coupling or. Similarly, while in leading order yj is zero, since the weak current is conserved in the massless theory, beyond leading order one could as well choose a renormalization prescription with yj # 0. For this reason we keep yj in eq. (2.13). Also, we have written all our formulae as if O± were multiplicatively renormalizable. The reason is that O+(O_) is symmetric (antisymmetric) under interchange of the two quarks and separately of the two antiquarks. As a consequence, they transform according to definite and different irreducible representations of the flavour group SU(f), which is a symmetry of the massless theory. This symmetry prevents any mixing of the two operators from occurring. Beyond leading order the statement that O± are multiplicatively renormalizable only means that it is possible to choose renormalization prescriptions such that this property is in fact true. As the properties under fermion exchange are related to the Fierz properties, such a renormalization prescription must, for example, treat in a symmetric way any two diagrams connected by a Fierz transformation. In the following sections we shall discuss different renormalization prescriptions, with or without 3'i = 0, with or without Ok multiplicatively renormalizable, and show that the differences are exactly compensated by the related variations of o~, 3;±(~), C±[a(t)] and the O± matrix elements in such a way that the physical results are not affected. By series expansions we define ~±(a)=~)a+~)

2+...,

C±(a) = 1 + C~) a +. • •,

(2.16) (2.17)

where the fact was used that in free field theory C+ = C_ = 1, as is clear from eqs.

468

G. Altarelli et al. / Corrections to weak decays

(2.2) and (2.15). Starting from eqs. (2.2) and (2.14)-(2.17) we can rearrange the expansion of HFI up to the next to leading corrections in the form

HFI=~/~GF Y. O/F'(a)(1 + i=±

Cl')a)Li(l){ 1 +a --a(t) pi+'" "/7"

"},

(2.18)

where O~ I (~) =

(2.19)

are the renormalized operator matrix elements and L±(t) - [ a (t)J

p±=b[~)-bC~

(2.20)

'

} -b'4/~)].

(2.21)

In L:~ we must, of course, take a (t) in its improved form with the b' term included: a (t) = a°(t){1 - b'a°(t) In [In (M~v/A2)]},

(2.22)

with a°(t) -

1

,. - ~ - - 2 . • b In t M ~ v / A )

(2.23)

We recall that b and b' both being independent of the definition of a, all changes in the meaning of o~ are reflected in a change of A. The factors L:dt) arise from the resummation of leading logarithms. The -(1) exponents y:~ are well known [2]: 3 ~ ) = w 3_.3_ U : ~ l 4,rr ( - - - ~ )

"

(2.24)

We now summarize the main formulae of the LLA. The effective hamiltonian for AC = 1 channels (and similarly for other cases) is given by L++L

,

,

(2.25) where, according to eqs. (2.20), (2.23) and (2.24) [in L_=[

[In ( M w / A ) ] 12/(33-2f, in(~-~A~ J ,

SU(3)colour],

L + = L 21/2 .

(2.26)

In charm decay/x = mc is a natural choice. The total non-leptonic rate for a c quark becomes in the L L A

FNL

NL/2L2+L2-\

LLA -----Ffroo~

~

-),

(2.27)

469

G. Altarelli et al. / Corrections to weak decays I

I

I

I

I

A-

-A "

/

f=4

TOTAL

LL~

125

"~

0

I

I

0.1

i

a(rn c ) rr

I

i

02

i

0.3

Fig. 3. The enhancement factor A _ / A + of relevance for the AT = ½ rule in strange-particle decays for t* = me and f = 4 as a function of a(rn¢). In general, A± = L±(t)(1 + { [ a - a ( t ) ] / z r } p ± ) as given in eq.

(2.18). In the leading logarithmic approximation (LLA~p. is dropped and L±(t)are expressed in terms of the leading form of a(t). The values of A are shown on the curves (in GeV).

where F NL free includes a factor of three for colour. Finally, the semileptonic branching ratio becomes in the same approximation SL

BLLA = (2 + 2L~ +L2-) -1

(2.28)

where no effort was m a d e to include phase-space distortions. Plots o f , /"NL// ,ir~NL f .... B S L and L_/L+ in the L L A are displayed in figs. 3, 4, and 1 2 5. In particular, as ~(2L+ +L2_) is sizeably larger than one at reasonable values of a(mc), we have the well-known increase with respect to the free theory (naive parton model) of the non-leptonic width and correspondingly the decrease of the semileptonic branching ratio. Going back to the improved form of eq. (2.18) a n u m b e r of remarks are in order. First of all, note that we kept as expansion p a r a m e t e r [ a - a ( t ) ] and not a(t), as is usually done for m o m e n t s of structure functions in deep inelastic scattering. This is because here we want to expose the difference between operators defined at Mw and at ix. As for the matrix elements in front [with the factors (1 + C~)a) included] they are to be taken as either non-perturbative (for example in strange particle

470

G. Altarelli et al. / Corrections to weak decays '4 I

i

I

f

i

i

i

["NON LEPTONIC 1-,(NAIVEPARTONS) " NONLEPTONIC

f =4

/OTAL

'1"

I

I

O

I

O.I

I

I

I

0.2

ct(mc )

0.3

7;" Fig. 4. The ratio of the OCD corrected non-leptonic width of a charm quark to the same quantity in the nai've parton model (a--> 0) as a function of a(mc) for f = 4. The corresponding values of A are shown on the curves (in GeV).

decays, where /x is to be chosen as the boundary of the perturbative region) or perturbative, as in studying a heavy quark decay in the parton approximation. A second observation is that p± are independent of the external states F and I. This is because/9± belong to the coefficients in the operator expansion. In particular, p± are infrared safe and independent of infrared regularization. As a consequence, i

i

i

~

i

BSL

0'O5I 0

I

i 0.11

I1(m c )

i

f =4

"~ TOTAL i 0.~2

013

7r

Fig. 5. T h e semileptonic branching ratio of a free c quark in Q C D as a function of o~(mc) for [ = 4. The values of A are shown on the curves (in GeV).

G. Altarelli et al. / Corrections to weak decays

471

p~ can be computed (and in fact will be) by taking arbitrary convenient (possibly non-physical) external states. In our calculation of p± we took four massless off-shell quarks with equal virtual masses p2. Moreover p~ are independent of the renormalization prescription for the operators Oz. In fact, this is the case for the quantities [3;~ ) - b C ~ ) ] and b ' y ~ ) [as opposed to 3,~ ~ and C~ ~ separately taken]. Of course p~ still depend on the definition of o~. We shall eventually give our final results for p± in the MS (modified minimal subtraction) definition [18] of a. With this single specification p± are completely determined. Once p± have been computed one is free to change the regularization method for the subsequent computation of the quantities 0v~~(~) = O~vI (o~)[1 + C ~ a ].

(2.29)

Since p± are well defined it follows that O~I(a) are well defined to the extent that H w is itself well defined. In the previous example of massless off-shell quarks as external states O~ ~ would depend on p2 acting as an infrared cut-off and also on the gauge chosen (contrary to p±). This is because HF~ is infrared divergent for a final state made up of only massless quarks and gauge dependent for off-shell external states. What we did was evaluate O±FI with a massive initial quark and a dimensional regularization both for the ultraviolet and infrared regions. Note that "FI O± (o~) is easily identified, because it contributes to HFI all terms of order a with no logarithms. Note in fact that in eq. (2.18) L ± ( t ) start as l + O ( a t ) + . . , and [ a - = ( t ) ] is of order o~2t. It is only after the complete calculation of real physical quantities that one can get rid of infrared cut-off and gauge parameters. For heavy quark decay this is only obtained after adding up the rates for the massive quark into qqq and qq~lG, where q and G are a real massless quark and gluon, respectively.

3. A choice of regularization: dimensional reduction Dimensional regularization [25] and minimal subtraction [26] provide a very practical framework for computations beyond leading accuracy (where problems of technical economy are essential) and in fact this method has been widely used in recent sophisticated Q C D calculations. However, for the particular problem of interest for us a straight application of this standard method leads to technical complications. Although not particularly significant in principle, they would in practice introduce further difficulties in already complicated calculations. In order to avoid this problem, we used a recently discussed variant of dimensional regularization called dimensional reduction (dim.red.) [12]. This method has already been studied and found successful for other two-loop calculations [13, 14]. Its importance in connection with supersymmetric theories has been stressed [13, 14]. However, it is still being discussed whether or not it provides a general and consistent approach.

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G. Altarelli et al. / Corrections to weak decays

The present p a p e r is the first to our knowledge where dim.red, is used for computing anomalous dimensions (and related more physical quantities) in two loops. At this level, at least, we found that the situation is completely under control as we shall discuss in the following sections. H e r e we first describe the problem that makes ordinary dimensional regularization impractical in our case and then introduce and summarize the alternative method of dim.red. The Dirac structure of the bare weak four-fermion operator is given by 3' = y~(1 - 3'5) ® 3,"(1 - ys),

(3.1)

where the tensor product makes explicit the presence of two distinct fermion lines. When one or m o r e gluons are exchanged between the two lines, products of several g a m m a matrices start appearing on each line. For example, with one-gluon exchange the following operator appears among others 0 ( 3 ) = 3'~3`a3', (1 - 3'5) ® 3`%/°'3'" (1 - 3'5)T~,,

(3.2)

where the indices /3 and /~' are contracted on some external a n d / o r internal momenta. In four dimensions this new operator is related to ~ in eq. (3.1) by the familiar relation

r,,rBr,(1-3`s)=(g~o3",-g,,,3"o +ga,,r~ +ie~,v3"~)(1-3,5).

(3.3)

However, in n < 4 dimensions 0 ( 3 ) is an independent operator. As far as one is interested in one-loop anomalous dimensions this feature is irrelevant. In this case, one is in fact interested in the residue of the 1/e leading singularity, where e = 2 - in.

(3.4)

Therefore one first factorizes the Dirac matrices, then performs the loop integration, picks up the 1/e singularity and finally evaluates the residue, including all g a m m a matrices, in four dimensions. However, when going to two-loop anomalous dimensions the above problem cannot be avoided any longer. Note that when two gluons are exchanged between the two-fermion lines products of up to 53'-matrices appear, as for example in the operator O(5)=3",,3"03`~3',3`,(1-3'5)®3` 3' 3' 3' 3' (1-3"5)TCw,

(3.5)

which is a new independent operator in n < 4 . As we are now interested in the residue of the subleading singularity behaving as 1/e (the leading singularity being l/e2), it is no longer allowed to neglect deviations of order e from relations valid in n = 4. The problem is clearly the lack of an extension of the completely antisymmetric tensor e,,o~ away from n = 4. Note that the same problem would also arise for no 3'5 in the operator at the start, i.e. for a product of vector currents, although in this case some operator mixing in Dirac space would be unavoidable anyway.

G. Altarelli et al. / Corrections to weak decays

473

In principle, one could take 3,, O(3), 0 ( 5 ) . . . . as independent, compute the renormalized 3' operator up to two loops as a combination of bare 3,, O(3), 0 ( 5 ) . . . . operators and only at the end go back to the real world with n = 4. But in practice this would imply performing the two-loop integrals in tensor form [integrals with up to four Lorentz indices corresponding to T~,88, in eq. (3.5)], which is a hard task. On the other hand, if one was allowed the use of Dirac algebra in four dimensions, the problem would easily be reduced to the calculation of relatively few scalar integrals which can be analytically evaluated directly. Dimensional reduction provides a simple and precise recipe for a practical way out of this problem. The idea is to only continue to n < 4 dimensions coordinates x~ and momenta p,, while leaving all other tensors in four dimensions. In particular, the gluon field G,, and the Dirac spinors are left with four components and the algebra of 3,-matrices is unaltered. The continuation prescription becomes well defined when formulated in terms of rules for computing Feynman diagrams. The Feynman rules are formally the same as in four dimensions. In particular, the gauge field • • (4) 2 (4) propagator is - t g g , / k , where g,,, is the metric tensor in four dimensions• However, all m o m e n t u m integrations (loop and phase-space integrals) are in n dimensions and are to be performed as in ordinary dimensional regularization. Thus, for example, symmetric integration transforms k , k , m o m e n t u m integrals into g(,'2 tensors (the metric tensor in n dimensions). One must, of course, carefully distinguish between the two metric tensors g~4~ and g~"2. It is interesting to compare the computing prescriptions of dim.red, and of ordinary dimensional regularization by formally splitting [13] in the lagrangian the range of the index /z into i <~ n and n ~
-- 4~'IJ l r'2-A P"z""~2-- ~\o[l aI~t'2 A'~2 , ) -i- c * A

otzj~AB

-F q a i J ~ a b q b ,

(3.6)

where G A and C A are the gluon and ghost fields, respectively, G Ap,v the non-abelian curl, and D , , / ) ~ , are covariant derivatives on quarks and ghosts [A, B = 1 . . . . . N are the colour indices in the adjoint representation of SU(N)~o~.... while a, b refer to the fundamental representation]. We can write the lagrangian (3•6) in the form 3? = 3?" + 37",

(3.7)

where 37n

37e

1

A

= -~(Gii)

2

1

-~(0

i

Gi

A

2

) -t-C*Aoil~IABC

B d-~tai3,'D~bqb,

(3.8)

1 A 2 ~C~i~A 1 2cABCcADEGBGCGiDG~E = -~(cgiGo, ) - - g t ~, . ~ A B C(, J. ~i Bt.r~O t.r,~ - - ~ g i ,~ 1 2..'~ABC..'~ADE..'~B..'~C~o'D~cr'E

-~g

t~

~

tx~,t_r

~

-

-gqa3,

cr A

~A

tabqb~.

(3.9)

3?n is the lagrangian density of ordinary dimensional regularization. G A are called e-scalars (under the Lorentz group in n dimensions). 3?¢ describes the interactions of e-scalars, which are not present in the usual procedure, and originates all

474

G. Altarelli et aL / Corrections to weak decays

differences between the ordinary dimensional regularization and dim.red. This is strictly true only provided the rules for the former are taken as stated for example in a paper by Marciano [27] (in particular Tr {1} = 4 in Dirac space). While the splitting of the e-scalar contribution to ~ is useful for a comparison of the two methods, of course in practical computations with dim.red, one directly operates with ~ , without separating the e-scalar contributions (otherwise the technical advantage of dim.red, would be lost). Note that local gauge invariance only applies to G A (because O~ is identically zero). Thus G a are the true gauge particles, while e-scalars act as matter fields transforming globally under the adjoint representation of SU(N). Also Lorentz invariance is only valid on the first n ~<4 indices. Thus, for example, starting from the bare operator 3' in eq. (3.1) we obtain from a computation of gluon corrections in dim.red, not only terms of the same form but also the new operator ~?= 3,.(1 - 3'5) ® 3'i(1 - 3'5)

(3.10)

when the sum is extended over the range i ~
"P, = g .(,~ v y d/

= 0,

(t* = o - ) ,

= 3"i,

(/~ = i).

(3.11)

Operators like 3 ' - ' p or y , , - 9 , which formally vanish for n = 4 will be called e -operato¢% A related phenomenon is that the Ward identities of gauge invariance only guarante¢~4 he equality of coupling constants for external gauge particles G A. This means that e-scalar couplings are renormalized independently. For example, the glyiqG~ and q3"~qG,~ couplings are not equally renormalized. Similarly, while for G~ particles the three-gluon coupling is guaranteed to be equal to the g l u o n - q u a r k antiquark coupling, it will, in general, be different from the e-scalar-gluon coupling present in ~ , eq. (3.9). In the present paper, as we are only concerned with two-loop amplitudes with external fermion lines, the renormalization of e - e - G couplings never enters into play (as well as all four-point couplings), so that we shall only explicitly consider the qqG and ~lqe couplings (this notation will be used in the following for gly'*qG,,). This apparent proliferation of couplings eventually makes no physical difference, but the existence of this feature must be taken properly into account. For example, it is known [13] that the correct two-loop ~8 function is only obtained in dim.red, provided one starts with external G/a gauge particles. On the other hand, e-particles can be allowed to run in all internal lines in that their total contribution to/~ is eventually seen to vanish. As is clear from the above description, dim.red, is less automatic than the usual method. But once a number of delicate points have been understood, then the advantage of a handier set of technical rules can be fully appreciated. The following

G. Altarelli et al. / Corrections to weak decays

475

sections are devoted to the clarification of these points and the discussion of the main aspects of dim.red, in the context of our particular computation.

4. Computing in dimensional reduction In this section we introduce the main features of dim.red, as they arise in practical computations. In particular, after introducing the renormalization group equations for local operator vertices and defining the related quantities of interest to this work, we survey first the calculations of relevant one-loop diagrams in dim.red. and point out the appearance of e-operators. A special discussion is devoted to the symmetry under Fierz rearrangement in n < 4 dimensions for four-fermion vertex functions. Next we consider a couple of typical examples of two-loop diagrams. The first diagram, involving coupling constant renormalization, will explicitly show the existence of several couplings in dim.red, and their different renormalization properties. The second diagram is selected so as to introduce the four-fermion counter terms and two possible renormalization schemes which arise from different choices of the operator basis in which minimal subtraction can be done. It is convenient to select at first a basis with a fixed ordering of the four fermions; for example, recalling eqs. (2.4) and (2.5), (YC)L(t~d)L and ~A(gtAc)L(~tAd)L. The ordering being fixed, we can from now on simply omit the external spinors and denote our operators as tensor products of Dirac and colour matrices (SC)L(/2d)L -'> "y~ -----0 1 ,

(4.1) ~'. ( s t A c ) L ( u t A d ) L -> 3/-8-= 0 2 , A

where y is defined in eq. (3.1) and we set in colour space ~=1®1, (4.2) -8- = ~ t A @ t A . A

The list of the two operators O1,2 covers the minimum number of linearly independent operators we need to consider. Upon renormalization up to two loops the previous list could possibly be found too limited in that new operators can get mixed with the above ones. There is no need to complete the list now. If there are mixings with additional operators these will show up from the calculation. The anomalous dimensions can be obtained from a study of the one-particle irreducible vertices corresponding to O~, Oz . . . . (with massless external quarks). We can take equal incoming and outgoing four-momenta (fig. 6) and, in order to prevent infrared divergences, a common virtual mass p2 is assigned to the external quarks. With a

476

G. AItarelli et al. / Corrections to weak decays

p

p

Fig. 6. The four-fermion irreducible vertex.

sufficiently extended set of bare operators OF and for a given renormalization prescription we associate a set of renormalized vertices o R O r = l' -h 'k~~ /- ~~,

OZ, ,~) O ~ .

(4.3)

Here A is the renormalized gauge parameter of covariant gauges (a = 1 is the Feynman gauge, A = 0 is the Landau gauge). It appears in eq. (4.3) because off-shell renormalized vertices of gauge-invariant operators are in general gauge dependent. The corresponding off-shell matrix elements of the hamiltonian will also be gauge dependent, while the coefficients in the operator expansion, being independent of the external states, are neither affected by p2 nor by A. In the following unless explicitly stated, all calculations are done in the Feynman gauge. The renormalized vertices o R satisfy the renormalization group equations

+ / 3 ~ + 8 A --+2"yFoh 6hk -{- ~,/hk O r = 0 ,

(4.4)

where y(a, A) is the anomalous dimension matrix, 2yF(a,A) is the contribution from the wave-function renormalization factor of the four external quarks, and we set

2 l = In /x 2. -p

(4.5)

The gauge-term function 6 is easy to compute to the required accuracy and is given by [see eq. (4.34)] 0 In A

6(a'h)=Ol--i~2=b~a+ . . . .

5 N - 2f 127r a + ' . . .

(4.6)

Non-leading terms are not needed in the following because (h 0 / 0 A ) O r are themselves of order a. Up to this point we have not introduced an ultraviolet regularization and, in particular, all previous formulae refer to n = 4 dimensions. We now go to n < 4 and regularize the theory by dim.red. We shall give a number of examples of

G. Altarelli et al. / Corrections to weak decays

a

477

b

Fig. 7. Fermion self-energy diagrams.

calculations of relevant Feynman diagrams by this method. This will also serve to discuss a number of features typical of dim.red, and very important in the following. In n dimensions coupling constants are dimensional. As usual one introduces the mass /z as a scale for the coupling constant and at each vertex makes the replacement •//" 2 . e/2 gdim = ( G ) gadim

(4.7)

where e is defined in eq. (3.4) and the factor (4~') -~/2 is introduced in order to make the formulae somewhat simpler (and in partial fulfilment of the MS prescription for the coupling). In the following g = gadim is always used. We start from one-loop diagrams, which we need to study in full including also non-leading terms. The simplest is the diagram in fig. 7a of the fermion self-energy in second order. One immediately obtains [CF = ( N 2 - 1)/2N] (2~)"

ipX(p) = g2CF

k 2 ( p - k) 2

(2~r)" k2(p - k) 2"

= g2CF

(4.8)

The two peculiarities of dim.red, used in eq. (4.8) are the gluon propagator - i g ~ / k 2 in terms of a four-dimensional metric tensor and the use of ordinary Dirac algebra. From this point on the calculation proceeds as in ordinary dimensional regularization. In particular, e

.

1

2

E

(~_~2) f (--~jT),(,/C.)/k dk 2( p - k ) 2 __ ( 4 ~i ) 2 ( _ ~ ) ( l + 2 - - y E ) ( ½ p ) 1+ O ( e ) ,

(4.9)

where Yu is the Euler-Mascheroni constant and O(e) terms will be omitted in the following. One thus obtains a 2 e 1 X(p) = ~ - - ~ - ( ~ ) C F ( 7 + 2 - - y E ) . (4.10) In ordinary dimensional regularization one would have got [(1/e)+ 1--yE]. The difference is precisely the contribution of e-scalars, as can easily be checked: Ot

/.£2 e

X~ (p) = ~--~ (----~) CF.

(4.11)

478

G. Altarelli et aL / Corrections to weak decays

For further reference the result in eq. (4.10) can be generalized to an arbitrary A gauge: O~ 2 e 1 2(p) =~-~ (_~--~)CF[(e+2--yE)--(1--A)(I+ 1--yE)] • (4.12) Note that the vanishing of X(p) at order a in the Landau gauge valid in dimensional regularization is not true in dim.red. The results in eqs. (4.10), (4.11), and (4.12) are reported in table 1, where a summary of one-loop diagrams is presented. TABLE 1 Results of the one-loop calculations

c~,~

colour

1

(CF--½CA)rJ 1

1

J~=J,,

Ca) (e)

2

, ', ~'(;+ ~): 4? ~)

(~) colour

[--CF(CF--ICA)'O~-+

2(CF-- 1CA) T]

- 4 y ( 1 + 2-4 In 2) Jb=J;

Ca)

_ y ( l + 2-4 In 2)

(e)

-23/ 1 3 In 2) -4y(~7+~-4

colour

[--CFCCF--'CA~+2/~--"CA~T] 1 E~ ~ ~1(~+ 2)+ 2~=~+~,~(~+ 2)+ 2.,

~=~,

1 1 y l _ (,y_ ,,?)+2P= y(~7+~)+ yf+ 2Pf

G. Altarelli et al. / Corrections to weak decays

479

TABLE 1 (continued) colour

CF

(7+2) 1

$(p)

(e)

1

The diagrams Ja . . . . , Jc' which are equal in pairs are those of fig. 8. The result of the diagram in fig. 7a is given by il~X(p). Under the entries labelled by "colour" the colour factors are given, for a four-fermion vertex of type ~ (upper entry) or of type ~- (lower entry) as defined in eq. (4.2). In all other entries a factor ( o t / 4 ~ ) ( l ~ 2 / - p 2 ) ~ is omitted and 1/e'=--(1/e)--yE. The label (3') refers to the results for the vertex y in eq. (3.1) in the Feynman gauge )t = 1. (A) refers to the coefficient of the gauge parameter h for the same results in a general gauge. (e) refers to the contribution of the e-scalar. (9) refers to a vertex -~ as defined in eq. (3.10) at the four-fermion vertex. All other symbols are defined in the text.

W e n o w go to o n e - l o o p d i a g r a m s for the g l u o n c o r r e c t i o n s to f o u r - f e r m i o n v e r t i c e s . C o l o u r f a c t o r s a r e left a s i d e a n d will b e r e i n s t a t e d at t h e e n d for b o t h o p e r a t o r s of i n t e r e s t 3,~ a n d 3/~-. W e c o n s i d e r first the v e r t e x c o r r e c t i o n g r a p h in fig. 8a, w h i c h w e d e n o t e b y Ja 2 ./.~2,

E J

J a = i g (~-'--~) J

d~k (2zr) n

3,°(P-JO3,"(1-3,s)(P-J()3,°®3,"(i-3/5). k 2 [ ( p - k)2] 2

(4.13)

B y o r d i n a r y a l g e b r a w e o b t a i n for a g e n e r a l f o u r - v e c t o r h . ( a c t u a l l y an n - v e c t o r ) -3'0 h3/, (1 - 3,5)h3, ° = ( 4 h , h - 2h 23/~)(1 - 3/5).

(4.14)

W h i l e t h e s e c o n d t e r m is o b v i o u s l y p r o p o r t i o n a l to 3/,, the first t e r m (with h = p - k ) a f t e r i n t e g r a t i o n g e n e r a t e s b o t h p~p/p2 a n d ~/, t e r m s [recall eq. (3.11)]. In fact h , h t r i v i a l l y v a n i s h e s f o r n ~
a

o'

2 - 3/E) -- 2pp--pP-T-~*P] (1 -- 3/5) (~) 3,u (1 -- 3/S) •

b

b'

c

Fig. 8. One-loop diagrams for four-fermion operators.

c'

(4.15)

G. Altarelli et al. / Corrections to weak decays

480

The diagram in fig. 8a' is identical. By defining P = [p(1 - 3'5) ® p ( 1

-

3's)]/p

(4.16)

2

and recalling the definitions in eqs. (3.1), (3.10), and (3.11) we obtain the result in the form

Ja=Ja,=~(~-~2p2)~[(2,-@)(l+2-,E)-2P]

.

(4.17)

This result, together with the corresponding colour factors, is reported in table 1. We make a number of observations on the result in eq. (4.17). We first note that a new operator appears that we can choose as ( 3 ' - 3~). This is an e - o p e r a t o r in the sense defined in the previous section. It does not affect the determination of the one-loop anomalous dimensions which are proportional to the residue of the 1/e pole. At the level of finite terms we also find P. This term is an artifact of the infrared regularization and in fact also appears, with an identical coefficient, in the usual dimensional regularization (in fact table 1 shows no such term in the contribution of the e-scalars). Being p dependent, this term does not contribute to the coefficient in the operator expansion, but clearly belongs to the matrix element. A second observation concerns current conservation. The diagrams Ja and J,, are vertex corrections for colour singlet or octet weak currents. As the colour singlet current is conserved one would expect the relevant combination of vertex and self-energy contributions to vanish. From eqs. (4.10) and (4.17) we see that J a -~(P)3' would in fact vanish if not for the ( 3 ' - ~2) and P terms. The P term is not surprising, being connected as usual to the failure of gauge invariance for off-shell fermions. The presence of the (3' - 9) term is instead typical of dim.red. Note that if we started from ~ we would have found the same result for Ja but with 3' = (see table 1). This feature will persist in two loops. The current ~, is automatically conserved, being associated with the true gauge particles (we are not worrying about the terms in p~p/p2), while the %, current is not identically conserved owing to the (3', - ~,) terms. We shall come back later to this point when discussing the renormalization procedure. At this stage the evaluation in dim.red, of the remaining one-loop diagrams in fig. 8, Jb = Jb' and Jc = Jc' shows no new features and we refer to table 1 for the results. Rather it is important to discuss the Fierz rearrangement properties of the results. We first consider the abelian case with all colour factors ignored. The Fierz rearranged operator has the two outgoing (or the two ingoing) legs interchanged. Thus Jb and Jb' are unaltered under this exchange, while Ja and Ja' go into Jc and J~,, respectively. A simple study shows that the "eigenvectors" of Fierz rearrangement in n dimensions are 3'

(eigenvalue + 1),

"¢f= ~ - ( 1 - ~ e ) 3 ' l Pf = P - ¼3' J

(eigenvalue - 1).

(4.18)

G. Aharelli et al. / Corrections to weak decays

481

Note that 3"3is another e-operator as it vanishes in four dimensions. It is seen from table 1 that starting from a 3' vertex (which is a Fierz eigenstate contrary to ~/) the results show an evident Fierz symmetry when expressed in terms of the basis % 3'f, and Pf. As a consequence, in the absence of colour, when the three pairs of diagrams are summed up, 3"~and Pf disappear, being of opposite Fierz symmetry to 3'. When colour is reinstated the Fierz rearrangement properties are affected by the reordering of colour indices according to eqs. (2.7) and (2.8). In colour space the eigenvectors of Fierz rearrangement O~: with eigenvalues +1, respectively, are given by [recall eqs. (2.4), (4.1), (4.2)] O±

=(N+I~ \ - - ~ - - 1 + ~-.

(4.19)

Note that with the present definition and eq. (3.1) for 3' the operators O± of eq. (2.4) can be expressed as O± = 0~:3'.

(4.20)

By taking into account that J~+Ja' = 2Ja and so on and the expressions of the Casimir operators CF and CA in terms of N[CA = N, CF = (N 2-1)/2N], one can write the results from the diagrams of fig. 8 in the form

(O~3")'=(----~)[(----~)(J~+Jc)~Jb

0~+

Ja--Jc)O~:,

(4.21)

where by a prime the result of applying the one-loop corrections to a given vertex was indicated. We see from table 1 that Ja + Jc and Jb only contain % while 3"f and Pf only contribute to Ja-Jc. Thus starting from O~-3, we obtain a combination of O~3", O~_3"f, and O~_Pf, which is correct because all the above operators are in fact Fierz symmetric; similarly when exchanging + with - and vice versa. We can easily check that the residues of the 1/e poles do indeed reproduce the correct results for the one-loop anomalous dimensions. In fact, by disregarding e-operators in the residue and by subtracting 227(p) from the diagonal terms we end up with

(0~3')'= 1:~--£~3 N- -~~I

1 (0~3")-+..., e

(4.2~)

which in view of eqs. (2.24) is the correct result. In table 1 a richer collection of results than discussed here is presented. The gauge dependence of the one-loop diagrams Ja to Jr', the contributions to these diagrams of e-scalars, and the results that would apply if one started from a operator vertex are also shown. These additional results will be useful in the following.

482

G. Altarelli et al. / Corrections to w e a k decays

We now consider two-loop diagrams. The relevant diagrams are shown in fig. 7 for the fermion self-energy and in fig. 2 for the four-fermion vertices. We need to evaluate the singular terms in 1/e 2 and 1/e of such diagrams, corresponding to quadratic and linear logarithmic terms [the relevant logarithm is l defined by eq. (4.5)] in the renormalized result as we shall discuss later. We consider as a first example the diagram K in fig. 9 which consists of a vertex correction to Jb. Except for colour factors, the quantity of interest, before subtracting counterterms, is given by 4 /. 2 2e

-

i2"rr)" (2=)"

×

"h, (1 - ~,5)(p - ~ ) ' / o (q - , k ' ) ' / ~ '° ® ~,~'(1 - ~/5)(g +,k')-/" kZq2(p _ q)2(/~ -- k)2(p + k)2(k _ q ) 2

(4.23)

The general procedure will be to first reduce the numerator to a single gamma matrix per fermion line [besides the inert ( 1 - y 5 ) factor] by using the fourdimensional Dirac algebra. This is most efficiently done by computer*. Then all scalar products produced in the first step are re-expressed in terms of factors already present in the denominators, in such a way as to reduce the computation to a sum of terms each being a sequence of two analytically computable integrals. In most cases each integral can be expressed in terms of only two Feynman parameters, but for a few very particular integrals with three Feynman parameters. One finds a very limit number of basic integrals which appear in all relevant diagrams. Once a table of these integrals has been prepared then the calculation is reduced to a pure algebraic manipulation. In the particular example of K in eq. (4.23) the numerator can be reduced to NUM(K)

-~ y [ 4 k 2 ( p

-

q)2_

8p2k2_

8q2k 2 +

4p2(k _ q)2 _

2(p

+ k)2(k - q)2

+4qE(k +p)2] + . . . ,

(4.24)

:•

P-q

P

k

P P Fig. 9. A n example of a two-loop diagram with vertex insertion. * The algebraic program S C H O O N S H I P written by M. Veltman was an essential device for the completion of this work.

G. Altarelliet al. / Correctionsto weakdecays

483

where the dots replace terms which by symmetry are bound to vanish upon integration. All the integrals can now easily be done. For example, the first term in eq. (4.24) [4k2(p _q)2] leads to an integral I: d"k

d~q

I°C l (p-k)2(p+k) 2 1 q2(q-k)2"

(4.25)

The integral over q is directly expressed in terms of two Feynman parameters and was already met in eq. (4.9). By dimensions it is proportional to (k2) -~, so that the next integral over k involves three denominators (k2) ~, ( p - k) 2, and (p + k) 2, i.e. three Feynman parameters. It is, however, a very particular one, because of the e exponent, so that it is not difficult to evaluate its singular and finite parts. This particular term contributes to K the quantity 2

2

2e

4yk2(p-q)2~(,a-ff--I ( G ~ \q--it/

\--p

2Y[4+1(5=41n(2)--2yE)] /

L8

•

(4.26)

e

The other terms can be similarly evaluated and the final result, before subtracting counterterms and without colour factors, is given by a 2 ~./,2 2~ - 1 1 K=(~-~) (--7) 2~'[--7+7(4--81n(2)--2~E)]"

(4.27)

We now consider the counterterm, associated with the diagram K, which is shown in fig. 10. In the minimal subtraction scheme only the pole part from the one-!oop vertex correction is to be subtracted. We recall that by subtracting the 1/e oneloop singularities from all subdiagrams, including those with vertex corrections and self-energy insertions, we obtain that in the final result all couplings, gauge parameters and masses can be directly identified with the renormalized values. The pole part of the vertex subdiagram in K, which essentially coincides with Ya, can be read from table 1 and is given by A

•

A C

1

Ol

Note that by subtracting this counterterm we are renormalizing both the true gauge-particle vertex ( ~ ) a n d the e-particle vertex ( y ~ - ~/.). Note that, since the contributions to Ju of both the gluon vertex y . and of the e-scalar vertex (y. - ~.)

y

a

=POLE P A R T N ~

b

Fig. 10. Counterterm associated with the diagram of fig. 9.

484

G. Altarelli et al. / Corrections to weak decays

can be read out from table 1, we are in a position to directly write down the counterterm K c to be subtracted from K (as for K the colour factor is omitted)

/ a ~2/ lz2"~~. [ 1 1 Kc=~-~--~] ~-27 ) , + y [ - ~ - ~ - ~ - ( 2 s - - 4 1 n ( 2 ) - y e ) ] .

(4.29)

By comparing with eq. (4.27) for K we observe the following salient features. While K is proportional to (tz2/-p2) 2~, Kc is proportional to (tx2/-p2) ~, because one factor of (iz2/-p2) ~ disappeared when taking the residue of the pole in 1/e of the subdiagram. Also note that the coefficient of the 1/e 2 singularity is twice as large in K c as in K. This is a general property of minimal subtraction, valid diagram by diagram, which amounts to a necessary and sufficient condition of renormalizability. In fact, owing to the previous observation on the powers of (Iz2/--p2) e in each diagram and its counterterm, it is seen that the latter relation on the double poles guarantees the absence o f / - d e p e n d e n t singular terms [see eq. (4.5)] in K - K c . A related feature is the systematic cancellation of "YE in differences like K - Kc. By reinstating colour factors we get the result [ O/ \ 2

K-Kc=[-~)

[ 1

1\1-

( C F - - 1 C A ) ~-

2y[-~+e)[(CF_½CA)[_CF(Cv_XCA)~+(2Cv_CA)~.],

(4.30)

where the upper (lower) entry in the colour factor bracket refers to the result obtained by starting from the operator y~(y~). We now make a short digression on the calculation in dim.red, of the/3 function in one loop. In a calculation of b defined in eq. (2.11) the contribution of W A,, in eq. (4.28) is to be rescaled by a suitable factor so that it becomes wqqG /.L = 4@ 2(CF -- ½CA)[~, + 2(y,, -- ~ , ) ] .

(4.31)

In fig. 11 the one-loop diagrams for a calculation of b are shown. A simple calculation for the three-gluon and the quark self-energy contributions leads to the results

y

W 3 G = 1 CA[3~,,+2(%,--~,,)] ~" 4¢r

(4.32)

wZ q

(4.33)

1 2cF[-~/.-(~,.-~?.)] 4rr

yy

Fig. 11. One-loop diagrams for coupling constant renormalization.

485

G. Altarelli et al. / Corrections to weak decays

Fig. 12. Vacuum polarization diagram. Finally from the complete expression of _/-/~,,(q2), the vacuum polarization of the gluon, i.e. the coefficient of iq28An in the diagrams of fig. 12, --

Of [ it/~2 "~e / [

(4)

ttg.~-g.v)tl-

-

A)t~EE ~--$'YE) /

,

(4.34)

one also obtains W zC

1

= ~---~[(~Cg--2f)?~ + (2CA-- f)("/~ -- *~)].

(4.35)

If one collects the total coefficient of ~ in eqs. (4.31) to (4.33) and (4.35) one obtains again the correct result for b as given in eq. (2.12). On the other hand, the sum of coefficients of ( y , - ~/,) in the same equations corresponds to the analogous p a r a m e t e r be for the e-scalar q u a r k - a n t i q u a r k coupling: bE =

2CF+ 2CA--f

4~r

(4.36)

We see a confirmation that the e-scalar coupling is renormalized differently from the gauge particle coupling, and that a new fake B function could be associated with this fictitious coupling. We shall come back to this point in the next section, where we shall m a k e use of the result for b, in eq. (4.36). As a second example of a two-loop diagram we consider the case where the four-fermion vertex itself is renormalized as in the diagram M of fig. 13 and its associated counterterm M c of fig. 14. Actually we are interested in pointing out two different options in the renormalization prescription for the four-fermion operators, so that we shall focus on M c and simply quote the result for M (except for the colour factor) ~¢

2

/d, 2

2¢

M=(~--~) (_-~) 1'~[2---~+1(~--',/E)]--~}.

Fig. 13. An example of a two-loop diagram with renormalization of the four-fermion vertex.

(4.37)

Fig. 14. Counterterm associated with the diagram of fig. 13.

486

G. Altarelli et al. / Corrections to weak decays

The internal loop whose leading singularity is to be subtracted away is clearly the diagram Jc of fig. 8c. Its singular part can be read from table 1 and is proportional to 35:

j~/~ - ?I~.

(4.38)

By inserting this term into the loop for Mc we obtain (by again looking at table i where the results for a 35 vertex are also shown)

a2lx2"~l

~},

(Mc)I : (~--~) (----~) { ~[~-'~+ 1 (2 -- ,E)] -- - -

(4.39)

where the index I is appended to indicate this first method of renormalization. By subtraction we obtain [ a\2_r 1 5q M-(Mc)I=~--~) ` / [ - - ~ E 2 -[- ~E J . (4.40) We observe the cancellation not only of `/E and l, but also of the other P - d e p e n d e n t terms proportional tO However, this first choice of renormalization, although perfectly legitimate, has some drawbacks in connection with Fierz rearrangement in that, for this prescription, the operators O± turn out not to be multiplicatively renormalizable in two loops. This is due to the fact that the singular four-fermion counterterm 35/e is not of definite parity under Fierz rearrangement, as already noticed [see eq. (4.18)]. In fact, the Fierz conjugate diagram of M is Mf, which is shown in fig. 15. The corresponding counterterm is proportional to the singular part of Ja

Pie.

j~/~

(2-/-35) e

(4.41)

As the two subtractions in eqs. (4.38) and (4.41) are not the Fierz conjugates of each other, as can be seen from eqs. (4.1 8), this prescription violates Fierz symmetry, at the level of non-leading terms, and as a consequence (O~-/) are not multiplicatively renormalizable because in general it is precisely Fierz symmetry that prevents mixing among them. We can keep O± multiplicatively renormalizable by turning to an explicitly Fierz symmetric prescription which amounts to doing minimal subtractions in the basis -/ and yf rather than 3, and (y-35). In fact, according to eqs. (4.18) 3/ and `/~ have definite parity under Fierz rearrangement. By making minimal subtractions in 3, and `/f the counterterm from Jc becomes (see table 1) j 1 / ~ ~ (`/4- -/F)/e •

Fig. 15. The Fierz conjugate diagram of that of fig. 13.

(4.42)

487

G. Altarelli et al. / Corrections to weak decays

while the Fierz conjugated diagram Mf gets the Fierz conjugated counterterm from

A, J~/~ ~ (y -

yf)/e.

(4.43)

A simple calculation leads to a 2 /z 2 ~ 1 1 (Mc)n=(4--~) ( - - - 7 ) { " / [ 1 + 1 ( 2 - - " / E ) ] + Y ' [ 7 - 2 + r ( 3 - - y E ) ] - - - ~ } '

(4.44)

and consequently, together with eq. (4.37), to a

2

1

1

1

7

a 2 1 7 1 = (~--~) { 9 ( - ~ e 2 + ~e-e) - y 7} .

(4.45)

By comparison with eq. (4.40) we see that the two prescriptions are different at the level of non-leading terms in 1/e. The advantage of the second prescription is that the corresponding result for Mr is exactly what is expected by Fierz symmetry according to eqs. (4.18) 1

M:-(Mec)n=(~-~){Y\ 2e e,+Yf(2:' 4~)}"

(4.46)

Both prescriptions are legitmate and both correspond to "minimal subtraction", because one must also specify on which basis the subtractions are performed. In sect. 6 we shall show that both prescriptions, although very different in intermediate steps, lead to the same physical conclusions. A list of the results for the singular parts of each two-loop diagram in the two regularization schemes which have been introduced above is presented in table 2. 5. Minimal subtraction in dimensional reduction

In the present section we discuss renormalization by minimal subtraction in dim.red, and the general procedure for extracting anomalous dimensions from the knowledge of double and single poles in e obtained from the calculation of two-loop diagrams along the lines described in the previous section. In ordinary dimensional regularization with minimal subtraction a direct proportionality relates the residue of the simple pole and the two-loop anomalous dimension. In dim.red, this simple rule is invalidated by two main facts. First the existence of spurious couplings. To two-loop accuracy and for the Green functions of interest in this paper only one additional coupling appears, the eqq coupling, and we indeed observed in the previous section that this coupling is renormalized by minimal subtraction in a

488

G. Altarelli et al. / Corrections to weak decays

TABLE 2 The singular terms from the diagrams of fig. 2 D

M

II

I

8

8 8

/ 16 16\ 2 1

11 12

/ 16 16\

4 y(4--4/+?(-2+1 \E

E/

-1)

2 2

E2

~

2 2

2 1

E/

13

4 3 , ( - 8 + 8)

,(__8+8)

14

2 -3,(4)

__,(4)

16

2 3'(-8)+p(4)

__(~,__~,f)(4)

17

1 - ~ ' ( ~4 ) + ~ ( ~ )3

3,(_1_1~+

3 1

18

1 y(2-3-/+ ~ ( - 3 + 2 )

1

3 1

19

1 y(- 1~62/

3'(-~2)

1

G. Altarelli et al. / Corrections to weak decays

489

TABLE 2 (continued) D

M

I

20

4

/ 24 20\ Y~-e-2+7)

21

4

yt-~-ff+~-)+~,t-~-~-e)

22

4

y(+4_8)__(10_7)

/

y ( _ ~24+ ~ 20\ _)

16 16\

/10

~\E

2

7\

y ~t" - ~4+ e )4\+ ~ , ~/- ~3 - ~ e3)\

24

2

2 2 - "~( - 3 + 2~) y(~-5-~-)

26

2

27

2

28

4

r

/

2

2",

/10

22\

y(~2_) _ ~2(~)

7\

6

4\

/10

7\

/2

"/(-~+e-)+Yr~,e"~-~e) 1 2\ / 3 5\ / 5",'1

4

++ 2 2 2 5 4° ~CA[3,(~-~+~e)+~?(--~--~ee)]--f~e-e (y-'~)

r~

/10

e2 e/+yf~-ff-~) 3'(

2

,

y(_6+4]

e]

23

,cA

II

2

2

1

2\

/3 2

,

Y,[½CA( ,,o

t +cAtT12

_ __ E

5\

5

2

4

5\ 22,

)+

1

4 /1\'1 ,

2

-yf

The symbols y, ,?, and yf are defined in the text. Colour factors are omitted and a common factor (as/47r) 2 is understood. For diagrams 25, 26, 27 of fig. 2, only the colour and flavour factors arising from the gluon self-energy are explicitly given. The number of each diagram, as indicated in fig. 2, is given in the first column. The multiplicity of each diagram (already included in the results) is given in the second column and the results according to the regularization prescriptions I and II explained in the text are given in the third and fourth columns.

different way from the Gq~l gauge coupling [recall eq. (4.36) for b+]. The second fact is the existence of spurious vertices, like 3'-~/ or 3"f in the previous section, which vanish in four dimensions but are seen to mix at e ~ 0 with the operators of interest. Both these features, related to e-scalars and e-operators, arise in dim.red. because Lorentz and gauge invariance only hold in n dimensions, while indices can also span the range between n and 4. In this section we show that, in spite of this, no difficulties are found up to two loops in the renormalization procedure and in the validity of the renormalization group equations. We shall also derive the relation that connects in dim.red, the anomalous dimension and the singular terms in the 1 / e expansion. We start by considering the renormalization of a set of vertices Oh in one loop [recall eq. (4.3)]. As in the previous section, we take external massless off-shell particles of virtual mass p2. We have seen that all diagrams of order g2 are

490

G. AltareUi et al. / Corrections to weak decays

proportional to a factor [see eqs. (4.5), (4.7)] =e

(5.1)

The unrenormalized vertex functions take the form

(5.2)

V~Ul~= 1 +,~ e~'(A + B / e ) .

As in eq. (4.3) F is to be taken as a matrix on a basis of bare operators O r . Thus the matrix 1 in trivial order means that each vertex reduces to a corresponding bare operator in that order. In eq. (5.2) all renormalized couplings are set equal. In particular, we have fixed equal values for the eqel and Gqel couplings (where G is the gauge particle). In order to obtain the renormalized vertices in the minimal subtraction scheme, we subtract the simple pole (and only that) from r'~) F~) = l + a

e~t(A +7B) - a B .

(5.3)

The subtracted pole also contributes the term of first order in a to the renormalization factor Z -1, Za~ = l - a - .

B

(5.4)

E

Z is a product of external leg and vertex renormalization factors which we are not interested at the m o m e n t to distinguish. As the quantity F~I) is free of singularities, one can take the limit e + O: (5.5)

F~) = 1 + c t ( A + BI) .

Note however that in dim.red, the operator basis for n < 4 can be larger than in n = 4 owing to e -operators like 3' - 3~ and yr. Thus while it can be formally important to consider F R and A , B . . . . in their complete forms, eventually we shall be interested in the restriction of the matrix [eq. (5.5)] to the basis of physical operators which survive in n = 4. We shall indicate this restriction by a superimposed bar (5.6)

P~I) = 1 + a ( A + B l ) .

As an example let us take Ja in table 1 [the entry labelled by (3')]. From . oe

J~U = 3"*4--rrrre

~t[;

]

[ t3"-3"e)(1+2-3'E)-2P',* , / '

(5.7)

{(3; - y,)(l + 2 - YE) -- 2P,},

(5.8)

{y(l + 2 - yz) - 2P,}.

(5.9)

we obtain a

j R = 3' + ~

= 3' + ~

G. Altarelli et al. / Corrections to weak decays

491

At order a 2 the computation of any diagram or sum of diagrams/, including the effect of the corresponding first-order counterterms Ic, leads to a total result of the form

I-Ic=a2

e2~t(F+O+

. . . ) - c ~ 2 e ~ t ( e ~ c + - D ~ + . . "),

(5.10)

where only the singular terms in the brackets were shown. The first term is the contribution to F u, i.e. without counterterm subtraction [analogous to K or M in eqs. (4.23), (4.27), and (4.37)]. The second term is from the insertion of first-order counterterms [as K c and M c in eqs. (4.29), (4.39), and (4.44)]. Note that the latter is proportional to e "~, because the first-order counterterms [for example o~B/e in eqs. (5.3) and (5.4)] have no such factor. In order to obtain the renormalized vertices in second order the double and single poles should be subtracted from eq. (5.10). The same poles also contribute to Z -1. By expanding the exponential in eq. (5.10) we find for the pole part

[

( I -- Ic)poles = a 2 ( F - F c ) E

~

(2F-Fc)l+ D - D c ]

.

(5.11)

E

Because Z cannot depend explicitly on l, but only on renormalized quantities like a (and possibly the gauge parameter A), a renormalizability condition emerges as in ordinary dimensional regularization with minimal subtraction: Fc = 2F.

(5.12)

In all the examples of the previous section the above equation was verified. This is in fact true diagram by diagram and in the course of our calculation we always took care to check it step by step. By using eq. (5.12) the pole part reduces to 2-F

D

[7+

Note that this pole part is precisely what we extract from each two-loop diagram, as seen in the examples of the previous section. Once eq. (5.12) is verified one can write F R and Z -1 to an accuracy of order 2

F(~) = 1 + a (A + Bl) + a 2[(2D - Dc) l + Fl2], B

Z~] = 1 -a---ae

2F-F

L~ -+

D-eDC ]

•

(5.14)

(5.15)

From eq. (5.15) it also follows that =

g

1_ g

2

+

•

(5.16)

The explicit form of D c and F c (hence of F ) can be determined by imposing

492

G. Altarelli et aL / Corrections to w e a k d e c a y s

the trivial fact that Z F R is independent of # once it is expressed in terms of bare couplings and gauge parameters. This is the starting point of the renormalization group approach which we shall impose first at n < 4. It is at this point that the presence of m o r e couplings shows up. In fact the renormalized coupling a is related to the dimensional bare couplings o~B and a ~ (the e - q - q bare coupling) by O' = (/d, 2 ) - e Z o t 0 f B

=

/ ~/z

2,,) e,'7 B , -%ocr~

(5.17)

with

ba

Z~ = 1 + - - , 8

(5.18) Z.~ = 1 + b~a , E

Note that at e # 0 we have fl(tx, e ) =

]~e (01~, E ) =

3a 2 _ _ e a + [ 3 ( o t ) = _ e o t _ b a 2 + . . . ,

(5.19)

aln/x

00~e 2 -- --EOI q- ~ e (Od) = - - C O / - b C a

2.

01n/x Similarly, the relation between the renormalized and bare gauge parameters can be written as

A=ZAAB = ( 1 - b~a + " " ) AB.

(5.20)

E

The explicit forms of b, b,, and bA are given in eqs. (2.12), (4.36), and (4.6), respectively. We shall not explicitly consider here other spurious couplings besides a , as they do not arise in the two-loop calculation of the G r e e n functions of interest for this paper. We can write the renormalization group equation in the form

Z-1 d (zFR)

dl

:

Z_ 1

E i ......

OZ OOli ~ R oFR -1 +--~--+ A OOdi Ol

E i...... X

oFR OOli oai Ol

0.

(5.21)

The last terms with no Z are certainly finite in e, thus also the first term must be non-singular. The first term to order a2 can be written as (apart from the factor F R which is finite) 2

Z _ 1 ~ OZ Oai_ Ba - 2 a 2 ( D - D e ) + a----[ 2 F - B 2 - b B ~ -b~B~ +b,Bx] i Oai O1 e '

(5.22)

where we used eqs. (5.15), (5.16), (5.19), (5.20), and (4.6). We also defined B , = A --0 B , 0A

(5.23)

orB = [aB~ + a~B~ ]. . . . .

(5.24)

493

G. Altarelli et al. / Corrections to weak decays

where in the last equation Be is the contribution of e-scalars. Note in fact that the derivative O/Oa in eqs. (5.21) and (5.22) is at fixed a~. We also used the Euler theorem (

OZ 3oe

et~ - - +

OZ rctB 2/-F D ;Dc)] ea~ OZ~ = e[-~--+ 2 a ~-TT+ 3ot~,l c,~=,,

(5.25)

in order to write down the contributions to eq. (5.22) of the terms - c a and - e a ~ in eqs. (5.19). By imposing the vanishing 9 f the residue of the 1/e singularity in eq. (5.22) we obtain [see eq. (5.12)] 2F = Fc

= B 2+

(5.26)

bB~ + b~B~ - bxBa.

This is the first of the required relations. To obtain the relation for D c we go back to eq. (5.21) and rewrite the finite terms for e ->0 in the form [ ~ + / 3 ( o t ) ~_aO+/3,(a) ¢3_3__+6(a,A)A_~A_aB_EOt2(D_Dc)]FR=0 Oa . (5.27) By inserting the explicit forms of/3, /3~, 8, and F R from eqs. (5.19), (4.6), and (5.14) we find that to order 2 all terms linear in l vanish because of eq. (5.26), while the cancellation of constant terms in l demands (5.28)

D c = B A + bA,, + b;A~ - bAA;~ ,

where A,,, A~, and Ax are defined in analogy with B~, B~, and Ba [see eqs. (5.23) and (5.24)]. Note that B A is the correct ordering when B and A are matrices. We have thus found the explicit expressions for F c and D c in terms of quantities that can be obtained from a study of one-loop diagrams. We have also seen from eq. (5.27) that F R satisfies a formal renormalization group equation with fake /3 functions, like/3,, and an "anomalous dimension function" simply related to the poles in the Z factor. This relation is the same as in the usual method of dimensional regularization and minimal subtraction. However, the formal eq. (5.27) is of no -R practical interest. Instead, we shall now show that the restriction -/-'(2) [in the sense of eq. (5.6)] does indeed satisfy a correct renormalization group equation as in eq. (4.4) and we derive the expression of the corresponding anomalous dimension function. Consider the ordinary renormalization group operator O d 3 = ~ + / 3 ( a ) ~-a + 6(a, A)A ~ - ,

(5.29)

--R

and apply this operator to F(2). Note that now O/Oa is the complete derivative with respect to a with no distinction between a and o~, contrary to the case of eq. (5.27). We have from eq. (5.14) up to order a 2 ~F~2) = a B + o t 2 [ ( 2 D - D c ) + 2 P l - b ( A + B l ) + b A ( . ' ~ x

+Bd)].

(5.30)

494

G. Altarelli et al. / Corrections to w e a k decays

If the equation =

(5.31)

is to be valid with y(O~, ~ ) = ~/(1)O/ + ')/(2)0/2 ,

(5.32)

then we should show that both the terms linear in l and those independent of l should separately vanish, i.e. it should be true that 2 t3 =/~/3 + b/~ - b ~ / ~ ,

(5.33)

- y (1) = / ~ ,

(5.34)

_y~2~ = 2 D - 1 9 c - B A

- bA + bxA~ .

(5.35)

If the first equation is proved to hold, then the remaining two provide the explicit form of the anomalous dimension. To prove eq. (5.33) we start by taking the restriction of eq. (5.26) for 2F: 2 F = ~-7+ b/3~ + b~/3~ - b~/~.

(5.36)

/3~ = 0 .

(5.37)

We first note that

In fact B~ is the contribution to B of e-scalars in internal lines (we never consider e-scalars as external particles). Obviously, the e-scalar contribution would vanish for e ~ 0 if not for the singularity of the integrations. Thus the only terms from e-scalars which behave like 1/e in one loop must necessarily multiply bare operators like y - 37 or yf which vanish in four dimensions. Then eq. (5.37) follows. Note that this implies that /~ =/~,, namely e-scalars do not contribute to the one-loop anomalous dimension y~l) given by eq. (5.34) which coincides with the usual L L A . It only remains for us to show that B-~ = ~2. Let us denote by P the set of physical operators and by V the set of e-operators, which vanish for n = 4. We need to show that (BB)pp, = ~ Bpp,,Bp,,p,. p,,

(5.38)

In general the left-hand side is equal to (BB)pp, = Y. Bpp,,Bp,,p, P"

+ Y~ BpvBvp,.

V

(5.39)

Eq. (5.38) is indeed valid because, in general Bvp = 0 .

(5.40)

This is true for similar reasons as f o r / ~ = 0: an operator which is the difference of two operators differing by terms of order e cannot mix at the leading level with finite operators. Thus eq. (5.33) is valid in general. Note that eqs. (5.37) and (5.40),

495

G. Altarelli et aL / Corrections to weak decays

which are necessary conditions for the validity of a correct renormalization group equation, can be checked by inspection of simple one-loop diagrams and can be verified in table 1 for our case. By using the restriction of eq. (5.28) for D c we can rewrite eq. (5.35) for y(2) in the form _ ~/~2)= 2(/9 - / 5 c ) + B A - B A + (b~ - b)~,~, (5.41) where we have taken into account that A = fi,~ +A~ identically. Of course [see eq. (5.39)]

( B A - B A ) p p , = Y. BpvAvp,

(5.42)

V

which in general is non-vanishing as well as fi,~. Note that the first term 2(D - D e ) is what we would have obtained from the usual rule valid in dimensional regularization with minimal subtraction which directly relates the two-loop anomalous dimension to the residue of the 1/e pole in Z. We see that one obtains additional terms in dim.red, which arise from the presence of e-scalars and e-operators. In the above analysis we did not separate wave-function and vertex renormalization factors, so that 3,(2) in eq. (5.41) also includes the contribution of external legs, which must be computed separately and subtracted as usual. Of course, the internal leg contribution to ~(2) must be computed by the same steps. For fermions this reduces to extracting the pole part 2(D ~ - D e z) from the self-energy diagrams and to adding (be - b ) A ~ [see eq. (6.2)]. In fact, for fermion external legs there are no e-operators, because p is the only bare operator in the result [see eq. (4.8)]. Note also that we phrased the discussion in the language of minimal subtraction. The changes which are needed in order to adopt the MS prescription are quite trivial and will be mentioned in the next section. 6. The anomalous dimensions of weak operators

In this section we apply the general formalism and the techniques described in the previous sections to the actual calculation of two-loop anomalous dimensions of weak four-fermion operators. We shall limit the details to a few points of particular interest, while for the rest we shall only give the final results. We consider first the fermion self-energy diagrams of fig. 7. The results in a one-loop approximation for ,~(p), defined in eq. (4.8), were already presented in sect. 4 and are shown in table 1. In two loops the pole part of Z ( p ) is found to be [,~(p)]poles=O/2

--7 -I

-De /-1

:(-~)2[C2(~e2--7)+CFCAt-~+-~e)J

13\q

.

(6.1)

G. Altarelli et al. / Corrections to weak decays

496

The double poles are in agreement with eq. (5.26), as can be verified from table 1 and eqs. (2.12), (4.6), and (4.36). Actually, separating the counterterm diagrams, we have verified eqs. (5.12) and (5.28) for F c~ and D c~. We stress again that we did these consistency checks for all Green functions of interest, although for brevity we shall not mention it every time. The function YF in eq. (4.4) is then given by --y
(6.2)

Recall that no e-operators are present in this case. Also from table i we have that A t = CF/47r.

(6.3)

We now consider the four-fermion vertices. In sect. 4 we described two definitions of renormalized four-fermion operators. The first does not preserve Fierz symmetry explicitly and consists in making minimal subtractions in 3' and y - ~ for all one-loop four-fermion insertions [the case of (Mc)x in eqs. (4.39) and (4.40)]. In the second definition one makes minimal subtractions in the basis 3' and ye of the Fierz eigenstates [as for (Mc)n in eqs. (4.44) and (4.45)]. Of course, the calculation of all diagrams without self-energy or vertex insertions is the same in the two cases [see K c in eqs. (4.29) and (4.30)]. We first state our results in the second scheme, which is simpler because O~T are multiplicatively renormalizable. Then we show that the first method, although m o r e complicated and quite different in the intermediate steps, eventually leads to the same physical results. 6.1. F I E R Z

SYMMETRIC

PRESCRIPTION

After computation of all diagrams as in the examples of sect. 4 and reported in table 2, we write the results for single poles in the form 2 OL

[o~,]

= _

single poles

o [a +,.+,(o+~,) + ~+,_,,(o_~,a], c

e

2

(6.4)

O/

[o~_~,]

= - - [A_~.+~,(OLw) +/t_~._~(OL~,~)].

single poles

e

Note that (OCT) and (OSyf) have the same symmetry under Fierz rearrangement. The matrix elements of A correspond to the matrix elements of D - D c . Actually we can directly subtract the contribution of the quark external legs by identifying dij = (D

-

Dc)ii

-

2(D x - D ~ )Sij.

(6.5)

The important elements for physics are those of the restriction ,~ii, so that the interesting results are 2 /N~I\ 2A:~v'±~'- (4~ 2 ) / - - - ~ )

4 7 N 2 ..b 15

[--~

7

2

~-N-~-~Nf].

(6.6)

G. Altarelli et al. / Corrections to weak decays

497

According to eqs. (5.41) and (5.42) the two-loop anomalous dimensions of (O~3,) defined in eqs. (2.13), (2.16), and (4.4) are given by

-3,~ ) =2A±v,±~+B±v.~v,A~v,,±~+(b~-b)(A~-2A~)±~,±v.

(6.7)

By using eqs. (4.18) and (4.21) and table 1, we can construct the relevant quantities

B±v'=v' :

(N~: 1) 4rr '

A~v"±v=

1 (N + 1) 2 4rr '

2E

(A~ -2A,)±~.±v=

(6.8)

1 (N:~ 1)(N+ 3) 4zr N

It is then easy to obtain _~/~)=

2 (-~--~)/ n,,r2_lm.T+~t:~57+~Nf+Sf}. (41r)2 --~1, + 7a-1'~ 8N

(6.9)

This concludes the calculation of anomalous dimensions of O± = (O~y) with this renormalization prescription. By themselves y ~ have no physical meaning. To make contact with physically interesting quantities we need also to compute y~2) and C~ ) [see eqs. (2.13), (2.16) and (2.21)], which appear in the operator expansion of two weak currents. We consider first the weak current operator J,, and its renormalization in order to fix y~2). In principle, we must first define J , and then consider the short-distance expansion of the product of two currents so defined in terms of the renormalized operator (O~:y) whose definition was independently given. The most natural way to define J , is to do it in such a way that it is conserved, i.e. y~2) = 0. This is obtained in dim.red, by starting from a bare y , and a renormalization through minimal subtraction in 3', and y ~ - ~ , (for a single current the factor 1 - 75 is inert and is omitted). With this definition of J , the single pole terms from the two-loop vertex (a subset of the two-loop diagrams in fig. 2 with gluons exchanged only within one single current) with the contribution of the external legs subtracted away, are given by 2 2 55 2 2(DS-X-D~-X)=(-~)2[-~-C~ +i~CvCA+~fCv](y~,-'~,).

(6.10)

We see that the restriction of eq. (6.10) to the physical basis vanishes. According to eqs. (5.41) and (5.42), in order to prove that 3,~2) = 0, with the present prescription, we also need to show that B J~,v_-vAJ~_~,~+ (b, - b)[(A~)~ - A ~ ] = 0.

(6.11)

498

G. Aharelli et al. / Corrections to weak decays

It is simple to verify from table 1 that this equation is indeed satisfied. Thus we conclude that 7(12) = 0.

(6.12) c

R

We have specified a definition of J ~ and of (O±7) . Therefore, we can now derive the consequent expressions of C~ ) which depend on both. This is easily done by considering the differences of one-loop diagrams of H v i , with the W boson propagator, and the corresponding diagrams with a four-fermion local vertex. Care must be taken for the different subtractions to be made in the two cases according to the definitions stated above for the renormalized current and the local operators. The final result is given by 1 N~I C~) =4---~ ( - - ~ - ) ( + 5 -

N).

(6.13)

It is at this point that we dropped the terms with 7E, the Euler constant, in order to formulate our results in the MS prescription. • • • R We have checked that starting by different definmons of J , the same results are obtained. For example, defining 3", in analogy with the treatment of four-fermion operators, i.e. by subtracting in 7, and (7f),, then 7~2) # 0, but its value exactly compensates the corresponding change of C~ ) in the relevant combination b C ~ ) 27(i2) [see eqs. (2.13) and (2.21)]. Alternatively we can define J ~ by starting 2~, instead of from 7,, with the same final results. From eqs. (2.13), (6.9), (6.12), and (6.13) we can state the final result of the lengthy calculation (47r)2[,~)

_(N=71) [

k/_~(1) 1__ JMS(red)--

--t,'l,..,±

~

57 !~/] W2q~N+E!~:4N+ "

(6.14)

We recall that this result, although independent of the operator renormalization prescription and of the regularization method (as we shall further demonstrate in the following), still depends on the definition of the renormalized coupling ce. This is made explicit by the index MS (red): later we shall give the relation between the coupling a defined by MS within dim.red, and its analogue in ordinary dimensional regularization. 6.2. AN ALTERNATIVE PRESCRIPTION We now report on the alternative calculation starting from a renormalization prescription based on minimal subtraction in 7 and 7-~2 [as in the example of ( M - Mc)i in sect. 4, eqs. (4.39) and (4.40)]. The novel feature is that Fierz symmetry is broken by the subtractions and the operators O± are no longer multiplicatively renormalizable in two loops• It is then instructive to follow how the same physical results are obtained.

G. Altarelli et al. / Corrections to weak decays

499

In the present case the anomalous dimension matrix in colour space is computed according to _y~2) = [2(/9 - / ) c h - 4 ( D

x - Dc)]v,v x + B v,v--vA v-'~,v + (b~ - b)[(A~)i - - 2A f]vv, (6.15)

when the index I distinguishes this prescription from the previous one. All matrices are now in the ~, y - ~ basis. The result for y~2) is in the form of a matrix that mixes Oi[i = +, 05: = ( O ~ ) ] :

['Y~2)O]i= (Y(2)),Oj •

(6.16)

Explicitly the diagonal elements turn out to be identical with those in eq. (6.9), (6.17)

.y I (2) x)5:5: = ,~ (j2)

while the non-diagonal terms are given by [see eq. (2.12) for b] (_y~m)5::~

1

( _ ~ _ 1 ) (4,rrb T 6) "

(6.18)

-- (4,rr)2

In the present case the matrix ,7(2)=-y~2) [see eqs. (2.13) and (6.12)] does not commute with the one-loop matrix ~(1). We then have to modify eqs. (2.14) and (2.18) by expanding an ordered exponential 3~(a')) da'} P{exp I~(') fl(od = 1+

da'+

=exp

ln--~] +

da'

da" ~(a')~(a") +...

d a ' exp

In ~--~] \ ---~-) exp [--if- In ~7] . (6.19)

With some straightforward manipulations to the required accuracy we can write the result in the form P {exp

[aL+(t) - a

f~(~)'a'

(t)L_(t)];/~)_-]

[ L + ( / ) [ 1 4 a--bad(t)3~(+2)+] ""

(t)L+(t)]~,~)+

/ =

L

-?2 ) -

)

+b (6.20)

where L:dt) are defined in eq. (2,20) and y± , 3'±5:, and " ~ are given by eqs. (2.24), (6.17), (6.9), and (6.18). At the same time the coefficients C~ ) are also modified. We keep the definition of J , given above, which leads to eq. (6.12) and -(1)

-(2)

-(2)

G. Altarelli et al. / Correctionsto weak decays

500

is based on minimal subtractions in 3', and y , - ~,. Note that this definition is completely natural in this case, because the four-fermion operators are also defined in a similar way. The corresponding coefficients are found to be [C7 ) ] I = C 7 ) + 1 t(N~:I'~ ~/,

(6.21)

where C~ ) are given in eq. (6.13). In order to show that the results obtained for y~2) and [C~)]I correspond to the same physics as those for 3/2) and C(~ ), we consider in general the structure of the transformation properties between the two prescriptions. The transformation matrix between the two sets of renormalized operators can be parametrized as follows

O~ = K[a ],sOR = [~,i + aKl 1' ]OR,

(6.22)

where the unit matrix in trivial order indicates that the two sets of operators reduce to the same bare operators at o~-->0. The short-distance expansion of the same operator product in the two different bases can be written as

Ci [~ (t)]Eij[a (t), ce}0 R (a ) = C[a (t)]E[a (t), a ] O n = C[a(t)]K-a[a(t)]K[a(t)]E[a(t),

a]K-'[a]K[a}O R

= C~[a (t)]E~[a (t), a ] o R (a),

(6.23)

where Ei[a(t), a] is the ordered exponential matrix in eqs. (6.19) and (6.20). We thus obtain the relations Ci[c~ (t)] = C [ a (t)]K-~[a (t)], (6.24) Ei{o/(t), ~] = K[c~ (t)]E[a (t), a ] K - l [ a ] . In the case of O R being multiplicatively renormalizable we have o

E[a(t), a] =

0

L-(t) [ 1 -~ ~a- a (t) ~ )

]1 '

(6.25)

and C~:[c~(t)] given by eq. (2.17). As a result, one obtains from eqs. (6.22), (6.24), (6.25), and (2.17) L+(t) [ 14 a-~---~-ba(t)(y~+ -bK~)+ )] El[Or(t), o~l =

[ot(t)L+(t)_olL_(t)]K~)+

[a(t)L_(t)-eL+(t)]K~ )-

]

L_(t)[l_bCe-~(t) (+~)__bK~) ] ' (6.26)

G. Altarelli et al. / Corrections to weak decays

[C±[ot(t)]]i= 1 +

a ( C ~ ) _ ,~,.(1) , ~ - , ,TO,-(1) ,± ).

501 (6.27)

On the other hand, a simple calculation of the renormalized matrix elements in one loop shows that in our case K ":k± ) = 0, (6.28) K(1) : __ I__I__(N± I~ :~:~ 4zr\ 2 ]" In the first place we see that the calculated relation between C~ 1) and C
~(2) _

_

Y~

(6.29)

These relations are indeed satisfied as can be seen from eqs. (6.18), (2.26), and (6.28) by recalling that 3~(2)= _y~z) in this case. This completes the proof of the equivalence of the two procedures. We have thus shown that starting from several different renormalization prescriptions within dim.red., for weak currents and four-fermion operators, we always end up with the same physical results. These different procedures, selected a m o n g others for which we also came to the same conclusions, have been described in relative detail in order to convince the reader that a complete control of the new technique of dim.red, was in fact achieved. W e close this section by studying the relation between the renormalized coupling a in dim.red, and in the conventional definition within dimensional regularization. The two definitions are in fact different even if modified minimal subtraction is adopted in both cases. Let a ' be the coupling in dimensional regularization with MS. Then the relation between the two couplings can, in general, be written as a =a'(l+ka'+"

• ").

(6.30)

As a consequence, eq. (2.16) can be recast in terms of a ' as ~(1)Og + ,)/(2)02 + ....

,y(1)o/, _{_[y(2) .4_ k]/(1))]o ,2 .+.. , . .

(6.31)

Thus the two-loop anomalous dimension in the primed scheme is given by ,]ff(2) : ~/(2) "4- k y

(1) .

(6.32)

In dim.red, a is defined as the coupling of the true gauge particle Gi (i ~ n ~<4). Thus the difference between dim.red, and dimensional regularization is due to the effect of the s-scalar internal lines, which we need to study in order to compute k

502

G. Altarelli et al. / Corrections to weak decays

[eq. (6.31)]. The simplest procedure is to select the gluon-ghost-ghost vertex. In fact this coupling being proportional to k,, the ghost n-momentum, e-scalars do not couple with ghosts, as is clear since e-scalars are matter fields. Thus, in this case, up to one loop the e-scalars influence the finite terms in the coupling o~ only through their contribution to the vacuum polarization diagram of the gluon. It is easy to isolate this particular contribution to the gauge particle vacuum polarization and one finds

k = ~--~

(6.33)

= 12rr"

Gauge invariance guarantees that the same result is obtained by starting from the gluon fermion or the three-gluon vertex (always with external gauge particles). We explicitly checked the equality of the result for the G;q~l vertex. Thus we find as a final result, from eqs (6.14), (6.32), and (6.33) ['Y(d:2) -- b C ~ ) ] ~

= ['Y(a:2) -- b C ~ 1) ] ~ ( r e d ) "{- k~(:t:1)

1 (Nq:l\[~:2qa~N+a =

(4rr)2 \ - N /

(6.34) 57

lO ]

:;:~~+*#fJ ,

which is now expressed in terms of the usually defined renormalized coupling a in MS. In the next section we shall discuss a number of physical implications that directly follow from eq. (6.34).

7. Discussion of the operator results and implications for strange particle decays The final results in eq. (6.33) [see also eqs. (2.18) and (2.21)] are operator results which are independent of the particular matrix element considered. We thus discuss in this section the structure of the results and their implications for strange particle decays, before going to the application to c and other heavy quark decays in the next section. We first observe the reflection symmetry of eqs. (2.24), (6.8), and (6.12) under N ~ - N , f ++ - f : (1) +~-Y (1) , Y+ (2) +.~ +y~2) Y+

(7.1)

C a ) ,--, _C¢1). Note that, according to eqs. (2.13) and (6.11), 3~ )(2) =_y(1)~2). As b and b' change sign under the same transformation so that o~~ - a , then eqs. (7.1) imply for the anomalous dimension functions up to order 2 and for the coefficients of

503

G. Altarelli et al. / Corrections to weak decays

order a [see eqs. (2.2) and (2.14)] the properties

v+(a)~/-(a), (7.2)

C+(t, a)~--~C_(t, a) . The origin of these properties is clarified by rewriting eq. (4.19) for bare operators in colour space in the form: O~:=\ 2N /~+g=

~'+~-'

(7.3)

where I ' = ~ / N = I ® I / N was rescaled so that its normalization (in the sense Tr 1 ® 1 = Tr 1 = N ) is N independent in analogy with eq. (2.6). Then for bare operators OC<-->O~ under N,~-~-N at fixed ~' and 27-. In order to preserve the validity of the same relations for renormalized operators, one has to add f ~ - f in order to endow a with a definite symmetry. Then eqs. (7.1) and (7.2) appear as a consequence of the bare operator reflection properties of the relation ~ = O~_ + O~ (the bare operator expansion of the bare hamiltonian). We also note that the leading terms in the limit N, f--> oo, i.e. terms of order N 2 and Nf, are absent in the final result of physical significance in eqs. (6.13) and (6.33), although such terms are present in all intermediate steps [see for example eq. (6.8)]. The cancellation of the leading terms in the large-N expansion is also true in L L A , in that y~) are of order 1, while their natural order should be N. This implies that in the limit N, f o ~ , with a N fixed, the quantities L±(t)(1 +{[a-e~(t)]/zr}p±) in eq. (2.18) approach 1, i.e. the free field value. The reasons and the meaning of this "soft" large-N limit of the weak four-fermion operator are obscure to us. The numerical importance of the results and their signs can be extracted by the values of p± and "y~)/b in eq. (2.18). For N = 3 and f = 4, 5, 6 we find --0.240

[a 1+°.48°[1 + a ~--~J

t

-a(t) (-0"469'/] zr \ +1.36 ] / '

f=4

-0.261

L±(t

)[

a-a(t) 1+ -77"

p±

] __ [ a 1+°'522

1 +a-a(t) (-0.510)] zr

\ +1.48 ]_1 '

f = 5,

(7.4)

--0.286

[ a I+0"572 1 + a

1 7 5J

[

-a(t) (-0.574~] zr \ +1.65] '

f=6.

Note that the improved form of a(t), as given by eq. (2.22), including the b' term, is to be used in the above equations, at least in L±(t). The magnitude of the corrections shows no unexpected features and their signs are such as to reinforce the pattern of enhancement and suppression found in the LLA. This trend is made

504

G. Altarelli et al. / Corrections to weak decays

manifest by observing that the most sizeable terms in 3;~ / - b C ~ ~ in eq. (6.33) are proportional to 3,~1), given in eq. (2.24) (in fact all terms in the bracket but for ~ ) so that they could be absorbed in a redefinition of a and are equivalent to a change of A in the leading term. For example, for f = 4 we could alter the correction term in eq. (7.4) by only including in p± the contribution of the term proportional to ~t which leads to [ a-a(t) L=~(t) 1 ~ --

~r

(+0.105~] \-0.210]_1 '

and at the same time scale up A by a factor of - 1 . 8 . The present result considerably reinforces the conclusion, previously based on the L L A , that gluon effects work in favour of the A T =½ rule in non-leptonic decays. We recall that in the case of AS = +1 transitions O_ is pure A T = ~, while O+ is a mixture of A T = 1 and 3. Thus enhancement of O and suppression of O+ goes in the direction of the A T = ½ rule. The massless theory (with some effective f between 4 and 6 as discussed in sect. 2) applies in the range of/1. between rnc and Mw. Below mc various effects [5] connected with the no longer negligible c mass appear, such as penguin diagrams and separate anomalous dimensions for operators transforming with different SU(3)r representations. However, the bulk of the interesting effect and the most reliably computed effect in perturbation theory arises from the region between mc and Mw. Then our result shows that non-leading corrections not only do not by their size upset our confidence in the perturbation expansion, but also further improve the calculated enhancement factor. Quantitatively, the results on the enhancement factor for Ix -~ mc and f = 4 are shown in fig. 3, where a comparison between the L L A and the corrected results is also shown. Note that the comparison is to be made at the same value of a (me) and not at fixed A, because the meaning of A in the leading and non-leading expression for a is different, and the connection between the two is precisely provided by the value of a at a given scale/x. We also recall that a further gain in the enhancement factor is also obtained from the region below mc as studied in detail within the L L A in ref. [5] (see also ref. [6]). While in the case of strange particle decays we cannot compute operator matrix elements in perturbation theory, such a calculation can be attempted for the decays of a c (or heavier) quark. This will be the subject of the next section, so that a discussion of the physical implications of our results for charmed particles or other heavy flavoured particles will be postponed to the last section.

8. Heavy quark decay In the present section we describe the calculation of the Q C D non-leading corrections to the non-leptonic decay rate of a heavy quark Q, in particular of a c quark, for massless final-state quarks and gluons. The result will be of general

505

G. Altarelli et al. / Corrections to weak decays

validity with the exception of the neglect of penguin contributions, which is quite justified for the total non-leptonic rate of the c quark but might be inadequate for some channels of b and t decays. Even in such cases the present calculation is complete in itself and the penguin contribution can be computed separately and added. According to the general description in sect. 2, we need to consider at the same time the decays Q ~ qlq2cl3 and Q ~ qlq2cl3G, because it is only after adding up the two contributions that the cancellation of infrared singularities is obtained in the total rate. Thus an infrared cut-off is needed in the intermediate steps of the calculation. An ultraviolet cut-off is also required for vertex and self-energy insertions. The choice of both cut-off methods is arbitrary and independent of the previous choice in the calculation of p±. Also in this case we have used dim.red. both as the infrared and the ultraviolet cut-off. Our starting point is eq. (2.18) for HF~, the general amplitude with non-leading corrections for arbitrary initial (I) and final (F) states. For F - q l q 2 c l 3 one must first evaluate the one-loop diagrams in fig. 16 with virtual gluon exchange and obtain the order a contribution to the quantities 0~-'3"(o~) in eqs. (2.18) and (2.29). Then the partial rate from no~3q is tO be computed. It is by interference with the amplitude in the L L A that both the p± terms and the virtual gluon corrections to 0 °-'3q (a) contribute to the rate. One must carefully distinguish tg± from O± as given by eq. (2.29). In order to identify tg± we observe that, in terms of the renormalized coupling a(/x) with/~-~ me, {[a-ot(t)]/Ir}p± in eq. (2.18) is of order a 2 t and L ± ( t ) - 1 is of order at. Thus, we can extract 0 ± by computing Ho--,3q at one loop and by separating terms of order a t from those of order c~. All terms of order o~t belong to L ± ( t ) , while all terms in a (without t) belong to 6±. It is because nQ~3q is the relevant quantity that in the diagrams with virtual gluon exchange in fig. 16 the W propagator is explicitly shown. For real gluon emission, i.e. for F = qlq2cl3G (see fig. 17) the situation is simpler in that the partial rate starts at order a and consequently one can drop the p± terms and also the difference between O± and tg±. Thus one is left with computing the partial rate from the sum of the contributions of the diagrams of fig. 17, where the light cone expansion in LLA, i.e. L ÷ O ± + L _ O is at the local four-fermion vertex. The calculation is organized as follows. Consider the virtual diagrams of figs. 16b to g. We first compute the interference of each one-loop diagram with the

o

b

c

d

e

f

g

Fig. 16. One-loop virtual gluon corrections to the c-quark decay rate. The dashed line indicates the W-boson propagator.

506

G. Altarelli et al. / Corrections to weak decays

C

s

u

d

Fig. 17. Real gluon emission diagrams contributing to the c ~ suaG rate. Each diagram is labelled by a letter which indicates the emitting quark. lowest order diagram, fig. 16a, and separate away the terms of order c~t because the sum of the whole leading logarithmic series will be included in L s ( t ) . Also we can at first disregard all colour matrices that can be taken care of at the end. We denote by Vab, Va. . . . . . Vag, V~z the interferences of the diagrams of figs. 16b to g with the diagram of fig. 16a, while Vaz is the interference of the diagram of fig. 16a with the self-energy insertion on the massive quark leg (all other self-energy insertions on the massless on-shell legs are zero, for similar reasons as in the ordinary method of dimensional regularization). As for the colour factors the sum of diagrams b + c + z of fig. 16 is to be multiplied by CF~ and the sum of the diagrams d + e + f + s of fig. 16 by ~-, as is seen from table 1. When reinstating the L L A factors L±(t), the previous operators are then replaced by CF~---~ CF(L+O+ + L _ O _ ) ,

(8.1) + +

~ 2N

j L-O-.

Finally the interference with the L L A term L + O + + L _ O _ is to be taken. By including the average over the initial quark colour we obtain for these interferences (in colour space) 1 I O N±I (05 5)= 2 '

(8.2) 1

Z(OsO~) = 0.

Finally, the complete expression for the contribution of virtual diagrams to the total rate is given by

(N2-l~{I(N~l)t2-'~(-~)t2](gab--[-

virtual=\

2N

gac-~- gaz)

/

+~(L+ -t2-)(Vad + Wac+ Vat+

Wag)

•

(8.3)

The calculation of real diagrams proceeds in a similar way. We first compute the absolute square of the diagrams of fig. 17 as if no colour and no L L A factors existed. We denote by Rcc, Rcs • • • the various total contributions to the rate from the squared modulus of the diagram of fig. 17c or the interference of the diagrams of figs. 17c and 17s . . . . . where c, s . . . . refer to the quark that emits the real gluon. Then we reinstate colour and L L A factors and in analogy with eq. (8.3) we can

G. Altarelli et al. / Corrections to weak decays

507

write the result in the form real

=(N2-1"~

+~(L+I 2 _ L 2 ) ( R c u + R s u + R c a + R s d ) l

(8.4) J The calculation of the V and R terms is long and complicated. It is done by using the techniques illustrated in detail in ref. [15] for the treatment of three- and four-body final-state phase space in n dimensions. Also algebraic manipulation by computer is essential. There is only one peculiarity of dim,red, in this part of the calculation which is worth mentioning. It was already observed that an ultraviolet cut-off is only required for vertex and self-energy corrections. While in dimensional regularization the sum of vertex and self-energy diagrams is ultraviolet convergent and needs no renormalization, in dim.red, this is not automatic because there are e-operator terms proportional to ( 1 / e ) ( y , - ~ ) in the vertex. The subtraction of this counterterm must be made (as already done for this same vertex in all previous instances) before taking the interference with the lowest order diagram. For example, for the diagram of fig. 16c, one obtains apart from an over-all non-singular factor V~

3"[--((~201)2]-e( t m-~o J

2 3 e2 e

)

1

7+~7r2 + ( 3 ' - Y ) e - "

(8.5)

It is precisely the last term which is subtracted, as it is exactly the same ultraviolet counterterm already met when discussing the weak-current renormalization in sect. 6. In table 3 the final results are reported for the contributions to the rate of both the real and virtual gluon diagrams. Sums of real and virtual gluon diagrams, which are separately infrared safe, were grouped together. The first two partial sums in table 3, i.e. ~ - 7 r 2 and 3 are old results and serve here as a check. As a further check we recomputed the entries - ~ and ~ - 7r2 by a different infrared regularization method, namely by giving a mass to the gluon (in n = 4 dimensions). This was an excess of caution motivated by the fact that one is using here, for the first time, dim.red, as an infrared regularization method. These particular diagrams were chosen because they are the only ones that could potentially involve subtleties connected with e-operators (3' - y). By replacing the results of table 3 in eqs. (8.3) and (8.4) with N = 3 and by adding up the anomalous dimension term p± appearing in eq. (2.18) we finally obtain for the total non-leptonic width of a heavy quark (with/x = too) { 20d (9 2 2 a ( 1 9 ] L2_-L2+ F NL = FNLLA 1 -4-~ g ----TF ) + 5 ~r \ 4 / 2L2++ L 2-

2 2 [ a - a (t)\ [2L+p+ + L _ p _ \ ) + z ~ ) l 2-~--+++L-~_ ) ,~

(8.6)

508

G. Altarelli et al. / Corrections to weak decays

TABLE 3 Detailed results of virtual and real gluon emission contributions to the inclusive rate of quark decay Vab

V~

1 e- 2

41 - 12e

3

117

1769 1 z ] -144 - ~ ' - ~ "3

/

R~ + R~ +R~

Vac

-1~ + - 59 -+ e 12e

3371 • _ -4~ 2 144 3

2 28 749 8 .rr2 2 "~e 3e 18 3 2 28 388 8 2

guu+gdd+g~d

-~+~+~----~r

Vad

1 19 2065 e--~+4e + 144

Red

Vae

Vaf

1 53 2 e 12e

~sd

3 2

"n"2 - - ~79

2 28 277 8 2 e~+ 3 - ~ + ' - ~ - - ~ ~"

Rus

Vag

I

1 19 3013 +4_ ~2 2 e 4e 144 3

2 28 2 e 3e

Rcu

1 2 3

h /

3

2069 blrr2_ 144 3

1 53 2765 e~+12e + 144 2 31 2 e 3e

4o

794 8 rr a ~-18 3

4 3

2

1573 +8_ 17-2 36 3

2 31 887 8 : e-"2+ 3~ + 1 - ~ - ~ ~r

67 12

An over-all factor (as/2~) is understood. With this normalization the entries in the table when inserted in eqs. (8.3) and (8.4) lead directly to the correction term in units of the uncorrected rate.

NL w a s d e f i n e d in eq. (2.27) a n d it c o n s i s t s of t h e n a i v e p a r t o n m o d e l w h e r e F LLA 1 2 r e s u l t t i m e s t h e L L A e n h a n c e m e n t f a c t o r ~(2L+ + L ~ ) . E q . (8.6), t o g e t h e r w i t h t h e e x p r e s s i o n f o r p± [or eq. (6.34)], is t h e m a i n r e s u l t of t h e p r e s e n t w o r k . I n ref. [15] o n l y t h e first c o r r e c t i o n t e r m w a s g i v e n . It arises f r o m g l u o n c o r r e c t i o n s w i t h i n t h e

G. Altarelli et al. / Corrections to weak decays

509

first and second current separately [the first term in eq. (8.3) and in eq. (8.4), or the first two subtotals in table 2]. This term can be obtained directly f r o m / x - d e c a y radiative corrections (2t~/3zr) [ ~ - 2 ] and the first-order correction to Re+e- (the ratio of the hadronic to the/z+/x - cross section in e+e - annihilation) given by o~/Tr. The second correction term in eq. (8.6) is from virtual gluon exchange between the two currents plus the corresponding interference terms from the real diagrams (the last contribution in eqs. (8.3) and (8.4), and the last four subtotals in table 3). It arises because r e s u m m a t i o n of leading logarithms invalidates the vanishing of the contribution from gluons exchanged between the two currents, which is valid 2 2 to order a. Note in fact that L_ - L + starts at order o~t, so that the second term in eq. (8.6) starts at order a2t. The last t e r m in eq. (8.6) also starts at order a2t and is from the two-loop anomalous dimension. Each of the three terms is separately gauge-invariant and prescription-independent, although the choice of a definition /,NL for a affects the value of LLA and is compensated by just the corresponding change in p±. A numerical evaluation of eq. (8.6) f o r / z = me = 1.5 G e V [so that o~~ a (rnc) and, as alwayS, a (t)---a(Mw)] shows that for all interesting values of A (and of f) the first and third correction terms, which are of opposite sign, almost cancel each other, so that the dominant contribution is given by the second term which is positive and substantial (although not abnormally large). The situation is displayed in fig. 4. The effect of Q C D in the L L A is an e n h a n c e m e n t of the non-leptonic 1 2 2 width over the naive parton result (with 3 colours) by a factor ~(2L+ + L _ ) . The Q C D prediction of an increase of F NL for a free quark is strengthened considerably by the present results on the non-leading corrections, which are found to add in the same direction and to be quite sizeable at least for charm. We recall that the corrections of order a to the semileptonic quark width (and also to the charged lepton spectrum) are already known. Actually the correction to the total semileptonic width in the limit of massless final-state partons can directly be obtained from our results, by restricting the case to that of a single quark current instead of two. We thus have from the first entry in table 3

(8.7) The correction term is negative in this case. Therefore the semileptonic branching SL [see eq. (2.28)], both ratio B sL of the c quark is depleted with respect to BLLA because of the decrease of F sL and because of the increase of F NL. As a result, it seems established that, at least for a free quark, B sL cannot exceed a value of 1 2 - 1 3 % (for A ~>0.25 GeV), as shown in detail in fig. 5. The implications of this result as regards the decays of charmed mesons will be further discussed in the next section.

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G. Altarelli et al. / Corrections to weak decays

9. Summary and conclusion In the present paper we have obtained the following results. First we computed in the massless theory the two-loop anomalous dimension of the four-fermion operators relevant to the non-lelStonic weak decays and the complete corrections of order o~ to the effective weak hamiltonian. This calculation was performed by using the regularization technique of dimensional reduction. A part of this paper was devoted to a study of the field theoretic subtleties associated with this variant of dimensional regularization. In particular, we elucidated the differences between minimal subtraction within ordinary dimensional regularization and within dimensional reduction with respect to the renormalization group equations and related quantities such as anomalous dimensions, (running) coupling constants and so on. Within two-loop accuracy we checked renormalizability and that renormalized quantities indeed satisfy the correct renormalization group equations. Our results are relevant to strange-particle non-leptonic decays and the AT = rule. The restriction to the massless theory can only be justified in this case if penguin diagrams are left aside and one considers the region of energy scales between mc and Mw (the effect of b and other heavy quarks can then be reabsorbed in the dependence of the various quantities on f, the number of excited flavours). In fact if we were to define renormalized operators a t / z ~ M w we would have no large logarithms to resum in the coefficient, but the matrix elements of such operators would be very different from those defined at the scale of energy typical of the class of process under study. By selecting/z --- mc we define operators which are, of course, more directly relevant to strange particle decays and ones whose coefficients can reasonably well be described in terms of the massless theory, because in most of the range between m~ and Mw the c mass is indeed negligible. Below m~ the effects of the "large" parameter In ( m ~ / i z ) become important and the massless theory is no longer sufficient. It is, however, clear that the most reliable class of perturbative predictions concerns the gluon effects in the region o f / z between m~ and Mw. We recall that penguin diagrams are not large because of large coefficients [their coefficients are of order (~/127r)In (mc//~)], but are believed to be large because of a plausible anomalously large size of the corresponding matrix elements. We also recall that the study of higher order gluon effects in penguin diagrams has already been carried out to some extent I-3, 22]. We found that the first subleading Q C D corrections, which are lower by a (me) with respect to the leading approximation, are of normal size, hence not at all negligible, because a (me) is not small. These corrections follow the same enhancement-suppression pattern of the leading term. That means that the A T = ½ amplitudes are further enhanced and the AT = 23amplitude further suppressed, see fig. 3. Although the numerical gain in the enhancement factor is not imposing in itself (also in view of the lack of detailed knowledge of matrix elements including those of penguin diagrams), we still find that the present result considerably strengthens the arguments for a Q C D effect

G. AItarelli et al. / Corrections to weak decays

511

in favour of the AT=½ rule. In fact, this result shows that to second order the convergence of the series is not endangered for current values of A. Furthermore, the pattern of non-leading corrections is also favourable to the AT =½ versus 3 amplitude ratio. We then derived the complete set of corrections of order a to the non-leptonic decay width of a charm quark into light quarks and gluons (all treated as massless). The same calculation, of course, also applies to the decays of other heavy quarks, provided again penguin diagrams (which are irrelevant to the Cabibbo allowed c decays) are treated separately, if they are at all necessary. This calculation supersedes the incomplete computation of ref. [15], which did not include either the two-loop anomalous dimension term or other terms from resummation of leading logs which only start at order a2 in perturbation theory. We actually find that the sign of the complete correction is such as to further increase the non-leptonic width of the decaying quark (opposite to what was found in ref. [15]). Thus, in this case also we find that the non-leading corrections act in the direction of strengthening the results obtained within the leading logarithmic approximation. It is interesting to remark that this second result does not automatically follow from the first one. In fact, in this case the corrections of order ot in the matrix elements are to be added to the corrections to the effective hamiltonian. The assumption is that for a quark as heavy as the c quark of heavier we can compute the relevant matrix elements in perturbation theory, according to [he parton model. This is evidently easier to accept for a c quark than for a charmed hadron, so that we shall take up the question of the relevance of the present calculation to the real world momentarily. Thus for a c quark the corrections of order o~(mc) are such as to increase (in addition to the leading effect) the non-leptonic width and to decrease the semileptonic width. The net result of these Q C D effects is a pronounced decrease of the semileptonic branching ratio of the decaying quark as seen in fig. 5. Until recently the results for a c quark were directly taken as relevant for charmed hadrons, the light constituent quarks being taken as inert spectators. After the experimental finding of a quite different lifetime for D O and D ÷, one is now led to a picture of charmed particle decays based not only on c quark decay, but also on the so-called annihilation diagrams with real gluon emission. However, the c quark decay prediction should remain essentially valid for D ÷, provided the spectator is really inert, because in D ÷ decays the annihilation process can only occur at the Cabibbo suppressed level. Since a value of D ÷ close to 20%' is being currently reported it was important to have a check of the validity of the Q C D prediction for a free c quark. If the high value of the semileptonic I~ranching ratio is confirmed then we can now conclude that this finding is to be attributed to non-partonic effects connected with the spectator. The best candidate for such an affect could be the destructive interference of the two a antiquarks present in the final state of D ÷ decay, i.e. the spectator and the c quark decay product as studied in ref. [10]. This effect tends to suppress the non-leptonic D ÷ width by an amount which

512 increases with

G. Altarelli et aL / Corrections to weak decays

I¢,o(0)12 [¢,D(0) b e i n g

t h e m e s o n w a v e f u n c t i o n at z e r o ] , i.e. w i t h t h e

i m p o r t a n c e o f t h e a n n i h i l a t i o n d i a g r a m s (in a q u a l i t a t i v e way). O f c o u r s e , t h e s a m e f o r m u l a e also h o l d f o r b q u a r k d e c a y s a n d an e x p e r i m e n t a l s t u d y of b o t t o m p a r t i c l e d e c a y s w o u l d h e l p c o n s i d e r a b l y in s e p a r a t i n g p e r t u r b a t i v e

effects f r o m w a v e -

f u n c t i o n effects. W e w o u l d l i k e to a c k n o w l e d g e a n u m b e r of v a l u a b l e d i s c u s s i o n s w e h a d w i t h N. C a b i b b o , L. M a i a n i a n d G . Parisi. O n e of us ( G . C . ) w i s h e s to t h a n k C E R N f o r t h e w a r m h o s p i t a l i t y a f f o r d e d h i m w h e n p a r t of this w o r k was b e i n g c a r r i e d o u t .

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