Non-leading mass effects in the effective weak Hamiltonian for Ac = 1 decays

GERRY MCKEON De/~~-'nrtt~letlt of Applied Motlzemotic.~, Utzive,:sity oJ We.stert~ Otltclrio, Lotdon, Otzt., Catzerdcl NhA 5B9

Rcceived January 18. 1982

Reccntly Altarelli, Curci, Martinelli, and Pctrarca have calculated thc second ordcr corrections induced by quantum chromodynamics to the effective weak Hamiltonian in A c = I processes. Wc cxtcnd thcir analysis by including the effects of the heavy t and b quarks. Thc technique used is that first introduced by Witten to study thc effccts of heavy quarks in deep inelastic scattering and used by Gilman and Wise and Gubcrina and Peccci to discuss how quark masses altcr thc effective weak Hamiltonian to leading order in lAsl = I proccsscs.

Altarelli, Curci, Martinelli et Petrarca ont rCccmmcnt calculC Ics corrections dc second ordrc apportees par la chromo- dynamique quantiquc h I'hamiltonien effectif d'interaction faiblc dans les proccssus Ac. = 1. Nous genkralisons lcur analysc cn incluant Ics effets dcs quarks t ct b lourds. La technique utilisee est ccllc que Wittcn a introduitc d'abord pour Ctudicr les cffcts dcs quarks lourds dans la diffusion inklastiquc 6 hautc Cnergie et que Gilman ct Wisc, ainsi quc Cube lha et Pcccei, ont utilisCe pour discuter la faqon dont les masscs des quarks altkrcnt I'liarniltonien effcctif d'intcraction faiblc, i I'ordrc dominant, dans Ics proccssus IAsl = I .

[Traduit par Ic journal] Can. J . Phys.. 60. 1163 (1982)

1. Introduction As was anticipated by Wilson ( I ) , the strong inter-

actions considerably alter the effective Hamiltonian of the weak interactions. Leading logarithm corrections with massless quarks were originally calculated by Al- tarelli and Maiani and Gaiilard and Lee (2). The leading logarithm effects of so-called "penguin" diagrams have been discussed by Shifman et 01. (3).

The effect of having quarks of non-negligible mass is perhaps most easily analyzed using a technique intro- duced by Witten (4) in the context of deep inelastic scattering and exploited by him in discussing the lAs( = 1 effective weak Hamiltonian in the four-flavour quark model (5). More recently this method has been used by Gilman and Wise (6) and Guberina and Peccei (7) in the six-flavour quark model.

Recently Altarelli et 01. (8) have extended the leading logarithm results of ref. 2 to next-to-leading order for the At. = 1 effective weak Hamiltonian. Quark masses have been neglected in this analysis. and the presence of the heavy b and t quarks is accounted for only by setting the number of flavours, j', equal to an "effective" value somewhere between 5 and 6.

The results of ref. 8 are extended in this article to include the effect of heavy quark masses. It is most convenient to use the method introduced by Witten (4). The basic idea behind this technique is to exploit the Appelquist Carrazone (9) theorem that states that to order l/M', the n-point Green's function satisfies the condition

A prime on a quantity refers to it being evaluated without the presence of the heavy quark of mass M. The mass independent renormalization scheme of 'tHooft and Weinberg (10) is used in these calculations.

These considerations are developed in the next sec- tion. The notation and conventions used are those of ref. 8.

2. Non-leading mass effects in the effective weak Hamiltonian

Under the influence of a weak charged current J p ( s ) , the transition matrix element to lowest order in the weak coupling g,, is given by

[2] H,, =

g;,, S dJ,v ~ , . (x ' , M ~ ~ ) ( F I T [ J , ( ~ ) J ' ~ ( O ) I ~ I ) Here D,,,(s2, M:,) denotes the W-boson propagator. As the boson mass M,,, approaches infinity, we can employ a Wilson expansion to re-express the time ordered prod- uct that appears in [2] as a sum of local operators. Operators of dimension four can be absorbed by renor- rnalization effects, so the leading contribution comes from operators of dimension six ( 1 1 ) . As a result, if we were to limit our consideration to charm decay pro- cesses, [2] reduces to

[ I ] r , ,(g, 1% M, P, 1);) = Zi'(g. M, t m , ( g l ( g , hI, where

0008-4204182108 1 163-05801 0010 01982 Nat~onal Rcscarch Council of CanaddConscll nat~onal dc rcchcrchcs du Canada

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1164 CAN. 1. PHYS. VOL. GO. 1982

t ~ . is the scale at which the strong coupling constant The solution to [4] is renormalized and

0, = 4 [(Ftc),(Gd'), ? ( i ~ ) ~ ( . ? ' d '),I [5] C,(r, a ) = C,(O, a(r)) exp

is given in terms of the "running coupling constant" a ( t ) defined by

E(a.1) dx Computing the c-number coefficients in the Wilson [6] In r = ja -

expansion can be done by solving the renormalization P(x)

group equation

a [41 {3 + P(a) - + ?.(a) d a A second order expansion of various terms in [5] where

?%(a) = a + ?(l"aZ + . . . , da

P(a) = p-- = -b a'(1 + ca + cZa2 + . . .) C,(a) = 1 + ~ y ' a + . . . d ,*.?

allows us to make the approximation

In refs. 2, 8, and 12 all of the parameters in [7] are calculated. The matrix element 0"a) in [7] is dependent on the masses of the heavy b and t quarks. Equation [ l ] will now

be systematically applied in order to determine the effect of those quark masses to non-leading order. In the following, an unprimed quantity will be evaluated in the six-quark model; for each prime one less quark

flavour will be considered. Equation [4] and the fact that H,, is independent of leads us to the equation

a a a [8] (2 + P(a) a + &(a) (mi 7 + ttz/ _) - ?*(a)) 07 (a , ,', mi, mi) = 0

am, a 171; where

is the anomalous mass dimension (10). By [l] we have, up to order l/mf

[9] 07(a, ,', mi, m') = Bi(a, m f / p 2 ) ~ ~ ' ( a f ( a , ml/p2), ,*.', mb2(a, 1712, mi, ,?))

(As we are restricting ourselves to Cabibbo allowed Ac = 1 decays, the problem encountered in refs. 6 and 7 of operator mixing partially due to the presence of penguin operators does not occur). The operator 01;" satisfies an equation similar to [8]

a a [lo] (2 + p l ( a l ) 7 + sl(al)m;'- - q ~ a ~ ) o7 ' (a1 , ,', tnb') = o da am;'

If we now substitute [9] into [8], we should arrive at an equation for 0;' that is equivalent to [lo]. This is true only if the following identifications are made

and

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MCKEON

If the renormalization of the mass m,?, that occurs in [9] is written

[I41 mb' = m f f ( a , m,'lP2)

then [13] reduces to

Power series expansions of the form ( K = In (nz:lF2)),

Bi(a , K ) = & ( K ) + Z I ( ~ ) a + . . .

can be inserted into [ l 1 1 , [12], and [I51 respectively. Upon equating corresponding powers of a in each of these equations, and using the appropriate boundary conditions, we can show that

- ( I " - - ( I ) + - " I 1 [16] B; = 1 + (5;"' - y j " ) ~ a + [(A(yj1) ' - yo') + (yj'" - y l " ) ) ~ + f ( -b(y; y ) yi (b' - b)

+ (? / ) I ) ' - y;l))')K*]a' + , , .

[I71 a ' = a + (b' - b ) ~ a ' + [(A(bf - b ) + (b'c' - b c ) ) ~ + (b' - b)'~']a' + . . . and

[I81 f = 1 + ( A - A ' ) K ~ + [((A? - A;) + A(A - A' ) )K + f ( (A - A')' + (-bA - blA' + 2bA1))~ ']a ' + . . . We can also solve [ l I ] by use of the auxiliary function & ( a , rn ' lF2) defined by

It is easily verified that [ I 1 1 is satisfied by

The same procedure that was used to discuss the effect of the t quark can be repeated to analyze the effect of the b quark. In analogy with [9] we have to order llm,',?

[21] ~ ~ " ( a ' , p.', mb') = B,!(al, mb2/p') O ~ ( a " ( a r , mb'lP2), ,,.')

The matrix elements 0I;"l obey the following renormalization group equation

Upon substituting [21] into [ l o ] , we arrive at an equation that should be equivalent to [22]. Consequently, we get the following conditions that are analogous to [ l I ] and [12]:

From these conditions we find that (K ' = In (m6'lF2)).

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1166 CAN. J . PHYS. VOL. 60. 1982

[26] a " = a' + ( u - bf)K'a'' + [(A1(b" - b') + (b"cU - b ' c ' ) ) ~ ' + (b" - b ' ) ' ~ " ] a ' ~ + . . . in analogy with [16] and [17]. It is also possible to solve [23], showing that

where

It is now possible to piece together the various contributions to 0;. Combining [9], [21], [20], and [27] we find that

An expansion to second order of the anonlalous dimensions and beta functions that appear in [29] leads us to the equation

In [30], the couplings a' and a" are expressed in terms of a , in,, and in; by [I71 and [26] respectively while [18] relates in; to m,,, a , and m,. The "running coupling constants" & and 6 are defined by [19] and [28]. To second order, we find that these equations give us

and

We consequently have, in principle, all of the ingre- dients necessary to calculate the terms on the right hand side of [30], save, of course, 0;'". This can now be substituted into [7] to give our final expression for the transition matrix element HI:,.

3. Discussion It is worth examining the procedure that must be

followed if [7] and [30] are to be used to provide defi- nite predictions for charmed quark decay.

First of all, the masses M,,, m,,, and m, are not known precisely. It is common practice to set M , = 85 GeV, in,, = 4.5 GeV, and m, 2 15 GeV.

Secondly, the coupling a must be fixed. This is in- timately connected with the problems involved with the QCD scaling variabIe A (13). This constant of integra- tion to the equation

must first of all be appropriately defined. For example, in ref. 14, A is selected so that an expansion of 6 does not contain a term I / In' (p'/A2). In ref. 15 A is chosen so that if one were restricted to second order, then Fi would equal infinity when CL = A. In other words, if P were evaluated only to second order then, using the definition of A given in ref. 15,

A comparison of [9] and [33] shows that

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MCKEON 1167

The value for A can be extracted from the data for deeu inelastic lepton-nucleon scattering. Equation [35] can then be used to determine the value of a.

There are still two other parameters whose values cannot be theoreticallv determined. First of all. the renormalization parameter p,I is not fixed. Since we are dealing with, charmed quark decay, we are inclined to set p,' = t f ; . However, this is not a fully justifiable choice; im; 1s in principle just as acceptable. The sec- ond parameter whose value is not fixed is the AI that appears in [18], [31], and [32]. This is a renor- malization scheme dependent parameter (16). Its im- plicit presence in [30] implies that an ambiguity resides in our final expression for H,,.

In principle, the second order coefficients yj2' and cj" are aiso renormalization scheme d e ~ e n d e n t . ow ever. as was po~nted out in ref. 17, the particular combination that appears in [7] is independent of the renormalization of the operators 0,. There is a dependency though on the definition of the coupling constant a. Additional dependence on also appears in [30].

Perturbation theory alone provides no criterion for choosing certain prefered values for ?:I1, AI, and p,? Stevenson (15) has suggested a principle that allows one to select an "optimal" value of these renor- malization scheme dependent parameters. His idea is that any physical quantity calculated in perturbation theory should be stable under small variations of the renormalization scheme dependent parameters. The "optimal" values of these parameters are thus those values under which the qiantity being calculated is unaltered by small changes. The justification for this procedure is that since the exact value of a physical quantity should be independent of the renormalization scheme used to calculate it, then its value as calculated in perturbation theory should share this property.

An extension of Stevenson's ideas to the mass- independent renormalization scheme (10) has been dis- cussed in refs. 16 and 18. It would be of interest to apply the results obtained there to the problem of opti-

mizing the values of yj", A2, and p,' used in [30] for charmed quark decay processes.

An evaluation ofAI has been made in refs. 18 and 19. The renormalization scheme used in ref. 18 is that of ref. 8 .

W e thus see that it is possible to determine the effec- tive Hamiltonian to second order in the strong coupling constant a and to leading order in l/m; and l/m;.

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