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Qcd enhancement of b → sγ decay for a heavy top quark
Nuclear Physics North-Holland
B365 (1991)
NUCLEAR PHYSICS
279-311
QCD ENHANCEMENT
OF b + sy DECAY Peter CHO
Lyman
Laboratory
of Physics,
and Benjamin Harvard
Received (Revised
B
FOR A HEAVY TOP QUARK
GRINSTEIN
University,
Cambridge,
MA 02138,
USA
13 May 1991 19 June 1991)
An effective five-quark hamiltonian for .b + s decay processes is derived by integrating out a heavy top quark with rn: > rnk from the minimal six-quark standard model. The leading strong interaction corrections to this hamiltonian are then determined in an effective field theory computation. The renormalization group running of the operators in the effective hamiltonian between a top quark scale of 250 GeV and the W-scale is found to be significant and enhances the inclusive rate for the rare weak decay B + X,y by about 14%.
1. Introduction
Determining the mass of the top quark remains one of ‘the most important outstanding challenges in high-energy physics. Despite substantial experimental effort, the top quark has continued to elude detection and is only currently known to weigh more than 89 GeV [l]. Complementing the direct experimental searches, there have been numerous theoretical attempts to indirectly measure the top mass rn, via its radiative corrections to physical observables such as the p-parameter. Such theoretical estimates typically yield central values for m, ranging from 100 to 200 GeV with large error bars [2]. Recently however, some of these radiative correction analyses have been called into question, and very heavy top masses up to 250 GeV have been proposed [3,4]. If one accepts the proposition that the top quark could indeed be quite massive, then it is necessary to reconsider some of the earlier theoretical estimates of its indirect effect upon various physical processes. We are interested in particular in re-examining previous effective field theory computations of rare weak decays and investigating the impact of a very heavy top quark on the results from such computations. The basic effective field theory idea is by now quite well established and has been frequently reported upon in the literature [5-91. Starting from some underlying full theory, one integrates out the heavy degrees of freedom and then runs the resulting effective field theory down to the appropriate scale for studying some process of interest using the renormalization group. If any additional heavy particle thresholds are crossed during the renormalization group running, then 0550.3213/91/$03.50
0 1991 - Elsevier
Science
Publishers
B.V. All rights
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280
P. Cho, B. Grinstein
/ b + sy decay
those particles should also be integrated out of the theory. Thus a succession of effective field theories is constructed which describes physics from small to large distance scales. In the particular case of the minimal six-quark standard model, one applies these ideas by first integrating out the top quark thereby generating an effective five-quark theory. This is subsequently run down to the W-scale at which point the weak gauge bosons are removed. In the past, the top quark and the W have usually been simultaneously integrated out from the full theory as a simplifying measure. This approximation is tantamount to neglecting the running of the strong interaction fine structure constant (Y, between the top and W scales and therefore necessarily incurs an error which grows with m,. We are interested in studying the dependence of this error on m, and determining its magnitude for especially heavy top mass. We will focus upon the strong interaction corrections to the decay of B-mesons to a strange hadronic state X, and a hard photon y. Strong corrections to this particular rare weak decay have previously been analyzed with the effective field theory formalism in the above-mentioned simplifying approximation [lo-131. Our results therefore represent a refinement of earlier work on this particular weak process. However, much of the discussion in this paper has much broader application to any effective theory computation. We therefore hope that it will prove useful to both novices and experts who wish to tackle other problems using the effective field theory approach.
2. Matching
Following the arguments presented in ref. [ill, we model the weak radiative meson decay B + X,-y by the quark decay process b + s-y. The validity of this approximation rests upon the belief that the dominant contribution to B-decay comes from an effective magnetic moment interaction generated at distances short compared to a typical hadronic scale A. Corrections to the b-quark decay picture are presumably suppressed by powers of the small ratio A/m,. We will thus analyze in detail the weak decay b + sy assuming the underlying fundamental theory describing this process is the minimal standard model. To begin, we summarize our standard model nomenclature conventions in table 1. We also adopt the mass-independent renormalization scheme of dimensional regularization plus modified minimal subtraction @IS). Thus we work in d = 4 - E dimensions and let ,u denote the renormalization scale. The covariant derivative associated with the unbroken color and electromagnetic subgroup H of the full standard model gauge group G therefore looks like D, = dp - ip’12
g,G,“X”
- ipE12 eApQ .
P. C/IO, B. Grinstein TABLE
Gaugegroup Generator Coupling Gauge field Field strength
G= SU(31, XR
x
($ G$
SU(21,
281
/ b + sy decay
x
1
U(l),
+
H = W(3),
x
U(l),, Q
T’
s
X”
62. T w:”
gl %
g
A’
B PV
G/f”
FPcu
The electric charge generator appearing in the covariant derivative’s photon term assumes the values Qt, = - $ and Q, = $ when acting on bottom and top quark fields, respectively. Following the effective field theory formalism, we integrate the top quark out of the full six-quark standard model and derive the effective hamiltonian A?& for b + s decays in the resulting five-quark theory. In effective theory computations, one evaluates Green functions in both the full and effective theories at a heavy mass scale m,, and requires that they agree to some specified order in l/m,. This process of matching Green functions fixes the coefficients of operators appearing in the hamiltonian Z&r at the heavy mass scale. Zerr generally contains all operators which are not prohibited by symmetry considerations. Such an effective hamiltonian is consequently non-renormalizable since it includes terms of arbitrarily large mass dimension. However, the coefficients multiplying operators of high dimension are necessarily suppressed by inverse powers of the heavy mass m,, so that the overall dimension of combinations of coefficients and operators equals d. In our calculation, we match off-shell 1PI Green functions first at the scale m,, = m,. We work in a background field version [14] of R, gauge in order to maintain explicit gauge invariance
the lagrangian
of our Green functions.
The gauge-fixing
part of
is
L& = - & c a
IPQz
+ g, f abc~~Q~c + 2isg,(
+t>Ta4 1’.
(2-l)
Here Qf: Epresents the quantum gauge fields for the standard model gauge group G while Q: stands for the classical background gauge fields for the unbroken subgroup H. For computational convenience, we will take 5 = 1. The gauge fixed by (2.1) then corresponds to the background field generalization of ‘t HooftFeynman gauge. While this background field gauge may not be quite so familiar as its conventional R, counterpart, the explicit gauge invariance of Green functions calculated in this gauge provides valuable checks on both matching condition and anomalous dimension computations. It is therefore the gauge of choice for lengthy off-shell calculations. We display in fig. 1 the few Feynman rules in 5 = 1 background field gauge that differ from the ones in ‘t Hooft-Feynman gauge which are relevant in our analysis.
P. Cho, B. Gtinstein
282
/ b +sy
decay
= ~"2gsfaac[2k,Yg“'X+ (k2 - k3)Xg”Y - 2k;g”x] hp
C,Y
=
-j~E/2e[2k;gpX
+ (k2 - k3)xgPy _ 2kfg”x]
Fig. 1. Relevant Feynman rules in ( = 1 background field gauge that differ from their counterparts in conventional ‘t Hooft-Feynman gauge. The closed circles at the ends of the external gluon and photon lines denote background gauge fields.
As illustrated in the figure, background field gauge yields a bonus simplification in our particular problem. For when we expand -9&r, the following trilinear interaction between the background photon field, W-boson, and would-be Goldstone boson L?--&= . . . -emwxpW$4-+
h.c.
cancels against the same term which arises in the Higgs kinetic energy. This cancellation cuts almost in half the number of diagrams we need to consider. The interaction terms in the full six-quark theory which contribute to the weak decay b + sy appear in the charged current sectors of the standard model
P. Cho, B. Grinstein
/ b -+ sy decay
283
lagrangian
where the left- and right-handed
quark fieIds are defined as
In this interaction lagrangian, V represents the 3 X 3 unitary Kobayashi-Maskawa matrix while Mu and Mu denote the diagonalized quark mass matrices,
Notice that the left-handed interaction terms involving the top quark and the would-be Goldstone bosons in (2.2) are enhanced by a factor of m,/m,,, relative to those involving the top and W, gauge bosons. Since we will only match Green functions to leading order in the expansion parameter 6 = m&/m:, we can thus ignore the charged current couplings to the W’s. In the effective five-quark theory, a complete basis for the local operators which contribute to weak radiative B-meson decay to leading order in 6 is listed below: Dimension
d + I: O[,
= -m,s,D*b,,
QLR= ,w&,4+4-~LbR.
(2.3)
284
Dimension
P. Cho, B. Grinstein
/ b + sy decay
d + 2: p1.A
L
=
-iSLT,A,,DwDYDubL,
Pz = p’/2eQ,SLypbL
a”Fp,,
p; = $12 eQhF’,fLYD”bL, Pz = i~“‘/2eQ,~,,SLy’“y5D”bL, pghost L
=
E.LEg,Zf,,,SLy,X”bL(a’“776)77C,
Rt = i$gf(
D”4+)4-SLymbL,
The tensor TFtIT appearing in Pk A assumes the following Lorentz structures as the index A ranges from 1 to 4:
A few points about these operators should be noted. Firstly, we have only displayed non-renormalizable operators in this list, for the effects from any renormalizable operators may simply be absorbed into various field and coupling constant redefinitions. Secondly, following the clever trick of Gilman and Wise [El, we have incorporated seemingly arbitrary factors of gf into the definitions of operators with would-be Goldstone boson fields. As we will see later, these factors facilitate combination of various parts of the leading-order anomalous dimension matrix for the basis operators. Finally, we have included the operator pghost = ~Egffobc~Ly~XabL(alL?7b)77= L with ghost fields 77 and Yj in our basis set for completeness. As pointed out in ref. [13], this ghost operator mixes at one-loop order with some of the other dimension(d + 2) operators containing gluon fields. However, we will see that no operator
P. Cho, B. Grinstein
285
/ b + s-y decay
mixes back into P[host. Thus this ghost operator turns out to have no effect upon the running between the top and W scales and could be neglected without loss. We now determine the coefficients of the operators appearing in the effective hamiltonian to leading order in 6 = m&/m: by matching Green functions in the full and effective field theories at the top scale. The coefficients of the operators which explicitly contain would-be Goldstone boson fields are the easiest to compute. As illustrated in fig. 2, only tree diagrams contribute to the rbs6++-, rbsgd+$- and rbsy4++- 1PI Green functions at leading order. Expanding the
Full
Intermediate
Theory
t b 9 jb:
EFT
+
.. .
+
...
+
...
s
Fig. 2. Leading-order matching conditions at the top quark scale for the IPI Green functions rbs8+‘-, +?l++and rbsud+&- m the full six-quark theory and in the intermediate five-quark effective field theory.
256
I? C/IO, B. Grinstein
/ b + sy decay
propagators of the virtual top quarks as
one easily finds that the full theory Green functions match onto effective theory Green functions with one insertion of the operator
= -Q,,+R:+R2, with coefficient
In decomposing 8 over the basis set (2.3), we have neglected terms proportional to the strange quark mass since m, -3xmb. Tree-level matching of Green functions is however generally insufficient to completely specify the matching conditions between operators in full and effective theories. One usually also has to include operators in the effective theory that arise as counterterms when external lines on Green functions are closed off into loops. Such new operators are generated in our particular problem when we close off the external Goldstone boson lines in fig. 2 into loops as shown in fig. 3. All of these one-loop graphs may be evaluated straightforwardly with the aid of the integrat formulas tabulated in appendix A. At this point, it is worthwhile noting some simple but important general relations between full and effective theory Green functions. In a full theory, the dependence of Green functions on heavy and light particles is intertwined and entangled. In an effective field theory on the other hand, all dependence on heavy particles is explicitly extracted out of Green functions and moved into coefficient functions. As previously mentioned, the coefficients are suppressed by inverse powers of heavy masses which are determined by naive dimensional analysis. Factoring out this trivial heavy mass dependence from the coefficient functions, one finds the foIlowing general relationship between Green functions in ful1 and effective theories:
where m,, and mr stand for heavy and light particle masses.
P. Cho, B. Grinstein
/ b -) s y decoy
htennediate
iW1 Theory
Fig. 3. One-loop
matching
287
conditions
at lz. = m, for the 1PI Green
functions
EFT
rhr,
rhsg and rhsy.
It is instructive to compare order by order in l/m, the logarithmic dependence of G,,, with that of the effective field theory terms in the infinite power series in (2.4), cfull
log(k/nz,)
+ciutl
lw(mdmd
=c,,ff
log(~/mh)
+ceffWmdPL).
(2.5)
Since only two independent dimensionless ratios can be formed from the arguments of G,,,, its logarithmic dependence upon the mass parameters /.L, mt, m,, can generally be expressed as the 1.h.s. of eq. (2.5). On the r.h.s. of this equation, the log@/m,) can only arise from the dimensionless coefficient function C(p)(~/m,> while the remaining log(m,/p) must come from Gig). Equating the coefficients of log mt and log m,, in eq. (2.5), one trivially concludes 4”ll = C,ff >
Cfull + &,I
= Ccoeff .
(2.6a, b)
These relations however convey nontrivial information. Eq. (2.6a) indicates that all infrared logarithms in Green functions computed in the full theory are reproduced in the effective theory. This must be the case since both the full and effective theories describe the same physics at low energies. Moreover, at the heavy mass scale, the leading logs in Gr,,, are identical to those in G,a. Since matching conditions are determined by taking differences between Green functions in the full and effective theories, there can consequently be no logarithms appearing in
288
P. C/IO, B. Grinsrein
/ b -t sy decay
matching conditions at p = m,,. Eq. (2.6b) indicates on the other hand that at the light mass scale, the full theory logarithms are reproduced in the effective field theory by the running of the coefficient function C’p’(~/m,). Therefore, all of the logarithm information in the full theory is recoverable in the effective theory. Returning to the problem at hand, we find the following leading-order values for the coefficients of the basis operators appearing in the effective hamikonian
(2.7) from the matching conditions at p = m,, c,
et
l/2
=--
167r* ’
OLR
3
=
l/2
=--
167~’ ’
OLR
11/18 cpp = cp,, = 167~’ ’
c P1.2
=
c
=
-
8/g -16T*
7
l/2 pp2
-
167~~ ’ 3/4 167r2Q, ’
c,: = c,:=o,
l/2 c,:=
-
16dQ,
’
(2.8)
t? Cho, B. Grinstein
/ b + sy decay
289
In accord with the preceding general discussion, we see that no logarithms appear in these coefficients at the top quark scale. Had we matched however at a different renormalization scale, we would have found a log in the Pt coefficient,
In sect. 3, we will verify that this logarithm which vanishes when p = m, is regenerated at lower scales by the renormalization group. 3. Running Starting from the fact that the effective hamiltonian (2.7) is independent of the renormalization scale II, one can readily derive the renormalization group equation satisfied by the coefficient functions C;(p), P$c;CP)
=
C
(3.1)
(YT)ijCj(P).
i
The anomalous dimension
matrix
yij = z, ‘p dZkj dp
appearing in eq. (3.1) is formally defined in terms of the divergent renormalization constants Zij which relate renormalized and bare composite operators, @J= I = Zij( /A)&$( p) . However, y is most conveniently calculated in practice by requiring renormalization group equations for Green functions with insertions of composite operators to be satisfied order by order in perturbation theory. Let r$) denote a renormalized n-point 1PI Green function with one insertion of the operator Hi. Then the anomalous dimension yij characterizing the mixing of Hi into Hj is simply determined from the renormalization group equation for r$‘), yijTgf)=
I
-
pa i
ap
a ag
+0-+-y
a mm --ny am
exf ry., 1
Here p = p.(d/dp)g, ‘y,, = &/m)(d/dp)m and nyext stands for the wave-function anomalous dimensions arising from radiative corrections to the Green function’s n external lines.
P. Cho,
290
El. Grinsteirl
/
b * sy decay
We first solve for the anomalous dimensions of the basis operators (2.3) to lowest order in the electroweak interactions and ignoring strong interactions for the moment. In this case, we can forget about the p, ‘y,,, and jz’yeX, terms in eq. (3.2) since they introduce additional powers of the electroweak coupling constants beyond those already present in the first p-derivative term. All the operators’ anomalous dimensions can be extracted from the single IPI Green function rbsy. Loop diagrams with insertions of Q and R Goldstone boson operators which contribute to rbsy are generally divergent and require 0 and P operators as counterterms. This consequently leads to mixing of the former operators with the latter. The converse mixing however does not occur. After evaluating the loop diagrams, we find the following mixing of the basis operators with would-be Goldstone fields into those without:
o,, P:'A QLR/oO y= R; R'L R”L
P; 0
\o
0
P;f
0
0 l/6&,
0 0 00
P;
- 1/6Q,,
01
00 0 0 0
! 53 “2
S'
(3.3)
0
Inserting the transpose of this anomalous dimension matrix back into the renormalization group equation (3.1) for the coefficient functions and observing that the coefficients on the r.h.s. do not run, one can trivially solve for the coefficient values at scales below the top. Only one is found to differ from its p = /n, value,
4GF CP&EL)
= G&m,)
+
-
fi
v,bv,:
1
2
log 4 . 96T2Qb 112,
As advertised, the log /12/m; term present in eq. (2.9) which originates from the full theory is recovered in the effective theory by the running of the coefficient function. We now reconsider the running of the basis operator coefficients taking into account strong interaction effects. We first need to know the quark, gluon and ghost wave-function anomalous dimensions yquark, ygluon and yshosl to leading order in the strong interactions. We will also need the quark mass anomalous dimension y,,, and QCD /?-function. In background field ‘t Hooft-Feynman gauge, these may all be simply computed from renormalization group equations for two-point Green
P. Cho, B. Grinstein
/ b + sy decay
291
functions [ 141, Yquark
=
2
3
g323
8,rr2
Ygluon
P
=
-
3
Yghost
g3
ym = -4g:
=
--- 3 g32 4 8rr2 ’
P=b$.
a,rr2 ’
The coefficient appearing in the p-function has the value b = - 11/2 + n,/3, where nF denotes the number of active quark flavors [16]. Integrating p, one obtains the well-known expression for the strong interaction fine structure constant in leading log approximation,
d(P) %(cL) = -= 47r
127r (33 - 2n,)log($/A&)
*
Since cu&) is continuous across quark mass thresholds, Am varies with nr. In our five-quark effective theory, the QCD scale is related to the one appropriate for three light quark flavors as
When the strong interactions are included, the solution to the coefficients’ renormalization group equation (3.1) is not quite so trivial as in the previous case where QCD was ignored. Since we are working in a mass-independent renormalization scheme, we can first eliminate the derivative with respect to the renormalization scale in favor of a derivative with respect to the coupling constant which yields
ps=
C(yT)ijCj.
3
i
The solution to this first-order differential notation as C(EL~)
equation then appears in obvious matrix
= exp[Jrf)dg -rT(d * P(s)I C(Pl)
Since we will only compute the anomalous
dimension
(3.4)
matrix y to leading order,
292
P, C/IO, B. Grinstein
/ b + sy decay
we decompose it as 2
Y=- 8227
+ o(d)
=g,,s-l +o(g$ where the eigenvalues hi of 9 are contained in the diagonal matrix A while the corresponding eigenvectors are arranged into the columns of matrix S. Our earlier adoption of the Gilman-Wise trick of incorporating strong coupling constants into some of our operator definitions results in the matrix 9 being scale independent and purely numerical, as will shortly become clear. The solution (3.4) to the coefficients renormalization group equation therefore becomes
C(p2) = (S-l)Tdiag
(3.5)
([~l*‘/2h)STC(pl).
We now calculate the anomalous dimensions for each of the operators in our basis set to O(gi). It is in this lengthy calculation that the tremendous utility of explicit gauge invariance in background field gauge becomes fully apparent. As a representative example, we describe the results for the QCD-induced mixing of the operator Pi ’ = -iS,D2@b,. To begin, we list the first few terms in the Feynman rule for this particular operator which is illustrated in fig. 4,
Ptl
Feynman rule = I’$ S,p2#b,
+ ip3’/2 9,( g,X”
+
eQb)
~(p’~y,+p,d+p;~)b,+....
p
(3.6)
p’
Here denotes the b-quark’s four-momentum entering into PL’ while is the s-quark’s momentum flowing out of the operator. The first term in eq. (3.6) corresponds to the piece of Pk’ with zero external gauge bosons emanating from the operator and is common to Pk’ and Pk” as well. Therefore, the identity of this operator is only uniquely established starting from the single gauge field terms. Inserting Pk ’ into the two-point function rbs, one may partially determine how it runs into itself and other basis operators under the action of QCD at one-loop order. The diagrams involved are displayed in fig. 5, and the divergent terms in
P. Cho, B. Grinstein
PjB1 Feynman
=
rule
293
/ b --) sy decay
!!+ b
+;y+
+pT’
Fig. 4. First few terms in the Feynman rule for the operator Pt’
+
= -iSLD2$bb,.
their sum are given by 4
c (0 external gauge-boson graph) j j=l
+&(A x
+logp2)
{ - (4m; + ;mbm:)SLb,
- ($msm; + 8mf)i,b,
- ($rnt + ~m~)S,@b, - $mbm,i,gbb, -6m,S,p’b,
- 2m,S,p2b, pl,A=1.2,3 L
+ f%p2$bL},
(3.7) where 2 A=;-y+log4r.
In the column on the right-hand side, we have listed the dimension-@ + 1) and -(d + 2) operators whose Feynman rules match particular terms in the sum. The information provided in (3.7) is however insufficient to completely specify the full mixing of P,‘vl. We therefore must consider Green functions with at least one external gauge boson. In fig. 6, we illustrate all the possible gluon corrections to the rbsg Green function with one insertion of the operator Pt’. The sum of the
Y AA \c+ b
5
Fig. 5. Diagrams with one insertion
+A+2 of the operator function I+.
Pk’
that contribute
to the 1PI Green
294
P. Cho, B. Grirrsfcin
/ b +sy
decay
Fig. 6. Diagrams with one insertion of P, ‘*I that contribute to the 1PI Green function rbsg.
infinite
parts of these one-loop diagrams is
12
c (1 external gluon graph)j j=l
= ip3E/2-l;i2 (A + log p2) x
- ($rni + $mf)S,g3X*yAbL
-6WLg3Xa(
p +p’)*bn - 2m,SRg3Xa(p +p’)*b,
+4m,S,g,X”i( -
%f,g,X”(
p’ -p)@a,,bR
+ 2m,S,g,X”i(
P12Yh+ Ph!! + Phqb,
+:~,~~X’(P.P’Y, +$,g,x”(
- ~m,m,S,g,X”y,b,
+~Ati+p;@‘)b~
p2.yA +~nd’ +ph”)b,
-- YSLg3X”( -iep,,rrAPp~‘p’yyu) bL)
p’ -p)‘u,,,b,
0L.R O& PI.1
L
PI.2 L
PI-3
L
P. C/IO, B. Grinsrein
/ b + sy decay
295
Again we have listed alongside the various terms in this result the operators to which they correspond. The fact that all the terms in eq. (3.8) precisely assemble into pieces of Feynman rules for various gauge-invariant operators constitutes a nontrivial check on the validity of the computation. It also stands in sharp contrast to results from similar non-abelian anomalous dimension computations performed in standard covariant gauges where ugly non-gauge-invariant terms typically arise. We further note that the coefficients multiplying the Feynman rules of operators appearing in both the zero-external gauge-boson and one-external gluon results are identical. The coefficients of the PL', Pk' and Pk' terms which are distinguishable in eq. (3.8) also add up to the coefficient in front of the dimension-cd + 2) term in eq. (3.7) where these operators are indistinguishable,
Thus we clearly see the explicit gauge invariance in background field gauge at work providing important consistency checks on these operator mixing results. After a long but straightfonvard computation of the mixing of all the other basis operators, one finds from the renormalization group equation (3.2) the following QCD-induced entries in the anomalous dimension matrix,
GR OZR 03LR pl.
L
PI.2 L
YE
I
0L.R 7.0 3
ohl 1
L
0”
pz 0
0
0
2 J
4 7
0
0
0
0
0
0
0
0
0
0
16 5
0
0
0
0
0
0
0
0
6
2
32 5
-194
-6
38 3
0
4
3 I
0
34 5
17 4
0
38 T
0
1
4 3
32 5
5 ;i
6
38 -ii-
0
27 T
aY
0
27 T
0
0
0
185 -Tc
209 Tr
6.5 -E
-+
p:
0
p:.
0
pghosl L
pghosl L
pf 0
-8
pk’
PI.4
PI.4
0
0
4 5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
247 ?K-
1
296
Fig. 7. Diagrams which potentially lead to one-loop mixing of P?” into PLM’. Recall that there is a relative minus sign difference between graph (a) and its crossed counterpart (b) due to the ghosts’ obeying Fermi statistics.
In the would-be Goldstone boson operator sector, the mixing is
Y
QLR = R;
I u 83
Rt R’L
0 ,o
0 39 -F
8
0
5
0’8 5;
0
$,
s32 8T2
(3.10) *
ago 0
Anomalous dimensions for operators in the (d + &dimension subsector of (3.9) have been reported before in the literature [17], and our results are consistent with previous findings. One sees in the last column of (3.9) the content of our earlier assertion that no operator mixes into the ghost operator Pp other than itself. The potential mixing of the Pt" operators into PL%os( via the diagrams shown in fig. 7 fails to materialize since their divergent pieces exactly cancel. Therefore, Pp starts and remains with zero coefficient at all renormalization scales and has no effect upon any other operator. We will consequently ignore it from now on. The y values in (3.9) and (3.10) must be combined with those given earlier in (3.3) to obtain the total leading-order anomalous dimension matrix. At this point, the reason for our having included factors of gz into the definitions of the Goldstone boson basis operators becomes evident. These factors were chosen so that all the entries in y would be proportional to gi. Consequently, diagonalizing the anomalous dimension matrix y = (gf/8?rIT)f necessitates finding only the eigenvalues and eigenvectors of the numerical matrix 9. Performing this diagonalL&ion, one verifies that the eigenvalues and eigenvectors are all real as required to maintain hermiticity of the effective harniltonian at all renormalization scales. Inserting the anomalous dimension results back into eq. (3.51, we can determine the running of the basis operator coefficients from the t-quark to W scales. To give some idea of the typical size of the strong interaction corrections when the top is
P. Cho,
B. Grinstein
/
291
b + sy decay
TABLE 2
The leading-order coefficients at or.= m, = 250 GeV and p = mw = 80.6 GeV with the QCD scale taken to be A$ = 300 MeV Operator
;” 0: QLR
PL’ PI.2 L 13 pi
PL” p: p1 2 R; R3L
C(m,)
- 0.0032 - 0.0032 0.0063 - 0.6895 0.0039 - 0.0056 0.0039 0.0032 - 0.0142 0 0.0095 0.6895 0.4754 0
Intermediate
C(mw)/C(m,)
C(m,)
- 0.0034 - 0.0031 0.0054 - 0.6785 0.0033 -0.0044 0.0033 0.0019 -0.0153 0 0.0081 0.5136 0.3666 - 0.0330
1.084 0.976 0.852 0.984 0.851 0.775 0.851 0.614 1.071 0.853 0.745 0.771 -
EE’T
EJ?T below
W scale
H =x +.” b
Fig. 8. Matching conditions at ~1= mw
.s
b
for four-quark operators and twoquark remain below the W-scale.
8
basis operators which
298
P. C/IO, B. Grinstein
/ b -‘sy
decay
quite heavy, we display in table 2 numerical values for the leading-order coefficientsat~=m,=250GeVand~=m, = 80.6 GeV with the QCD scale taken to be A(&= 300 MeV. From the entries in the last column of this table, one sees that the QCD-induced running between the two scales is substantial for several operators. In order to continue to run the basis operator coefficients down to lower scales, one must integrate out the weak gauge bosons and would-be Goldstone bosons at p = m,. There is consequently a new set of matching conditions relating operators in the intermediate five-quark effective theory with those in a five-quark theory without the W, and 4* fields. As illustrated in fig. 8, these matching conditions involve the two-quark operators in eq. (2.3) that originated from diagrams with virtual top quarks and remain below the W-scale. In addition, there are new four-quark operators which arise from graphs containing charm and up quarks. Neglecting small terms proportional to mf or ntt in these new matching conditions, one finds the following relations between coefficient functions just above and below p = m,:
C,:.l(m,)=C,ki(m&)
4/18 + s,
Cpt2(mi)
=C&m&)
- 2,
C,p(
m,)
= C,k3( m&) + $$,
C,p(
m,)
= C,p(
Cpf(m;)
m&) + j-$
=C,&m$)
+
y
l/2
167r2Q, ’
C,:( mG> = C,;( m&) , Cp&m,)
= C,:(m&)
-
3 167?Q,
*
(3.11)
P. C/IO, B. Grinsrein
/
b + sy decay
299
In sect. 4, we will use the results of ref. [ll] to continue running the coefficients of operators in the effective hamiltonian down to the b-quark scale. But in order to match onto those results, we first need to use equations of motion to reduce all the remaining two-quark operators to the gluon and photon magnetic moment operators Otn and Ot, and to four-quark operators. Again neglecting the strange quark mass in comparison to the bottom quark mass, we obtain the on-shell equivalence relations
p’.’L = - +O&
- f(&
pl.2 = -to2 L LR - f O:,
Py
= fo&
+ fo:,
Pf = four-quark
The effective hamiltonian
1
+ four-quark operators,
+ four-quark operators, operators,
then appears just below the W-scale as
+ (four-quark
operators))
As a check, one can momentarily ignore QCD and evaluate and O& coefficients given above with the values specified conditions (2.8) and (3.11). The coefficients arc then found leading terms in the on-shell matching conditions for these
the on-shell Ot, in the matching to agree with the magnetic moment
P. Cho, B. Grinstein / b + sy decay
300
c-
/-
/-
--
0.0008
F z YI u
0.0004
0.0002
0.0000
100
150
300
250
200 m, (dev)
Fig. 9. Dependence of the gluon magnetic moment operator’s coeffkient Co;,(m~) on m, for At&= 300 MeV. The coefficient values with and without the leading-order strong corrections are indicated by the solid and dashed curves, respectively.
operators when the top and W are integrated
C,&(MW) = -
1
167~’
Co&G)
1 = -16rr2
out together [12,181,
; - ;a - $2 (l-q3
-1-$6+;62 (l-q3
:a2 - (l-q41°g6
i ’
;a - 32 -
(l-q4
log6
1 *
(3.12)
Since we have only calculated the strong interaction corrections to leading order in 6, we will exercise our perturbative freedom to add on the subleading terms in (3.12) so that our Ok and 03, coefficients will reduce to those in (3.12) as m, + m,. The QCD-induced deviations away from these uncorrected coefficient values become significant for heavy top masses, as can be seen in the plots of figs. 9 and 10, and are on the order of 18% for m, = 250 GeV. This leads to an enhancement of rare B-decays as demonstrated in sect. 4.
l? Cho, B. Grinstein
0.000
t ’
’ ’ 100
’ ’
’ ’
150
’
/ b + sy decay
’ ’
’ ’
200
’
’
’ ’
’ ’
250
’ ’
’ 1
300
m, (dev) Fig. 10. Dependence of the photon magnetic moment operator’s coefficient Co$m,) on m, for k&=300 MeV. The coefficient values with and without the leading-order strong corrections are indicated by the solid and dashed curves, respectively.
4. B + X,y decay rate
Since we are interested in studying the process B --f X,y, running the photon magnetic moment operator down from scale. In ref. [ll], this running was calculated up to two-loop in terms of the coefficients of our operators O& and Ota,,
we need to continue the W to the bottom order. When written the result looks like
The magnetic moment coefficient Co&+,) then directly enters into the inclusive rate for B-meson decay modelled upon the quark process b + sy,
P. Cho, B. G&stein
302
/ b + sy decay
0.006
0.005
0.004
0.003
0.002
0.001
0.000
200
150
250
300
m, KW Fig. 11. Inclusive decay rate for B MeV. The rate which includes the indicated by the solid cutve. Rates strong interaction corrections at all
+ X;v normalized to the semileptonic decay rate with A’&= 300 leading-order QCD running between the top and bottom scales is including only the running between the W and bottom scales or no are illustrated by the dashed and dot-dashed curves, respectively.
The sensitive dependence of this decay rate upon the b-quark mass and the KM matrix elements can be removed by normalizing it to the rate for semileptonic B-decay. We therefore show in fig. 11 the ratio
r(B + KY) T(B+X,ei;,)
512~~ =
3
aEM f(mc/mtJ
1
Ic,&pb)I’
(4.2)
plotted as a function of m, rather than the absolute prediction (4.1). In this last expression, the function f(m,/m,) represents a phase space suppression factor which equals 0.44 for mb = 4.5 GeV and m, = 1.5 GeV. The electromagnetic fine structure constant also appearing in eq. (4.2) is evaluated at the b-quark scale where it has the numerical value cr,,(m,) = 1.032 aEM = l/132.7. For comparison, we have plotted alongside our result in fig. 11 the decay rate which includes the QCD running between the W- and b-scales but not that between the t and W. We have also displayed the completely uncorrected rate which neglects the strong interactions altogether. The rumring between the top quark and the weak gauge bosons is seen to enhance T(B 4 X,y) by approximately 14% for m, = 250 GeV when compared to the partially corrected rate.
P. Cho, B. Grinsrein
/ b --) sy decqv
303
We believe that our estimates for the enhancement of the rare weak decay B --, X,y are most likely on the conservative side since we have only included the QCD running of the lowest-dimension operators in the effective hamiltonian. The standard procedure for systematically determining higher-order power corrections in the effective field theory program is clear. One must retain operators of higher dimension in the matching conditions and renormalization group running. The effort required to actually carry through with such computations however appears formidable and is most likely unwarranted in our case given current experimental uncertainties. Recently though, there have been attempts to develop more elegant approaches for dealing with the power corrections which arise in all effective theory problems [19]. Perhaps these efforts will soon yield new calculational tools that will broaden the scope of the effective field theory formalism. It is a pleasure to acknowledge helpful discussions with Howard Georgi. B.G. would like to thank the Alfred P. Sloan Foundation for their support of this project. This work was also supported in part by NSF contract PHY-87-14654.
Appendix A INTEGRAL
TABLE
We tabulate here general expressions for several integrals which frequently arise in the evaluation of one-loop Feynman diagrams. Our notation follows that of refs. [20,21]. The closed-form expressions for integrals with one and two internal propagators listed in this table have been transcribed from almost identical expressions appearing in these two references. Significantly more complicated solutions for integrals with three and four propagators may also be found in these papers. While such exact, closed-form expressions are in principle desirable, one is often more interested in practice in the first few terms of series expansions of these integrals. This is the case for instance in effective field theory computations when one matches non-local contributions to Green functions onto local operators. We therefore provide series expansions for all of the integrals listed below in various commonly encountered limits. All of our integrals are in d-dimensional Minkowski space, and the signature of our metric is (+, -, -, -, . . . 1. After dimensional regularization, all divergent pieces in these integrals appear in the combination 4s
1 -y+log4r, E
where E = 4 -d. In the MS renormalization scheme, this term is cancelled by counterterms. Thus 4 may simply be discarded in each of the integrals below.
304
B. Grinstein
P. Cho,
/
b * sy decay
Integral with one propagator: =-
2
im*
A+log$+l
167-r* Integrals with two propagators
=-
(2~)~
1
1 1 ddq qP A +logK
m:
- /IdxlogS(x) 0
2
i 167~~
8,=/fj-
(q*-mt+ia)[(q-P)‘-m$+ie] 2
i 167r*
=-
.
:
ddq
BO=$j-
1
A +lo&
-F(l,x,)
-F(l,x,)
>
4
(27~)~ (q*-mf+k)[(q-p)*-m$+i&]
q/44, (2~)~
(q*-mf+ie)[(q-p)‘-m$+ie] CL2 - 2 -3i*d
ml
xx* logS(x) /
A +logcL2 -F(3,x,) m:
o’dxS(x)logS(x)
I
-F(3,x2) 2
+y
(3mf+3m;-p*)
A+logk+l i
+3(mf-
m~+p*)(V,x,)
- 2p2(F(3,
x,) + F(3, X,))
m2
+F(2,x2))-6m:(F(1,x,)
+F(Lx,))
,
P. Cho, B. Grinstein
/ b * sy decay
305
where
F(l,x)
= -xlog-
X-l
-l=&+$+-$+...,
X
x-l 1 F(2,x)=-x*Iogx-Z-X=~+-$+-$+.... F(3,x)
=
x-l
1
-x310g-
-
1
-
-
3
X
1
-x-x2=
-
2
1
4x
1
+ -
+
5x2
-
6x3
+ . .. ,
are the roots of mfS(x>/p2. x1.2 In the limit where the mass appearing in the first propagator is large compared to the mass and external momentum in the second, the expansions up to quadratic order in the external momentum of these closed-form expressions for the Bintegrals appear as follows: and
Limit
1: m f = rni > rn$ = rns,
p2;
6 =
rns/rni.
+ (1-6)3 2
B=:i P
[ A+logcc+2 167~‘~” 4
$-
l-6
+s
i )I 4
6
log6
+ 0 !4
26-iY2 + (l-6)2
)
log 6
P2 +z
’
P. Cho,
306
B. Grinstein
/
b + sy decay
In a second case where both masses are equal and large compared to the external momentum, the expansions for these integrals look like Limit 2:
mf=m~=m2>p2.
-p2)(A
+logs)
+6m’) 4 +o (
zm2
)I
’
Integrals with three propagators: lx, (2~)~
(q2-m~+ie)[(q-p)2-m~+ie][(q-p’)2-m:+iE]
= -m!--~l~x/ol-x,y
“;;r”, 3
ddq
q”=Pfj-
=-
C
(2i~)~
qLl4” (q2-m~+ia)[(q-p)2-m:+is][(q-p’)2-m~+ie]
i
-/oldXk’-Xdy
16a2
ddq =pu’jP”U (2~)~
X2PFP” -tXY( P,P: +P;P”)
+Y2P;PI
S(X> Y)
9,9”4,
(q2-m:+is)[(q-p)2-m$+is][(q-p’)2-m:+ie] P+P’)o+gyo(P+P’)~+g,,(P+P’)“}
-~dx~l-xd,(
[x3p,p,p,+x2y(p,p,pb+p,plP,+p;P,P,) +Xy2(P;PlP~+P;P,Pb.+P,P:Pb)
+Y”P;P:P:]/s(GY S(x, Y)
+ f[gp”(xP+YP’)~+g”~(Xp+YP’)~+g~~(Xp+YP’)~l~og~
>
P. Cho, B. Grinstein
/ b + sy decay
307
where S( x, y) = x2p2 + 2xyp .p’ + y2pt2 -x(p2+m:-m:)-y(p’*+m;-m~)+m:. As mentioned previously, the closed-form solutions for these C-integrals are quite complicated. So we will not write them down here. Instead, we give the expansions of these integrals up to quadratic order in the external momenta in the limit in which the masses in the second and third internal propagators are equal and are either heavy or light compared to the first propagator’s mass. Limit 1: p2,p~*,p~p’
-
1
6
1-S
+-
4
+-
l 16rr’mt . iip,
; + is - is2
- ; + $3 + $32
P .P’
L- gi
- (l-q’
1 i II 4
is2
log6
+ 0 !4
s2
(1-s)4
- (1-#
-&+++*-+~3
P.P’ 4 r2
& -
(1-S)”
i
gs
log6
563
i
+f-
,
log6
+&s-&s2+&s3
2
+f4
+o($)]
I
is2
(&
4
log 6
+ (l-q4
(l-6)’
i
+-
s
+ (1-6)4
(1-S)”
i
4
(--=-
log 6
+ (1 -8)’
p2 + PI2
6=m;/mi.
-
$2
(l-q4
i(P^PY],
- (1-S)’ + +p
92
-
-
(1-Q
log 6
$3
log 6
I
30s
P. Cho, B. Grinstein / b --) sy decay 1
2
4
/j+,0g-t4-+r-
f-
Ji3
li2
2
l-6
(l-6)
5-226+56’
p2 + pr2
18m2,
l&S2 - 6a3
(l-Q3
-
2-7tS+11ti2
p@pV +-p’Pp’y
+ (1 -q4
2 - 76 + 11s2
18mi
-
2 - 76 + 11a2
36mi
[
log 6
log6
6S3 + (l-6)”
(1 -Q3
ppptv +p”P’
(l-q4 6cS3
(1 -q3 -
log6
(1 -cq3
6ti3 + (l-q4
log6
log6
,
g,,pcr + g”,Pp + &JLP” 12
xiLt+logY+a3 2
f
+-
--
24mi i t2
- (l-S)3
13s3
(l-q4 3-13s+2362-25s3
p-p’
--- 3 P,P”P, 144 rni
(1- q4
log6
48a3 - 126” -
5 - 236 + 49s2 - 19ti3
12mZ, i
+ (l-f+
L&p
(l-6)”
24m2,
12a4
+
7 - 33s + 756’-
P2
P
;a
(1 +j)2
rni
+-
-
(l+$
log6 1
24?i3- 12ti4 -
(l-csy
12s4 + (1 -*)S
3-136+23S’-2563
log6
log6
12a4
(1 -ql
+ (l-Q5
)i
log 6
5 log6 +(p++p'++O !f( )I. i
m”h
P. Cho, B. Grinstein / b +sy decay Limit
2:
p2,pr2,p
co=
*p’ < rn: = rni = rn: << rnf = mf;
6
6
1-S
+ (1-6~~
-
I 16r2mz/
.
[
--
4 +-
6
(1 -q3
;a -
I 16r’rns pp . [i
- (l-q4
f+++p
m’h i
c,=
62
$2
(l-q3
i
P-P?
log6 i log6
+ (l-q4
$62
(l-6)’
6 = m$/mt.
log 6
; - 2s -
p= +p12
309
r2
- (1 -S)’
$))
p2 +pr2
18mt
+
pwp”+p’pp’v
18mg p”p”
+
+p’“p’ 36mz
log8 i
$8 + (l-6)5
logs
I
gj - $2
+ (l-q5
(1 -q4
i
2
-~
)
a2
++!?p+L&
-- P m;
4
p*
;a
P -PI ~+~s-&62+~63 f 1 (1 -a>* 5 i
A+logIr-+L
+o
+ (1 -s)~ log6
+$i-$i=++33 -- P2 (1 -q4 4 i
+o(
1 i411
log6
1
I(P^P.)]. 3
$8
26-ij2
l-6 5-22s+562 (1 -q3
6-186 + (1 -a)4
log6
11 - 7s + 2a2 (1 -q3 11 - 76 + 2zi2 (1 -q3
6 + (l-q4
,
310
P. Cho, B. Grinsrein / b -) sy decay
i cww =- 16~’
&wP, + g,, PN + L-p P”
12
A&! - gj + f&2
2
x 4 + log c” 4 i --
P’
(l-6)’
-
-
19 - 496 + 236’ - 563 (l-q1
p-p’ 25-236-b136’-36” (1 -q4 + 12mi i 3 + --144 PpP,.P, rn:
(1 -q5
logs
12 - 246 - (1 -sy
log6
12 - (1 -q5
log6
25 - 236 + 136; - 3a3
i)
12 - (l-6)”
(l+j)J
log6
12
25--~6+13s2-3~”
+ p~P,.P:,+P,pIP,+P~P,.P,
144mi
:; i 4
log 6
(1 -q3 12 - 486
(l-q4
-a
+(p*p’)+O
2-36+3a2-6’
13 - 756 + 336’ - 76j
24m2, P,2
+
(l-6)4
i
- (1-q”
log6
i
. )I
As a reminder, we note that g,,g P’ = n = 4 - E in &dimensions. Therefore, if C,,. is contracted with gPy, an additional finite constant arises from the product of the 0(1/c) piece of 4 with the O(E) piece of gMyga” in the integral expression,
=-
2
i 16~’
4-f+logL+...
rnt
1 .
References 111 G.P. Yeh (CDF Collab.), talk given at La Thuile, March 1990; K. Sliwa (CDF Collab.), talk given at Moriond, March 1990 121U. Amaldi et al., Phys. Rev. D36 (19S7) 13sS [31 E. Jenkins and A.V. Manohar, Phys. Lett. B237 (1990) 259 [4] W.A. Bardeen, C.T. Hill and M. Lindner. Phys. Rev. D41 (1990) 1647 151 E. Witten. Nucl. Phys. B122 (1977) 109
P. C/IO, B. Grinstein
/ b + sy decay
311
[6] S. Weinberg, Phys. Lett. B91 (19SO) 51 [7] A. Cohen, H. Georgi and B. Grinstein, Nucl. Phys. 8232 (1984) 61 [S] M Wise. in Proc. Banff Summer Institute, ed. A.N. Kamal and F.C. Khanna (World Scientific. Singapore. 19881 p. 124 [!I] H. Georgi, Nucl. Phys. B363 (1991) 301 [lo] B. Grinstein. R. Springer and M. Wise. Phys. Lett. 8202 (198Sl 138; B. Grinstein and M. Wise. Phys. Lett. B274 (19S8) 274 [ll] B. Grinstein. R. Springer and M. Wise, Nucl. Phys. B339 (19901 269 [12] R. Grigjanis, P. O’Donnell and M. Sutherland, Phys. Lett. B213 (1988) 355 [13] R. Grigjanis, P. O’Donnell and M. Sutherland, Phys. Lett. B224 (1989) 209 [14] L. Abbott, Nucl. Phys. B185 (19811 189 1151 F. Gilman and M. Wise, Phys. Rev. D21 (19SOl3150 [16] D.J. Gross and F. Wilaek, Phys. Rev. Lett. 30 (1973) 1343: H.D. Poker, Phys. Rev. Lett. 30 (1973) 1346 [17] R.K. Ellis, Nucl. Phys. B106 (1976) 239: M.A. Shifman. AI. Vainshtein and V.I. Zakharov, Phys. Rev. D18 (197512583 [lS] T. Inami and C.S. Lim, Prog. Theor. Phys. 6S (1981) 297 [19] H. Georgi. Nucl. Phys. B361 (19911339 [?O] G. ‘t Hooft and M. Veltman. Nucl. Phys. B153 (1979) 365 [21] G. Passarino and M. Veltman, Nucl. Phys. B160 (1979) 151
B365 (1991)
NUCLEAR PHYSICS
279-311
QCD ENHANCEMENT
OF b + sy DECAY Peter CHO
Lyman
Laboratory
of Physics,
and Benjamin Harvard
Received (Revised
B
FOR A HEAVY TOP QUARK
GRINSTEIN
University,
Cambridge,
MA 02138,
USA
13 May 1991 19 June 1991)
An effective five-quark hamiltonian for .b + s decay processes is derived by integrating out a heavy top quark with rn: > rnk from the minimal six-quark standard model. The leading strong interaction corrections to this hamiltonian are then determined in an effective field theory computation. The renormalization group running of the operators in the effective hamiltonian between a top quark scale of 250 GeV and the W-scale is found to be significant and enhances the inclusive rate for the rare weak decay B + X,y by about 14%.
1. Introduction
Determining the mass of the top quark remains one of ‘the most important outstanding challenges in high-energy physics. Despite substantial experimental effort, the top quark has continued to elude detection and is only currently known to weigh more than 89 GeV [l]. Complementing the direct experimental searches, there have been numerous theoretical attempts to indirectly measure the top mass rn, via its radiative corrections to physical observables such as the p-parameter. Such theoretical estimates typically yield central values for m, ranging from 100 to 200 GeV with large error bars [2]. Recently however, some of these radiative correction analyses have been called into question, and very heavy top masses up to 250 GeV have been proposed [3,4]. If one accepts the proposition that the top quark could indeed be quite massive, then it is necessary to reconsider some of the earlier theoretical estimates of its indirect effect upon various physical processes. We are interested in particular in re-examining previous effective field theory computations of rare weak decays and investigating the impact of a very heavy top quark on the results from such computations. The basic effective field theory idea is by now quite well established and has been frequently reported upon in the literature [5-91. Starting from some underlying full theory, one integrates out the heavy degrees of freedom and then runs the resulting effective field theory down to the appropriate scale for studying some process of interest using the renormalization group. If any additional heavy particle thresholds are crossed during the renormalization group running, then 0550.3213/91/$03.50
0 1991 - Elsevier
Science
Publishers
B.V. All rights
reserved
280
P. Cho, B. Grinstein
/ b + sy decay
those particles should also be integrated out of the theory. Thus a succession of effective field theories is constructed which describes physics from small to large distance scales. In the particular case of the minimal six-quark standard model, one applies these ideas by first integrating out the top quark thereby generating an effective five-quark theory. This is subsequently run down to the W-scale at which point the weak gauge bosons are removed. In the past, the top quark and the W have usually been simultaneously integrated out from the full theory as a simplifying measure. This approximation is tantamount to neglecting the running of the strong interaction fine structure constant (Y, between the top and W scales and therefore necessarily incurs an error which grows with m,. We are interested in studying the dependence of this error on m, and determining its magnitude for especially heavy top mass. We will focus upon the strong interaction corrections to the decay of B-mesons to a strange hadronic state X, and a hard photon y. Strong corrections to this particular rare weak decay have previously been analyzed with the effective field theory formalism in the above-mentioned simplifying approximation [lo-131. Our results therefore represent a refinement of earlier work on this particular weak process. However, much of the discussion in this paper has much broader application to any effective theory computation. We therefore hope that it will prove useful to both novices and experts who wish to tackle other problems using the effective field theory approach.
2. Matching
Following the arguments presented in ref. [ill, we model the weak radiative meson decay B + X,-y by the quark decay process b + s-y. The validity of this approximation rests upon the belief that the dominant contribution to B-decay comes from an effective magnetic moment interaction generated at distances short compared to a typical hadronic scale A. Corrections to the b-quark decay picture are presumably suppressed by powers of the small ratio A/m,. We will thus analyze in detail the weak decay b + sy assuming the underlying fundamental theory describing this process is the minimal standard model. To begin, we summarize our standard model nomenclature conventions in table 1. We also adopt the mass-independent renormalization scheme of dimensional regularization plus modified minimal subtraction @IS). Thus we work in d = 4 - E dimensions and let ,u denote the renormalization scale. The covariant derivative associated with the unbroken color and electromagnetic subgroup H of the full standard model gauge group G therefore looks like D, = dp - ip’12
g,G,“X”
- ipE12 eApQ .
P. C/IO, B. Grinstein TABLE
Gaugegroup Generator Coupling Gauge field Field strength
G= SU(31, XR
x
($ G$
SU(21,
281
/ b + sy decay
x
1
U(l),
+
H = W(3),
x
U(l),, Q
T’
s
X”
62. T w:”
gl %
g
A’
B PV
G/f”
FPcu
The electric charge generator appearing in the covariant derivative’s photon term assumes the values Qt, = - $ and Q, = $ when acting on bottom and top quark fields, respectively. Following the effective field theory formalism, we integrate the top quark out of the full six-quark standard model and derive the effective hamiltonian A?& for b + s decays in the resulting five-quark theory. In effective theory computations, one evaluates Green functions in both the full and effective theories at a heavy mass scale m,, and requires that they agree to some specified order in l/m,. This process of matching Green functions fixes the coefficients of operators appearing in the hamiltonian Z&r at the heavy mass scale. Zerr generally contains all operators which are not prohibited by symmetry considerations. Such an effective hamiltonian is consequently non-renormalizable since it includes terms of arbitrarily large mass dimension. However, the coefficients multiplying operators of high dimension are necessarily suppressed by inverse powers of the heavy mass m,, so that the overall dimension of combinations of coefficients and operators equals d. In our calculation, we match off-shell 1PI Green functions first at the scale m,, = m,. We work in a background field version [14] of R, gauge in order to maintain explicit gauge invariance
the lagrangian
of our Green functions.
The gauge-fixing
part of
is
L& = - & c a
IPQz
+ g, f abc~~Q~c + 2isg,(
+t>Ta4 1’.
(2-l)
Here Qf: Epresents the quantum gauge fields for the standard model gauge group G while Q: stands for the classical background gauge fields for the unbroken subgroup H. For computational convenience, we will take 5 = 1. The gauge fixed by (2.1) then corresponds to the background field generalization of ‘t HooftFeynman gauge. While this background field gauge may not be quite so familiar as its conventional R, counterpart, the explicit gauge invariance of Green functions calculated in this gauge provides valuable checks on both matching condition and anomalous dimension computations. It is therefore the gauge of choice for lengthy off-shell calculations. We display in fig. 1 the few Feynman rules in 5 = 1 background field gauge that differ from the ones in ‘t Hooft-Feynman gauge which are relevant in our analysis.
P. Cho, B. Gtinstein
282
/ b +sy
decay
= ~"2gsfaac[2k,Yg“'X+ (k2 - k3)Xg”Y - 2k;g”x] hp
C,Y
=
-j~E/2e[2k;gpX
+ (k2 - k3)xgPy _ 2kfg”x]
Fig. 1. Relevant Feynman rules in ( = 1 background field gauge that differ from their counterparts in conventional ‘t Hooft-Feynman gauge. The closed circles at the ends of the external gluon and photon lines denote background gauge fields.
As illustrated in the figure, background field gauge yields a bonus simplification in our particular problem. For when we expand -9&r, the following trilinear interaction between the background photon field, W-boson, and would-be Goldstone boson L?--&= . . . -emwxpW$4-+
h.c.
cancels against the same term which arises in the Higgs kinetic energy. This cancellation cuts almost in half the number of diagrams we need to consider. The interaction terms in the full six-quark theory which contribute to the weak decay b + sy appear in the charged current sectors of the standard model
P. Cho, B. Grinstein
/ b -+ sy decay
283
lagrangian
where the left- and right-handed
quark fieIds are defined as
In this interaction lagrangian, V represents the 3 X 3 unitary Kobayashi-Maskawa matrix while Mu and Mu denote the diagonalized quark mass matrices,
Notice that the left-handed interaction terms involving the top quark and the would-be Goldstone bosons in (2.2) are enhanced by a factor of m,/m,,, relative to those involving the top and W, gauge bosons. Since we will only match Green functions to leading order in the expansion parameter 6 = m&/m:, we can thus ignore the charged current couplings to the W’s. In the effective five-quark theory, a complete basis for the local operators which contribute to weak radiative B-meson decay to leading order in 6 is listed below: Dimension
d + I: O[,
= -m,s,D*b,,
QLR= ,w&,4+4-~LbR.
(2.3)
284
Dimension
P. Cho, B. Grinstein
/ b + sy decay
d + 2: p1.A
L
=
-iSLT,A,,DwDYDubL,
Pz = p’/2eQ,SLypbL
a”Fp,,
p; = $12 eQhF’,fLYD”bL, Pz = i~“‘/2eQ,~,,SLy’“y5D”bL, pghost L
=
E.LEg,Zf,,,SLy,X”bL(a’“776)77C,
Rt = i$gf(
D”4+)4-SLymbL,
The tensor TFtIT appearing in Pk A assumes the following Lorentz structures as the index A ranges from 1 to 4:
A few points about these operators should be noted. Firstly, we have only displayed non-renormalizable operators in this list, for the effects from any renormalizable operators may simply be absorbed into various field and coupling constant redefinitions. Secondly, following the clever trick of Gilman and Wise [El, we have incorporated seemingly arbitrary factors of gf into the definitions of operators with would-be Goldstone boson fields. As we will see later, these factors facilitate combination of various parts of the leading-order anomalous dimension matrix for the basis operators. Finally, we have included the operator pghost = ~Egffobc~Ly~XabL(alL?7b)77= L with ghost fields 77 and Yj in our basis set for completeness. As pointed out in ref. [13], this ghost operator mixes at one-loop order with some of the other dimension(d + 2) operators containing gluon fields. However, we will see that no operator
P. Cho, B. Grinstein
285
/ b + s-y decay
mixes back into P[host. Thus this ghost operator turns out to have no effect upon the running between the top and W scales and could be neglected without loss. We now determine the coefficients of the operators appearing in the effective hamiltonian to leading order in 6 = m&/m: by matching Green functions in the full and effective field theories at the top scale. The coefficients of the operators which explicitly contain would-be Goldstone boson fields are the easiest to compute. As illustrated in fig. 2, only tree diagrams contribute to the rbs6++-, rbsgd+$- and rbsy4++- 1PI Green functions at leading order. Expanding the
Full
Intermediate
Theory
t b 9 jb:
EFT
+
.. .
+
...
+
...
s
Fig. 2. Leading-order matching conditions at the top quark scale for the IPI Green functions rbs8+‘-, +?l++and rbsud+&- m the full six-quark theory and in the intermediate five-quark effective field theory.
256
I? C/IO, B. Grinstein
/ b + sy decay
propagators of the virtual top quarks as
one easily finds that the full theory Green functions match onto effective theory Green functions with one insertion of the operator
= -Q,,+R:+R2, with coefficient
In decomposing 8 over the basis set (2.3), we have neglected terms proportional to the strange quark mass since m, -3xmb. Tree-level matching of Green functions is however generally insufficient to completely specify the matching conditions between operators in full and effective theories. One usually also has to include operators in the effective theory that arise as counterterms when external lines on Green functions are closed off into loops. Such new operators are generated in our particular problem when we close off the external Goldstone boson lines in fig. 2 into loops as shown in fig. 3. All of these one-loop graphs may be evaluated straightforwardly with the aid of the integrat formulas tabulated in appendix A. At this point, it is worthwhile noting some simple but important general relations between full and effective theory Green functions. In a full theory, the dependence of Green functions on heavy and light particles is intertwined and entangled. In an effective field theory on the other hand, all dependence on heavy particles is explicitly extracted out of Green functions and moved into coefficient functions. As previously mentioned, the coefficients are suppressed by inverse powers of heavy masses which are determined by naive dimensional analysis. Factoring out this trivial heavy mass dependence from the coefficient functions, one finds the foIlowing general relationship between Green functions in ful1 and effective theories:
where m,, and mr stand for heavy and light particle masses.
P. Cho, B. Grinstein
/ b -) s y decoy
htennediate
iW1 Theory
Fig. 3. One-loop
matching
287
conditions
at lz. = m, for the 1PI Green
functions
EFT
rhr,
rhsg and rhsy.
It is instructive to compare order by order in l/m, the logarithmic dependence of G,,, with that of the effective field theory terms in the infinite power series in (2.4), cfull
log(k/nz,)
+ciutl
lw(mdmd
=c,,ff
log(~/mh)
+ceffWmdPL).
(2.5)
Since only two independent dimensionless ratios can be formed from the arguments of G,,,, its logarithmic dependence upon the mass parameters /.L, mt, m,, can generally be expressed as the 1.h.s. of eq. (2.5). On the r.h.s. of this equation, the log@/m,) can only arise from the dimensionless coefficient function C(p)(~/m,> while the remaining log(m,/p) must come from Gig). Equating the coefficients of log mt and log m,, in eq. (2.5), one trivially concludes 4”ll = C,ff >
Cfull + &,I
= Ccoeff .
(2.6a, b)
These relations however convey nontrivial information. Eq. (2.6a) indicates that all infrared logarithms in Green functions computed in the full theory are reproduced in the effective theory. This must be the case since both the full and effective theories describe the same physics at low energies. Moreover, at the heavy mass scale, the leading logs in Gr,,, are identical to those in G,a. Since matching conditions are determined by taking differences between Green functions in the full and effective theories, there can consequently be no logarithms appearing in
288
P. C/IO, B. Grinsrein
/ b -t sy decay
matching conditions at p = m,,. Eq. (2.6b) indicates on the other hand that at the light mass scale, the full theory logarithms are reproduced in the effective field theory by the running of the coefficient function C’p’(~/m,). Therefore, all of the logarithm information in the full theory is recoverable in the effective theory. Returning to the problem at hand, we find the following leading-order values for the coefficients of the basis operators appearing in the effective hamikonian
(2.7) from the matching conditions at p = m,, c,
et
l/2
=--
167r* ’
OLR
3
=
l/2
=--
167~’ ’
OLR
11/18 cpp = cp,, = 167~’ ’
c P1.2
=
c
=
-
8/g -16T*
7
l/2 pp2
-
167~~ ’ 3/4 167r2Q, ’
c,: = c,:=o,
l/2 c,:=
-
16dQ,
’
(2.8)
t? Cho, B. Grinstein
/ b + sy decay
289
In accord with the preceding general discussion, we see that no logarithms appear in these coefficients at the top quark scale. Had we matched however at a different renormalization scale, we would have found a log in the Pt coefficient,
In sect. 3, we will verify that this logarithm which vanishes when p = m, is regenerated at lower scales by the renormalization group. 3. Running Starting from the fact that the effective hamiltonian (2.7) is independent of the renormalization scale II, one can readily derive the renormalization group equation satisfied by the coefficient functions C;(p), P$c;CP)
=
C
(3.1)
(YT)ijCj(P).
i
The anomalous dimension
matrix
yij = z, ‘p dZkj dp
appearing in eq. (3.1) is formally defined in terms of the divergent renormalization constants Zij which relate renormalized and bare composite operators, @J= I = Zij( /A)&$( p) . However, y is most conveniently calculated in practice by requiring renormalization group equations for Green functions with insertions of composite operators to be satisfied order by order in perturbation theory. Let r$) denote a renormalized n-point 1PI Green function with one insertion of the operator Hi. Then the anomalous dimension yij characterizing the mixing of Hi into Hj is simply determined from the renormalization group equation for r$‘), yijTgf)=
I
-
pa i
ap
a ag
+0-+-y
a mm --ny am
exf ry., 1
Here p = p.(d/dp)g, ‘y,, = &/m)(d/dp)m and nyext stands for the wave-function anomalous dimensions arising from radiative corrections to the Green function’s n external lines.
P. Cho,
290
El. Grinsteirl
/
b * sy decay
We first solve for the anomalous dimensions of the basis operators (2.3) to lowest order in the electroweak interactions and ignoring strong interactions for the moment. In this case, we can forget about the p, ‘y,,, and jz’yeX, terms in eq. (3.2) since they introduce additional powers of the electroweak coupling constants beyond those already present in the first p-derivative term. All the operators’ anomalous dimensions can be extracted from the single IPI Green function rbsy. Loop diagrams with insertions of Q and R Goldstone boson operators which contribute to rbsy are generally divergent and require 0 and P operators as counterterms. This consequently leads to mixing of the former operators with the latter. The converse mixing however does not occur. After evaluating the loop diagrams, we find the following mixing of the basis operators with would-be Goldstone fields into those without:
o,, P:'A QLR/oO y= R; R'L R”L
P; 0
\o
0
P;f
0
0 l/6&,
0 0 00
P;
- 1/6Q,,
01
00 0 0 0
! 53 “2
S'
(3.3)
0
Inserting the transpose of this anomalous dimension matrix back into the renormalization group equation (3.1) for the coefficient functions and observing that the coefficients on the r.h.s. do not run, one can trivially solve for the coefficient values at scales below the top. Only one is found to differ from its p = /n, value,
4GF CP&EL)
= G&m,)
+
-
fi
v,bv,:
1
2
log 4 . 96T2Qb 112,
As advertised, the log /12/m; term present in eq. (2.9) which originates from the full theory is recovered in the effective theory by the running of the coefficient function. We now reconsider the running of the basis operator coefficients taking into account strong interaction effects. We first need to know the quark, gluon and ghost wave-function anomalous dimensions yquark, ygluon and yshosl to leading order in the strong interactions. We will also need the quark mass anomalous dimension y,,, and QCD /?-function. In background field ‘t Hooft-Feynman gauge, these may all be simply computed from renormalization group equations for two-point Green
P. Cho, B. Grinstein
/ b + sy decay
291
functions [ 141, Yquark
=
2
3
g323
8,rr2
Ygluon
P
=
-
3
Yghost
g3
ym = -4g:
=
--- 3 g32 4 8rr2 ’
P=b$.
a,rr2 ’
The coefficient appearing in the p-function has the value b = - 11/2 + n,/3, where nF denotes the number of active quark flavors [16]. Integrating p, one obtains the well-known expression for the strong interaction fine structure constant in leading log approximation,
d(P) %(cL) = -= 47r
127r (33 - 2n,)log($/A&)
*
Since cu&) is continuous across quark mass thresholds, Am varies with nr. In our five-quark effective theory, the QCD scale is related to the one appropriate for three light quark flavors as
When the strong interactions are included, the solution to the coefficients’ renormalization group equation (3.1) is not quite so trivial as in the previous case where QCD was ignored. Since we are working in a mass-independent renormalization scheme, we can first eliminate the derivative with respect to the renormalization scale in favor of a derivative with respect to the coupling constant which yields
ps=
C(yT)ijCj.
3
i
The solution to this first-order differential notation as C(EL~)
equation then appears in obvious matrix
= exp[Jrf)dg -rT(d * P(s)I C(Pl)
Since we will only compute the anomalous
dimension
(3.4)
matrix y to leading order,
292
P, C/IO, B. Grinstein
/ b + sy decay
we decompose it as 2
Y=- 8227
+ o(d)
=g,,s-l +o(g$ where the eigenvalues hi of 9 are contained in the diagonal matrix A while the corresponding eigenvectors are arranged into the columns of matrix S. Our earlier adoption of the Gilman-Wise trick of incorporating strong coupling constants into some of our operator definitions results in the matrix 9 being scale independent and purely numerical, as will shortly become clear. The solution (3.4) to the coefficients renormalization group equation therefore becomes
C(p2) = (S-l)Tdiag
(3.5)
([~l*‘/2h)STC(pl).
We now calculate the anomalous dimensions for each of the operators in our basis set to O(gi). It is in this lengthy calculation that the tremendous utility of explicit gauge invariance in background field gauge becomes fully apparent. As a representative example, we describe the results for the QCD-induced mixing of the operator Pi ’ = -iS,D2@b,. To begin, we list the first few terms in the Feynman rule for this particular operator which is illustrated in fig. 4,
Ptl
Feynman rule = I’$ S,p2#b,
+ ip3’/2 9,( g,X”
+
eQb)
~(p’~y,+p,d+p;~)b,+....
p
(3.6)
p’
Here denotes the b-quark’s four-momentum entering into PL’ while is the s-quark’s momentum flowing out of the operator. The first term in eq. (3.6) corresponds to the piece of Pk’ with zero external gauge bosons emanating from the operator and is common to Pk’ and Pk” as well. Therefore, the identity of this operator is only uniquely established starting from the single gauge field terms. Inserting Pk ’ into the two-point function rbs, one may partially determine how it runs into itself and other basis operators under the action of QCD at one-loop order. The diagrams involved are displayed in fig. 5, and the divergent terms in
P. Cho, B. Grinstein
PjB1 Feynman
=
rule
293
/ b --) sy decay
!!+ b
+;y+
+pT’
Fig. 4. First few terms in the Feynman rule for the operator Pt’
+
= -iSLD2$bb,.
their sum are given by 4
c (0 external gauge-boson graph) j j=l
+&(A x
+logp2)
{ - (4m; + ;mbm:)SLb,
- ($msm; + 8mf)i,b,
- ($rnt + ~m~)S,@b, - $mbm,i,gbb, -6m,S,p’b,
- 2m,S,p2b, pl,A=1.2,3 L
+ f%p2$bL},
(3.7) where 2 A=;-y+log4r.
In the column on the right-hand side, we have listed the dimension-@ + 1) and -(d + 2) operators whose Feynman rules match particular terms in the sum. The information provided in (3.7) is however insufficient to completely specify the full mixing of P,‘vl. We therefore must consider Green functions with at least one external gauge boson. In fig. 6, we illustrate all the possible gluon corrections to the rbsg Green function with one insertion of the operator Pt’. The sum of the
Y AA \c+ b
5
Fig. 5. Diagrams with one insertion
+A+2 of the operator function I+.
Pk’
that contribute
to the 1PI Green
294
P. Cho, B. Grirrsfcin
/ b +sy
decay
Fig. 6. Diagrams with one insertion of P, ‘*I that contribute to the 1PI Green function rbsg.
infinite
parts of these one-loop diagrams is
12
c (1 external gluon graph)j j=l
= ip3E/2-l;i2 (A + log p2) x
- ($rni + $mf)S,g3X*yAbL
-6WLg3Xa(
p +p’)*bn - 2m,SRg3Xa(p +p’)*b,
+4m,S,g,X”i( -
%f,g,X”(
p’ -p)@a,,bR
+ 2m,S,g,X”i(
P12Yh+ Ph!! + Phqb,
+:~,~~X’(P.P’Y, +$,g,x”(
- ~m,m,S,g,X”y,b,
+~Ati+p;@‘)b~
p2.yA +~nd’ +ph”)b,
-- YSLg3X”( -iep,,rrAPp~‘p’yyu) bL)
p’ -p)‘u,,,b,
0L.R O& PI.1
L
PI.2 L
PI-3
L
P. C/IO, B. Grinsrein
/ b + sy decay
295
Again we have listed alongside the various terms in this result the operators to which they correspond. The fact that all the terms in eq. (3.8) precisely assemble into pieces of Feynman rules for various gauge-invariant operators constitutes a nontrivial check on the validity of the computation. It also stands in sharp contrast to results from similar non-abelian anomalous dimension computations performed in standard covariant gauges where ugly non-gauge-invariant terms typically arise. We further note that the coefficients multiplying the Feynman rules of operators appearing in both the zero-external gauge-boson and one-external gluon results are identical. The coefficients of the PL', Pk' and Pk' terms which are distinguishable in eq. (3.8) also add up to the coefficient in front of the dimension-cd + 2) term in eq. (3.7) where these operators are indistinguishable,
Thus we clearly see the explicit gauge invariance in background field gauge at work providing important consistency checks on these operator mixing results. After a long but straightfonvard computation of the mixing of all the other basis operators, one finds from the renormalization group equation (3.2) the following QCD-induced entries in the anomalous dimension matrix,
GR OZR 03LR pl.
L
PI.2 L
YE
I
0L.R 7.0 3
ohl 1
L
0”
pz 0
0
0
2 J
4 7
0
0
0
0
0
0
0
0
0
0
16 5
0
0
0
0
0
0
0
0
6
2
32 5
-194
-6
38 3
0
4
3 I
0
34 5
17 4
0
38 T
0
1
4 3
32 5
5 ;i
6
38 -ii-
0
27 T
aY
0
27 T
0
0
0
185 -Tc
209 Tr
6.5 -E
-+
p:
0
p:.
0
pghosl L
pghosl L
pf 0
-8
pk’
PI.4
PI.4
0
0
4 5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
247 ?K-
1
296
Fig. 7. Diagrams which potentially lead to one-loop mixing of P?” into PLM’. Recall that there is a relative minus sign difference between graph (a) and its crossed counterpart (b) due to the ghosts’ obeying Fermi statistics.
In the would-be Goldstone boson operator sector, the mixing is
Y
QLR = R;
I u 83
Rt R’L
0 ,o
0 39 -F
8
0
5
0’8 5;
0
$,
s32 8T2
(3.10) *
ago 0
Anomalous dimensions for operators in the (d + &dimension subsector of (3.9) have been reported before in the literature [17], and our results are consistent with previous findings. One sees in the last column of (3.9) the content of our earlier assertion that no operator mixes into the ghost operator Pp other than itself. The potential mixing of the Pt" operators into PL%os( via the diagrams shown in fig. 7 fails to materialize since their divergent pieces exactly cancel. Therefore, Pp starts and remains with zero coefficient at all renormalization scales and has no effect upon any other operator. We will consequently ignore it from now on. The y values in (3.9) and (3.10) must be combined with those given earlier in (3.3) to obtain the total leading-order anomalous dimension matrix. At this point, the reason for our having included factors of gz into the definitions of the Goldstone boson basis operators becomes evident. These factors were chosen so that all the entries in y would be proportional to gi. Consequently, diagonalizing the anomalous dimension matrix y = (gf/8?rIT)f necessitates finding only the eigenvalues and eigenvectors of the numerical matrix 9. Performing this diagonalL&ion, one verifies that the eigenvalues and eigenvectors are all real as required to maintain hermiticity of the effective harniltonian at all renormalization scales. Inserting the anomalous dimension results back into eq. (3.51, we can determine the running of the basis operator coefficients from the t-quark to W scales. To give some idea of the typical size of the strong interaction corrections when the top is
P. Cho,
B. Grinstein
/
291
b + sy decay
TABLE 2
The leading-order coefficients at or.= m, = 250 GeV and p = mw = 80.6 GeV with the QCD scale taken to be A$ = 300 MeV Operator
;” 0: QLR
PL’ PI.2 L 13 pi
PL” p: p1 2 R; R3L
C(m,)
- 0.0032 - 0.0032 0.0063 - 0.6895 0.0039 - 0.0056 0.0039 0.0032 - 0.0142 0 0.0095 0.6895 0.4754 0
Intermediate
C(mw)/C(m,)
C(m,)
- 0.0034 - 0.0031 0.0054 - 0.6785 0.0033 -0.0044 0.0033 0.0019 -0.0153 0 0.0081 0.5136 0.3666 - 0.0330
1.084 0.976 0.852 0.984 0.851 0.775 0.851 0.614 1.071 0.853 0.745 0.771 -
EE’T
EJ?T below
W scale
H =x +.” b
Fig. 8. Matching conditions at ~1= mw
.s
b
for four-quark operators and twoquark remain below the W-scale.
8
basis operators which
298
P. C/IO, B. Grinstein
/ b -‘sy
decay
quite heavy, we display in table 2 numerical values for the leading-order coefficientsat~=m,=250GeVand~=m, = 80.6 GeV with the QCD scale taken to be A(&= 300 MeV. From the entries in the last column of this table, one sees that the QCD-induced running between the two scales is substantial for several operators. In order to continue to run the basis operator coefficients down to lower scales, one must integrate out the weak gauge bosons and would-be Goldstone bosons at p = m,. There is consequently a new set of matching conditions relating operators in the intermediate five-quark effective theory with those in a five-quark theory without the W, and 4* fields. As illustrated in fig. 8, these matching conditions involve the two-quark operators in eq. (2.3) that originated from diagrams with virtual top quarks and remain below the W-scale. In addition, there are new four-quark operators which arise from graphs containing charm and up quarks. Neglecting small terms proportional to mf or ntt in these new matching conditions, one finds the following relations between coefficient functions just above and below p = m,:
C,:.l(m,)=C,ki(m&)
4/18 + s,
Cpt2(mi)
=C&m&)
- 2,
C,p(
m,)
= C,k3( m&) + $$,
C,p(
m,)
= C,p(
Cpf(m;)
m&) + j-$
=C,&m$)
+
y
l/2
167r2Q, ’
C,:( mG> = C,;( m&) , Cp&m,)
= C,:(m&)
-
3 167?Q,
*
(3.11)
P. C/IO, B. Grinsrein
/
b + sy decay
299
In sect. 4, we will use the results of ref. [ll] to continue running the coefficients of operators in the effective hamiltonian down to the b-quark scale. But in order to match onto those results, we first need to use equations of motion to reduce all the remaining two-quark operators to the gluon and photon magnetic moment operators Otn and Ot, and to four-quark operators. Again neglecting the strange quark mass in comparison to the bottom quark mass, we obtain the on-shell equivalence relations
p’.’L = - +O&
- f(&
pl.2 = -to2 L LR - f O:,
Py
= fo&
+ fo:,
Pf = four-quark
The effective hamiltonian
1
+ four-quark operators,
+ four-quark operators, operators,
then appears just below the W-scale as
+ (four-quark
operators))
As a check, one can momentarily ignore QCD and evaluate and O& coefficients given above with the values specified conditions (2.8) and (3.11). The coefficients arc then found leading terms in the on-shell matching conditions for these
the on-shell Ot, in the matching to agree with the magnetic moment
P. Cho, B. Grinstein / b + sy decay
300
c-
/-
/-
--
0.0008
F z YI u
0.0004
0.0002
0.0000
100
150
300
250
200 m, (dev)
Fig. 9. Dependence of the gluon magnetic moment operator’s coeffkient Co;,(m~) on m, for At&= 300 MeV. The coefficient values with and without the leading-order strong corrections are indicated by the solid and dashed curves, respectively.
operators when the top and W are integrated
C,&(MW) = -
1
167~’
Co&G)
1 = -16rr2
out together [12,181,
; - ;a - $2 (l-q3
-1-$6+;62 (l-q3
:a2 - (l-q41°g6
i ’
;a - 32 -
(l-q4
log6
1 *
(3.12)
Since we have only calculated the strong interaction corrections to leading order in 6, we will exercise our perturbative freedom to add on the subleading terms in (3.12) so that our Ok and 03, coefficients will reduce to those in (3.12) as m, + m,. The QCD-induced deviations away from these uncorrected coefficient values become significant for heavy top masses, as can be seen in the plots of figs. 9 and 10, and are on the order of 18% for m, = 250 GeV. This leads to an enhancement of rare B-decays as demonstrated in sect. 4.
l? Cho, B. Grinstein
0.000
t ’
’ ’ 100
’ ’
’ ’
150
’
/ b + sy decay
’ ’
’ ’
200
’
’
’ ’
’ ’
250
’ ’
’ 1
300
m, (dev) Fig. 10. Dependence of the photon magnetic moment operator’s coefficient Co$m,) on m, for k&=300 MeV. The coefficient values with and without the leading-order strong corrections are indicated by the solid and dashed curves, respectively.
4. B + X,y decay rate
Since we are interested in studying the process B --f X,y, running the photon magnetic moment operator down from scale. In ref. [ll], this running was calculated up to two-loop in terms of the coefficients of our operators O& and Ota,,
we need to continue the W to the bottom order. When written the result looks like
The magnetic moment coefficient Co&+,) then directly enters into the inclusive rate for B-meson decay modelled upon the quark process b + sy,
P. Cho, B. G&stein
302
/ b + sy decay
0.006
0.005
0.004
0.003
0.002
0.001
0.000
200
150
250
300
m, KW Fig. 11. Inclusive decay rate for B MeV. The rate which includes the indicated by the solid cutve. Rates strong interaction corrections at all
+ X;v normalized to the semileptonic decay rate with A’&= 300 leading-order QCD running between the top and bottom scales is including only the running between the W and bottom scales or no are illustrated by the dashed and dot-dashed curves, respectively.
The sensitive dependence of this decay rate upon the b-quark mass and the KM matrix elements can be removed by normalizing it to the rate for semileptonic B-decay. We therefore show in fig. 11 the ratio
r(B + KY) T(B+X,ei;,)
512~~ =
3
aEM f(mc/mtJ
1
Ic,&pb)I’
(4.2)
plotted as a function of m, rather than the absolute prediction (4.1). In this last expression, the function f(m,/m,) represents a phase space suppression factor which equals 0.44 for mb = 4.5 GeV and m, = 1.5 GeV. The electromagnetic fine structure constant also appearing in eq. (4.2) is evaluated at the b-quark scale where it has the numerical value cr,,(m,) = 1.032 aEM = l/132.7. For comparison, we have plotted alongside our result in fig. 11 the decay rate which includes the QCD running between the W- and b-scales but not that between the t and W. We have also displayed the completely uncorrected rate which neglects the strong interactions altogether. The rumring between the top quark and the weak gauge bosons is seen to enhance T(B 4 X,y) by approximately 14% for m, = 250 GeV when compared to the partially corrected rate.
P. Cho, B. Grinsrein
/ b --) sy decqv
303
We believe that our estimates for the enhancement of the rare weak decay B --, X,y are most likely on the conservative side since we have only included the QCD running of the lowest-dimension operators in the effective hamiltonian. The standard procedure for systematically determining higher-order power corrections in the effective field theory program is clear. One must retain operators of higher dimension in the matching conditions and renormalization group running. The effort required to actually carry through with such computations however appears formidable and is most likely unwarranted in our case given current experimental uncertainties. Recently though, there have been attempts to develop more elegant approaches for dealing with the power corrections which arise in all effective theory problems [19]. Perhaps these efforts will soon yield new calculational tools that will broaden the scope of the effective field theory formalism. It is a pleasure to acknowledge helpful discussions with Howard Georgi. B.G. would like to thank the Alfred P. Sloan Foundation for their support of this project. This work was also supported in part by NSF contract PHY-87-14654.
Appendix A INTEGRAL
TABLE
We tabulate here general expressions for several integrals which frequently arise in the evaluation of one-loop Feynman diagrams. Our notation follows that of refs. [20,21]. The closed-form expressions for integrals with one and two internal propagators listed in this table have been transcribed from almost identical expressions appearing in these two references. Significantly more complicated solutions for integrals with three and four propagators may also be found in these papers. While such exact, closed-form expressions are in principle desirable, one is often more interested in practice in the first few terms of series expansions of these integrals. This is the case for instance in effective field theory computations when one matches non-local contributions to Green functions onto local operators. We therefore provide series expansions for all of the integrals listed below in various commonly encountered limits. All of our integrals are in d-dimensional Minkowski space, and the signature of our metric is (+, -, -, -, . . . 1. After dimensional regularization, all divergent pieces in these integrals appear in the combination 4s
1 -y+log4r, E
where E = 4 -d. In the MS renormalization scheme, this term is cancelled by counterterms. Thus 4 may simply be discarded in each of the integrals below.
304
B. Grinstein
P. Cho,
/
b * sy decay
Integral with one propagator: =-
2
im*
A+log$+l
167-r* Integrals with two propagators
=-
(2~)~
1
1 1 ddq qP A +logK
m:
- /IdxlogS(x) 0
2
i 167~~
8,=/fj-
(q*-mt+ia)[(q-P)‘-m$+ie] 2
i 167r*
=-
.
:
ddq
BO=$j-
1
A +lo&
-F(l,x,)
-F(l,x,)
>
4
(27~)~ (q*-mf+k)[(q-p)*-m$+i&]
q/44, (2~)~
(q*-mf+ie)[(q-p)‘-m$+ie] CL2 - 2 -3i*d
ml
xx* logS(x) /
A +logcL2 -F(3,x,) m:
o’dxS(x)logS(x)
I
-F(3,x2) 2
+y
(3mf+3m;-p*)
A+logk+l i
+3(mf-
m~+p*)(V,x,)
- 2p2(F(3,
x,) + F(3, X,))
m2
+F(2,x2))-6m:(F(1,x,)
+F(Lx,))
,
P. Cho, B. Grinstein
/ b * sy decay
305
where
F(l,x)
= -xlog-
X-l
-l=&+$+-$+...,
X
x-l 1 F(2,x)=-x*Iogx-Z-X=~+-$+-$+.... F(3,x)
=
x-l
1
-x310g-
-
1
-
-
3
X
1
-x-x2=
-
2
1
4x
1
+ -
+
5x2
-
6x3
+ . .. ,
are the roots of mfS(x>/p2. x1.2 In the limit where the mass appearing in the first propagator is large compared to the mass and external momentum in the second, the expansions up to quadratic order in the external momentum of these closed-form expressions for the Bintegrals appear as follows: and
Limit
1: m f = rni > rn$ = rns,
p2;
6 =
rns/rni.
+ (1-6)3 2
B=:i P
[ A+logcc+2 167~‘~” 4
$-
l-6
+s
i )I 4
6
log6
+ 0 !4
26-iY2 + (l-6)2
)
log 6
P2 +z
’
P. Cho,
306
B. Grinstein
/
b + sy decay
In a second case where both masses are equal and large compared to the external momentum, the expansions for these integrals look like Limit 2:
mf=m~=m2>p2.
-p2)(A
+logs)
+6m’) 4 +o (
zm2
)I
’
Integrals with three propagators: lx, (2~)~
(q2-m~+ie)[(q-p)2-m~+ie][(q-p’)2-m:+iE]
= -m!--~l~x/ol-x,y
“;;r”, 3
ddq
q”=Pfj-
=-
C
(2i~)~
qLl4” (q2-m~+ia)[(q-p)2-m:+is][(q-p’)2-m~+ie]
i
-/oldXk’-Xdy
16a2
ddq =pu’jP”U (2~)~
X2PFP” -tXY( P,P: +P;P”)
+Y2P;PI
S(X> Y)
9,9”4,
(q2-m:+is)[(q-p)2-m$+is][(q-p’)2-m:+ie] P+P’)o+gyo(P+P’)~+g,,(P+P’)“}
-~dx~l-xd,(
[x3p,p,p,+x2y(p,p,pb+p,plP,+p;P,P,) +Xy2(P;PlP~+P;P,Pb.+P,P:Pb)
+Y”P;P:P:]/s(GY S(x, Y)
+ f[gp”(xP+YP’)~+g”~(Xp+YP’)~+g~~(Xp+YP’)~l~og~
>
P. Cho, B. Grinstein
/ b + sy decay
307
where S( x, y) = x2p2 + 2xyp .p’ + y2pt2 -x(p2+m:-m:)-y(p’*+m;-m~)+m:. As mentioned previously, the closed-form solutions for these C-integrals are quite complicated. So we will not write them down here. Instead, we give the expansions of these integrals up to quadratic order in the external momenta in the limit in which the masses in the second and third internal propagators are equal and are either heavy or light compared to the first propagator’s mass. Limit 1: p2,p~*,p~p’
-
1
6
1-S
+-
4
+-
l 16rr’mt . iip,
; + is - is2
- ; + $3 + $32
P .P’
L- gi
- (l-q’
1 i II 4
is2
log6
+ 0 !4
s2
(1-s)4
- (1-#
-&+++*-+~3
P.P’ 4 r2
& -
(1-S)”
i
gs
log6
563
i
+f-
,
log6
+&s-&s2+&s3
2
+f4
+o($)]
I
is2
(&
4
log 6
+ (l-q4
(l-6)’
i
+-
s
+ (1-6)4
(1-S)”
i
4
(--=-
log 6
+ (1 -8)’
p2 + PI2
6=m;/mi.
-
$2
(l-q4
i(P^PY],
- (1-S)’ + +p
92
-
-
(1-Q
log 6
$3
log 6
I
30s
P. Cho, B. Grinstein / b --) sy decay 1
2
4
/j+,0g-t4-+r-
f-
Ji3
li2
2
l-6
(l-6)
5-226+56’
p2 + pr2
18m2,
l&S2 - 6a3
(l-Q3
-
2-7tS+11ti2
p@pV +-p’Pp’y
+ (1 -q4
2 - 76 + 11s2
18mi
-
2 - 76 + 11a2
36mi
[
log 6
log6
6S3 + (l-6)”
(1 -Q3
ppptv +p”P’
(l-q4 6cS3
(1 -q3 -
log6
(1 -cq3
6ti3 + (l-q4
log6
log6
,
g,,pcr + g”,Pp + &JLP” 12
xiLt+logY+a3 2
f
+-
--
24mi i t2
- (l-S)3
13s3
(l-q4 3-13s+2362-25s3
p-p’
--- 3 P,P”P, 144 rni
(1- q4
log6
48a3 - 126” -
5 - 236 + 49s2 - 19ti3
12mZ, i
+ (l-f+
L&p
(l-6)”
24m2,
12a4
+
7 - 33s + 756’-
P2
P
;a
(1 +j)2
rni
+-
-
(l+$
log6 1
24?i3- 12ti4 -
(l-csy
12s4 + (1 -*)S
3-136+23S’-2563
log6
log6
12a4
(1 -ql
+ (l-Q5
)i
log 6
5 log6 +(p++p'++O !f( )I. i
m”h
P. Cho, B. Grinstein / b +sy decay Limit
2:
p2,pr2,p
co=
*p’ < rn: = rni = rn: << rnf = mf;
6
6
1-S
+ (1-6~~
-
I 16r2mz/
.
[
--
4 +-
6
(1 -q3
;a -
I 16r’rns pp . [i
- (l-q4
f+++p
m’h i
c,=
62
$2
(l-q3
i
P-P?
log6 i log6
+ (l-q4
$62
(l-6)’
6 = m$/mt.
log 6
; - 2s -
p= +p12
309
r2
- (1 -S)’
$))
p2 +pr2
18mt
+
pwp”+p’pp’v
18mg p”p”
+
+p’“p’ 36mz
log8 i
$8 + (l-6)5
logs
I
gj - $2
+ (l-q5
(1 -q4
i
2
-~
)
a2
++!?p+L&
-- P m;
4
p*
;a
P -PI ~+~s-&62+~63 f 1 (1 -a>* 5 i
A+logIr-+L
+o
+ (1 -s)~ log6
+$i-$i=++33 -- P2 (1 -q4 4 i
+o(
1 i411
log6
1
I(P^P.)]. 3
$8
26-ij2
l-6 5-22s+562 (1 -q3
6-186 + (1 -a)4
log6
11 - 7s + 2a2 (1 -q3 11 - 76 + 2zi2 (1 -q3
6 + (l-q4
,
310
P. Cho, B. Grinsrein / b -) sy decay
i cww =- 16~’
&wP, + g,, PN + L-p P”
12
A&! - gj + f&2
2
x 4 + log c” 4 i --
P’
(l-6)’
-
-
19 - 496 + 236’ - 563 (l-q1
p-p’ 25-236-b136’-36” (1 -q4 + 12mi i 3 + --144 PpP,.P, rn:
(1 -q5
logs
12 - 246 - (1 -sy
log6
12 - (1 -q5
log6
25 - 236 + 136; - 3a3
i)
12 - (l-6)”
(l+j)J
log6
12
25--~6+13s2-3~”
+ p~P,.P:,+P,pIP,+P~P,.P,
144mi
:; i 4
log 6
(1 -q3 12 - 486
(l-q4
-a
+(p*p’)+O
2-36+3a2-6’
13 - 756 + 336’ - 76j
24m2, P,2
+
(l-6)4
i
- (1-q”
log6
i
. )I
As a reminder, we note that g,,g P’ = n = 4 - E in &dimensions. Therefore, if C,,. is contracted with gPy, an additional finite constant arises from the product of the 0(1/c) piece of 4 with the O(E) piece of gMyga” in the integral expression,
=-
2
i 16~’
4-f+logL+...
rnt
1 .
References 111 G.P. Yeh (CDF Collab.), talk given at La Thuile, March 1990; K. Sliwa (CDF Collab.), talk given at Moriond, March 1990 121U. Amaldi et al., Phys. Rev. D36 (19S7) 13sS [31 E. Jenkins and A.V. Manohar, Phys. Lett. B237 (1990) 259 [4] W.A. Bardeen, C.T. Hill and M. Lindner. Phys. Rev. D41 (1990) 1647 151 E. Witten. Nucl. Phys. B122 (1977) 109
P. C/IO, B. Grinstein
/ b + sy decay
311
[6] S. Weinberg, Phys. Lett. B91 (19SO) 51 [7] A. Cohen, H. Georgi and B. Grinstein, Nucl. Phys. 8232 (1984) 61 [S] M Wise. in Proc. Banff Summer Institute, ed. A.N. Kamal and F.C. Khanna (World Scientific. Singapore. 19881 p. 124 [!I] H. Georgi, Nucl. Phys. B363 (1991) 301 [lo] B. Grinstein. R. Springer and M. Wise. Phys. Lett. 8202 (198Sl 138; B. Grinstein and M. Wise. Phys. Lett. B274 (19S8) 274 [ll] B. Grinstein. R. Springer and M. Wise, Nucl. Phys. B339 (19901 269 [12] R. Grigjanis, P. O’Donnell and M. Sutherland, Phys. Lett. B213 (1988) 355 [13] R. Grigjanis, P. O’Donnell and M. Sutherland, Phys. Lett. B224 (1989) 209 [14] L. Abbott, Nucl. Phys. B185 (19811 189 1151 F. Gilman and M. Wise, Phys. Rev. D21 (19SOl3150 [16] D.J. Gross and F. Wilaek, Phys. Rev. Lett. 30 (1973) 1343: H.D. Poker, Phys. Rev. Lett. 30 (1973) 1346 [17] R.K. Ellis, Nucl. Phys. B106 (1976) 239: M.A. Shifman. AI. Vainshtein and V.I. Zakharov, Phys. Rev. D18 (197512583 [lS] T. Inami and C.S. Lim, Prog. Theor. Phys. 6S (1981) 297 [19] H. Georgi. Nucl. Phys. B361 (19911339 [?O] G. ‘t Hooft and M. Veltman. Nucl. Phys. B153 (1979) 365 [21] G. Passarino and M. Veltman, Nucl. Phys. B160 (1979) 151