Normalization of QCD corrections in top quark decay

8 December 1994

PHYSICS LETTERS B ELSEVIER

Physics Letters B 340 (1994) 176-180

Normalization of QCD corrections in top quark decay Brian H. Smith a, M.B, Voloshin b a School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA b Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA and Institute for Theoretical and Experimental Physics, Moscow 117259, Russia Received 23 May 1994; revised manuscript received 24 September 1994 Editor: H. Georgi

Abstract

We discuss the effects of QCD corrections to the on-shell decay t --~ bW. We resolve the scale ambiguity using the Brodsky-Lepage-Mackenzie scheme, and find that the appropriate coupling constant is asMs(0.122mt). The largest long distance contribution comes from the definition of the on-shell mass of the top quark. We note that QCD corrections to the electroweak p parameter are extremely small when the p parameter is expressed in terms of the top quark width.

There has been much discussion in the literature of QCD corrections to electroweak processes involving the top quark [ 1-4]. One of the motivations for this has been an attempt to extract as much information as possible from electroweak radiative corrections to measurements such as F ( Z --~ bb) and the p parameter. With the measurement of possible signals of top production at the Tevatron [ 5] and the observation of electroweak radiative corrections at LEP [6], there is now the exciting possibility of using a measured value of mt to extract information about the Higgs boson or new physics beyond the Standard Mode from precise electroweak data. In a previous paper [2], we have discussed QCD effects on electroweak corrections, specifically the p parameter. We showed that the QCD corrections were dominated by perturbative effects involving momentum transfers on the order of rnt. Since any top quarks appearing in radiative loops have a high virtuality, there are no sizable ff threshold corrections. The largest nonperturbative effect is associated with expressing the p parameter through the top quark

mass measured at the pole, mr. This is an uncertainty that arises when a quantity which is well defined at short distances is written in terms of a long distance parameters and brings about corrections on the order of AQCD/mt 1. When the p parameter is written in terms of quantities defined at short distances, all nonperturbative effects are suppressed by a factor of (AQcD/mt) 4. In the perturbation theory the use of the perturbatively defined on-shell mass mt results in a numerical decrease in the appropriate momentum scale for the QCD coupling as. In particular by using the method of Brodsky, Lepage, and Mackenzie [ 8] it is found [2] that the relevant coupling constant is aMsS(o.154mt) and the numerically small factor 0.154 is traced to the on-shell renormalization of the top quark mass. It was also argued [ 2 ] that a similar behavior should be expected for other processes, which are determined by the parameter mtl Alterna!

1 The intrinsic ambiguity of order AQCD in the definition of the quark on-shell mass was also recently emphasized in detail by Bigi et al. [7].

0370-2693/94/$07.00 Q 1994 Elsevier Science B.V. All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 1 2 7 0 - 9

177

B.H. Smith, M.B. Voloshin ~Physics Letters B 340 (1994) 176-180

tively, a somewhat low in units of mt normalization scale should disapper from relations between such quantities. The purpose of this letter is to illustrate this behavior by analyzing within the BLM scheme the QCD corrections for the on-shell decay rate of t --+ b W, which at the tree level is proportional to mt3. All calculations are performed in the limit mtz >> rn~. Therefore the present calculation is intended for the purpose of illustrating the relation of the normalization scale to the usage of the on-shell mass of quark, rather than for a precision phenomenological analysis of data on the top quark decay, may such data appear in future. A practical application would require more theoretical analysis, including accounting for finite ratio mw/m t2 2 and, if it comes to that, a complete two-loop calculation of the QCD correction. The basic idea of the BLM prescription is that when one changes the normalization point from as(tXl) to Crs(~2), the amplitude will gain a higher order term proportional to the number of light quark flavors. When a result is expressed in two different scales, the difference is an nf dependent higher order in o~ term. A poor choice of scale will contain a large nf dependent higher order corrections. If one wants to find an appropriate scale to use at order a~, one should calculate the nf dependent part of the o~2 correction. There exists a scale for which the nf dependent asz correction is zero. This scale roughly corresponds to the weighted average of gluon momenta. It was argued by BLM that this scale is the physically relevant scale for any given problem. It should be noted however that the BLM scheme involves only a (gauge invariant) subset of higher order graphs, and does not amount to a full-scale higher order calculation. A recent example, is the full three-loop calculation [9] of the as2 correction to the electroweak parameter p, where the BLM prescription does not completely eliminate the large coefficient of the o~2 term. The leading in as correction for the on-shell decay rate of t --, bW can be found in the literature for finite rnw [3].In all calculations, the Cabibbo-KobyashiMaSkawa matrix element Vtb is taken to be unity. In thelimit rnt2 >> rn2, the width is given to the leading order in as by [3] F o 2 - F (a) = GFm3 [ 1 - 2Ols

8--;

(1)

where F0 = v~ ~ is the tree-level width. We wish to resolve the scale ambiguity in this correction by calculating the nf dependent part of the O(ot~2) correction, and quantify this scale by using the BLM prescription. The nf dependent virtual correction can be found by considering all one gluon contributions with an additional vacuum polarization insertion made in the gluon propagators. There will also be additional bremsstrahlung contributions arising from the emission of any one of nf pairs of soft quarks. The calculation of the ny dependent two-loop amplitude is technically simplified by writing the amplitude as an integral over a fictitious gluon mass. We will now show how this is done for both the virtual loop and bremsstrahlung corrections. To simplify the virtual correction, consider writing the one-loop virtual contribution, F(. 1) in terms of an --vlrt integral over the virtual gluon momenta,

F(1) virt =

fd4ka, w(k;pi)D ( k 2)

(2)

where D ( k 2) is the gluon propagator and w(k;pi) is a weight function depending on the gluon momentum, k, and the external momenta, Pi, that can be calculated from ordinary Feynman diagrammatic techniques. The one-loop gluon vacuum polarization insertion can be made in Eq. (2) by replacing the gluon propagator with D(k2)T',.(k2), where 79r(k2), is the dimensionless gluon vacuum polarization renormalized at k 2 = -m~. It is defined in terms of its unrenormalized counterpart, 79(k2), by a subtraction at the Euclidean point k2 -- - m L ~Dr(k 2) = ~E)(k2) -- ~°(--mt2 ) .

(3)

Renormalizing in this way corresponds to the Brodsky-Lepage-Mackenzie so-called V scheme (in which as(Q) is normalized by the Coulomb potential between two infinitely massive color objects at momentum transfer Q). The nf dependent part of the two-loop virtual correction is, when expressed in the same form as Eq. (3),

~r(2) ol via =

d4kces(mt) w(k;pi)

~Or ( k 2 )

k2

•

(4)

The vacuum polarization of the gluon can be written as an integral over its imaginary part through the dispersion relation with a subtraction at k 2 = - m r2,

178

B.H. Smith, M.B. Voloshin / Physics Letters B 340 (1994) 176-180 oo

1f 79r(k2) = ¢t"

The four body phase space can be written as the integral over the product of a three body phase space and a two body phase space, so that the first bremsstrahlung contribution to the width is given by

im790,2)

0 ×

/z 2 _ k 2

/z 2 + m r 2' d/z 2.

(5)

f'~' .M2

By adding and subtracting 1//z 2 in the weight function, the dispersion relation in Eq. (5) becomes oo

k2 f

79,(k2) =

/m79(/z2)

el*2

0 oo

m2t f

Im79(~ 2)

+ ¢r J / z 2 ( / z 2 + m t2) d/'*2" 0

(6)

The first term in the dispersion integral (6) when taken together with Eq. (4) has the form of an integral over the gluon mass. The denominator in the first integrand in Eq. (6) is a massive gluon propagator. The nf dependent part of the virtual two-loop contribution to the width can be rewritten as an integral over the gluon mass in the one-loop contribution,

F~(~)(/*) o

x

d72(~) d r 3 ( m t ) diz2

27r '

0

Im79(/-.2) * 2

(8)

where A//2 is the square of the matrix element, dr2 (/~) is the two body phase space of the two tag quarks with center of mass energy /z, and d'r3(mt) is the three body phase space of the W boson, the b quark, and an on-shell "gluon" of mass/z. The phase space integral can be simplified from the form of Eq. (8) . Integration over the three body phase space will result in a factor of Fb(~)m(g), the width of the process t --+ W b g ( t z ) , where g(/z) is a gluon of mass/~. The integral over the two body phase space of the light quarks can be written in terms of the imaginary part of the gluon polarization operator. The quark hremsstrahlung contribution is

,g (2) 1 fF(1 ) 2 2 rq-brem = - 7 d brem(/Z)ImP(/~ )d/~.

o~

aF(vi~ = - 7

F q _ b r e m=. ,] ~

(9)

0

#2-~Tm,2-vi~,~, ! t 2 (7)

where r(vilr{(/~) is the virtual contribution of the

one-loop width calculated with a gluon of mass ~. Although F(v]2 (/z) is divergent as/,2 _+ 0, the second term in the integrand of Eq. (7) should be understood as having a small regularizing gluon mass. This infrared divergence will be cancelled by the bremsstrahlung terms, below. We now show that an analogous relation is true for the bremsstrahlung contribution. The ny dependent part of the O(as) bremsstrahlung correction receives contributions from two sources. The first corresponds to the emission of two soft tag quarks. The second comes from making a vacuum polarization insertion in the final state gluon propagator of normal one gluon bremsstrahlung, and corresponds to the renormalization of the gluon wave function in the process t --+ bWg. The former type of process, quark bremsstrahlung, shall be discussed first.

(It should be noted in this connection that in the standard definition of the vacuum polarization 79 (k 2) as entering the exact propagator as D(k 2) = (k 2 (1 79(k2))) -1, the contribution of a quark pair to the imaginary part Im79 is negative.) The final contribution comes from the gluon wave function renormalization in the single gluon bremsstrahlung. The n f dependent piece is calculated by making a vacuum polarization insertion in the final gluon propagator appearing in the soft bremsstrahlung process t --* bWg. The correction to the width is Fb(~e)m(0)79(0) and can be written using the dispersion integral (6) evaluated at k 2 = O. m2 m,(2 ) 1 7F(1 ) Im79" 2, VXg--brem = - ,g/-J brem I,!.1, )tz2(iz-£~mzt)alz2, 0 (10)

where Fbrem(0) (1) is the tree-level single gluon bremsstrahlung partial width. Like the virtual contribution, Eq. (7), the quark bremsstrahlung contribution, Eq. (9), has the form of

B.H. Smith, M.B. Voloshin /Physics Letters B 340 (1994) 176-180

an integral over the gluon mass. Note that the upper limit of the integration in Eq. (9) is rn] instead of in(l) ( / z2) , lS • finity. However, if the decay rate, Ybrem understood as containing a step function, i.e. Fb(~e)m(/z 2) = 0 for all /z2 > m 2, the total n f dependent two-loop contribution becomes a single integral over the gluon mass,

179

(1) 2 ~es 1 2 Fbrem(/Z) = F0~-~'[~( 5 - 18x - 7x 2) - 6(1 + x) (arctan [.-V~ +x-L~ ~ ]

x(20+2x-x 2) +

2 v ~ 4 - x)

[(1-x)x/x(4-x)] arctan

~-x--- 3-)

+¼(lO+x2)logx+½(l+x)(logx) 2. ~r(2) = - 7

1 f@2m(.)

o)

0 --

mt2

F O)

~2 + rn~

)

dlz2

(0) ImP(~ 2) ~2 •

(11)

While individual contributions may be infrared divergent; the integral in Eq. (11) is well behaved both in the infrared and ultraviolet regions. The procedure for calculating the nf dependent correction is now clear. First, we calculate the leading virtual and bremsstrahlung corrections with a fictitious gluon mass, /x. This gluon mass will also serve the purpose of regularizing any infrared divergences that might otherwise arise in intermediate results. Next, we perform a weighted integration over the gluon mass. The weight function is given in Eq. (11). The virtual massive gluon correction to the width, F(1)r, virt ~t ~ 2~ J ' is found by calculating diagrams corresponding to the dressing of the vertex, the initial state top quark propagator, and the final state b quark propagator with a massive gluon. The explicit result of this calculation is

x

- 5-(2 - x 2) logx 4 x log [ ½ ( x - ~ / ~ -

The sum of the expressions (12) and (13) is finite both in the infrared and ultraviolet regions. It is a simple matter to carry out the integration numerically. For a single light tag fermion in an Abelian theory, the imaginary part of the ghion vacuum polarization, 79(/~2) is -a/3. The proper coefficient for QCD can be obtained by replacing ImT~(k2) with a,b/4, where b = 11 - 2nf is the first coefficient in the QCD beta function. The integral in Eq. (11) can be evaluated numerically, which gives the decay rate as Ft = F0

[1

2aV(mt) (~7r2_ 5)(1

4))]

4))] ],

+ ~2.54)].

(14,

Within the BLM prescription, the appropriate norrealization point for as is chosen so that there are no ny dependent terms of O ( ~ ) . The normalization point can be found by rescaling as Eq. (14) by

aV(mt)=aV(Q) [l + 4@as(Q)log-~-2t] ,

r, = r0 [1 -

- 2 ( 1 + x ) log [ ½ ( x + V ~ -

(13)

(15)

and choosing a scale Q such that the ny dependent term in Eq. (14) is cancelled by the nf dependent term coming from the rescaling of as. Numerically, this scale is 0.281mt. Eq. (14) can be written with this new scale,

r(.~) Oz) = ro2-~ [-5(3 + 2x) vlrt + x(28 + 10x - 5x 2) ? 2x/~x) arctan

' ] ~j 2

(12)

where x is the dimensionless gluon to top mass ratio,

iz2/m~, and F0 is the tree level top width.

The bremsstrahlung term is calculated by considering the emission of a massive gluon from either of the quarks in t decay. The partial width of this process is

2

- I) aV(0.281mt)] .

(16)

The coupling constant in the above equation is written in terms of a v, which is derived from the potential between two infinitely massive quarks. This coupling constant can be related to the more conventional MS scheme by o~V(k) = aM--]-g(e-5/6k)[8]. The coupling constant in Eq. (16) is then equal to a~l~s(0.122mt). It is interesting to note that this scale is close to the scale of 0.154mr which was shown in a previous

B.H. Smith, M.B. Voloshin/ Physics Letters B 340 (1994) 176-180

180

work [2] to characterize QCD correction to the electroweak p parameter. We recall the result that the p parameter is, with a properly normalized one gluon correction, ap -

3Gem28~r2V~[1

2

+ 1)a~(0.154mt)]

(17) to leading order in m t2/mw2 QCD corrections to the top width and the p parameter are physically characterized by momenta transfer scales on the order of mt. When these physical quantities are written in terms of a mass defined at long distances (like the pole mass used above), corrections are introduced corresponding to gluon effects on momenta scales smaller than rot. This has the effect of bringing the normalization point of the coupling constant down to a lower scale. As discussed by Sirlin [4], the QCD correction in Eq. (17) can be rewritten in terms of the running MS mass, m Ms, normalized at the scale mt. The corrections to the p parameter, when expressed through this running MS mass, contain only quantities defined at short distances, and any long distance nonperturbative effects are suppressed by a factor of .

(9((AQcolmt)4). While this form may be theoretically more palatable, the quantity m Ms is not easily interpreted in a physical way. The numerical closeness of the QCD corrections to Ft and Ap suggests that it may be prudent to express Ap in terms of the top quark decay rate. F_x1. (16) and (17) can be combined in the relation

mt 7r

~

(7 _ ½~r2) c~-(0.91mt) , (18)

which illustrates our point that a numerically low momentum scale disappears from the relation between

physical quantifies sensitive only to the short-distance dynamics, unlike the case when each of these quantifies is separately related to the on-shell top mass. If these short distance corrections are written in terms of quantities defined at longer distances like the pole mass, there is also a non-perturbative uncertainty of (.9(AQCD~rot) associated with relating a long range parameter to a short range one. This uncertainty is minimized by writing short distance parameters in terms of each other, like writing the p parameter in terms of

Ft/mt. This work was supported in part by DOE grant DEAC-02-83ER40105. BHS is supported by the University of Minnesota Doctoral Dissertation Fellowship program.

References [1] A. Djouadi and C. Verzegnassi,Phys. Lett. B 195 (1987) 265. [2] B.H. Smith and M.B. Voloshin, Univ. Minnesota report, UMN-TH-1241/94, TPI-MINN-94/5-T, January 1994, submitted to Phys. Rev. D. [3] N. Cabibbo, G. Corbo and L. Maiani, Nucl. Phys. B 155 (1979) 93; M. Jezabekand J.H. Kuhn, Nncl. Phys. B 314 (1989) 1; B 320 (1989) 20; Phys. Lett. B 207 (1988) 91; C.S. Li, R.J. Oakes and T.C. Yuan, Phys. Rev. D 47 (1991) 3759. [4] A. SMin, QCD Corrections to 6p, HEPPH-9403282,March 1994. [5] CDF Collaboration, Fermilab report, FERMILAB-PUB-94097-E, April 1994. [6] V.A.Novikov,L.B. Okun, A.N. Rozanovand M.I. Vysotsky, CERN report, CERN-TH.7217/94, April 1994. [7] I. Bigi et al., Univ. Minnesota report TPI-MINN-94/4-T, 1994. [8] S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, Phys. Rev. D 28 (1983) 228. [9] L. Avdeevet al., Univ. Bielefeld report BI-TP-93-60, 1994.