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QED bremsstrahlung in semileptonic B and leptonic τ decays
Physics Letters B 303 (1993) 163-169 North-Holland
P H YSIC S LETT ER S B
QED bremsstrahlung in semileptonic B and leptonic z decays E. Richter-Wos 1 Theory Division, CERN, CH-1211 Geneva 23, Switzerland and Chair of Computer Science, Jagellonian University, ul. Reymonta 4, Cracow, Poland
Received 16 December 1992
In this paper we present semianalytical and Monte Carlo calculations of radiative corrections in semileptonic decays. For two decay channels, B +-~D°e +P(y) and T~e +vg(y), we compare numerical results on the electron spectrum from the exact analytical O (a) formula, the leading-logformula, and the approximate formula presently used in experimental analyses and Monte Carlo simulations. We estimate the physical precision of the presented results and propose a new prescription on how to treat QED bremsstrahlung effects in the B decays.
The semileptonic branching fractions play a crucial role in the data analysis of heavy flavour decays at LEP [ 1 ]. Their uncertainty affects the determination of the partial width of Z decay to the bb- pair Fbs, elements of the C K M matrix or the forwardbackward asymmetry A ~ . A typical way of precisely determining semileptonic branching fractions is to perform the comparison and fitting between data and Monte Carlo for the leptonic spectrum in the (P, PT) plane. In this procedure radiative corrections to the leptonic spectrum have to be included in the data analysis a n d / o r included in the experimental error. Up to now radiative corrections to semileptonic decays were usually not included in the experimental analysis of heavy flavour decays. The requirements for the precision o f the theoretical predictions in this case are not so high because o f large experimental errors. However, encouraged by an increasing precision in the measurement of heavy flavour processes, we have decided to discuss and test a semianalyticalMonte Carlo technique for calculating radiative corrections to leptonic spectra. In this paper, we will discuss only pure Q E D radiative corrections. Other effects such as hadronic structure effects, Q C D and weak corrections, and Coulomb threshold corrections are outside the scope o f this paper. Work supported in part by KBN grant PB 2295/2/91 and Convention IN2P3 of the French-Polish Collaboration.
The exact analytical description of radiative corrections to decays is in general rather complex, because every decay mode of every resonance or particle requires, in principle, an independent study. In certain decay channels, exact O ( o 0 calculations have been performed as, under certain assumptions, in the semileptonic decay K + ~Tr°e+~ [2] or in the pure leptonic decay z + -~e + z,# [ 3 ]. In the case of heavy flavour decays, process-independent approximate approaches for radiative corrections are still used. The universal approximate formula for the radiative corrections to the electron spectrum in heavy flavour decays was proposed in ref. [4]; it has been applied in ref. [5] to estimate the size of the bremsstrahlung effect. To calculate radiative corrections for any observable different from the electron energy spectrum, e.g. lepton PT distribution or even energy spectrum in the presence o f experimental cuts the Monte Carlo technique ( M C ) has to be applied instead of the analytical formula. Such a Monte Carlo program, which simulates bremsstrahlung in the decay of resonances in approximate O ( a ) , is available [6]; however, to our knowledge it has so far not been used or tested for heavy flavour decay processes. The aim of this paper is twofold: First, to verify the physical precision of the Monte Carlo [ 6 ] and semianalytical formula [4] for radiative corrections in decays. Secondly, to construct the leading-log-type
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
163
formula that is consistent with the Monte Carlo [ 6 ] and present them as a tool for calculating bremsstrahlung effects in experimental analysis. There are two reasons for us to compare numerical calculations for the decay channels B + ~ D ° e ±~(7) and r ± ~ e ± v~(7). First, the shapes of their lowest order decay distributions are completely different. Secondly, the exact analytical formulas for the O (o~) decay distribution for these processes are available. This makes it possible to validate different approximations. Let us consider two decay processes B + ~D°e+-O and r -+-,e ± ell The lowest order decay distributions dF°/dx, where x denotes the fraction of available energy carried by the lepton x = E f f E m , x , have the following form ~ in the decaying particle rest frame
[2,31: dF o -(B+__,DOe +~) dx G u2m 82 2q 5 x 2 ( l _ x ) 2 - ~327t ]V~bI2[fD+[ 1--rlX ' d/-O
2
s
Gum~ x 2 ( 6 - 4 x )
- ~ - ( r + - ~ e + v g ) = 192zc3
( 1)
(2)
w h e r e q = l - r n 2o / m B2, G u is the Fermi coupling constant, and If D+I denotes the decay form factor (it is assumed to be x-independent)./3m~x = ½m~ in the case of r decay and Emax= ( m ~ - m 2 ) / 2 m ~ in the case of B decay. The emission of a photon reduces the energy of the lepton and changes the shape of the leptonic spectrum. The first order approximate exponentiated forexp mula O (C~)approx for radiative corrections to the leptonic spectrum in semileptonic decays had been proposed in ref. [4]. In the case of the generic decay R + - - , Y ° I ± O ( 7 ) , the radiative corrections for the electron/muon spectrum are implemented in the form of a nearly process-independent radiationdumping factor, F ( x ) : dF dx
_
dF ° F(x), dx
(3)
~2 Inref. [2] the analyticalformulashave been calculatedfor the K+----,n°e±~,(7)processand we have applied it, as in ref. [4], to B+---,D°e +-u(?) with the necessary changesin masses and couplings. 164
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PHYSICSLETTERSB
Volume 303, number 1,2
/
\ 2 ( o ' / n ) lln
(mgx/ml)
-- 1 ]
(3 cont'd)
The parameter c is chosen in such a way that ( 1 - x) / cx becomes unity at x=F~ffEmax, where the average lepton energy Et is determined by the lowest order spectrum. The spectrum is suppressed for E l > / ~ and enhanced for E~ 0.6. The agreement is not so good in the case of B decays. For nearly the whole x region the agreement between the exact and approximate formula is rather poor, of the order of 30% of the QED correction itself. This overall precision is sufficient for a rough estimate only. Note also (figs. la, lb), that formula (3) expanded to the first order is nearly indistinguishable from the "exponentiated" one. The difference is much below 0.5%, and the exponentiation cannot explain the disagreement between the exact and approximate results. Another approximate approach for radiative corrections in decays was presented in ref. [6], where the universal algorithm for the Monte Carlo simulation of QED single-photon emission in decays was implemented in the package called PHOTOS. This program is based on the leading-log approximation for a bremsstrahlung matrix element conserving proper soft photon behaviour. It provides four-momenta of all final state particles including effects due to massive particles. Combining this program with any Born level MC the bremsstrahlung correction can be simulated on an event-by-event basis. This program has been tested before on ~ decay [7], where it was shown to work well within the required precision. However, to our knowledge it has not been tested for heavy flavour decays. In figs. 2a and 2b we compare results of radiative corrections to the electron spectrum generated with
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PHYSICS LETTERS B
Volume 303, number I,2
B*
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(4
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dx
. . . . . PHOTOS C?(CI&,,, I
0.10
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i
,
T* rest frame
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~~,~~o~,__-,~,_.-~~-~~~~“‘~.‘....’...”.’.’
0.25
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0.50
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1.0
Fig. 1. (a) Radiative correction to the decay distribution (dI’/ d_x- dI’O/dx) for B * -+D0e’8( y) in the 3 k rest frame. Open circles are from the exact analytical formula [ 21, filled circles from the approximate formula [4] expanded to O(a), Points denote results from the approximate, exponentiated formula [4]. The results are given in units of (G:mi/32n3)N,,I Vcbl*jf”, 1’ where Nn=$JA [.x2( 1 -x)~/ (1 -t]x) ]d_x. The value of the coefftcientc=(l-f)/~where~=J~[x3(l-~x)2/(1-x)]dx/ Jh[x*( 1 -qx)/( 1 -x)] dx and q= 1 -m$/mi. (b) Radiative correction to the decay distribution (dI’/dx-dT’“fdx) for r * + e * uT(y ) in the r * rest frame. Open circles are from the exact analytical formula [ 31, filled circles from the approximate formula [4] expanded to O(a). Points denote results from the approximate exponentiated formula [ 41. The results are given in units ofCgm:/192n3, with c=$.
0.25
0.50
2 = &I&w.
1.00
Fig. 2. (a) Radiative correction to the decay rate (U/dx-drol dx) for B’ -Doe%(y) in the B’ rest frame. Open circles are from the exact analytical formula [ 21, points with the marked statistical errors from PHOTOS applied to JETSET 7.3. A total of lo7 events have been generated. The results are given in units of (Gzmi/32n3)N ]V,b]2]fT I*, where N,,=q5J~[~2(1-~)2/ ( 1 - qx) ] dx and q=“l - mb/m$. (b) Radiative correction to the decay distribution (fl/dx-d.f’/d_x) for r’+e’urr(y) in the T* rest frame. Open circles are from the exact analytical formula [ 31, points with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 5 x IO6events have been generated. The results are given in units of Gzm:/192n’.
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PHOTOS and obtained from exact analytical calculations for the decay modes B+-~D°e+-g(y) and r-+~ e + v O(7). To generate decay events we have used JETSET 7.3 [8 ] ~2. PHOTOS works remarkably well for these two decay channels ~3. The slight disagreement between the analytical formula [ 2 ] and PHOTOS for the low-x region in the case of B decay is the effect of subleading terms not included in PHOTOS. The leading-log technique for QED and QCD radiative corrections has been known for a long time [ 9 ] and used successfully for many purposes [ 10]. The general idea of this approach for final-state radiation is to describe the corrected distribution in the form of a convolution of the lowest-order energy spectrum with the Lipatov-Altarelli-Parisi kernel of lepton splitting into itself and a photon. Let us consider the leptonic spectrum in the semileptonic decay of the parent charged particle R +--, Y°I+O(),), where yo is a neutral particle. The decay spectrum dFLL/dy with leading-log O (or) radiative corrections can be written in the form d F LL
I
1
0
o
-
dk~-f~pr(k)d((1-k)x-y ) ,
(4)
where the photon radiation density function reads as follows ~4.
pr(k) = (1 +71n e+ ~7)d(k) l+(1-k)
+O(k-e).½7
k
2
(5)
The x represents the lepton energy fraction before emission, and kx is the energy fraction carried by the photon. The y is the lepton energy fraction after emission,Y= E ff Emax and Emax=( m~ -- m2 ) / 2mR. There is always a freedom of choice of the nonleading contribution to 7. For our comparison we have taken two definitions for ~,:
#2 For this comparison, in the case of B decay we have replaced the matrix element actually used in JETSET 7.3 by the one used in ref. [2]. ~3 To reproduce these results, the PHOTOS version 1.06 or higher has to be used. ~4 This formula can be applied not only to the leptonic spectrum. If the spin of a radiating particle is not equal to ½one should change the splitting kernel.
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m21/2y 2 In
Y=~r
m2
) --2
(6)
'
and
a-{lnm~ - 2 )
Yc~ae = ~r \
m~
(7)
'
where t/= l - m]ffrn~. The first one resembles the result from the exact O ( a ) formula [2]. The second one comes from the direct integration over angles of the soft photon factor (for El= ½mR). After simple integration the following leading-log convolution formula is obtained from (4): dFL L dy
dF o
- (1 + y l n ~+ ~ 7 ) - 1
+ ½y
] y/(1-~)
dx
dy
l + (y/x)2 dF ° x-y dx
(8)
It is interesting to realize that this formula works equally well in the decaying panicle rest frame as well as in the laboratory frame. If the emission is to be described in the laboratory frame it is more convenient to use the variable zt=EffER to define the lepton spectrum in this frame. The lowest-order decay distribution can be easily transformed from the rest frame to the laboratory frame through the following integral: q
dF°a~=o(½(l+fl)-z,) dzt
[2/( 1+fl) ]Zl
dx/dF~°es, , xl dxt
(9)
where xt=r/x a n d f l = ~ 2 2 After this transmR/ER. formation on dF°/dx, formula (8) can be applied to calculate the radiative correction in the laboratory frame. Formula (8) can be used if the lowest-order distribution is known analytically or numerically. The appropriate simple code can be obtained from ref. [ 11 ]. It can also be applied to modify the decay spectrum of any charged particle (not necessarily a lepton). In figs. 3a, 3b we compare radiative corrections from the exact analytical formulas [3,2] and the leading-log (LL) formula (8) calculated in the decaying panicle rest frame. Two different choices of definition for y are shown to illustrate the magnitude of subleading terms, which depends on the decay
Volume 303, number 1,2 (a)
0.20
dF
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dx
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PHYSICS LETTERS B .
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Fig. 3. (a) Radiative correction to the decay distribution (dF/ d x - d F ° / d x ) for B +-~D°e +-O(y)in the B +-rest frame. Open circles are from the exact analytical formula [2], filled circles from the leading-log formula (8) with y = ( a / n ) [ l n (mB~ 2 2x /2m , )z - 2 ] . Points denote results for 7~.~d,=(a/n) [In (m~/m~) -21• The results are given in units of (G2umSs/32n3)N,l Vcb121f°+12, where N,=tlsf~[x2(1-x)2/ ( l - q x ) ] dx and q= l-mZD/rn2n. (b) Radiative correction to the decay distribution (dF/dx-d/'°/ dx ) for z -+-, e + vO(y) in the T+ rest frame. Open circles are from the exact analytical formula [ 3 ], filled circles from the leadinglog formula (8) with 7= (a/n) [In (m~x2/m~) - 2 ] . Points denote results for 7~.~ae=(a/n) [In (m~/m~) -2]• The results are given in units of G ~m ~/ 192n ~.
channel. One could easily minimize subleading terms by the p r o p e r choice o f scale in y. The i m m e d i a t e question that arises is the physical precision o f the above formula. We expect it to miss
8 April 1993
non-logarithmic O ( a ) and higher than first order leading-log corrections ~ [ ( a / n ) In (mR~me)]2 2 n, n = 2, .... The question o f the size o f subleading O (or) terms can be answered rigorously only by c o m p a r i son with exact O ( a ) results. The effect o f higher-order leading-log terms can be calculated by iteration. We have checked, following the prescription in ref. [ 12 ], that for our example decay modes, this effect is always smaller than 0.5%. In the case where there is more than one charged particle in the final state, O ( a ) interference terms affect the distributions through non-leading terms. Also for more charged particles in the final state an extra non-leading but sizeable correction factor from an additional C o u l o m b static interaction should be included [ 13 ]. We do not include or discuss these corrections here as not being within the subject o f this paper. They are expected anyway to enter only as an overall factor in the decay rate [4 ]. Note also that there is no agreement in the literature on their actual form [ 14]. In the real experimental analysis any k i n d o f analytical formulas are o f limited use in a complicated detector/cut-offs environment. Therefore, any Monte Carlo p r o g r a m which simulates radiative effects/correction is welcome. To this end P H O T O S Monte Carlo can be used. However, it is also convenient to have semianalytical formulae for necessary tests on the consistency o f the obtained results, quick calculations or fits. In this case the semi-analytical leading-log prescription can be applied. In the previous sections we have shown that these two a p p r o x i m a t e approaches work very well. F o r the considered decay channels, both o f them r e p r o d u c e d exact results with a precision o f ~ 1% on the leptonic distribution or better. These two approaches, Monte Carlo and semianalytical, can be used together to calculate and rem o v e Q E D radiative corrections in a d a t a analysis. As both o f them are based on the leading-log approximation, they can be treated as a self-checking set when the exact analytical formula is not available. In figs. 4a and 4b, we show a c o m p a r i s o n / c h e c k of P H O T O S and formula ( 8 ) for B + and z + decays in the decay rest frame, and in figs. 5a and 5b in the laboratory frame. The excellent agreement between these two approaches is observed in every case. Figs. 4 and 5 show that we control a n d u n d e r s t a n d the relations between the semi-analytical formula and 167
Volume 303, number 1,2 (a}
dr
dx
020
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Fig. 4. (a) Radiative correction to the decay distribution ( d F / d x - d / m / d x ) for B+---,DOe+-~(y) in the B -+ rest frame. Points are from the leading-log formula (8) with 7 = ( c ~ / n ) × [ I n (mstl 2 2 x 2 /m~) 2 - 2 ]. Filled circles with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 107 events have been generated. The results are given in units of (Gums~2 5 32n3)N~l Vd,121f°+ 12, where N,l=tlsf~[x2(1 - x ) 2 / (1-rlx) ] dx and q = 1 m o2/ m ~2. (b) Radiative correction to the decay distribution ( d F / d x - d / ~ / d x ) for z ± --,e ± v#(7) in the z ± rest frame. Points are from the leading-log formula (8) with y=(o~/ n ) [ In ( m,2 x Z / m 2 ) _ 2 ]. Filled circles with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 5 × 106 events have been generated. The results are given in units of GZm~ / 192n a.
Fig. 5. (a) Radiative correction to the decay distribution ( d F / dzt - dF°/dz/) for B ± --,D°e + g( y ) in the laboratory frame. Points are from the leading-log formula (8) with y = ( a / n ) × [ln ( m $ z ] / m ~ ) - 2 ] . Filled circles with the marked statistical error from PHOTOS applied to JETSET 7,3. A total of 107 events have been generated for fixed Es = 45 GeV. The results are given in units of (G2um~/32n3)N, d Vcb121f°+] z, where N,l=f~[x ~ × ( q _ x t ) 2 / ( l _ x t ) ] d x t a n d t l . . = l - t o o2~ m s 2. (b) Radiative correction to the decay distribution (dl"/d&-dI'°/dzt) for z-+--+ e ± m*(y) in the laboratory frame. Points are from the leading-log formula (8) with y= ( a / a ) [In (m*2z~/mZ,)-2]. Filled circles with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 5X 106 events have been generated for fixed E, = 45 GeV. The results are given in units of G~ m ~/ 192n 3.
PHOTOS. This comparison constitutes also a test of the technical precision of PHOTOS. The size of subleading terms could be estimated from a comparison
of the different choices of definition for y. Encouraged by the above results we propose the following prescription for calculating and removing
-
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the radiative correction in the analysed data: ( 1 ) Simulate decay events without radiative correction. (2) Simulate decay events a n d apply the P H O T O S M o n t e Carlo to generate bremsstrahlung photons in decay. ( 3 ) F i t relevant distributions with the lowest-order analytical formula and the leading-log i m p r o v e d one. ( 4 ) Take care o f the configurations where a detector does not separate the photons from the electrons. Through the redefinition o f scale in ~, include detector-induced non-leading corrections in the semi-analytical formula. (5) Whenever possible, address the question o f the genuine O ( a ) and higher order corrections and in this way estimate the size o f the corresponding physical uncertainty o f the calculated corrections. In this p a p e r we have studied the radiative corrections to the electron energy spectrum in decays. The semi-analytical leading-log formula for this correction was constructed. The discrepancy o f this form u l a with respect to the known analytical results is smaller by a factor 3 than that found with the formula from ref. [ 4 ]. In the case where analytical results are available it agrees with t h e m within an assumed precision o f ~ 1% on the electron energy distribution. A M o n t e Carlo simulation with the algorithm o f P H O T O S [ 6 ] was c o m p a r e d with the exact analytical results for the semileptonic decays o f heavy flavour hadrons for the first time. G o o d agreement (1%) has also been found. P H O T O S a n d the semi-analytical formula work in the leading-log a p p r o x i m a t i o n . Therefore, a M o n t e Carlo can be c o m b i n e d with a semi-analytical formula to b u i l d a tool for the calculation and removal o f Q E D corrections in the decay analysis. The question o f physical precision o f this technique was also addressed.
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I a m grateful to J. Carr for his initial encouragem e n t to start this study a n d to Ch. Benchouk for useful discussions. I a m also grateful to T. Sj6strand and Z. W~s for help in using their programs.
References [ 1] P. Roudeau, in: Proc. LP-HEP Conf. (Geneva, 1991 ), Vol. 2 (World Scientific, Singapore, 1992) p. 301. [2] E.S. Ginsberg, Phys. Rev. 142 (1966) 1035. [3] T. Kinoshita and A. Sidin, Phys. Rev. 113 (1959) 1652; L.M. Lifshitz and L.P. Pitaevskii, Relativistic quantum theory, Vol. 4 (Pergamon, Oxford, 1974) part 2. [4] D. Atwood and J.W. Marciano, Phys. Rev. D 41 (1990) 1736. [ 5] OPAL Collab., P.D. Acton et al., Z. Phys. 55 (1992) 191; CLEO Collab., S. Henderson et al., Phys. Rev. D 45 (1992) 2212. [6] E. Barberio, B. van Eijk and Z. W~s, Comput. Phys. Commun. 66 (1991) 115. [7] M. Je~abek, Z. Wos, S. Jadach and J.H. Kiihn, Comput. Phys. Commun. 70 (1992) 69. [8] T. Sj6strand, Comput. Phys. Commun. 39 (1986) 347; T. SjiSstrandand M. Bengtsson, Comput. Phys. Commun. 43 (1987) 367. [9] V. Gribov and L. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 675; L.N. Lipatov, Yad. Fiz. 20 (1974) 1981. [ 10 ] Yu. Dokshitzer, D.I. Dyakonov and S.I. Troyan, Phys. Rep. 58 (1980) 269; G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298. [11] E. Richter-W~s, unpublished (can be obtained from ERICHTER at CERNVM). [12 ] S. Jadach and B.F.L. Ward, The exclusive exponentiation in the Monte Carlo: The case of initial state radiation, in: Proc. Ringberg Workshop on Radiative corrections (Ringberg, 1989) p. 118. [13] E. Ginsberg, Phys. Rev. 171 (1968) 1675; 174 (1968) 2169 (E). [14] D. Atwood and J.W. Marciano, Phys. Rev. D 41 (1990) 1736; G.P. Lepage, Phys. Rev. D 42 (1990) 3251.
169
P H YSIC S LETT ER S B
QED bremsstrahlung in semileptonic B and leptonic z decays E. Richter-Wos 1 Theory Division, CERN, CH-1211 Geneva 23, Switzerland and Chair of Computer Science, Jagellonian University, ul. Reymonta 4, Cracow, Poland
Received 16 December 1992
In this paper we present semianalytical and Monte Carlo calculations of radiative corrections in semileptonic decays. For two decay channels, B +-~D°e +P(y) and T~e +vg(y), we compare numerical results on the electron spectrum from the exact analytical O (a) formula, the leading-logformula, and the approximate formula presently used in experimental analyses and Monte Carlo simulations. We estimate the physical precision of the presented results and propose a new prescription on how to treat QED bremsstrahlung effects in the B decays.
The semileptonic branching fractions play a crucial role in the data analysis of heavy flavour decays at LEP [ 1 ]. Their uncertainty affects the determination of the partial width of Z decay to the bb- pair Fbs, elements of the C K M matrix or the forwardbackward asymmetry A ~ . A typical way of precisely determining semileptonic branching fractions is to perform the comparison and fitting between data and Monte Carlo for the leptonic spectrum in the (P, PT) plane. In this procedure radiative corrections to the leptonic spectrum have to be included in the data analysis a n d / o r included in the experimental error. Up to now radiative corrections to semileptonic decays were usually not included in the experimental analysis of heavy flavour decays. The requirements for the precision o f the theoretical predictions in this case are not so high because o f large experimental errors. However, encouraged by an increasing precision in the measurement of heavy flavour processes, we have decided to discuss and test a semianalyticalMonte Carlo technique for calculating radiative corrections to leptonic spectra. In this paper, we will discuss only pure Q E D radiative corrections. Other effects such as hadronic structure effects, Q C D and weak corrections, and Coulomb threshold corrections are outside the scope o f this paper. Work supported in part by KBN grant PB 2295/2/91 and Convention IN2P3 of the French-Polish Collaboration.
The exact analytical description of radiative corrections to decays is in general rather complex, because every decay mode of every resonance or particle requires, in principle, an independent study. In certain decay channels, exact O ( o 0 calculations have been performed as, under certain assumptions, in the semileptonic decay K + ~Tr°e+~ [2] or in the pure leptonic decay z + -~e + z,# [ 3 ]. In the case of heavy flavour decays, process-independent approximate approaches for radiative corrections are still used. The universal approximate formula for the radiative corrections to the electron spectrum in heavy flavour decays was proposed in ref. [4]; it has been applied in ref. [5] to estimate the size of the bremsstrahlung effect. To calculate radiative corrections for any observable different from the electron energy spectrum, e.g. lepton PT distribution or even energy spectrum in the presence o f experimental cuts the Monte Carlo technique ( M C ) has to be applied instead of the analytical formula. Such a Monte Carlo program, which simulates bremsstrahlung in the decay of resonances in approximate O ( a ) , is available [6]; however, to our knowledge it has so far not been used or tested for heavy flavour decay processes. The aim of this paper is twofold: First, to verify the physical precision of the Monte Carlo [ 6 ] and semianalytical formula [4] for radiative corrections in decays. Secondly, to construct the leading-log-type
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
163
formula that is consistent with the Monte Carlo [ 6 ] and present them as a tool for calculating bremsstrahlung effects in experimental analysis. There are two reasons for us to compare numerical calculations for the decay channels B + ~ D ° e ±~(7) and r ± ~ e ± v~(7). First, the shapes of their lowest order decay distributions are completely different. Secondly, the exact analytical formulas for the O (o~) decay distribution for these processes are available. This makes it possible to validate different approximations. Let us consider two decay processes B + ~D°e+-O and r -+-,e ± ell The lowest order decay distributions dF°/dx, where x denotes the fraction of available energy carried by the lepton x = E f f E m , x , have the following form ~ in the decaying particle rest frame
[2,31: dF o -(B+__,DOe +~) dx G u2m 82 2q 5 x 2 ( l _ x ) 2 - ~327t ]V~bI2[fD+[ 1--rlX ' d/-O
2
s
Gum~ x 2 ( 6 - 4 x )
- ~ - ( r + - ~ e + v g ) = 192zc3
( 1)
(2)
w h e r e q = l - r n 2o / m B2, G u is the Fermi coupling constant, and If D+I denotes the decay form factor (it is assumed to be x-independent)./3m~x = ½m~ in the case of r decay and Emax= ( m ~ - m 2 ) / 2 m ~ in the case of B decay. The emission of a photon reduces the energy of the lepton and changes the shape of the leptonic spectrum. The first order approximate exponentiated forexp mula O (C~)approx for radiative corrections to the leptonic spectrum in semileptonic decays had been proposed in ref. [4]. In the case of the generic decay R + - - , Y ° I ± O ( 7 ) , the radiative corrections for the electron/muon spectrum are implemented in the form of a nearly process-independent radiationdumping factor, F ( x ) : dF dx
_
dF ° F(x), dx
(3)
~2 Inref. [2] the analyticalformulashave been calculatedfor the K+----,n°e±~,(7)processand we have applied it, as in ref. [4], to B+---,D°e +-u(?) with the necessary changesin masses and couplings. 164
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PHYSICSLETTERSB
Volume 303, number 1,2
/
\ 2 ( o ' / n ) lln
(mgx/ml)
-- 1 ]
(3 cont'd)
The parameter c is chosen in such a way that ( 1 - x) / cx becomes unity at x=F~ffEmax, where the average lepton energy Et is determined by the lowest order spectrum. The spectrum is suppressed for E l > / ~ and enhanced for E~
0.20r”““““““““‘““‘“““““‘“~
(4
c _ dx
dlT”
75
1
0.10 i
-0.10
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PHYSICS LETTERS B
Volume 303, number I,2
B*
--f
O.ZOL’,
(4
“-g
D’e*v(y)
Atwood-Marciano O(a)~$,,, . . . . . Atwood-Mar&no O(a).,,, 0 0 0 D* Ginsberg exact t)(a)
,“““,,,“,““,‘,“““,,““““‘~
B* -+ D’e*v(y)
dx
. . . . . PHOTOS L?(~),~or 0tiD00 Ginsberg exact 0(a)
0.10-
-
1
-CJ. 10
!;
-0.211
I,,,,
,,,“Ipb;,f;y:l 0.25
Ibl
(b)
Atwood-Mar&no O(CI)~&~, s. . s. Atwood-Mar&no O(a),,pp., (I0 n D0 Kinoshita-Sirlin exact 0(a)
I
0.50
T = Eel&w
1.00
T* -+ e*vP(y)
dx
. . . . . PHOTOS C?(CI&,,, I
0.10
L
Tf rest flame
i
n 0 0 0 0 Kinoshita-Sirlin exact 0(a)
i
,
T* rest frame
-0.20 L&--I_I
~~,~~o~,__-,~,_.-~~-~~~~“‘~.‘....’...”.’.’
0.25
,,,,,,,,,,,,,,,,,,,;
o.20 I
“-dg
?* --+ e*m(y)
-
0.50
2 = E,/J%,,
1.0
Fig. 1. (a) Radiative correction to the decay distribution (dI’/ d_x- dI’O/dx) for B * -+D0e’8( y) in the 3 k rest frame. Open circles are from the exact analytical formula [ 21, filled circles from the approximate formula [4] expanded to O(a), Points denote results from the approximate, exponentiated formula [4]. The results are given in units of (G:mi/32n3)N,,I Vcbl*jf”, 1’ where Nn=$JA [.x2( 1 -x)~/ (1 -t]x) ]d_x. The value of the coefftcientc=(l-f)/~where~=J~[x3(l-~x)2/(1-x)]dx/ Jh[x*( 1 -qx)/( 1 -x)] dx and q= 1 -m$/mi. (b) Radiative correction to the decay distribution (dI’/dx-dT’“fdx) for r * + e * uT(y ) in the r * rest frame. Open circles are from the exact analytical formula [ 31, filled circles from the approximate formula [4] expanded to O(a). Points denote results from the approximate exponentiated formula [ 41. The results are given in units ofCgm:/192n3, with c=$.
0.25
0.50
2 = &I&w.
1.00
Fig. 2. (a) Radiative correction to the decay rate (U/dx-drol dx) for B’ -Doe%(y) in the B’ rest frame. Open circles are from the exact analytical formula [ 21, points with the marked statistical errors from PHOTOS applied to JETSET 7.3. A total of lo7 events have been generated. The results are given in units of (Gzmi/32n3)N ]V,b]2]fT I*, where N,,=q5J~[~2(1-~)2/ ( 1 - qx) ] dx and q=“l - mb/m$. (b) Radiative correction to the decay distribution (fl/dx-d.f’/d_x) for r’+e’urr(y) in the T* rest frame. Open circles are from the exact analytical formula [ 31, points with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 5 x IO6events have been generated. The results are given in units of Gzm:/192n’.
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PHOTOS and obtained from exact analytical calculations for the decay modes B+-~D°e+-g(y) and r-+~ e + v O(7). To generate decay events we have used JETSET 7.3 [8 ] ~2. PHOTOS works remarkably well for these two decay channels ~3. The slight disagreement between the analytical formula [ 2 ] and PHOTOS for the low-x region in the case of B decay is the effect of subleading terms not included in PHOTOS. The leading-log technique for QED and QCD radiative corrections has been known for a long time [ 9 ] and used successfully for many purposes [ 10]. The general idea of this approach for final-state radiation is to describe the corrected distribution in the form of a convolution of the lowest-order energy spectrum with the Lipatov-Altarelli-Parisi kernel of lepton splitting into itself and a photon. Let us consider the leptonic spectrum in the semileptonic decay of the parent charged particle R +--, Y°I+O(),), where yo is a neutral particle. The decay spectrum dFLL/dy with leading-log O (or) radiative corrections can be written in the form d F LL
I
1
0
o
-
dk~-f~pr(k)d((1-k)x-y ) ,
(4)
where the photon radiation density function reads as follows ~4.
pr(k) = (1 +71n e+ ~7)d(k) l+(1-k)
+O(k-e).½7
k
2
(5)
The x represents the lepton energy fraction before emission, and kx is the energy fraction carried by the photon. The y is the lepton energy fraction after emission,Y= E ff Emax and Emax=( m~ -- m2 ) / 2mR. There is always a freedom of choice of the nonleading contribution to 7. For our comparison we have taken two definitions for ~,:
#2 For this comparison, in the case of B decay we have replaced the matrix element actually used in JETSET 7.3 by the one used in ref. [2]. ~3 To reproduce these results, the PHOTOS version 1.06 or higher has to be used. ~4 This formula can be applied not only to the leptonic spectrum. If the spin of a radiating particle is not equal to ½one should change the splitting kernel.
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m21/2y 2 In
Y=~r
m2
) --2
(6)
'
and
a-{lnm~ - 2 )
Yc~ae = ~r \
m~
(7)
'
where t/= l - m]ffrn~. The first one resembles the result from the exact O ( a ) formula [2]. The second one comes from the direct integration over angles of the soft photon factor (for El= ½mR). After simple integration the following leading-log convolution formula is obtained from (4): dFL L dy
dF o
- (1 + y l n ~+ ~ 7 ) - 1
+ ½y
] y/(1-~)
dx
dy
l + (y/x)2 dF ° x-y dx
(8)
It is interesting to realize that this formula works equally well in the decaying panicle rest frame as well as in the laboratory frame. If the emission is to be described in the laboratory frame it is more convenient to use the variable zt=EffER to define the lepton spectrum in this frame. The lowest-order decay distribution can be easily transformed from the rest frame to the laboratory frame through the following integral: q
dF°a~=o(½(l+fl)-z,) dzt
[2/( 1+fl) ]Zl
dx/dF~°es, , xl dxt
(9)
where xt=r/x a n d f l = ~ 2 2 After this transmR/ER. formation on dF°/dx, formula (8) can be applied to calculate the radiative correction in the laboratory frame. Formula (8) can be used if the lowest-order distribution is known analytically or numerically. The appropriate simple code can be obtained from ref. [ 11 ]. It can also be applied to modify the decay spectrum of any charged particle (not necessarily a lepton). In figs. 3a, 3b we compare radiative corrections from the exact analytical formulas [3,2] and the leading-log (LL) formula (8) calculated in the decaying panicle rest frame. Two different choices of definition for y are shown to illustrate the magnitude of subleading terms, which depends on the decay
Volume 303, number 1,2 (a)
0.20
dF
dP °
dx
dx
PHYSICS LETTERS B .
.
.
.
.
.
.
.
i ~
.
.
.
.
.
.
.
.
I
.
.
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B + __. D ° e ± p ( @ LL O(a) LL crude O(a) Ginsberg exact O(c~)
..... ..... .....
0.10
,.o
0.00
...:o .°o .°~ •
":!::..
.'
t
-0.10
B ± rest frame -0.20
(b)
0.20
dP
dF °
dx
I
0.25
0.50
T
F
x = E~/E~
l.O0
]
dx 010
i
.. .. .. .. .. .....
LLcrude O ( a ) O(ct)
Kinoshita-Sirlin exact O(a)
F 000 ~
[
-., •e "*o .0 .o
0.10 i
r
r ± rest frame
0.20 i 0.25
0. 0
:r = E ~ / E ~ ,
1.00
Fig. 3. (a) Radiative correction to the decay distribution (dF/ d x - d F ° / d x ) for B +-~D°e +-O(y)in the B +-rest frame. Open circles are from the exact analytical formula [2], filled circles from the leading-log formula (8) with y = ( a / n ) [ l n (mB~ 2 2x /2m , )z - 2 ] . Points denote results for 7~.~d,=(a/n) [In (m~/m~) -21• The results are given in units of (G2umSs/32n3)N,l Vcb121f°+12, where N,=tlsf~[x2(1-x)2/ ( l - q x ) ] dx and q= l-mZD/rn2n. (b) Radiative correction to the decay distribution (dF/dx-d/'°/ dx ) for z -+-, e + vO(y) in the T+ rest frame. Open circles are from the exact analytical formula [ 3 ], filled circles from the leadinglog formula (8) with 7= (a/n) [In (m~x2/m~) - 2 ] . Points denote results for 7~.~ae=(a/n) [In (m~/m~) -2]• The results are given in units of G ~m ~/ 192n ~.
channel. One could easily minimize subleading terms by the p r o p e r choice o f scale in y. The i m m e d i a t e question that arises is the physical precision o f the above formula. We expect it to miss
8 April 1993
non-logarithmic O ( a ) and higher than first order leading-log corrections ~ [ ( a / n ) In (mR~me)]2 2 n, n = 2, .... The question o f the size o f subleading O (or) terms can be answered rigorously only by c o m p a r i son with exact O ( a ) results. The effect o f higher-order leading-log terms can be calculated by iteration. We have checked, following the prescription in ref. [ 12 ], that for our example decay modes, this effect is always smaller than 0.5%. In the case where there is more than one charged particle in the final state, O ( a ) interference terms affect the distributions through non-leading terms. Also for more charged particles in the final state an extra non-leading but sizeable correction factor from an additional C o u l o m b static interaction should be included [ 13 ]. We do not include or discuss these corrections here as not being within the subject o f this paper. They are expected anyway to enter only as an overall factor in the decay rate [4 ]. Note also that there is no agreement in the literature on their actual form [ 14]. In the real experimental analysis any k i n d o f analytical formulas are o f limited use in a complicated detector/cut-offs environment. Therefore, any Monte Carlo p r o g r a m which simulates radiative effects/correction is welcome. To this end P H O T O S Monte Carlo can be used. However, it is also convenient to have semianalytical formulae for necessary tests on the consistency o f the obtained results, quick calculations or fits. In this case the semi-analytical leading-log prescription can be applied. In the previous sections we have shown that these two a p p r o x i m a t e approaches work very well. F o r the considered decay channels, both o f them r e p r o d u c e d exact results with a precision o f ~ 1% on the leptonic distribution or better. These two approaches, Monte Carlo and semianalytical, can be used together to calculate and rem o v e Q E D radiative corrections in a d a t a analysis. As both o f them are based on the leading-log approximation, they can be treated as a self-checking set when the exact analytical formula is not available. In figs. 4a and 4b, we show a c o m p a r i s o n / c h e c k of P H O T O S and formula ( 8 ) for B + and z + decays in the decay rest frame, and in figs. 5a and 5b in the laboratory frame. The excellent agreement between these two approaches is observed in every case. Figs. 4 and 5 show that we control a n d u n d e r s t a n d the relations between the semi-analytical formula and 167
Volume 303, number 1,2 (a}
dr
dx
020
[
~
dr ° !
r ~ .
{a)
, , ~
i
..... .....
d~1
PHOTOS O(a)~vu~o~ LL O(cQ
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0.20
dF
B:~ ~ D0e:~P(7)
dx !
0A0
PHYSICS LETTERS B
dP° -
-
d'4
B ± -+ D ° e ± ~ ( 7 )
0.10
..... ..... ,
PHOTOS O(a)a~,o, LL O(a)
t t
! .,°~
0.00
.
.
.
.
.
.
0.00
+' ~ t
-0.10[
i
010
I
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~ 0.25
.....
~
........
x = E~/E=~,
0.50
{b}
0.20 ~ ' '
I ''~'
dP dx
dF°
7-m -._+ e±m>(@
....
~l
..... .....
010
~ ~ 4 " v
~
laboratory frame
B =~rest frame
-0.20 L~. . . . . . .
dx
'~"~"
n IQ
...........
1.00
J
PHOTOS Cg(a)~n,~o, LL CO(a)
{b)
020
dF
dF°
dz I
dz l
J 0.25
T± ~
OlO ;
0.~50
zl = E~/EB
1.00
~:%~(7) . . . . . PHOTOS O(a)~n,~o~ . . . . . LL O(a)
,~t..°,''°°'.,,°.t .°
0.00
0.00
a
"°*~'.~.~...I.......°t.....*"
l
-0.10
-0.10
r ± rest frame -020
~A 0.2.5
l 0.50
L
-0.20 ~ z = E,/E,~,~
1.00
laboiatory frame
0,25
~
0.50
z~ = Ee/E,
'''
1.00
Fig. 4. (a) Radiative correction to the decay distribution ( d F / d x - d / m / d x ) for B+---,DOe+-~(y) in the B -+ rest frame. Points are from the leading-log formula (8) with 7 = ( c ~ / n ) × [ I n (mstl 2 2 x 2 /m~) 2 - 2 ]. Filled circles with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 107 events have been generated. The results are given in units of (Gums~2 5 32n3)N~l Vd,121f°+ 12, where N,l=tlsf~[x2(1 - x ) 2 / (1-rlx) ] dx and q = 1 m o2/ m ~2. (b) Radiative correction to the decay distribution ( d F / d x - d / ~ / d x ) for z ± --,e ± v#(7) in the z ± rest frame. Points are from the leading-log formula (8) with y=(o~/ n ) [ In ( m,2 x Z / m 2 ) _ 2 ]. Filled circles with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 5 × 106 events have been generated. The results are given in units of GZm~ / 192n a.
Fig. 5. (a) Radiative correction to the decay distribution ( d F / dzt - dF°/dz/) for B ± --,D°e + g( y ) in the laboratory frame. Points are from the leading-log formula (8) with y = ( a / n ) × [ln ( m $ z ] / m ~ ) - 2 ] . Filled circles with the marked statistical error from PHOTOS applied to JETSET 7,3. A total of 107 events have been generated for fixed Es = 45 GeV. The results are given in units of (G2um~/32n3)N, d Vcb121f°+] z, where N,l=f~[x ~ × ( q _ x t ) 2 / ( l _ x t ) ] d x t a n d t l . . = l - t o o2~ m s 2. (b) Radiative correction to the decay distribution (dl"/d&-dI'°/dzt) for z-+--+ e ± m*(y) in the laboratory frame. Points are from the leading-log formula (8) with y= ( a / a ) [In (m*2z~/mZ,)-2]. Filled circles with the marked statistical error from PHOTOS applied to JETSET 7.3. A total of 5X 106 events have been generated for fixed E, = 45 GeV. The results are given in units of G~ m ~/ 192n 3.
PHOTOS. This comparison constitutes also a test of the technical precision of PHOTOS. The size of subleading terms could be estimated from a comparison
of the different choices of definition for y. Encouraged by the above results we propose the following prescription for calculating and removing
-
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the radiative correction in the analysed data: ( 1 ) Simulate decay events without radiative correction. (2) Simulate decay events a n d apply the P H O T O S M o n t e Carlo to generate bremsstrahlung photons in decay. ( 3 ) F i t relevant distributions with the lowest-order analytical formula and the leading-log i m p r o v e d one. ( 4 ) Take care o f the configurations where a detector does not separate the photons from the electrons. Through the redefinition o f scale in ~, include detector-induced non-leading corrections in the semi-analytical formula. (5) Whenever possible, address the question o f the genuine O ( a ) and higher order corrections and in this way estimate the size o f the corresponding physical uncertainty o f the calculated corrections. In this p a p e r we have studied the radiative corrections to the electron energy spectrum in decays. The semi-analytical leading-log formula for this correction was constructed. The discrepancy o f this form u l a with respect to the known analytical results is smaller by a factor 3 than that found with the formula from ref. [ 4 ]. In the case where analytical results are available it agrees with t h e m within an assumed precision o f ~ 1% on the electron energy distribution. A M o n t e Carlo simulation with the algorithm o f P H O T O S [ 6 ] was c o m p a r e d with the exact analytical results for the semileptonic decays o f heavy flavour hadrons for the first time. G o o d agreement (1%) has also been found. P H O T O S a n d the semi-analytical formula work in the leading-log a p p r o x i m a t i o n . Therefore, a M o n t e Carlo can be c o m b i n e d with a semi-analytical formula to b u i l d a tool for the calculation and removal o f Q E D corrections in the decay analysis. The question o f physical precision o f this technique was also addressed.
8 April 1993
I a m grateful to J. Carr for his initial encouragem e n t to start this study a n d to Ch. Benchouk for useful discussions. I a m also grateful to T. Sj6strand and Z. W~s for help in using their programs.
References [ 1] P. Roudeau, in: Proc. LP-HEP Conf. (Geneva, 1991 ), Vol. 2 (World Scientific, Singapore, 1992) p. 301. [2] E.S. Ginsberg, Phys. Rev. 142 (1966) 1035. [3] T. Kinoshita and A. Sidin, Phys. Rev. 113 (1959) 1652; L.M. Lifshitz and L.P. Pitaevskii, Relativistic quantum theory, Vol. 4 (Pergamon, Oxford, 1974) part 2. [4] D. Atwood and J.W. Marciano, Phys. Rev. D 41 (1990) 1736. [ 5] OPAL Collab., P.D. Acton et al., Z. Phys. 55 (1992) 191; CLEO Collab., S. Henderson et al., Phys. Rev. D 45 (1992) 2212. [6] E. Barberio, B. van Eijk and Z. W~s, Comput. Phys. Commun. 66 (1991) 115. [7] M. Je~abek, Z. Wos, S. Jadach and J.H. Kiihn, Comput. Phys. Commun. 70 (1992) 69. [8] T. Sj6strand, Comput. Phys. Commun. 39 (1986) 347; T. SjiSstrandand M. Bengtsson, Comput. Phys. Commun. 43 (1987) 367. [9] V. Gribov and L. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 675; L.N. Lipatov, Yad. Fiz. 20 (1974) 1981. [ 10 ] Yu. Dokshitzer, D.I. Dyakonov and S.I. Troyan, Phys. Rep. 58 (1980) 269; G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298. [11] E. Richter-W~s, unpublished (can be obtained from ERICHTER at CERNVM). [12 ] S. Jadach and B.F.L. Ward, The exclusive exponentiation in the Monte Carlo: The case of initial state radiation, in: Proc. Ringberg Workshop on Radiative corrections (Ringberg, 1989) p. 118. [13] E. Ginsberg, Phys. Rev. 171 (1968) 1675; 174 (1968) 2169 (E). [14] D. Atwood and J.W. Marciano, Phys. Rev. D 41 (1990) 1736; G.P. Lepage, Phys. Rev. D 42 (1990) 3251.
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