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Z boson decay into a fermion-antifermion pair and an arbitrary number of hard photons
Physics Letters B 271 (1991) 261-266 North-Holland
PHYSICS LETTERS B
Z boson decay into a fermion-antifermion pair and an arbitrary number of hard photons W.J. Stirling Theory Division, CERN. CH-1211 Geneva 23, Switzerland
Received 30 August 1991
The partial widths for the decay ofa Z boson into a massless fermion pair and an arbitrary number of hard acollinear photons, F ( Z - - . f ( + n T ) , are calculated. Such decays are an important background to a variety of new physics processes involving new
particles decaying into photons and light fermions. The appropriate matrix elements are calculated exactly using "spinor techniques". Some sample numerical results are presented for n~~ 4.
1. Introduction Q E D radiative corrections to the basic e+e - ~ Z - , f ( p r o c e s s e s at LEP and SLC are of f u n d a m e n t a l i m p o r t a n c e in deriving tests o f the s t a n d a r d model. Techniques, both M o n t e Carlo and analytic, have been developed for analyzing the emission o f an a r b i t r a r y n u m b e r o f soft and collinear photons emitted from the initial and final state fermions [ 1 ]. These have an i m p o r t a n t influence on, for example, the Z line shape, the f o r w a r d - b a c k w a r d asymmetries, small-angle Bhabha scattering etc. In the absence o f cuts, these m u l t i p h o t o n soft and collinear configurations do indeed d o m i n a t e the cross section. However, at a much smaller rate energetic, well-separated ( " h a r d " ) photons can also be emitted. F o r such processes, soft and collinear leading-logarithm a p p r o x i m a t i o n s - as i m p l e m e n t e d for example in Monte Carlos and exponential form factors - are clearly no longer appropriate, a n d the correct theoretical tools are exact matrix elements. Processes involving the emission of hard photons are i m p o r t a n t not so much as a test o f QED, but rather because they are backgrounds to possible new physics processes. F o r example, the decay Z--, Z* ( --, f f ) X ( ~ TT), where X is a new massive state (for example, the standard model Higgs b o s o n ) , would give rise to a final state with a f e r m i o n - a n t i f e r m i o n pair and two well-separated photons. The same is true for the pair p r o d u c t i o n and radiative decay o f " e x c i t e d fermions": Z ~ f*(*~ ff77. In this p a p e r we calculate the matrix elements for Z - , i f + nT. By normalizing to the leading order Z - , f f d e c a y we obtain the probabilities for the emission o f an arbitrary n u m b e r o f hard photons. Obviously each p h o t o n emission "costs" a power o f c~ << 1, and so in practice we restrict our numerical estimates to n ~<4. W i t h (n + 1 ) ! F e y n m a n diagrams for an n-photon process, the calculation is only tractable using " s p i n o r techniques", where the complex a m p l i t u d e s are calculated numerically for each s p i n / h e l i c i t y configuration [2,3]. The m e t h o d we use is in fact very similar to that for the calculation o f e + e - - , n T presented in ref. [ 4 ]. C o m p a c t expressions for the matrix elements are presented in the next section. In section 3, we illustrate our results by calculating the decay probabilities for the case when the photons are kept separate from the fermions by an invariant mass cut. We consider both ~t+g - a n d h a d r o n i c (i.e., ~ q q ) final states. Permanent address: Department of Physics, University of Durham, Durham DH 1 3LE, UK. 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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Volume 271, number 1,2
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14 November 1991
Note that the one-photon calculation is trivial and can be performed analytically. Exact two-photon bremsstrahlung corrections to e+e - ~ f f w e r e first calculated in ref. [5 ], from which the final-state photon emission contributions on the Z pole could in principle be extracted.
2. Calculation of the matrix elements
The calculation of the partial decay width F ( Z ~ i f + n 7) is performed using the "spinor techniques" described in ref. [2], which should be consulted for more details about the definition of the fundamental spinors, polarization vectors, etc. Here we simply list the result. Although we are interested in the partial decay width of an on-shell Z boson, the quantity we actually calculate is the peak cross section, i.e., a(e+e -,Z--,ff+ nT) with x/~=Mz. This allows us to correctly include the correlations between the initial and final state particle momenta, as manifest for example in the forward-backward lepton asymmetry. The peak cross section is trivially related to the decay width by a ( e + e - - ~ Z ~ X ) I../~=Mz =
127rFeeF(Z-, X) 2 2 MzFto~
( 1)
By restricting ourselves to the peak cross section we can neglect the effects of initial state radiation and s-channel photon-Z interference. We can assume that in practice additional cuts will be applied to the final state particles to keep them well separated from the beam direction. Of course the actual value of the peak cross section is very sensitive to electroweak radiative corrections, but these will tend to cancel when we form the ratios of the nphoton to zero-photon cross sections. We set all fermion masses to zero, which is certainly adequate for final states where all pairs of invariant masses are required to be large. The massive fermion calculation, although more complicated, could be performed using the generalization of massless-fermion spinor techniques described in ref. [41. The momentum and helicity labelling is e - (Pi, vj ) + e + (P2, P2)~f(P3, v3) + f-(P4, 124)"Jr" 7(ki, 21) +...+ 7(kn, 2n) ,
(2)
and because of helicity conservation at the fermion-gauge-boson vertices, the independent helicity labels can be chosen to be v~, v3 and the 2i. The n-photon cross section has the generic form
,f
an=2M~n !
d~n~
~
IJ/hi 2 ,
(3)
Pl ,tJ3,)~s = ±
with
Y Pl ,/)3,,~i = +
1.I¢n[2= g4e~ n 2 2
Mz/'tot
~
[(Ve-ae)2(vf-af)21Z',(ki,2i;Pl,P2,P3,P4)l
2
2i = ±
+ (re +a¢)2(vr-af)21Z,,(ki, 2i; P2, Pl, P3, P4) 124- (Fe - a e ) Z ( v f + a f ) 2 1 Z , , ( k i , 2i; Pl, P2, P4, P3) I z 4- (re 4-ae)2(vf4-af)XlZn(ki, )~i; P2, Pl, P4, P3) [2] .
(4)
Note that g2 = x/~ G y M 2 and ae = - l, Ve= -- ~ + 2 sin20w, etc. The four terms in the above are the amplitudes for vl, v3= 4-. The complex scattering amplitude Z~ corresponds to definite helicity incoming and outgoing fermions, and arbitrary helicity photons. The (n + 1 )! Feynman diagrams for Z, are simply the n! permutations of the photons attached to the final state fermion line, with n + 1 ways of partitioning the photons on either side of the Z f f vertex, i.e., 262
Volume 271, number 1,2
PHYSICS LETTERS B
Z~(ki,).i;pl,Pz,P3,P4)=
Z
14 November 1991
~ X( m , n - m ; q i , K , ; p l , p 2 , p 3 , p 4 ) "
(5)
pe'rms.{k,2} = {q,~<} rn = 0
The amplitude X contains spinor products and propagators:
X ( m, n _ m; qi, lci ; p, , pz, p3 , p4 ) = ( _ l ) . . . . , =lTI.. , ~
1
D3 D 4 Y3 Y4
(6)
where
D3 = l~
P3 +
,
D4 =
i=1
Y3=s(p3, bl) I ]
t(a,,p3)s(p3, b,+l)+
i=i
P4 +
I~
j
i=1
q.+ l -j
j=l
t(ai, qj)s(qj, bi+l) j=J
X(t(am,P3)S(p3,P~)+~ t(am,qj)s(qj,Pl)), j=l
Y4=t(a.,P.)
I]
t(a._i, P4)S(p4, b.+,_~)+
i=1
t(a~_i,q.+,_j)s(qn+,_j,b.+,_,) d=l
n- m
X
t(p2,p4)s(P4, b,.+l)+
/
~. t(pz, q.+l j)s(q.+j_j, bm+l)
•
(7)
j=l
The massless four-vectors a~, b, are defined as follows: ~:/=+:
ai=Po, bi=q,,
i¢i=-:
ai=qi, bi=Po.
(8)
The four-vector p~ is an arbitrary, massless m o m e n t u m vector which specifies the photon gauge. The matrixelement squared is independent of Po, and any choice not parallel to the qi is acceptable. The s and t spinor products are defined by [2 ] /
0
x \|/2
s(pi, Pj) = (p~'+ipf) [P; -P)" ~ k pO _p~; j
-
(pO _p~,.']~/2
(py + ip;) ~ p O _ p ~ j
,
t(pi, pj) = - s* (pi, pj) ,
(9)
for any pair of massless four-momenta appearing in the problem. There is some arbitrariness in the definition o f s and t, but the above - with z labelling the beam direction and x and y perpendicular to it - is convenient in practice. The above expression for the cross section, although at first sight unwieldy, is in fact not difficult to program. The most attractive feature is that the same general expression is valid for any number of photons. For the phase space integration we use the R A M B O event generator [ 6 ], which returns constant phase-space weights for massless multiparticle final states. Since we use photon cuts which deliberately avoid phase-space regions where the matrix elements are singular, the overall efficiency of the numerical calculation is quite satisfactory ~. Several cross-checks on the calculation have been performed: (i) gauge invariance, (ii) comparison ~a The self-contained FORTRAN program used for these calculations is available from VXCEREN: :WJS. 263
Volume 271, number 1,2
PHYSICS LETTERSB
14 November 1991
with the analytic (n = 1 ) result, and (iii) comparison with a QED version of an existing "four-jet" program ~2 (n=2).
3. Phenomenology In practice, the exact cuts on the fermions and photons will be determined by the nature of the expected signal for which the present analysis provides a background. However, to illustrate the overall size of the decay rates it is sufficient to use a simple set of minimal cuts on the photons only. The typical cuts appropriate for this type of calculation are photon energy and photon-fermion angle cuts. These can be implemented simultaneously by using a "JADE algorithm" type invariant-mass cut [ 7 ]. Thus we introduce a dimensionless parameter Your,and require that the invariant mass of each photon-fermion pair be larger than Yx/~t Mz, i.e., (P3 q-ki)2>Ycu,M2 ,
(P4 +ki)Z>YcutM2 ,
(10)
where the k/ are the four-momenta of the photons and p~, p4u are the momenta of the final-state (massless) fermion, antifermion respectively. Note that the single parameter Ycutprotects the matrix elements from the soft and collinear singular configurations. The maximum value of Ycut for a non-zero decay rate depends on the number of photons:
1 y cm~X(n ) = 2~'
(11)
Note that we do not impose any isolation cut between the photons themselves - we assume that the photonphoton resolution is in practice much smaller than the Ycut separation of the photons and fermions. To begin with, we also do not impose any separation cut between the fermion and antifermion, as would be appropriate, for example, for f = ~t. For this reason the one-photon region corresponds to 0 ~
R.(y~u0=
F(Z-~t+~t- + nT) F(Z~.+ _) ,
(12)
as a function of Yourfor n ~ 4 . The values o~= i-~,~ Mz=91.175 OeV and sin20w=0.230 are assumed. For n = 1 the calculation can in fact be performed analytically:
el(y)= ~[log2y+½(1-y)(y_B) log(~-Y) +~(1-2y)+ 2 Cia(y)-~Tz2].
(13)
As expected, the rates decrease rapidly for increasing numbers of photons. Away from the edges of phase space, for example at y~,, = 0.01, the rates fall roughly geometrically, each additional photon reducing the decay rate by a factor of about 100, i.e. O ( 1/ a ) . At larger Yc~t,the phase space constraints are more severe for the multiphoton rates. In the limit of small Yeut, the soft-photon region dominates the R. fractions, and it is a straightforward exercise to show - using for example the techniques of ref. [ 8 ] - that the leading logarithmic behaviour is ~2 We are grateful to Z. Kunszt for providingthe computercode. 264
1 0 -a
I
14 November 199l
PHYSICS LETTERS B
Volume 271, number 1,2 ' =- I ' ''~1
I
'
' I '
'a
10 z
~
10 3
\
\
10 3
n
y ~
!
\
2
10-5
n
i
10 -~
=2
i
10_7 I
\ n =
10 7
-
10 -4
\ 10_4
q q + ""~'--.....
10~e[
n =
'
3
\ 1 0 ~9
10 -~ = 4 10 -9 ,01
[ ,02
I 05
,,,I
10
.2
.I
~
o .01
~ .02
.05
Yeut
.i
.2
5
1
ycut
Fig. 2. As for fig. 1, but for the sum of five massless quark flavours q = u, d, s, c, b. The dashed lines show the decay rates when an additional invariant mass cut is imposed between the quarks.
Fig. 1. The ratio R~=F(Z-~p.+~ - + nT)/F(Z-qa+/a - ) as a function of the invariant mass cut between the muons and photons.
Table 1 Ratios of approximate (soft-photon) to exact decay rates for different Ycutand different photon multiplicities (n).
n
Ycut
1 2 3
R . ( Y c u t ) ~ n.T
l°g2ycut
y
0.001
0.01
0.1
0.3
0.79 0.59 0.52
0.68 0.49 0.42
0.43 0.16 -
0.13 -
(14)
"
A s s t r e s s e d earlier, at large Yc.t o n l y a n e x a c t m a t r i x e l e m e n t c a l c u l a t i o n m a k e s s e n s e . W e c a n i l l u s t r a t e t h i s b y repeating the calculation using the soft-photon matrix element approximation: 2 1 i=~ I~#, [ 2-~ 1,.¢[0 I G ] - [ e
m =
{
2P3'P4
P3" ki P4" ki
.
T a b l e 1 s h o w s t h e ratio o f t h e a p p r o x i m a t e
(15) to e x a c t p h o t o n f r a c t i o n s o b t a i n e d in t h i s w a y , for d i f f e r e n t Yc,t
v a l u e s . W e s e e that, as e x p e c t e d , t h e a p p r o x i m a t i o n fails at large Yc,,. Fig. 2 s h o w s t h e p h o t o n f r a c t i o n s c a l c u l a t e d for h a d r o n i c f i n a l states, i.e., for f i v e f l a v o u r s o f ( m a s s l e s s ) q u a r k s . T h e full c u r v e is for n o s e p a r a t i o n c u t b e t w e e n t h e q u a r k s ( o n e - j e t + t w o - j e t c o n f i g u r a t i o n s ) , w h i l e t h e 265
Volume 271, number 1,2
PHYSICS LETTERS B
14 November 1991
dashed curve shows the effect of imposing the additional cut (P3 q"P4)2> YcutM~ (two-jet configuration only). Not surprisingly, the difference is largest at large );cut - at small Yourthe emission of soft photons leads naturally to a two-jet configuration. Note also that because of the factor 4 difference in electric charge squared, the multiphoton final states are d o m i n a t e d by the u a n d c quarks. Unlike fig. 1, the results for the hadronic widths in fig. 2 are subject to higher order perturbative Q C D corrections, and care must be taken in the definition of "isolated" photons. We would expect the Q C D corrections to be comparable in size to the next-to-leading order corrections to multi-jet final states. The O(o~s) corrections to the one-photon final state have recently been calculated in ref. [ 9 ]. Finally, we note that the matrix elements for the decays Z ~ q q g + nT, from which one could for example calculate the decay into a "three-jet + isolated photons" final state, can be trivially obtained from the q0 + (n + 1 )7 matrix element by adjusting the coupling constants, e 2(~+ ~) Cvgs = e 2n, and the symmetry factor, 1/ (n + 1 ) !
1/n!. Acknowledgement Useful discussions with Vincenzo Innocente, Ronald Kleiss, D a v i d Stickland and Bolek Wyslouch are gratefully acknowledged. I would like to t h a n k the Theory Division at CERN for its kind hospitality.
References [ 1] See for example D.R. Yennie, S.C. Frautschi and H. Suura, Ann. Phys. 13 ( 1961 ) 379; R. Kleiss et al., Z physicsat LEP1, CERN Yellow Report 89-08 (1989), Vol. 3, p. 1; S. Jadach, B.F.L. Ward and Z. W~ts,preprint CERN-TH.5994/91 ( 1991 ); R. Kleiss, preprint CERN-TH.6155 ( 1991 ), and references therein. [2] R. Kleiss and W.J. Stirling, Nucl. Phys. B 262 (1985) 235. [3] J.F. Gunion and Z. Kunszt, Phys. Len. B 161 (1985) 333. [4 ] R. Kleiss and W.J. Stirling, Phys. Lett. B 179 (1986) 159. [ 5 ] CALKULCollab., F.A. Berends et al., Nucl. Phys. B 264 ( 1986) 243. [6] S.D. Ellis, R. Kleiss and W.J. Stirling, Comput. Phys. Commun. 40 (1986) 359. [ 7 ] JADE Collab., S. Bethke et al., Phys. Len. B 213 (1988) 235. [8] N. Brown and W.J. Stirling, Phys. Lett. B 252 (1990) 657. [9 ] G. Kramer and B. Lampe, preprint DESY-91-078 (1991 ).
266
PHYSICS LETTERS B
Z boson decay into a fermion-antifermion pair and an arbitrary number of hard photons W.J. Stirling Theory Division, CERN. CH-1211 Geneva 23, Switzerland
Received 30 August 1991
The partial widths for the decay ofa Z boson into a massless fermion pair and an arbitrary number of hard acollinear photons, F ( Z - - . f ( + n T ) , are calculated. Such decays are an important background to a variety of new physics processes involving new
particles decaying into photons and light fermions. The appropriate matrix elements are calculated exactly using "spinor techniques". Some sample numerical results are presented for n~~ 4.
1. Introduction Q E D radiative corrections to the basic e+e - ~ Z - , f ( p r o c e s s e s at LEP and SLC are of f u n d a m e n t a l i m p o r t a n c e in deriving tests o f the s t a n d a r d model. Techniques, both M o n t e Carlo and analytic, have been developed for analyzing the emission o f an a r b i t r a r y n u m b e r o f soft and collinear photons emitted from the initial and final state fermions [ 1 ]. These have an i m p o r t a n t influence on, for example, the Z line shape, the f o r w a r d - b a c k w a r d asymmetries, small-angle Bhabha scattering etc. In the absence o f cuts, these m u l t i p h o t o n soft and collinear configurations do indeed d o m i n a t e the cross section. However, at a much smaller rate energetic, well-separated ( " h a r d " ) photons can also be emitted. F o r such processes, soft and collinear leading-logarithm a p p r o x i m a t i o n s - as i m p l e m e n t e d for example in Monte Carlos and exponential form factors - are clearly no longer appropriate, a n d the correct theoretical tools are exact matrix elements. Processes involving the emission of hard photons are i m p o r t a n t not so much as a test o f QED, but rather because they are backgrounds to possible new physics processes. F o r example, the decay Z--, Z* ( --, f f ) X ( ~ TT), where X is a new massive state (for example, the standard model Higgs b o s o n ) , would give rise to a final state with a f e r m i o n - a n t i f e r m i o n pair and two well-separated photons. The same is true for the pair p r o d u c t i o n and radiative decay o f " e x c i t e d fermions": Z ~ f*(*~ ff77. In this p a p e r we calculate the matrix elements for Z - , i f + nT. By normalizing to the leading order Z - , f f d e c a y we obtain the probabilities for the emission o f an arbitrary n u m b e r o f hard photons. Obviously each p h o t o n emission "costs" a power o f c~ << 1, and so in practice we restrict our numerical estimates to n ~<4. W i t h (n + 1 ) ! F e y n m a n diagrams for an n-photon process, the calculation is only tractable using " s p i n o r techniques", where the complex a m p l i t u d e s are calculated numerically for each s p i n / h e l i c i t y configuration [2,3]. The m e t h o d we use is in fact very similar to that for the calculation o f e + e - - , n T presented in ref. [ 4 ]. C o m p a c t expressions for the matrix elements are presented in the next section. In section 3, we illustrate our results by calculating the decay probabilities for the case when the photons are kept separate from the fermions by an invariant mass cut. We consider both ~t+g - a n d h a d r o n i c (i.e., ~ q q ) final states. Permanent address: Department of Physics, University of Durham, Durham DH 1 3LE, UK. 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
261
Volume 271, number 1,2
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14 November 1991
Note that the one-photon calculation is trivial and can be performed analytically. Exact two-photon bremsstrahlung corrections to e+e - ~ f f w e r e first calculated in ref. [5 ], from which the final-state photon emission contributions on the Z pole could in principle be extracted.
2. Calculation of the matrix elements
The calculation of the partial decay width F ( Z ~ i f + n 7) is performed using the "spinor techniques" described in ref. [2], which should be consulted for more details about the definition of the fundamental spinors, polarization vectors, etc. Here we simply list the result. Although we are interested in the partial decay width of an on-shell Z boson, the quantity we actually calculate is the peak cross section, i.e., a(e+e -,Z--,ff+ nT) with x/~=Mz. This allows us to correctly include the correlations between the initial and final state particle momenta, as manifest for example in the forward-backward lepton asymmetry. The peak cross section is trivially related to the decay width by a ( e + e - - ~ Z ~ X ) I../~=Mz =
127rFeeF(Z-, X) 2 2 MzFto~
( 1)
By restricting ourselves to the peak cross section we can neglect the effects of initial state radiation and s-channel photon-Z interference. We can assume that in practice additional cuts will be applied to the final state particles to keep them well separated from the beam direction. Of course the actual value of the peak cross section is very sensitive to electroweak radiative corrections, but these will tend to cancel when we form the ratios of the nphoton to zero-photon cross sections. We set all fermion masses to zero, which is certainly adequate for final states where all pairs of invariant masses are required to be large. The massive fermion calculation, although more complicated, could be performed using the generalization of massless-fermion spinor techniques described in ref. [41. The momentum and helicity labelling is e - (Pi, vj ) + e + (P2, P2)~f(P3, v3) + f-(P4, 124)"Jr" 7(ki, 21) +...+ 7(kn, 2n) ,
(2)
and because of helicity conservation at the fermion-gauge-boson vertices, the independent helicity labels can be chosen to be v~, v3 and the 2i. The n-photon cross section has the generic form
,f
an=2M~n !
d~n~
~
IJ/hi 2 ,
(3)
Pl ,tJ3,)~s = ±
with
Y Pl ,/)3,,~i = +
1.I¢n[2= g4e~ n 2 2
Mz/'tot
~
[(Ve-ae)2(vf-af)21Z',(ki,2i;Pl,P2,P3,P4)l
2
2i = ±
+ (re +a¢)2(vr-af)21Z,,(ki, 2i; P2, Pl, P3, P4) 124- (Fe - a e ) Z ( v f + a f ) 2 1 Z , , ( k i , 2i; Pl, P2, P4, P3) I z 4- (re 4-ae)2(vf4-af)XlZn(ki, )~i; P2, Pl, P4, P3) [2] .
(4)
Note that g2 = x/~ G y M 2 and ae = - l, Ve= -- ~ + 2 sin20w, etc. The four terms in the above are the amplitudes for vl, v3= 4-. The complex scattering amplitude Z~ corresponds to definite helicity incoming and outgoing fermions, and arbitrary helicity photons. The (n + 1 )! Feynman diagrams for Z, are simply the n! permutations of the photons attached to the final state fermion line, with n + 1 ways of partitioning the photons on either side of the Z f f vertex, i.e., 262
Volume 271, number 1,2
PHYSICS LETTERS B
Z~(ki,).i;pl,Pz,P3,P4)=
Z
14 November 1991
~ X( m , n - m ; q i , K , ; p l , p 2 , p 3 , p 4 ) "
(5)
pe'rms.{k,2} = {q,~<} rn = 0
The amplitude X contains spinor products and propagators:
X ( m, n _ m; qi, lci ; p, , pz, p3 , p4 ) = ( _ l ) . . . . , =lTI.. , ~
1
D3 D 4 Y3 Y4
(6)
where
D3 = l~
P3 +
,
D4 =
i=1
Y3=s(p3, bl) I ]
t(a,,p3)s(p3, b,+l)+
i=i
P4 +
I~
j
i=1
q.+ l -j
j=l
t(ai, qj)s(qj, bi+l) j=J
X(t(am,P3)S(p3,P~)+~ t(am,qj)s(qj,Pl)), j=l
Y4=t(a.,P.)
I]
t(a._i, P4)S(p4, b.+,_~)+
i=1
t(a~_i,q.+,_j)s(qn+,_j,b.+,_,) d=l
n- m
X
t(p2,p4)s(P4, b,.+l)+
/
~. t(pz, q.+l j)s(q.+j_j, bm+l)
•
(7)
j=l
The massless four-vectors a~, b, are defined as follows: ~:/=+:
ai=Po, bi=q,,
i¢i=-:
ai=qi, bi=Po.
(8)
The four-vector p~ is an arbitrary, massless m o m e n t u m vector which specifies the photon gauge. The matrixelement squared is independent of Po, and any choice not parallel to the qi is acceptable. The s and t spinor products are defined by [2 ] /
0
x \|/2
s(pi, Pj) = (p~'+ipf) [P; -P)" ~ k pO _p~; j
-
(pO _p~,.']~/2
(py + ip;) ~ p O _ p ~ j
,
t(pi, pj) = - s* (pi, pj) ,
(9)
for any pair of massless four-momenta appearing in the problem. There is some arbitrariness in the definition o f s and t, but the above - with z labelling the beam direction and x and y perpendicular to it - is convenient in practice. The above expression for the cross section, although at first sight unwieldy, is in fact not difficult to program. The most attractive feature is that the same general expression is valid for any number of photons. For the phase space integration we use the R A M B O event generator [ 6 ], which returns constant phase-space weights for massless multiparticle final states. Since we use photon cuts which deliberately avoid phase-space regions where the matrix elements are singular, the overall efficiency of the numerical calculation is quite satisfactory ~. Several cross-checks on the calculation have been performed: (i) gauge invariance, (ii) comparison ~a The self-contained FORTRAN program used for these calculations is available from VXCEREN: :WJS. 263
Volume 271, number 1,2
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14 November 1991
with the analytic (n = 1 ) result, and (iii) comparison with a QED version of an existing "four-jet" program ~2 (n=2).
3. Phenomenology In practice, the exact cuts on the fermions and photons will be determined by the nature of the expected signal for which the present analysis provides a background. However, to illustrate the overall size of the decay rates it is sufficient to use a simple set of minimal cuts on the photons only. The typical cuts appropriate for this type of calculation are photon energy and photon-fermion angle cuts. These can be implemented simultaneously by using a "JADE algorithm" type invariant-mass cut [ 7 ]. Thus we introduce a dimensionless parameter Your,and require that the invariant mass of each photon-fermion pair be larger than Yx/~t Mz, i.e., (P3 q-ki)2>Ycu,M2 ,
(P4 +ki)Z>YcutM2 ,
(10)
where the k/ are the four-momenta of the photons and p~, p4u are the momenta of the final-state (massless) fermion, antifermion respectively. Note that the single parameter Ycutprotects the matrix elements from the soft and collinear singular configurations. The maximum value of Ycut for a non-zero decay rate depends on the number of photons:
1 y cm~X(n ) = 2~'
(11)
Note that we do not impose any isolation cut between the photons themselves - we assume that the photonphoton resolution is in practice much smaller than the Ycut separation of the photons and fermions. To begin with, we also do not impose any separation cut between the fermion and antifermion, as would be appropriate, for example, for f = ~t. For this reason the one-photon region corresponds to 0 ~
R.(y~u0=
F(Z-~t+~t- + nT) F(Z~.+ _) ,
(12)
as a function of Yourfor n ~ 4 . The values o~= i-~,~ Mz=91.175 OeV and sin20w=0.230 are assumed. For n = 1 the calculation can in fact be performed analytically:
el(y)= ~[log2y+½(1-y)(y_B) log(~-Y) +~(1-2y)+ 2 Cia(y)-~Tz2].
(13)
As expected, the rates decrease rapidly for increasing numbers of photons. Away from the edges of phase space, for example at y~,, = 0.01, the rates fall roughly geometrically, each additional photon reducing the decay rate by a factor of about 100, i.e. O ( 1/ a ) . At larger Yc~t,the phase space constraints are more severe for the multiphoton rates. In the limit of small Yeut, the soft-photon region dominates the R. fractions, and it is a straightforward exercise to show - using for example the techniques of ref. [ 8 ] - that the leading logarithmic behaviour is ~2 We are grateful to Z. Kunszt for providingthe computercode. 264
1 0 -a
I
14 November 199l
PHYSICS LETTERS B
Volume 271, number 1,2 ' =- I ' ''~1
I
'
' I '
'a
10 z
~
10 3
\
\
10 3
n
y ~
!
\
2
10-5
n
i
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=2
i
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\ n =
10 7
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\ 10_4
q q + ""~'--.....
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,,,I
10
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1
ycut
Fig. 2. As for fig. 1, but for the sum of five massless quark flavours q = u, d, s, c, b. The dashed lines show the decay rates when an additional invariant mass cut is imposed between the quarks.
Fig. 1. The ratio R~=F(Z-~p.+~ - + nT)/F(Z-qa+/a - ) as a function of the invariant mass cut between the muons and photons.
Table 1 Ratios of approximate (soft-photon) to exact decay rates for different Ycutand different photon multiplicities (n).
n
Ycut
1 2 3
R . ( Y c u t ) ~ n.T
l°g2ycut
y
0.001
0.01
0.1
0.3
0.79 0.59 0.52
0.68 0.49 0.42
0.43 0.16 -
0.13 -
(14)
"
A s s t r e s s e d earlier, at large Yc.t o n l y a n e x a c t m a t r i x e l e m e n t c a l c u l a t i o n m a k e s s e n s e . W e c a n i l l u s t r a t e t h i s b y repeating the calculation using the soft-photon matrix element approximation: 2 1 i=~ I~#, [ 2-~ 1,.¢[0 I G ] - [ e
m =
{
2P3'P4
P3" ki P4" ki
.
T a b l e 1 s h o w s t h e ratio o f t h e a p p r o x i m a t e
(15) to e x a c t p h o t o n f r a c t i o n s o b t a i n e d in t h i s w a y , for d i f f e r e n t Yc,t
v a l u e s . W e s e e that, as e x p e c t e d , t h e a p p r o x i m a t i o n fails at large Yc,,. Fig. 2 s h o w s t h e p h o t o n f r a c t i o n s c a l c u l a t e d for h a d r o n i c f i n a l states, i.e., for f i v e f l a v o u r s o f ( m a s s l e s s ) q u a r k s . T h e full c u r v e is for n o s e p a r a t i o n c u t b e t w e e n t h e q u a r k s ( o n e - j e t + t w o - j e t c o n f i g u r a t i o n s ) , w h i l e t h e 265
Volume 271, number 1,2
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dashed curve shows the effect of imposing the additional cut (P3 q"P4)2> YcutM~ (two-jet configuration only). Not surprisingly, the difference is largest at large );cut - at small Yourthe emission of soft photons leads naturally to a two-jet configuration. Note also that because of the factor 4 difference in electric charge squared, the multiphoton final states are d o m i n a t e d by the u a n d c quarks. Unlike fig. 1, the results for the hadronic widths in fig. 2 are subject to higher order perturbative Q C D corrections, and care must be taken in the definition of "isolated" photons. We would expect the Q C D corrections to be comparable in size to the next-to-leading order corrections to multi-jet final states. The O(o~s) corrections to the one-photon final state have recently been calculated in ref. [ 9 ]. Finally, we note that the matrix elements for the decays Z ~ q q g + nT, from which one could for example calculate the decay into a "three-jet + isolated photons" final state, can be trivially obtained from the q0 + (n + 1 )7 matrix element by adjusting the coupling constants, e 2(~+ ~) Cvgs = e 2n, and the symmetry factor, 1/ (n + 1 ) !
1/n!. Acknowledgement Useful discussions with Vincenzo Innocente, Ronald Kleiss, D a v i d Stickland and Bolek Wyslouch are gratefully acknowledged. I would like to t h a n k the Theory Division at CERN for its kind hospitality.
References [ 1] See for example D.R. Yennie, S.C. Frautschi and H. Suura, Ann. Phys. 13 ( 1961 ) 379; R. Kleiss et al., Z physicsat LEP1, CERN Yellow Report 89-08 (1989), Vol. 3, p. 1; S. Jadach, B.F.L. Ward and Z. W~ts,preprint CERN-TH.5994/91 ( 1991 ); R. Kleiss, preprint CERN-TH.6155 ( 1991 ), and references therein. [2] R. Kleiss and W.J. Stirling, Nucl. Phys. B 262 (1985) 235. [3] J.F. Gunion and Z. Kunszt, Phys. Len. B 161 (1985) 333. [4 ] R. Kleiss and W.J. Stirling, Phys. Lett. B 179 (1986) 159. [ 5 ] CALKULCollab., F.A. Berends et al., Nucl. Phys. B 264 ( 1986) 243. [6] S.D. Ellis, R. Kleiss and W.J. Stirling, Comput. Phys. Commun. 40 (1986) 359. [ 7 ] JADE Collab., S. Bethke et al., Phys. Len. B 213 (1988) 235. [8] N. Brown and W.J. Stirling, Phys. Lett. B 252 (1990) 657. [9 ] G. Kramer and B. Lampe, preprint DESY-91-078 (1991 ).
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