- Email: [email protected]
On W+W− production near threshold
Physics Letters B 311 ( 1993 ) 311-316 North-Holland
PHYSICS LETTERS B
O n W + W - p r o d u c t i o n near threshold V.S. Fadin, V.A. Khoze and A.D. Martin Department of Physics, University of Durham, Durham DH1 3LE, UK Received 28 January 1993; revised manuscript received 23 April 1993 Editor: P.V. Landshoff
We study the effects of radiative corrections and W boson finite width effects on the energy dependence of the process e+e - ~ W + W - ~ flf2f3f4 in the W + W - threshold region. We pay particular attention to the effect of the W finite width on the Coulomb correction. We present a "master" formula for the cross section which is relevant to a precision determination of M w by an energy scan of W + W - production through the threshold region.
It is well known that the process e+e - ~ W + W provides a unique opportunity to probe the heart of the standard electroweak model. Indeed, observation of this process at the LEP 200 e+e - collider has long been advocated as a means of probing the triple gauge boson vertex, and of allowing a precise determination of the W boson mass, M w , via a detailed energy scan in the region of the W + W - threshold [1,2]. When combined with the known values of ~, GF and Mz, a precise measurement of M w is a classic and most stringent test of the standard model (particularly if the top quark has been found). Experiments scanning across the threshold should even yield a measurement of the W boson width l-'w, although here the precision is not expected to rival the existing (indirect) determinations. Such a precision determination of M w relies on an accurate theoretical knowledge of the energy dependence of the e+e - ~ W + W - cross section in the threshold region. That is, electroweak radiative and finite width (Fw) effects must be reliably calculated to (more than) match the expected experimental precision. There is quite an extensive literature treating various aspects of the subject, see for example refs. [ 1-6 ]. We first briefly outline the general structure of radiative corrections to W + W - production and then we draw attention to important subtleties in the Coulomb attraction between the outgoing nonrelativistic W bosons which must be included in a precision analysis. With reference to fig. 1, we may
y, S1
S2
/ F(~, s)
a0(1 +-~r a 6ww + _c~v6tout)
p(si)
Fig. 1. Pictorial description of the "master" formula (I) for e+e - ~ W + W - X ~ 4 f X'. (The radiative emissions can, of course, materialise as e + e- pairs etc.)
write the cross section for the semi-inclusive process e +e - ~ W + W - X ~ 4 f X ' in the form s a(s) = f
(v'7- v'Ti-)2 dSlp(Sl)
0
Xmax × f
/
ds2p(s2)
0
dxF(x's)cr°(s'sl's2)
0
X (1 + -~ww + ~Coul),
(1)
where v/3S and v/kS denote the c.m. energies of the decay products of the W bosons and
0370-2693/93/$ 06.00 © 1993-Elsevier Science Publishers B.V. All rights reserved
3 11
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
e+
W +
e-
W-
e+
e-
~W +
/
\W-
Fig. 2. The contributions which give the lowest order cross section, ao, for e+e - ---, W + W - .
p (si) -
x
B ( W --. f f ) 7C
v~i Fw ( si ) (si - M ~ ) 2 + s i r t ~ ( s i )
(2) '
with Fw (si) = x / ~ , F w / M w . The contributions to the lowest order cross section, or0, are shown in fig. 2. We assume that the v exchange (S wave) contribution, and therefore the threshold cross section itself, is not suppressed by a choice o f the e + beam polarizations. The finite width formula based on ao(J, si,s2) was originally presented in ref. [3] but in the absence o f all radiative corrections. Some insight into the structure of the "master" formula for a can be obtained from fig. 1. Below we will elaborate on the origin o f the various terms, as well as point out contributions that are not incorporated in (1). However, first we note the following: (i) the integral over F represents the corrections from initial state radiation; (ii) dCoul arises from the Coulomb interaction between the outgoing W bosons o f relative velocity v (in units of c); (iii) ~ w w contains all the (one-loop) radiative corrections to the lowest order cross section for e + e - ~ W + W - - apart from those already included in F and 6cout;in practice, because 6 w w is associated with short-distance physics it is well a p p r o x i m a t e d by the O ( a ) corrections to the on-shell e+e - ~ W + W - process [4]; (iv) strictly speaking we should also include radiative emissions from the virtual W bosons but they give negligible contributions in the threshold region #~ provided, o f course, we choose a physical gauge; (v) the radiative effects associated with the W ~ f f decays are included by taking the physical width Fw in the numerators of p ( s , ) ; (vi) the finite width effects of the W We thus avoid double counting, which may otherwise occur for photons from W lines which may be considered to be emitted at either the production or the decay stage. 312
29 July 1993
boson propagators are contained in the denominators of p(si), where, to simplify the treatment of gauge invariance, it is possible to approximate siF~ (si) by M 2 F 2 to O ( F 2 /M2w ) accuracy. The finite width effects lead to events below the nominal W + W - threshold and reduce or0 in the region of the peak (which occurs near v q = 200 GeV) by some 5% [3,7]. There are two reasons for this reduction. First, as compared to the zero-width approximation, the integrations over p (s,)ds~ led to a reduction due to the finite limits of integration and due to the si dependence of v ~ Fw (si). Second, it can be shown that negative corrections, also of O ( F w / M w ) , come from the collective effect of the higher order contributions in the expansion of o ( s ) / v in terms of V2" Just as for the Z line shape in e + e - collisions, initial state radiation with its logarithmic enhancements gives the largest corrections to the lowest order (Born) cross section a0. These corrections result in the integration over F (x, s) in (1). Due to the initial state radiation we sample the "hard" reaction over a range of depleted energies .2 , v/~, with =-
XIX2 S
(1--X)S,
=
(3)
where x is in the interval 0 < x < Xmax and (1 Xmax)S = (V/~ -'F t~-~)2. TO be precise, F is the "luminosity" function for producing a sub-system of energy v~: 1
1
F(x,s)=f/dxldx26(l-x-xlx2) 0
0
xe(xl,s)e(x2,s)
,
(4)
where, just as in QCD, the logarithmically enhanced terms are absorbed in electron (and positron) density functions e ( x i , s ) . These structure functions e ( x , s ) , and hence F ( x , s ) , have been calculated in refs. [ 9,10 ]. The explicit expression, to second order in r , is #2 We note that the point J = M 2 occurs in the integration region of (1). However, the contribution of the Z exchange diagram is negligibly small [5,8], assuming the proper treatment of the Z pole region.
Volume 311, n u m b e r 1,2,3,4
F(x,s)
PHYSICS LETTERS B
29 July 1993
= )~x ~-~
n2 Vo
c~ww = 81(s)
x [1 + 3 f l _ lf12 (½1n (simS) + 2zc2 - ~ ) ]
v02/4 n [ ,a0(~)_ c~ a [ F O ) ( x ' s ) - lla--M-~°X o
-fl(1-
½x) + ~flz [ 4 ( 2 - x ) l n ( 1 / x ) z~--~z- 2 (In (s/m2e) - 1) { 3 + In (~ vg)
l+3(l-x)21n(l--~)-6+x] +
(5)
= (~1 (S) -- 'U0
X
vgl4
where fl = _~__2a(In (s/m2e) - 1) .
(6)
Contributions from radiated e+e - pairs have been omitted, but they could be included if necessary. The details are given, for example, in refs. [ 9,10 ]. Their inclusion would ensure that terms of the type f12 ln(m 2) cancel out in the fully inclusive cross section. However, the e+e - pair contribution depends on the particular experimental criteria; moreover it is numerically small in the W + W - threshold region, where the available phase space for e+e - pair production is very limited. The first term in square brackets in F dominates. The double logarithmic effects appear because of the electron mass singularities, ln(vG/me), and infrared singularities, ln(x/~/AE), where AE characterises the maximal energy of initial state radiation. In fact F includes, through the exponential factor x/~ = e x p ( - f l I n ( l / x ) ), the effects o f soft photon emission to all orders in perturbation theory. We emphasize this treatment [9] of soft initial state radiation is quite universal*~3 and can be used to determine such corrections to any hard e+e - process. In summary, initial state radiation is accounted for by a function F which includes all the logarithmically enhanced terms except those connected with charge renormalization of the Born cross section tr0. Provided a0 is expressed in terms of the running coupling c~ at a scale ~ Mw, it follows that ~ww does not contain electron mass singularities and that it is given by the O(a/z~) corrections to e+e - --+ W + W - apart from those already included in 8couJ and in the initial state radiation. To be precise #3 Well a b o v e t h r e s h o l d t h e applicability o f the universal s-
channel treatment of initial state radiation is not proven for e+e - --+ W + W -, due, inter alia, to the precise choice of scale for the t-channel u-exchange contribution [11].
+
*
2
'
o
where v0 = 2V/(1 - 4 M 2 / s ) is the mass-shell value of the relative velocity v, where al represents the full one-loop electroweak corrections to the on-shell e+e - --+ W + W - process [4] as defined by
a(e+e ----, W + W - ) = ao(S) (1 + -~8,(s)) ,
(8)
and where F °) (x, s) is the first-order approximation for F (x, s). The term 7~2/Vo in (7) corresponds to the Coulomb correction if we assume that the W bosons are stable (see below), and the final term comes from initial state radiation. The strong s dependence seen on the right hand side of (7) cancels between the various terms leaving a slowly varying function aww (s). We can therefore safely assume the threshold value of 8ww in the region below threshold. We turn now to the sizeable Coulomb attraction between the slowly moving W bosons in the threshold region. Because the underlying Coulomb physics is different from the other radiative corrections it is possible to treat the Coulomb corrections separately. It was originally discovered in QED [12] that, when oppositely charged particles have low relative velocity, v << c, Coulomb effects enhance the cross section by a factor which, to leading order in c~/v, is (1 + (~z/v), provided the particles are stable. Now it has been shown [ 13] that the Coulomb effects may be radically modified when the interacting particles are short-lived rather than stable. This is the case for W bosons. We would anticipate the modification to be significant when the characteristic distance of the Coulomb interaction (dc ~ 1 / v M w ) is greater than the typical spatial separation when the diverging W bosons decay (dr ~ v/Fw). The more that dr is less than dc the more we expect the Coulomb attraction to be suppressed. If we note that ½v = v/-E/Mw, 313
Volume 311, n u m b e r 1,2,3,4
PHYSICS LETTERS B
then we see that the condition d~ ~ dc translates into E < Fw, where E is the non-relativistic energy of the W bosons. It is sufficient to calculate the Coulomb correction in the non-relativistic approximation. In the W W c.m. frame we have ~ fiCoul
_2Re{/
d 3k (27r)3k 2
4~ × [(19 + k ) 2 / M w
- E - iPw]
} '
(9)
where E is the non-relativistic energy of the W bosons, E = ~/~ - 2 M w , p is the m o m e n t u m of a virtual W , p = ½ M w v , and v is the relative velocity of the virtual W bosons, v = ((~ - sl - s2) 2 - 4sls2)ll2/2M2w.
(10)
On integrating (9) over d3k we find ~Cou¿ = 2 Re { - i l n ( ~ i - + - ip2 ip-' ~ --~-- ~p } i P) =2arctan(p~
2pp'
+ p~ _ p2
)
'
(11)
where Pl - ipz = ~ ¢ / M w ( - E - i / w ) , that is
In (11) we define arctan(0) such that 0 ~< 0 <~ n. The result (11 ) giving the Coulomb correction for unstable particles should be contrasted with the (leading order) value 6coui = n for stable particles. To calculate the Coulomb corrections for interactions between stable particles it becomes increasingly necessary to include higher order terms in ~ / v as the threshold is approached, that is, as v ~ 0. However, in the finite width case the instability of the particles prevents the large distance effects from contributing and so the first order term in a / v is sufficient. In fact from ( 11 ) we see that 6coul ---* 0 as v ~ 0. An estimate of the Coulomb contribution to the cross section (1) can be obtained by neglecting the and si dependence of ao/v in the threshold region. Then we can perform the si integrations and obtain 314
29 July 1993 oo
×
2 arctan
X / ~ ++ Fw2 +
.
(12)
For unstable particles 6Coul is a function of both the m o m e n t u m and the energy of the virtual W bosons. Therefore the quantity in square brackets, obtained after the integrations over si, is the most analogous to 6coul ( = n) for stable particles. We see that it increases monotonically from 0 well below the nominal threshold, through n / 2 at threshold, to n well above threshold. As a rough estimate we have Aacoul
a(s)
6~rc
~/(E 2 + F2)I/2/Mw
(13)
in the threshold region, where a (s) is given by (1). When IEI < Fw we see that Aacoul gives an enhanced fraction of W + W - production, which emphasizes the importance of the finite width effects of the Coulomb interaction in a precision determination of M w by an energy scan. So far our discussion of the radiative and finite width effects in e+e - ~ 4 f X ' in the W + W - threshold region has been based on the "master" formula, eq. ( 1 ). However, there are contributions which cannot be included in the factorised form of ( 1 ). We can divide these into those arising from diagrams containing a W + W - pair in an intermediate state and background contributions which do not contain two intermediate W bosons (together with their interference with the W W diagrams). We consider first the additional (interference) radiative corrections to the W W contributions. To our knowledge these interference effects have not been adequately studied so far, particularly in the important region of the W W threshold. There are three different types of interference corrections that are not, and cannot be, included in the factorised form of the master formula (1) for a ( e + e - ~ 4 f X ' ) . They are illustrated by the photons denoted by (a), (b) and (c) on fig. 3. It is easy to see, that in all cases, only soft photons (virtual as well as real) of energy k0 < Fw can lead to significant contributions. If k0 >> l"w then the photon emission pushes the final state f 7 pair far off the W resonance energy and the contribution of the diagrams shown in fig. 3 is negligible. If, on the
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
29 July 1993
frontation of ( 1 ) with W + W - production data to determine M w . The background can be reduced by an appropriate choice of the W decay channels. We conclude that the radiative and finite width corrections can be calculated with sufficient accuracy, via the master formula ( 1 ), to allow a precise comparison with W + W - production data collected in an energy scan across threshold at the LEP 200 e+e - collider and hence allow an accurate measurement of M w .
(a)
Fig. 3. Three types of interference contributions to W + W - X ---, 4 f X ' illustrated by photons (a), (b) and (c), which, in turn, represent initial-final, final-final and intermediate-final state interference, respectively. The photons are real or virtual depending on the position of the vertical dot-dashed line. e+e - ~
One of us (V.A.K.) thanks D. Bardin, M. Peskin and M. Swartz for discussions. This work was supported in part by the U K Science and Engineering Research Council. V.S.F. thanks the Centre for Particle Theory at the University of Durham for hospitality.
References +
e+
W-: e
v
Fig. 4. An example of a background contribution to e+e - --+ W+ W - - - 4 f . contrary, ko < Fw, then it appears that there will be interference contributions from real as well as virtual photons. Fortunately we have been able to explicitly verify [ 14 ] that all these "soft" real emissions are cancelled in the inclusive cross section by virtual contributions up to at least order ~ F w / M w . Therefore, it is possible to neglect the interference terms completely. Hence the contributions of all diagrams with a W W pair in an intermediate state are described to high accuracy by the master equation, eq. ( l ), for a (s). As mentioned above, there are also background contributions to e + e - --* 4 f X ' which do not contain two intermediate W bosons. An example is shown in fig. 4. These contributions (and their interference with the W W diagrams) cannot be neglected to O ( F w / M w ) , but fortunately they do not give rise to the characteristic W W threshold behaviour of eq. ( 1 ). We can therefore perform a background subtraction in a con-
[1] A. B6hm and W. Hoogland, eds., Proc. ECFA Workshop on LEP 200 (Aachen, 1986), CERN 8708, ECFA 78/108 (1987), in particular J. Bijnens et al., p. 49. [2] M.L. Swartz, in: Proc. NATO Adv. Studies Inst. on Z ° physics (Cargese, 1990), eds. M. Levy et al., p. 141. [3] T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. A 1 (1986) 203. [4] M. Lemoine and M. Veltman, Nucl. Phys. B 164 (1980) 445; M. Bohm, A. Denner, T. Sack, W. Beenacker, F. Berends and H. Kuijf, Nucl. Phys. B 304 (1988) 463; S. Dittmaier, M. B6hm and A. Denner, Nucl. Phys. B 376 (1992) 29; J. Fleischer, J.L. Kneur, K. Kolodziej, M. Kuroda and D. Schildknecht, Nucl. Phys. B 378 (1992) 443; and references therein. [5] A. Aeppli and D. Wyler, Phys. Lett. B 262 (1991) 125; A. Aeppli, in: Proc. Workshop on e+e - collisions at 500 GeV: The physical potential, ed. P.M. Zerwas, DESY report 92-123A (1992) p. 203. [6] D. Bardin, M. Bilenky, A. Olchevski and T. Riemann, DESY preprint 93-035 (March 1993). [7] G. Barbiellini et al., in: Physics at LEP, CERN Yellow Report 86-02, Vol. 2, eds. J. Ellis and R. Peccei, p. 1. [8] W. Marciano and D. Wyler, Z. Phys. C 3 (1979) 181. [9] E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. 41 (1985) 466. [I0] V.S. Fadin and V.A. Khoze, Sov. J. Nucl. Phys. 47 (1988) 1073. [ 11 ] W. Beenakkcr, K. Kolodziej and T. Sack, Phys. Lett. B 258 (1991) 469;
315
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
J. Fleischer, K. Kotodziej and F. Jegerlehner, Phys. Rev. D 47 (1993) 830; M. Cacciari, A. Deandrea, G. Montagna and O. Nicrosini, Z. Phys. C 52 (1991) 421. [12] A. Sommerfeld, Atombau und Spektrallinien, Bd. 2 (Vieweg, Braunschweig, 1939);
316
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A.D. Sakharov, JETP 18 (1948) 631. [13] V.S. Fadin and V.A. Khoze, JETP Lett. 46 (1987) 525; Sov. J. Nucl. Phys. 48 (1988) 309; in: Proc. 24th LNPI Winter School (Leningrad, 1989), Vol. 1, p. 3. [14] V.S. Fadin, V.A. Khoze and A.D. Martin, Durham preprint DTP/93/18.
PHYSICS LETTERS B
O n W + W - p r o d u c t i o n near threshold V.S. Fadin, V.A. Khoze and A.D. Martin Department of Physics, University of Durham, Durham DH1 3LE, UK Received 28 January 1993; revised manuscript received 23 April 1993 Editor: P.V. Landshoff
We study the effects of radiative corrections and W boson finite width effects on the energy dependence of the process e+e - ~ W + W - ~ flf2f3f4 in the W + W - threshold region. We pay particular attention to the effect of the W finite width on the Coulomb correction. We present a "master" formula for the cross section which is relevant to a precision determination of M w by an energy scan of W + W - production through the threshold region.
It is well known that the process e+e - ~ W + W provides a unique opportunity to probe the heart of the standard electroweak model. Indeed, observation of this process at the LEP 200 e+e - collider has long been advocated as a means of probing the triple gauge boson vertex, and of allowing a precise determination of the W boson mass, M w , via a detailed energy scan in the region of the W + W - threshold [1,2]. When combined with the known values of ~, GF and Mz, a precise measurement of M w is a classic and most stringent test of the standard model (particularly if the top quark has been found). Experiments scanning across the threshold should even yield a measurement of the W boson width l-'w, although here the precision is not expected to rival the existing (indirect) determinations. Such a precision determination of M w relies on an accurate theoretical knowledge of the energy dependence of the e+e - ~ W + W - cross section in the threshold region. That is, electroweak radiative and finite width (Fw) effects must be reliably calculated to (more than) match the expected experimental precision. There is quite an extensive literature treating various aspects of the subject, see for example refs. [ 1-6 ]. We first briefly outline the general structure of radiative corrections to W + W - production and then we draw attention to important subtleties in the Coulomb attraction between the outgoing nonrelativistic W bosons which must be included in a precision analysis. With reference to fig. 1, we may
y, S1
S2
/ F(~, s)
a0(1 +-~r a 6ww + _c~v6tout)
p(si)
Fig. 1. Pictorial description of the "master" formula (I) for e+e - ~ W + W - X ~ 4 f X'. (The radiative emissions can, of course, materialise as e + e- pairs etc.)
write the cross section for the semi-inclusive process e +e - ~ W + W - X ~ 4 f X ' in the form s a(s) = f
(v'7- v'Ti-)2 dSlp(Sl)
0
Xmax × f
/
ds2p(s2)
0
dxF(x's)cr°(s'sl's2)
0
X (1 + -~ww + ~Coul),
(1)
where v/3S and v/kS denote the c.m. energies of the decay products of the W bosons and
0370-2693/93/$ 06.00 © 1993-Elsevier Science Publishers B.V. All rights reserved
3 11
Volume 311, number 1,2,3,4
PHYSICS LETTERS B
e+
W +
e-
W-
e+
e-
~W +
/
\W-
Fig. 2. The contributions which give the lowest order cross section, ao, for e+e - ---, W + W - .
p (si) -
x
B ( W --. f f ) 7C
v~i Fw ( si ) (si - M ~ ) 2 + s i r t ~ ( s i )
(2) '
with Fw (si) = x / ~ , F w / M w . The contributions to the lowest order cross section, or0, are shown in fig. 2. We assume that the v exchange (S wave) contribution, and therefore the threshold cross section itself, is not suppressed by a choice o f the e + beam polarizations. The finite width formula based on ao(J, si,s2) was originally presented in ref. [3] but in the absence o f all radiative corrections. Some insight into the structure of the "master" formula for a can be obtained from fig. 1. Below we will elaborate on the origin o f the various terms, as well as point out contributions that are not incorporated in (1). However, first we note the following: (i) the integral over F represents the corrections from initial state radiation; (ii) dCoul arises from the Coulomb interaction between the outgoing W bosons o f relative velocity v (in units of c); (iii) ~ w w contains all the (one-loop) radiative corrections to the lowest order cross section for e + e - ~ W + W - - apart from those already included in F and 6cout;in practice, because 6 w w is associated with short-distance physics it is well a p p r o x i m a t e d by the O ( a ) corrections to the on-shell e+e - ~ W + W - process [4]; (iv) strictly speaking we should also include radiative emissions from the virtual W bosons but they give negligible contributions in the threshold region #~ provided, o f course, we choose a physical gauge; (v) the radiative effects associated with the W ~ f f decays are included by taking the physical width Fw in the numerators of p ( s , ) ; (vi) the finite width effects of the W We thus avoid double counting, which may otherwise occur for photons from W lines which may be considered to be emitted at either the production or the decay stage. 312
29 July 1993
boson propagators are contained in the denominators of p(si), where, to simplify the treatment of gauge invariance, it is possible to approximate siF~ (si) by M 2 F 2 to O ( F 2 /M2w ) accuracy. The finite width effects lead to events below the nominal W + W - threshold and reduce or0 in the region of the peak (which occurs near v q = 200 GeV) by some 5% [3,7]. There are two reasons for this reduction. First, as compared to the zero-width approximation, the integrations over p (s,)ds~ led to a reduction due to the finite limits of integration and due to the si dependence of v ~ Fw (si). Second, it can be shown that negative corrections, also of O ( F w / M w ) , come from the collective effect of the higher order contributions in the expansion of o ( s ) / v in terms of V2" Just as for the Z line shape in e + e - collisions, initial state radiation with its logarithmic enhancements gives the largest corrections to the lowest order (Born) cross section a0. These corrections result in the integration over F (x, s) in (1). Due to the initial state radiation we sample the "hard" reaction over a range of depleted energies .2 , v/~, with =-
XIX2 S
(1--X)S,
=
(3)
where x is in the interval 0 < x < Xmax and (1 Xmax)S = (V/~ -'F t~-~)2. TO be precise, F is the "luminosity" function for producing a sub-system of energy v~: 1
1
F(x,s)=f/dxldx26(l-x-xlx2) 0
0
xe(xl,s)e(x2,s)
,
(4)
where, just as in QCD, the logarithmically enhanced terms are absorbed in electron (and positron) density functions e ( x i , s ) . These structure functions e ( x , s ) , and hence F ( x , s ) , have been calculated in refs. [ 9,10 ]. The explicit expression, to second order in r , is #2 We note that the point J = M 2 occurs in the integration region of (1). However, the contribution of the Z exchange diagram is negligibly small [5,8], assuming the proper treatment of the Z pole region.
Volume 311, n u m b e r 1,2,3,4
F(x,s)
PHYSICS LETTERS B
29 July 1993
= )~x ~-~
n2 Vo
c~ww = 81(s)
x [1 + 3 f l _ lf12 (½1n (simS) + 2zc2 - ~ ) ]
v02/4 n [ ,a0(~)_ c~ a [ F O ) ( x ' s ) - lla--M-~°X o
-fl(1-
½x) + ~flz [ 4 ( 2 - x ) l n ( 1 / x ) z~--~z- 2 (In (s/m2e) - 1) { 3 + In (~ vg)
l+3(l-x)21n(l--~)-6+x] +
(5)
= (~1 (S) -- 'U0
X
vgl4
where fl = _~__2a(In (s/m2e) - 1) .
(6)
Contributions from radiated e+e - pairs have been omitted, but they could be included if necessary. The details are given, for example, in refs. [ 9,10 ]. Their inclusion would ensure that terms of the type f12 ln(m 2) cancel out in the fully inclusive cross section. However, the e+e - pair contribution depends on the particular experimental criteria; moreover it is numerically small in the W + W - threshold region, where the available phase space for e+e - pair production is very limited. The first term in square brackets in F dominates. The double logarithmic effects appear because of the electron mass singularities, ln(vG/me), and infrared singularities, ln(x/~/AE), where AE characterises the maximal energy of initial state radiation. In fact F includes, through the exponential factor x/~ = e x p ( - f l I n ( l / x ) ), the effects o f soft photon emission to all orders in perturbation theory. We emphasize this treatment [9] of soft initial state radiation is quite universal*~3 and can be used to determine such corrections to any hard e+e - process. In summary, initial state radiation is accounted for by a function F which includes all the logarithmically enhanced terms except those connected with charge renormalization of the Born cross section tr0. Provided a0 is expressed in terms of the running coupling c~ at a scale ~ Mw, it follows that ~ww does not contain electron mass singularities and that it is given by the O(a/z~) corrections to e+e - --+ W + W - apart from those already included in 8couJ and in the initial state radiation. To be precise #3 Well a b o v e t h r e s h o l d t h e applicability o f the universal s-
channel treatment of initial state radiation is not proven for e+e - --+ W + W -, due, inter alia, to the precise choice of scale for the t-channel u-exchange contribution [11].
+
*
2
'
o
where v0 = 2V/(1 - 4 M 2 / s ) is the mass-shell value of the relative velocity v, where al represents the full one-loop electroweak corrections to the on-shell e+e - --+ W + W - process [4] as defined by
a(e+e ----, W + W - ) = ao(S) (1 + -~8,(s)) ,
(8)
and where F °) (x, s) is the first-order approximation for F (x, s). The term 7~2/Vo in (7) corresponds to the Coulomb correction if we assume that the W bosons are stable (see below), and the final term comes from initial state radiation. The strong s dependence seen on the right hand side of (7) cancels between the various terms leaving a slowly varying function aww (s). We can therefore safely assume the threshold value of 8ww in the region below threshold. We turn now to the sizeable Coulomb attraction between the slowly moving W bosons in the threshold region. Because the underlying Coulomb physics is different from the other radiative corrections it is possible to treat the Coulomb corrections separately. It was originally discovered in QED [12] that, when oppositely charged particles have low relative velocity, v << c, Coulomb effects enhance the cross section by a factor which, to leading order in c~/v, is (1 + (~z/v), provided the particles are stable. Now it has been shown [ 13] that the Coulomb effects may be radically modified when the interacting particles are short-lived rather than stable. This is the case for W bosons. We would anticipate the modification to be significant when the characteristic distance of the Coulomb interaction (dc ~ 1 / v M w ) is greater than the typical spatial separation when the diverging W bosons decay (dr ~ v/Fw). The more that dr is less than dc the more we expect the Coulomb attraction to be suppressed. If we note that ½v = v/-E/Mw, 313
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then we see that the condition d~ ~ dc translates into E < Fw, where E is the non-relativistic energy of the W bosons. It is sufficient to calculate the Coulomb correction in the non-relativistic approximation. In the W W c.m. frame we have ~ fiCoul
_2Re{/
d 3k (27r)3k 2
4~ × [(19 + k ) 2 / M w
- E - iPw]
} '
(9)
where E is the non-relativistic energy of the W bosons, E = ~/~ - 2 M w , p is the m o m e n t u m of a virtual W , p = ½ M w v , and v is the relative velocity of the virtual W bosons, v = ((~ - sl - s2) 2 - 4sls2)ll2/2M2w.
(10)
On integrating (9) over d3k we find ~Cou¿ = 2 Re { - i l n ( ~ i - + - ip2 ip-' ~ --~-- ~p } i P) =2arctan(p~
2pp'
+ p~ _ p2
)
'
(11)
where Pl - ipz = ~ ¢ / M w ( - E - i / w ) , that is
In (11) we define arctan(0) such that 0 ~< 0 <~ n. The result (11 ) giving the Coulomb correction for unstable particles should be contrasted with the (leading order) value 6coui = n for stable particles. To calculate the Coulomb corrections for interactions between stable particles it becomes increasingly necessary to include higher order terms in ~ / v as the threshold is approached, that is, as v ~ 0. However, in the finite width case the instability of the particles prevents the large distance effects from contributing and so the first order term in a / v is sufficient. In fact from ( 11 ) we see that 6coul ---* 0 as v ~ 0. An estimate of the Coulomb contribution to the cross section (1) can be obtained by neglecting the and si dependence of ao/v in the threshold region. Then we can perform the si integrations and obtain 314
29 July 1993 oo
×
2 arctan
X / ~ ++ Fw2 +
.
(12)
For unstable particles 6Coul is a function of both the m o m e n t u m and the energy of the virtual W bosons. Therefore the quantity in square brackets, obtained after the integrations over si, is the most analogous to 6coul ( = n) for stable particles. We see that it increases monotonically from 0 well below the nominal threshold, through n / 2 at threshold, to n well above threshold. As a rough estimate we have Aacoul
a(s)
6~rc
~/(E 2 + F2)I/2/Mw
(13)
in the threshold region, where a (s) is given by (1). When IEI < Fw we see that Aacoul gives an enhanced fraction of W + W - production, which emphasizes the importance of the finite width effects of the Coulomb interaction in a precision determination of M w by an energy scan. So far our discussion of the radiative and finite width effects in e+e - ~ 4 f X ' in the W + W - threshold region has been based on the "master" formula, eq. ( 1 ). However, there are contributions which cannot be included in the factorised form of ( 1 ). We can divide these into those arising from diagrams containing a W + W - pair in an intermediate state and background contributions which do not contain two intermediate W bosons (together with their interference with the W W diagrams). We consider first the additional (interference) radiative corrections to the W W contributions. To our knowledge these interference effects have not been adequately studied so far, particularly in the important region of the W W threshold. There are three different types of interference corrections that are not, and cannot be, included in the factorised form of the master formula (1) for a ( e + e - ~ 4 f X ' ) . They are illustrated by the photons denoted by (a), (b) and (c) on fig. 3. It is easy to see, that in all cases, only soft photons (virtual as well as real) of energy k0 < Fw can lead to significant contributions. If k0 >> l"w then the photon emission pushes the final state f 7 pair far off the W resonance energy and the contribution of the diagrams shown in fig. 3 is negligible. If, on the
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frontation of ( 1 ) with W + W - production data to determine M w . The background can be reduced by an appropriate choice of the W decay channels. We conclude that the radiative and finite width corrections can be calculated with sufficient accuracy, via the master formula ( 1 ), to allow a precise comparison with W + W - production data collected in an energy scan across threshold at the LEP 200 e+e - collider and hence allow an accurate measurement of M w .
(a)
Fig. 3. Three types of interference contributions to W + W - X ---, 4 f X ' illustrated by photons (a), (b) and (c), which, in turn, represent initial-final, final-final and intermediate-final state interference, respectively. The photons are real or virtual depending on the position of the vertical dot-dashed line. e+e - ~
One of us (V.A.K.) thanks D. Bardin, M. Peskin and M. Swartz for discussions. This work was supported in part by the U K Science and Engineering Research Council. V.S.F. thanks the Centre for Particle Theory at the University of Durham for hospitality.
References +
e+
W-: e
v
Fig. 4. An example of a background contribution to e+e - --+ W+ W - - - 4 f . contrary, ko < Fw, then it appears that there will be interference contributions from real as well as virtual photons. Fortunately we have been able to explicitly verify [ 14 ] that all these "soft" real emissions are cancelled in the inclusive cross section by virtual contributions up to at least order ~ F w / M w . Therefore, it is possible to neglect the interference terms completely. Hence the contributions of all diagrams with a W W pair in an intermediate state are described to high accuracy by the master equation, eq. ( l ), for a (s). As mentioned above, there are also background contributions to e + e - --* 4 f X ' which do not contain two intermediate W bosons. An example is shown in fig. 4. These contributions (and their interference with the W W diagrams) cannot be neglected to O ( F w / M w ) , but fortunately they do not give rise to the characteristic W W threshold behaviour of eq. ( 1 ). We can therefore perform a background subtraction in a con-
[1] A. B6hm and W. Hoogland, eds., Proc. ECFA Workshop on LEP 200 (Aachen, 1986), CERN 8708, ECFA 78/108 (1987), in particular J. Bijnens et al., p. 49. [2] M.L. Swartz, in: Proc. NATO Adv. Studies Inst. on Z ° physics (Cargese, 1990), eds. M. Levy et al., p. 141. [3] T. Muta, R. Najima and S. Wakaizumi, Mod. Phys. Lett. A 1 (1986) 203. [4] M. Lemoine and M. Veltman, Nucl. Phys. B 164 (1980) 445; M. Bohm, A. Denner, T. Sack, W. Beenacker, F. Berends and H. Kuijf, Nucl. Phys. B 304 (1988) 463; S. Dittmaier, M. B6hm and A. Denner, Nucl. Phys. B 376 (1992) 29; J. Fleischer, J.L. Kneur, K. Kolodziej, M. Kuroda and D. Schildknecht, Nucl. Phys. B 378 (1992) 443; and references therein. [5] A. Aeppli and D. Wyler, Phys. Lett. B 262 (1991) 125; A. Aeppli, in: Proc. Workshop on e+e - collisions at 500 GeV: The physical potential, ed. P.M. Zerwas, DESY report 92-123A (1992) p. 203. [6] D. Bardin, M. Bilenky, A. Olchevski and T. Riemann, DESY preprint 93-035 (March 1993). [7] G. Barbiellini et al., in: Physics at LEP, CERN Yellow Report 86-02, Vol. 2, eds. J. Ellis and R. Peccei, p. 1. [8] W. Marciano and D. Wyler, Z. Phys. C 3 (1979) 181. [9] E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. 41 (1985) 466. [I0] V.S. Fadin and V.A. Khoze, Sov. J. Nucl. Phys. 47 (1988) 1073. [ 11 ] W. Beenakkcr, K. Kolodziej and T. Sack, Phys. Lett. B 258 (1991) 469;
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J. Fleischer, K. Kotodziej and F. Jegerlehner, Phys. Rev. D 47 (1993) 830; M. Cacciari, A. Deandrea, G. Montagna and O. Nicrosini, Z. Phys. C 52 (1991) 421. [12] A. Sommerfeld, Atombau und Spektrallinien, Bd. 2 (Vieweg, Braunschweig, 1939);
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A.D. Sakharov, JETP 18 (1948) 631. [13] V.S. Fadin and V.A. Khoze, JETP Lett. 46 (1987) 525; Sov. J. Nucl. Phys. 48 (1988) 309; in: Proc. 24th LNPI Winter School (Leningrad, 1989), Vol. 1, p. 3. [14] V.S. Fadin, V.A. Khoze and A.D. Martin, Durham preprint DTP/93/18.