Bhabha scattering at high energy

Bhabha scattering at high energy

Computer Physics Communications 48 (1988) 335—351 North-Holland, Amsterdam 335 BHABHA SCATFERING AT HIGH ENERGY s~ S. KURODA b a T. KAMITANI b K...

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Computer Physics Communications 48 (1988) 335—351 North-Holland, Amsterdam

335

BHABHA SCATFERING AT HIGH ENERGY

s~

S. KURODA b

a T. KAMITANI b K. TOBIMATSU c KAWABATA Physics Department, Nagoya University, Chikusa.ku Nagoya 464, Japan Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan

and Y. SHIMIZU

Faculty of General Education, Meiji-Gakuin University, Totsuka.ku Yokohama 244, Japan and National Laboratory for High Energy Physics (KEK), Oho, Tsukuba 305, Japan Received 1 August 1987

Bhabha scattering in the energy region around the Z° boson has been studied. The calculation is performed in the standard electroweak theory and contains all tV(a) radiative corrections, i.e. virtual, soft photon and hard photon corrections. Some realistic kinematical Cuts are imposed. It is found that the correction around the Z° pole is sensitive to Mz. An event generator of the Bhabha scattering has also been developed. By using this, experimentally interesting distributions are generated.

1. Introduction At the energy of PEP and PETRA the Bhabha scattering has been used to check QED up to 3). Now the energy of the colliding beam t12(a machines TRISTAN, SLC and LEP will be around the Z° mass and in this energy region the weak interaction becomes important. Our interest is whether the standard electroweak theory can correctly describe the Bhabha process. In order to compare the theory with experiments the radiative correction to the process is required. The radiative correction to the Bhabha scattering has been calculated by many authors in this decade. The complete calculations in QED have been given in refs. [1—3].In the electroweak theory, several works are published in refs. [4—9]. The calculation of £!2(a) radiative corrections including hard photon emission is completed by two of the present authors [9]. We studied the Z° mass dependence of the radiative corrections and the corrected total cross sections around the Z°pole, the important quantities to be measured by experiments at SLC and LEP. In the calculation the on-shell renormaliza-

tion scheme [10] was adopted, where we used the masses of Z° and W ± bosons, M~and M~,as the parameters of the weak interaction. This choice of parameters is quite natural because these masses will be directly measured with high precision in future experiments. By using the Fermi coupling constant GF, however, M~ can be calculated from M~. In ref. [11] a table of the relations between M~and M~with the one loop electroweak corrections is given. We use this table for fixing M~in our calculation. From the experimental point of view, event generators are indispensable, because in data analysis one must know about the detector acceptance, detection efficiency of photon and electron, position and energy resolutions of detected photons and electrons etc. These quantities cannot be studied without the event generator of the process. We have developed an event generator for Bhabha scattering. In sections 2 and 3 the elastic process and the hard photon emission process are described. The results and method of the event generation are given in section 4. In the last section discussion and conclusion are presented.

0010-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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2. The elastic process

CL

ct,.

N~ZNN~(7 The lowest order Feynman diagrams of the process

~

;z

(

(1)

are shown in fig. 1. The differential cross section is given by _

dG° d cos 9

2

(s

‘ira

—~-—(T

=

0 ~+ T~t)+

(2)

Tjmflt))

where

Fig. 1. The Feynman diagrams of Bhabha scattering.

2s(s—M~) (s—Mfl2+M~F~

28+ T~~1+cos

and

X(C~(1+cos9)+2C~cos9}

s=(p++p_)2=w2,

The angle 0 is defined as the polar angle of the scattering positron momentum with respect to the initial positron. In the above expression we neglect the electron mass me and include the finite decay width of the Z° boson, F~.In order to fix the renormalization point we use the fermion masses, Z° boson mass M~,W~boson mass M~,Higgs boson mass MH and fine structure constant a as parameters. C~and CA are the vector and axial vector couplings at the eeZ° vertex, respectively and defined by ____

+

(s

~



)2

+

M~F~

x {(c~ + C~)2(1+ cos28) +8C~C~ cos 2

T~t) =

{

— ~

t2

e},

I 1 + cos ~ ~



2 /

+

(

t t

2s2 Mfl



48 x((C~_C~)+(C~+C~)cos

~) \

((c~_c~)2

s2

+

C~M~(3_ 4M/(4M~M2M2) M~j CA=M~/(4MZ~M~-M~).

(t-Mfl2 +

Td int)

=

{

6C~C~ + C~)cos4~)}~

4 s-+1I t

t

s —

M~+

s

t

(s

\

M~) Mfl2 + M~F~)

S (s —

t=(q~—p~)2.



The one loop diagrams of the process (1) give the virtual corrections which come from vertex corrections, box diagrams and vacuum polarizations. In the framework of the electroweak theory the cornplete formulas of all these corrections have appeared for the process e~e ~ in ref. [121, and are used in our calculation. The resulting —

X (C~.+ C~)

(s M~) t—M~ (s—Mfl2+M~F~ .~

+



x(C~+6c~C~+ C2)}

4.~. 2

virtual correction to the lowest order cross section is denoted by dcos 0

=

dcos o~(0, X),

(4)

where X is a fictitious photon mass. The explicit expression of S~is too long and complicated to be

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Bhabha scattering at high energy

described here. It has been derived by using the method given in ref. [12]. In order to regularize the infrared divergence we have to include the soft photon correction ~ with an energy cut k~.The cross section of the soft photon emission is given by da d cos 0

=

.1

~sa0 ~(9 d cos 0

A, k~)

and dt

Sp(z)

f —~—ln(1



(5)

written as da (elastic) d cos 0

2~ —~

0

d a° d ~

=

2k

d cos 0(1





0(1 + &.~(0, A)

+

8~~(0, k~)).

+

andmassesoftheu,d,s,c,bandtquarktobe 0.1, 0.1, 0.5, 1.5, 5.0 and 40.0 GeV, respectively.

.

+Sp(cos2~) Sp(sin2~) —



terms

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W=6OGeV

k~=1GeV~

-0.5

0.5 GeV

~O.1GeV

—1.5 -

—1

(6)

Note ~ k~)does not depend on the photon mass A. The k~dependence of ~ is shown in fig. 2. Throughout this paper we use MH = 200 GeV

1 —1n-~ 2 me

0 0 —2 ln(cot~)ln(cos~s1n~)

+ additional

~(0, A~ku))

da°

~2(ln_~_ —2 ln(cot~) i) ln—f

+ln—~ 1 me

t).



Unless k~<
A, k~)

=

=

0

where ~

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0.5

cos6 Fig. 2. The cos 8 distributions of ~ with (a) W = 60 GeY, (b) 93 GeV and (c) 150 GeV.

-

1

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Bhabha scattering at high energy

X

3. The process of hard photon emission

:~ (7) are given in fig. 3. Denoting the amplitude by we define the squared matrix element T~1~ ~ T~’°T~1~~,

e6r~’ñ=

~

~—

~

~

~

~

~

(8)

spin p01.

where e is the electric charge. In the calculation of traces over Dirac matrices the symbolic manipulation program REDUCE [14] was used. We retain all the electron mass me. The one photon emission cross section is given by

I

~

\/\A1

~

1

Vrei(2P+o)(2P_ 0) 3

3

3

Fig. 3. The Feynman diagrams of the process e~e

—~e~e

y.

dq~ 32q dq_ 3 2k(21T)3 dk 2q~0(2’ir)

0(2~) f~ \4~4/ X i,.crr, ~ ‘~p+±p——

(9~ / ~/ where Vrel = 2. The integrand T (1~ has several singularities in the phase volume. The first type of them is the mass singularity of electron and positron. For example, when the initial electron emits the photon along its direction, the denominator of the electron propagator takes its minimum value proportional to m~.The second type is the s ‘-singularity. When the photon is emitted with its maximum energy from the initial electron or positron, s’ (q~+ q)2 takes its minimum value 4m~.When the positron or electron is scattered into the initial direction, the third type of singularity appears, where the invariant t (q~—p~)2 (t’ (q_ —p)2) is exactly 0 in the limit of zero photon energy. We call this I ‘)-singularity. This is just the Coulomb singularity. The last type is the infrared singularity, which is avoided by introducing a photon energy cut k~but still gives a sharp rise at k = k~. As the independent variables for the phase —



kle6T~~ ‘

.

space integration, variables k, cos 0.~,,cos Q and c~ are taken, which are illustrated in fig. 4 (we do not consider the polarized beams). By this set of variables we can easily describe the locations of singularities, i.e., we find the initial mass singularities at cos = ±1, the final ones at cos 9~ ±1, the s’-singularity at k = km~= (W2 4m~)/2Wand the infrared one at k = k~.Eq. (9) can be rewrit—

k

X

Fig. 4. A schematic figure to illustrate integral variables.

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0.75

4—

0.50





0.25



~1—

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a 0

~O

--~

—0.25

~T

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—0.5

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---

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-~

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b—

0.5

cos6 Fig. 5. The cos 8 distributions of b~,jat (a) W = 60 GeV, (b) 93 GeV and (c) 150 GeV. Dotted, solid and dashed histograms are the cases Mz = 90 GeV, 93 GeV and 96 GeV, respectively.

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0.00

-~

—0.05

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-



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‘0

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cos6 Fig. 5. Continued.

ten as follows:

which we obtain the differential cross section

a3

klq÷I q÷ 0 2 w k (i —cos ~± 0 4’rrW Xdkdcos~dcos0vdwT~’~, —

da~~/dcos0.



(10)

Then the corrected differential cross section is da(t0t5~ dat55t~ da~’~~ dcos0dcos0+dcos0

where =

do° d cos 0(1

+

~

(12)

q± 0= {ww_ k)(W— 2k)

)

8~(o) = 6~~(0, k~)+ ~h(0, k~), 2_4m~(W_k)2

±kcos 0~[W2(W_2k) 1/2 +4m~k2cos2O~]

/2{(W_ k)2



k2 cos2Oy}.

(11)

Here, the minus sign should be taken for kc ~ k ~ W( W 2 me )/ 2( W m~) and both signs for W(W— 2m~)/2(W— m~)
6h(0, k~)~da~/da°.

(13) (14)

Here 8h is the hard photon correction and 6~a1 is sum of three parts, i.e., virtual, soft and hard photon corrections. Note ~total(0) should not depend on k~.



4. The results and event generation In order to study the M~dependence of the radiative corrections, we choose values 90, 93, 96

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(c) 2.0

(b)

0. 1

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90

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100

110

V~ (GeV) Fig. 6. The total cross section as a function of ~ with (a) Mz = 90 GeV, (b) 93 GeV and (c) 96 GeV. The dotted and solid curve are the lowest and corrected cross section, respectively. 60

-

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40—

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‘—

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20—

fl

0—

IIIi



0 E-~

—20









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60

“.----_~“.-.

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~ Fig. 7. The total radiative correction ~ TOT as a function of

~

with M

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(GeV)

5 = 90 GeV (dotted curve), 93 GeV (solid curve) and 96 GeV (dashed curve).

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0eV for M~ and 2.5 GeV for F~.Further we impose a polar angle cut 300 ~ 0 ~ 1500 and an acollinearity angle cut ~ = 100 upon the final electron and positron. The acollinearity angle is defined as the complementary angle between out-

of Born cross sections. One can see the radiative tail in the figure. We show the total radiative correction ~ TOT in fig. 7. ~ TOT is defined by ~

going electron and positron. Fig. 5 shows the cos dependence the radiative correction 6total 8(cos 0) at three of different CM energies (a) 60 GeV, (b) 93 GeV, (c) 150 GeV. The M~dependence is remarkable in fig. Sb but not in fig. Sa and fig. Sc. In fig. Sb the percentage correction for M~= 90 GeV is fairly large due to the radiative tail. In fig. 6 we show the total cross section a (total) as a function of CM energy from 80 GeV to 110 GeV with (a) M~= 90 GeV, (b) M~= 93 GeV and (c) M~= 96 GeV. The dotted lines and the solid lines show the total Born cross sections and corrected ones, respectively. The peak of the total Born cross section is located below the Z° pole, about 240, 220 and 190 MeV lower for M~= 90, 93 and 96 GeV. The peak of corrected ones slightly shifts (about 150 MeV higher) compared to those

0.4

~ (/1 0 C)

I

0.3



0.2



0.1



I

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I I

I

TOT =

d d~i° cos 0 d cos 0,

0,,,~

0(total)/f

where we use 0~ = 300 and °max = 1500 the same as for a(totaO. The total radiative correction is almost flat —3% outside of the pole region. The effect of changing M~turns out not only to shift the pole position but to increase the correction with increasing M~. For the event generation, we divide the process into two parts, the elastic part and the hard photon part. The event samples of each part are generated independently. The generation of the elastic event is performed in the following manner. From eq. (6) we calculate numerically

a5ti~(cos 0) =

cos B

f

da(e~stw) d cos ~ d cos 0,

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cos

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~

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(15)

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cos6 Fig. 8. The generation function for elastic event

(yc

=

60 GeV and k~= 0.5 GeV).

I

1

(16)

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a —

Cl) ~4~)

U) U) 0

-

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-~

U)

-

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--

z

.

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2

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io

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-



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cos6

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III

-~

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3000

U) U)

2000

0 U)

j

H

--

800 700

-

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600 500

..L

—1

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0.5

cos9 Fig. 9. The cos 8 distributions of the generated events with (a) W= 60 GeV, (b) 93 GeV and (c) 150 GeV. Solid and dotted histograms show the distribution of the generated and calculated ones, respectively.

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W=1SOGeV

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Cl) U) U) ‘II

0

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U) --

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~-

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cosO Fig. 9. Continued.

10~

-

-

W= ~

60GeV

a

io~

-

U) U)

10

-

0 l-~ U)

~

-

-

‘I

H 0

I..

30

60

I

90

120

150

180

~ (degree) Fig. 10. The acollinearity angle distributions of the generated events at (a) W

=

60 GeV, (b) 93 GeV and (c) 150 GeV.

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U) U) 0

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io~

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io2

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-,

o

30

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150

180

~ (degree)

10~



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C

io~

-

U)

~

to3

-

-

1(12

-

-

101 100

-,

0 ~-4

U)

E ~

0

I

30

.1,

I.,

60

90 t~

(degree)

Fig. 10. Continued.

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150

180

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Fig. 11. The electron energy distributions of the generated events at (a) W = 60 GeV, (b) 93 GeV and (c) 150 GeV.

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Ee (GeV) Fig. 11. Continued.

where cos

Umax

~

cos

u

~

cos

pole. Figs. ha, b and c show the electron energy distributions of the event samples.

u~ 0.

In fig. 8 we show this distribution at = 60 GeV with k~= 0.5 GeV. The generating function for the elastic event can be determined by this distribution. To generate the hard photon event, the event generation program SPRING [15] is used together with BASES. The final sample of events is made by mixing the events of each part according to the relative weight of each cross section. It is noted that the cross sections a astic) and ~ depend on the soft photon energy cut k~,but the sum of these cross sections, a (total) = a (elastic) + ~ does not, As an example we generate 50000 events under the condition; 300 ~ 0 ~ 150~, M~= 93 0eV and F~= 2.5 GeV. Fig. 9 shows the scattering angle distributions of the event samples (solid histogram) which reproduce the calculated one (dashed histogram). We also give the acollinearity angle distributions of the samples in fig. 10. One can see a bump in fig. lOc corresponding to the Z°

5. Discussions and conclusions The M~dependence of the radiative correction has been studied around the Z° pole. The result shows that the correction is fairly large in the energy region above the Z° pole. This does not mean the breakdown of perturbative calculation. In this region the correction is dominated by the radiative tail of the Z° peak. The percentage correction is large because we compare the tail with the Born cross section in the off-pole region. The generator gives both “elastic” and “radiative Bhabha events. Here, the “elastic” contains not only the elastic events but also those events with a soft photon of energy less than k~.The “radiative” or the “hard photon” events are the rest. Thus the ratio of “elastic” and “radiative” depends on k~and the separation is artificial because the sum of two parts gives a cross section

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independent of kc if the soft photon approximation is valid. Therefore, in principle, we can choose k~of any value, if it is smaller than the threshold energy of the detector. The method used for event generation, however, gives rise to a restriction on k~,since the “elastic” cross section is not positive definite for small kc (6~less than 1 in fig. 2). One solution to get rid of the negative cross section is to include some of the “radiative” events into the “elastic” events. For example, when the photon is emitted along the final electron or positron, one cannot observe them separately, due to the finite resolution for determining the position in the detector. This type of events can be regarded as the “elastic” one. In order to see how does the correction change by this modification, we calculate the radiative Bhabha scattering with collinear photon in QED. Here the word “collinear” means that the photon is emitted into the direction of e~or e within a small cone as shown in fig. 12. We set 1 degree for half of its opening angle. The results are shown in table 1 together with contributions of other parts, namely, soft photon and virtual corrections. In the table we also include the contribution from the initial collinear photon (a collinear photon to the initial electron or positron). What one can see from the result is the following: 1. About half of the negative soft photon comes from the inside of the collinear cone (collinear soft photon). 2. The contribution of the hard collinear photon amounts to be 30% of the lowest cross section. 3. The total collinear photon contribution (HARD(CONE)) is approximately 15%. Thus inclusion of the hard collinear photon into the “elastic” reduces the negative cross section. Another way to solve the problem is the exponentiation of the soft photon contributions. The negative cross section comes from the double logarithmic term in ~ ~8 ln(2 k~/W), where /3 is the Bond factor. The probability to generate the “elastic” event is calculated by

e

+



\ \







1

+ ô.,~=

1

+

/3 ln(2 kc/ W) + 6residual

\ beam axis



—~ -—

— — — —

:5:..-

,—.~:

\



— —

_______



\\ \

\

e

-

Fig. 12. A figure to illustrate collinear photon. The sphere of radius k~shows the soft photon region. The region of the collinear photon is inside the cones. Two larger and two smaller cones show final and initial collinear photon, respectively. The hard andinside the soft photon are the collinear one collinear outside and the collinear sphere, respectively.

The negative probability is avoided by summing up contributions of the double logalithmic terms of all order. This is accomplished by the modification, 1

+ 8vs

(2k~/W)~(1 + 6residual).

By this way we have a positive definite “elastic” cross section for any kc because ~residual ~S fV((a/iT) 1n(s/m~)). In conclusion we have finished coding a Bhabha event generator and obtained various distributions by using it. The details of the program will be described in another paper. We used the symbolic manipulation program REDUCE in calculating the exact squared matrix element of the hard photon emission diagrams. Since the result contains about 3000 lines in FORTRAN code, it requires very long CPU time to integrate them over the phase volume by BASES. To overcome the problem one of the authors made

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Table 1 The contributions from collinear photon, soft and virtual corrections in QED at = 70 GeV. HARD(INIT) and HARD(FINAL) are the initial and final collinear hard photon corrections, respectively. SOF1’(OUT) means the contribution from outside of collinear cones in the soft photon contribution (SOFT). HARD(CONE) is sum of collinear hard photon and collinear soft photon. The virtual (VIRTUAL) cross section is devided into three parts, vertex (VERTEX), vacuum polarization (LOOP) and box (BOX). TOTAL is sum of all component. Conditions used are 35 ° <8 <145 °, acollinearity angle cut ~ = 10 ° and k~= 1 GeV. All numerical values are the ratio to the Born cross section cos(9)

HARD(INIT)

HARD(FINAL) SOFT(OUT)

SOFT HARD(CONE)

VERTEX

LOOP VIRTUAL

BOX

TOTAL

—0.7750

0.1070

0.1614 —0.3816

—0.8165 —0.1668

0.1561

0.1079 0.2648

0.0008

—0.2833

—0.7250

0.1216

0.1628 —0.3740

—0.8091 —0.1509

0.1554

0.1078 0.2652

0.0020

—0.2596

—0.6750

0.1225

0.1634 —0.3675

—0.8021 —0.1493

0.1545

0.1076 0.2650

0.0030

—0.2511

—0.6250

0.1195

0.1642 —0.3618

—0.7968 —0.1516

0.1537

0.1074 0.2649

0.0038

—0.2482

—0.5750

0.1164

0.1638 —0.3566

—0.7921 —0.1550

0.1528

0.1072 0.2646

0.0046

—0.2472

—0.5250

0.1145

0.1648 —0.3519

—0.7873 —0.1559

0.1520

0.1069 0.2642

0.0053

—0.2438

—0.4750

0.1135

0.1651 —0.3475

—0.7829 —0.1566

0.1511

0.1067 0.2638

0.0060

—0.2405

—0.4250

0.1123

0.1645 —0.3433

—0.7784 —0.1585

0.1501

0.1063 0.2631

0.0067

—0.2386

—0.3750

0.1096

0.1642 —0.3394

—0.7745 —0.1614

0.1492

0.1060 0.2625

0.0073

—0.2382

—0.3250

0.1104

0.1659 —0.3356

—0.7705 —0.1589

0.1482

0.1056 0.2617

0.0079

—0.2325

—0.2750

0.1086

0.1642 —0.3320

—0.7668 —0.1624

0.1473

0.1052 0.2610

0.0085

—0.2330

—0.2250

0.1090

0.1650 —0.3285

—0.7637 —0.1613

0.1464

0.1048 0.2602

0.0090

—0.2296

—0.1750

0.1090

0.1660 —0.3250

—0.7603 —0.1603

0.1454

0.1043 0.2593

0.0095

—0.2261

—0.1250

0.1068

0.1619 —0.3216

—0.7567 —0.1665

0.1444

0.1038 0.2583

0.0101

—0.2297

—0.0750

0.1056

0.1634 —0.3183

—0.7533 —0.1662

0.1435

0.1033 0.2573

0.0106

—0.2269

—0.0250

0.1084

0.1650 —0.3150

—0.7500 —0.1618

0.1425

0.1027 0.2563

0.0111

—0.2203

0.0250

0.1061

0.1641 —0.3117

—0.7468 —0.1651

0.1415

0.1021 0.2551

0.0116

—0.2215

0.0750

0.1087

0.1661 —0.3084

—0.7435 —0.1604

0.1405

0.1014 0.2540

0.0121

—0.2146

350

S. Kuroda el al.

/

Bhabha scattering at high energy

Table 1 (continued) cos(O)

HARD(INIT)

HARD(FINAL) SOFT(OUT)

SOFT HARD(CONE)

VERTEX

LOOP VIRTUAL

BOX

TOTAL

0.1250

0.1066

0.1615 —0.3050

—0.7401 —0.1671

0.1395

0.1007 0.2528

0.0126

—0.2192

0.1750

0.1081

0.1638 —0.3017

—0.7367 —0.1634

0.1384

0.1000 0.2516

0.0132

—0.2132

0.2250

0.1096

0.1648 —0.2982

—0.7332 —0.1609

0.1374

0.0992 0.2503

0.0137

—0.2085

0.2750

0.1083

0.1658 —0.2947

—0.7298 —0.1611

0.1363

0.0948 0.2490

0.0142

—0.2067

0.3250

0.1117

0.1650 —0.2911

—0.7262 —0.1585

0.1353

0.0976 0.2477

0.0148

—0.2018

0.3750

0.1133

0.1656 —0.2873

—0.7224 —0.1563

0.1341

0.0967 0.2462

0.0155

—0.1972

0.4250

0.1133

0.1655 —0.2833

—0.7184 —0.1564

0.1329

0.0957 0.2448

0.0161

—0.1948

0.4750

0.1150

0.1652 —0.2792

—0.7140 —0.1551

0.1316

0.0947 0.2431

0.0169

—0.1908

0.5250

0.1172

0.1660 —0.2748

—0.7097 —0.1521

0.1302

0.0935 0.2414

0.0177

—0.1851

0.5750

0.1203

0.1645 —0.2701

—0.7050 —0.1504

0.1287

0.0924 0.2396

0.0186

—0.1806

0.6250

0.1228

0.1652 —0.2649

—0.7000 —0.1472

0.1269

0.0910 0.2376

0.0197

—0.1744

0.6750

0.1253

0.1653 —0.2592

—0.6942 —0.1447

0.1248

0.0895 0.2354

0.0210

—0.1682

0.7250

0.1225

0.1610 —0.2527

—0.6877 —0.1518

0.1223

0.0878 0.2327

0.0226

—0.1716

0.7750

0.0988

0.1652 —0.2451

—0.6802 —0.1713

0.1193

0.0858 0.2297

0.0246

—0.1866

a preprocessor for REDUCE output which produced a FORTRAN code suitable for the vector processor. Integration is performed by VBASES [16], the supercomputer version of BASES. We used the supercomputer HITAC S810/10 at KEK. The CPU time for the vectorized program is about 46 times shorter than that for the original version on a scalar computer FACOM M382 so that the integration of a full squared matrix element finishes typically in 5 mm. Further we made a numerical comparison between our exact squared matrix element and an approximate formula given

in ref. [17] and obtained excellent agreement. A program using this formula was also coded, run and we found that even on a scalar machine like FACOM M382 the CPU time becomes comparable to the supercomputer. Acknowledgement This work was made in course of the theoretical studies for TRISTAN e~e project. We would like to express our sincere gratitude to the members of the project for useful discussions.

S. Kuroda eta!.

/ Bhabha scattering at high energy

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[10] K-I. Aoki, Z. Hioki, R. Kawabe, M. Konuma and T. Muta, Suppl. Prog. Theor. Phys. 73 (1982). [11] Z. Hioki, Prog. Theor. Phys. 68 (1982) 2134. [12] M. Igarashi, N. Nakazawa, T. Shimada and Y. Shimizu, Nucl. Phys. B263 (1986) 347. [13] F.A. Berends, R. Kleiss and S. Jadach, Nuci. Phys. B202 (1982) 63. [14] A.C. Hearn, REDUCE 2 User’s Mannual, UCP-19, Univ. of Utah, 1973. [15] S. Kawabata, Comput. Phys. Cominun. 41 (1986) 127. [16] S. Kawabata and T. Kaneko, Comput. Phys. Commun. 48 (1988) 353. [17] F.A. Berends, R. Kreiss, P. De Causmaecker, R. Gastmans, W. Troost and Tai Tsun Wu, NucI. Phys. B206 (1982) 61.