- Email: [email protected]
The complex four point function for arbitrary masses
Physics Letters B 282 (1992) 185-189 North-Holland
PHYSICS LETTERS B
The complex four point function for arbitrary masses G.J. van Oldenborgh Sektion Physik, Ludwig-Maximilians-Universitiit, Theresienstrafie 37, W-8000 Munich 2, FRG
Received 10 February 1992; revised manuscript received 6 March 1992
A numerically stable algorithm is given for the analytical continuation of the four point function of arbitrary masses for the case that one or more of the masses are taken complex (to account for the finite width of an unstable particle). It is valid for all mass combinations needed in present calculations.
1. Introduction In some processes which will be studied in high-precision experiments one cannot separate the initial and final state radiative corrections in a gauge invariant way (Bhabha scattering, W pair production), in others these corrections may be important in their own right (near threshold). To evaluate the radiative corrections to reactions with unstable particles in an intermediate state which is close to on-shell it is necessary to resum the propagator of this particle. The resulting integrals can be kept calculable by evaluating the Q2-dependent width at a fixed point, normally the physical mass. This amounts to taking the mass o f this particle as a constant complex quantity in the Feynman integrals. In this letter we show some algorithms to evaluate these integrals. We will not discuss the subject of the consistency and gauge invariance o f this method, as a large body o f literature exists on this problem, which appears unsolved in the case o f charged particles. The analytic continuation of the general two and three point functions for complex masses is given in ref. [ 1 ]. These have been implemented in a numerically stable way in the FF package [2,3 ]. Of the complex four point function only special cases have until now been calculated explicitly (see for instance refs. [ 1,4] ). A way to evaluate this function for arbitrary masses with a finite imaginary part is given in this letter. The higher point functions are easily reducible to the four point functions with some propagators left out [ 5-7 ], so these do not pose any additional problems. The layout of this letter is as follows. We first discuss the nature of the problem and a possible solution which turns out to be unsatisfactory. Next a suitable method to evaluate these functions is given and some cases which require special care are discussed. Throughout this paper we will use the notation of refs. [2,7 ].
2. The problem The scalar four point function simply is the integral 1 f d4Q Do=i~ 5 (a2-m2)[(O+p~)2-m22][(a+pl+p2)2-m
]][(a-p4)2-m
2]
1
= f dul du2 du3 du4 0
O(1-Ul-u2-u3-u4
)
[ (UlSI "~- U2S 2 ~'- U3S 3 -t-/,/4S4)2--i~ ] 2"
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
(1)
185
Volume 282, n u m b e r 1,2
PHYSICS LETTERS B
21 May 1992
In the last equation we have introduced vectors si with s~ = m 2, p~ =s~_~ - s ~ . Solutions for the case that all masses are real have long been known [ 1 ]. A compact formula is obtained by performing a projective transformation Aixi
(2)
Ui = E¢=l Ajxj (no summation over i implied) to eliminate for instance the terms involving x~x3, XzX3 and x3z while maintaining the integration boundaries. The four point function then simplifies to the difference of two three point functions with transformed arguments s~ =A~s~. We prefer to use this solution with 24 dilogarithms, as the possible cancellations between the dilogarithms have been studied to a great depth ~ If one tries to use this method to evaluate the four point function for a complex mass parameter the arguments of these three point functions become in general complex. However, because the parameters of the projective transformation A~ have a non-zero imaginary part not only the masses m~ but also the momenta squared p2 are complex. The integration parameters also have to be taken along complex paths (although the boundaries are unchanged). The results for the usual complex three point function thus cannot be taken over. As a result it is not at all obvious where the branch cuts of the logarithms and dilogarithms of the final answer have to be taken to get correct continuation of the real result. In principle this continuation only gives rise to some extra logarithms times 2zfi and terms n 2. We will sketch a way to define these extra logarithms which does not give rise to an analytical result for most cases, unfortunately this method is useless to evaluate the four point function numerically. We restrict ourselves to the case that the integration region does not contain any singularities in the projective transformation, ~j= 4 ~AT lxj ¢ 0 for x~> 0. This is the equivalent of the condition Ai> 0 for the real case. One then integrates over the first two Feynman parameters without specifying the branch cut of the logarithm to obtain the three point function as a sum of three functions of the type [ note that x is integrated over a complex path given by eq. (2) ] I
Si =
{log [p~2(x - z +) ( x - z / - ) ] -log[p'i2(y,-z + )(Yi-z~- ) ] } .
~
(3)
0
The roots y+and z + are defined as in ref. [ 2 ]. We then transform back to real variables by performing the inverse projective transformation. This gives an expression of very similar form, but now the integration variable is real: 1
&=
~
(
1
du uSv~
1 )
u-Sai {l°g[p2(u-w+)(u-w[-)l-21°g[(A[-l-A[-+ll)u+AT+ll]
0
- log [p,? (vi - w~- ) (v~- wF ) ] + 2 log[ (AF l -AT+ ~,)vi q-/i+ll ] }.
(4)
In this equation u=A~xJ [ (A,-Ai+ ~)xi+A~+l ] is the back-transformed integration variable, v~and w + depend similarly on y~ and z + . (Note that the w + are roots of the same equation as the z + but with untransformed vectors sl', this does not hold for the vg.) Finally, ai=A~+ 1/(A~+~-A~) is a second pole, which, however, lies outside the integration region. We can now fix the branch cuts of the first two logarithms by demanding that no branch cuts are crossed over the integration paths (which join to form a triangle). The Riemann sheet on which the other two (subtraction) logarithms lie is specified by demanding that the straight line to the point u = 0 in $2 does not cross any branch cuts, this condition can be transformed to the proper variables of S~ and $3. After rewriting the (di)logarithms back to functions of Yi and z + this procedure gave an expression with 144 t/× log terms ~2 for each three point function. It is, however, useless in practice. #~ We have not yet investigated the solution with 16 dilogarithms found recently [ 8 ]. ~2 May be this could be optimized somewhat.
186
Volume 282, number 1,2
PHYSICS LETTERSB
21 May 1992
The r/function consists of step functions of the imaginary parts of its arguments. These are often numerically very unstable. A frequent problem is that they arise as Im (zlz2) with z2 ~ - z } . The algorithms given in ref. [ 2 ] guarantee that the numbers as a whole are stable, but the imaginary part need not be. In the limit Im (m 2)__, 0 these imaginary parts vanish and their role is taken over by the ie prescription, which, however, can be very different. - These t/× log terms cause large cancellations among each other, more often than not in fact adding up to zero in non-trivial ways. The problems arising from the pole in the transformation (the equivalent of the extra terms for Aelj< 0 in the real case) have thus not been investigated. -
3. A solution
A very easy solution follows from the fact that the four point function is an analytic function of its arguments and the observation that we have no problems computing it for real arguments. A Taylor expansion in the widths seems thus feasible. For the W and Z particles F / m ,,~0.025, hence neglecting the fourth derivative will give an error of order 10-8 for well-behaved functions. With a required relative accuracy in the final answer of 10-3 this allows for five digits cancellations between objects containing these four point functions (gauge cancellations such as may occur near the edge of phase space). This should suffice for most calculations. The derivatives can be obtained numerically by varying the real input mass; if the variation is chosen to be of the same magnitude as the expansion parameter the error is the same as the one introduced by neglecting higher order terms in the Taylor expansion. Convergence is improved by noticing that the four point function is a sum of dilogarithms and logarithms squared. A Taylor expansion in log (m 2) will thus be much better suited to the problem than one in m 2. One thus obtains for one complex parameter
D o ( m 2 _ i m F ) = O o ( m 2 ) + zD0(l) + ½z200(2)+l(z3z)O~3) q _ l (Z4 z2)D0(4) + O ( F / m ) 5 ,
(5)
with
DO(') = ½[Do(m2x) - D o ( m 2 / x ) l ,
(6)
DO(2) = D o ( m 2 x ) - 2 D o ( m 2 ) + D o ( m 2 / x ) ,
(7)
00(3) = [ O o ( m 2 / ( ) _ 3Do (rn 23~) + 3 D o ( m 2 / 3 ~ ) _ D o ( m 2 / x ) ] ( 3 )3,
(8)
00(4)
=
[ D0 (m 2/() --4Do(m2x/~) + 6Do(m 2 ) - 4 D o ( m 2 / v / x )
--Oo(m2/t¢)]'2
4 ,
(9)
with x = 1 + F / m , z = l o g ( 1 - i F / m ) / l o g x ~ - i . One function evaluation of the complex scalar four point function will thus cost three, five or seven times as much as a four point function of real arguments, depending on how many terms are required in the Taylor expansion. Considering that complex arithmetic is much more expensive than real arithmetic this does not seem excessive. An estimate of the precision of the truncated Taylor series is obtained by an estimate of the next term 1 ID0(~) I2 A D o - n! IDU-~)I
(10)
When there is more than one unstable particle involved the scheme is easily extended. One then also needs the cross derivatives, which are obtained as straightforward combinations of differentials except
O (0 l . l ) -_- LnI O n . r t~i t 21~t~'l , m ~ x 2 ) - D o ( m ~ x l , m 2 ) - O o ( m 2, m~K2) + 2Do(m21, m22) _Do(m2/t¢l, m2) 2 --Do(m,,2 m2/x2) + Do( m2/xl , m2/x2) .
( 11 )
This saves two function evaluations compared with the more obvious formula. The cost of computing a four 187
Volume 282, number 1,2
PHYSICSLETTERSB
21 May 1992
point function with more complex parameters increases rapidly, it is therefore worthwhile to continue the Taylor series only in those parameters in which it is necessary. Often only one needs to be taken beyond the second derivative.
4. Subtracting singularities The procedure sketched above of course only works if the function is analytical in the neighbourhood of the point around which it is expanded. A singularity even at a moderate distance completely spoils convergence. Singularities exist in two kinds of configurations. The first type occurs when the diagram would be infra-red divergent if the mass of the unstable particle where to be taken real. The width now functions as the intra-red regulator and the answer is proportional to log F. This is immediately apparent as the expansion parameter in this region is (m 2)]. We first discuss the case in which there is only a single pole of this type. Then this divergence is easily isolated and subtracted before doing the Taylor expansion. The function
F/[s-Re
/)0 =
f d4Q(Qa_22)[Q2+p.Q+(p~_m~)][Q2_p4.Q+(p~_m~)] ,
× ((Q+p~ 1 +p2)2-m~
1 (p, + p 2 ) 2 - m~)
=Do - Co/[(p~ + p 2 ) 2 - m~] ,
(12)
QU
= 0. It can be withp 2~~ m 22 and p2 ~ m 24, is well behaved in the neighbourhood of the infra-red singularity at approximated by a Taylor expansion and the three point function added again with complex arguments - the continuation of the three point function is known. This also means that the divergent terms, which will most likely cancel against similar terms in the real diagrams, are known to a much higher precision than the finite terms which remain. Doing these cancellations numerically thus does not affect the precision of the final answer very much. When there are two separate infra-red singularities, for instance in the WTW7 box in off-shell WW scattering, one can just subtract two infra-red divergent three point functions to obtain a well-behaved function. The case that remains is the one in which the two poles overlap. In the limit that the widths go to zero one would have a linear rather than logarithmic divergence. The linear part is easily subtracted; however, the remaining logarithmic divergence is not. This situation occurs in two cases. The first case is an infrared-divergent diagram in which the fourth propagator contains an unstable particle, which is close to being on-shell. An example is the configuration shown in fig. 1 which occurs in W pair production. Fortunately the complete solution for this first case, including the analytic continuation for a complex mass m, is given in ref. [4]. The second case is the same diagram with one or both of the other particles unstable as well. This type of configuration occurs in a chain decay of two unstable particles. An example is also shown in fig. 1. This case is as yet unsolved; as these configurations are not needed at present they have not been investigated further.
Fig. 1. Examplesof the linearly divergentconfigurationswhich cannot be evaluated with the method described in this letter. 188
Volume 282, number 1,2
PHYSICS LETTERS B
21 May 1992
81 ,~ TYt~V
u
s2 >> m~v
Fig. 2. Example of a diagram with a threshold singularity.
There exists a second type o f singularity which has to be subtracted before the Taylor expansion can be performed. W h e n a vertex with a complex mass is near threshold one obtains F l o g F terms. These show up in the b e h a v i o u r o f the four p o i n t function (as a function o f a real mass p a r a m e t e r ) as a singularity in the first derivative o f the real part and a d i s c o n t i n u i t y in the derivative o f the imaginary part at threshold. An example is the box d i a g r a m given in fig. 2. Suppose that the (rn~, m2, p~) vertex is near threshold, so x / 2 ( m 2, m~,p 2) << m2, m2, IP~I. In case x / ~ t2 ,~m~ + m 2 the singularity is at Q " = -pfm~/v/~, when the first two propagators both vanish. The c o m b i n a t i o n / ) 0 = f d4Q
1
( O2- m2) [ ( a+ pl )2-m2 ]
( x
l -[ ( Q + p , + p 2 ) 2 _ m~] [ ( Q - p 4 ) 2 - m42 ] -
=Do
-Bo/[(p, m2/.,/~l + p 2 ) 2 -
m~] [ ( p ,
[(p2
+p, m2/x/~
, ) 2 _ m~] [(p4
+p~m~/x/~
) )2 - m l ]
m , / ~ l +p4)2-rnl]
is thus again well behaved. Similar expressions hold for the other thresholds (m2 ~ m ~+ x / ~ 2, m~ ~ m2 + ~ The complex two p o i n t function is easily evaluated and a d d e d again.
(13) ).
5. Conclusion A m e t h o d to evaluate the scalar four p o i n t function for arbitrary complex masses has been given, which in almost all cases will give sufficient precision. The exception is a chain decay o f two unstable particles. The m e t h o d has been i m p l e m e n t e d a n d is being used in the calculation o f the electro-weak radiative corrections to W pair p r o d u c t i o n including decay.
References [ 1] G. 't Hooft and M. Veltman, Nucl. Phys. B 153 (1979) 365. [2] G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. C 46 (1990) 425. [ 3 ] G.J. van Oldenborgh, Comput. Phys. Commun. 66 ( 1991 ) 1. [4] W. Beenakker and A. Denner, Nucl. Phys. B 338 (1990) 349. [ 5 ] D.B. Melrose, Nuovo Cimento 40 ( 1965 ) 181. [ 6 ] W.U van Neerven and J.A.M. Vermaseren, Phys. Lett. B 137 ( 1984 ) 241. [7] G.J. van Oldenborgh, Ph.D. thesis, Universiteit van Amsterdam ( 1991 ). [8] A. Denner, U. Nierste and R. Scharf, Nucl. Phys. B 367 ( 1991 ) 637.
189
PHYSICS LETTERS B
The complex four point function for arbitrary masses G.J. van Oldenborgh Sektion Physik, Ludwig-Maximilians-Universitiit, Theresienstrafie 37, W-8000 Munich 2, FRG
Received 10 February 1992; revised manuscript received 6 March 1992
A numerically stable algorithm is given for the analytical continuation of the four point function of arbitrary masses for the case that one or more of the masses are taken complex (to account for the finite width of an unstable particle). It is valid for all mass combinations needed in present calculations.
1. Introduction In some processes which will be studied in high-precision experiments one cannot separate the initial and final state radiative corrections in a gauge invariant way (Bhabha scattering, W pair production), in others these corrections may be important in their own right (near threshold). To evaluate the radiative corrections to reactions with unstable particles in an intermediate state which is close to on-shell it is necessary to resum the propagator of this particle. The resulting integrals can be kept calculable by evaluating the Q2-dependent width at a fixed point, normally the physical mass. This amounts to taking the mass o f this particle as a constant complex quantity in the Feynman integrals. In this letter we show some algorithms to evaluate these integrals. We will not discuss the subject of the consistency and gauge invariance o f this method, as a large body o f literature exists on this problem, which appears unsolved in the case o f charged particles. The analytic continuation of the general two and three point functions for complex masses is given in ref. [ 1 ]. These have been implemented in a numerically stable way in the FF package [2,3 ]. Of the complex four point function only special cases have until now been calculated explicitly (see for instance refs. [ 1,4] ). A way to evaluate this function for arbitrary masses with a finite imaginary part is given in this letter. The higher point functions are easily reducible to the four point functions with some propagators left out [ 5-7 ], so these do not pose any additional problems. The layout of this letter is as follows. We first discuss the nature of the problem and a possible solution which turns out to be unsatisfactory. Next a suitable method to evaluate these functions is given and some cases which require special care are discussed. Throughout this paper we will use the notation of refs. [2,7 ].
2. The problem The scalar four point function simply is the integral 1 f d4Q Do=i~ 5 (a2-m2)[(O+p~)2-m22][(a+pl+p2)2-m
]][(a-p4)2-m
2]
1
= f dul du2 du3 du4 0
O(1-Ul-u2-u3-u4
)
[ (UlSI "~- U2S 2 ~'- U3S 3 -t-/,/4S4)2--i~ ] 2"
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
(1)
185
Volume 282, n u m b e r 1,2
PHYSICS LETTERS B
21 May 1992
In the last equation we have introduced vectors si with s~ = m 2, p~ =s~_~ - s ~ . Solutions for the case that all masses are real have long been known [ 1 ]. A compact formula is obtained by performing a projective transformation Aixi
(2)
Ui = E¢=l Ajxj (no summation over i implied) to eliminate for instance the terms involving x~x3, XzX3 and x3z while maintaining the integration boundaries. The four point function then simplifies to the difference of two three point functions with transformed arguments s~ =A~s~. We prefer to use this solution with 24 dilogarithms, as the possible cancellations between the dilogarithms have been studied to a great depth ~ If one tries to use this method to evaluate the four point function for a complex mass parameter the arguments of these three point functions become in general complex. However, because the parameters of the projective transformation A~ have a non-zero imaginary part not only the masses m~ but also the momenta squared p2 are complex. The integration parameters also have to be taken along complex paths (although the boundaries are unchanged). The results for the usual complex three point function thus cannot be taken over. As a result it is not at all obvious where the branch cuts of the logarithms and dilogarithms of the final answer have to be taken to get correct continuation of the real result. In principle this continuation only gives rise to some extra logarithms times 2zfi and terms n 2. We will sketch a way to define these extra logarithms which does not give rise to an analytical result for most cases, unfortunately this method is useless to evaluate the four point function numerically. We restrict ourselves to the case that the integration region does not contain any singularities in the projective transformation, ~j= 4 ~AT lxj ¢ 0 for x~> 0. This is the equivalent of the condition Ai> 0 for the real case. One then integrates over the first two Feynman parameters without specifying the branch cut of the logarithm to obtain the three point function as a sum of three functions of the type [ note that x is integrated over a complex path given by eq. (2) ] I
Si =
{log [p~2(x - z +) ( x - z / - ) ] -log[p'i2(y,-z + )(Yi-z~- ) ] } .
~
(3)
0
The roots y+and z + are defined as in ref. [ 2 ]. We then transform back to real variables by performing the inverse projective transformation. This gives an expression of very similar form, but now the integration variable is real: 1
&=
~
(
1
du uSv~
1 )
u-Sai {l°g[p2(u-w+)(u-w[-)l-21°g[(A[-l-A[-+ll)u+AT+ll]
0
- log [p,? (vi - w~- ) (v~- wF ) ] + 2 log[ (AF l -AT+ ~,)vi q-/i+ll ] }.
(4)
In this equation u=A~xJ [ (A,-Ai+ ~)xi+A~+l ] is the back-transformed integration variable, v~and w + depend similarly on y~ and z + . (Note that the w + are roots of the same equation as the z + but with untransformed vectors sl', this does not hold for the vg.) Finally, ai=A~+ 1/(A~+~-A~) is a second pole, which, however, lies outside the integration region. We can now fix the branch cuts of the first two logarithms by demanding that no branch cuts are crossed over the integration paths (which join to form a triangle). The Riemann sheet on which the other two (subtraction) logarithms lie is specified by demanding that the straight line to the point u = 0 in $2 does not cross any branch cuts, this condition can be transformed to the proper variables of S~ and $3. After rewriting the (di)logarithms back to functions of Yi and z + this procedure gave an expression with 144 t/× log terms ~2 for each three point function. It is, however, useless in practice. #~ We have not yet investigated the solution with 16 dilogarithms found recently [ 8 ]. ~2 May be this could be optimized somewhat.
186
Volume 282, number 1,2
PHYSICS LETTERSB
21 May 1992
The r/function consists of step functions of the imaginary parts of its arguments. These are often numerically very unstable. A frequent problem is that they arise as Im (zlz2) with z2 ~ - z } . The algorithms given in ref. [ 2 ] guarantee that the numbers as a whole are stable, but the imaginary part need not be. In the limit Im (m 2)__, 0 these imaginary parts vanish and their role is taken over by the ie prescription, which, however, can be very different. - These t/× log terms cause large cancellations among each other, more often than not in fact adding up to zero in non-trivial ways. The problems arising from the pole in the transformation (the equivalent of the extra terms for Aelj< 0 in the real case) have thus not been investigated. -
3. A solution
A very easy solution follows from the fact that the four point function is an analytic function of its arguments and the observation that we have no problems computing it for real arguments. A Taylor expansion in the widths seems thus feasible. For the W and Z particles F / m ,,~0.025, hence neglecting the fourth derivative will give an error of order 10-8 for well-behaved functions. With a required relative accuracy in the final answer of 10-3 this allows for five digits cancellations between objects containing these four point functions (gauge cancellations such as may occur near the edge of phase space). This should suffice for most calculations. The derivatives can be obtained numerically by varying the real input mass; if the variation is chosen to be of the same magnitude as the expansion parameter the error is the same as the one introduced by neglecting higher order terms in the Taylor expansion. Convergence is improved by noticing that the four point function is a sum of dilogarithms and logarithms squared. A Taylor expansion in log (m 2) will thus be much better suited to the problem than one in m 2. One thus obtains for one complex parameter
D o ( m 2 _ i m F ) = O o ( m 2 ) + zD0(l) + ½z200(2)+l(z3z)O~3) q _ l (Z4 z2)D0(4) + O ( F / m ) 5 ,
(5)
with
DO(') = ½[Do(m2x) - D o ( m 2 / x ) l ,
(6)
DO(2) = D o ( m 2 x ) - 2 D o ( m 2 ) + D o ( m 2 / x ) ,
(7)
00(3) = [ O o ( m 2 / ( ) _ 3Do (rn 23~) + 3 D o ( m 2 / 3 ~ ) _ D o ( m 2 / x ) ] ( 3 )3,
(8)
00(4)
=
[ D0 (m 2/() --4Do(m2x/~) + 6Do(m 2 ) - 4 D o ( m 2 / v / x )
--Oo(m2/t¢)]'2
4 ,
(9)
with x = 1 + F / m , z = l o g ( 1 - i F / m ) / l o g x ~ - i . One function evaluation of the complex scalar four point function will thus cost three, five or seven times as much as a four point function of real arguments, depending on how many terms are required in the Taylor expansion. Considering that complex arithmetic is much more expensive than real arithmetic this does not seem excessive. An estimate of the precision of the truncated Taylor series is obtained by an estimate of the next term 1 ID0(~) I2 A D o - n! IDU-~)I
(10)
When there is more than one unstable particle involved the scheme is easily extended. One then also needs the cross derivatives, which are obtained as straightforward combinations of differentials except
O (0 l . l ) -_- LnI O n . r t~i t 21~t~'l , m ~ x 2 ) - D o ( m ~ x l , m 2 ) - O o ( m 2, m~K2) + 2Do(m21, m22) _Do(m2/t¢l, m2) 2 --Do(m,,2 m2/x2) + Do( m2/xl , m2/x2) .
( 11 )
This saves two function evaluations compared with the more obvious formula. The cost of computing a four 187
Volume 282, number 1,2
PHYSICSLETTERSB
21 May 1992
point function with more complex parameters increases rapidly, it is therefore worthwhile to continue the Taylor series only in those parameters in which it is necessary. Often only one needs to be taken beyond the second derivative.
4. Subtracting singularities The procedure sketched above of course only works if the function is analytical in the neighbourhood of the point around which it is expanded. A singularity even at a moderate distance completely spoils convergence. Singularities exist in two kinds of configurations. The first type occurs when the diagram would be infra-red divergent if the mass of the unstable particle where to be taken real. The width now functions as the intra-red regulator and the answer is proportional to log F. This is immediately apparent as the expansion parameter in this region is (m 2)]. We first discuss the case in which there is only a single pole of this type. Then this divergence is easily isolated and subtracted before doing the Taylor expansion. The function
F/[s-Re
/)0 =
f d4Q(Qa_22)[Q2+p.Q+(p~_m~)][Q2_p4.Q+(p~_m~)] ,
× ((Q+p~ 1 +p2)2-m~
1 (p, + p 2 ) 2 - m~)
=Do - Co/[(p~ + p 2 ) 2 - m~] ,
(12)
QU
= 0. It can be withp 2~~ m 22 and p2 ~ m 24, is well behaved in the neighbourhood of the infra-red singularity at approximated by a Taylor expansion and the three point function added again with complex arguments - the continuation of the three point function is known. This also means that the divergent terms, which will most likely cancel against similar terms in the real diagrams, are known to a much higher precision than the finite terms which remain. Doing these cancellations numerically thus does not affect the precision of the final answer very much. When there are two separate infra-red singularities, for instance in the WTW7 box in off-shell WW scattering, one can just subtract two infra-red divergent three point functions to obtain a well-behaved function. The case that remains is the one in which the two poles overlap. In the limit that the widths go to zero one would have a linear rather than logarithmic divergence. The linear part is easily subtracted; however, the remaining logarithmic divergence is not. This situation occurs in two cases. The first case is an infrared-divergent diagram in which the fourth propagator contains an unstable particle, which is close to being on-shell. An example is the configuration shown in fig. 1 which occurs in W pair production. Fortunately the complete solution for this first case, including the analytic continuation for a complex mass m, is given in ref. [4]. The second case is the same diagram with one or both of the other particles unstable as well. This type of configuration occurs in a chain decay of two unstable particles. An example is also shown in fig. 1. This case is as yet unsolved; as these configurations are not needed at present they have not been investigated further.
Fig. 1. Examplesof the linearly divergentconfigurationswhich cannot be evaluated with the method described in this letter. 188
Volume 282, number 1,2
PHYSICS LETTERS B
21 May 1992
81 ,~ TYt~V
u
s2 >> m~v
Fig. 2. Example of a diagram with a threshold singularity.
There exists a second type o f singularity which has to be subtracted before the Taylor expansion can be performed. W h e n a vertex with a complex mass is near threshold one obtains F l o g F terms. These show up in the b e h a v i o u r o f the four p o i n t function (as a function o f a real mass p a r a m e t e r ) as a singularity in the first derivative o f the real part and a d i s c o n t i n u i t y in the derivative o f the imaginary part at threshold. An example is the box d i a g r a m given in fig. 2. Suppose that the (rn~, m2, p~) vertex is near threshold, so x / 2 ( m 2, m~,p 2) << m2, m2, IP~I. In case x / ~ t2 ,~m~ + m 2 the singularity is at Q " = -pfm~/v/~, when the first two propagators both vanish. The c o m b i n a t i o n / ) 0 = f d4Q
1
( O2- m2) [ ( a+ pl )2-m2 ]
( x
l -[ ( Q + p , + p 2 ) 2 _ m~] [ ( Q - p 4 ) 2 - m42 ] -
=Do
-Bo/[(p, m2/.,/~l + p 2 ) 2 -
m~] [ ( p ,
[(p2
+p, m2/x/~
, ) 2 _ m~] [(p4
+p~m~/x/~
) )2 - m l ]
m , / ~ l +p4)2-rnl]
is thus again well behaved. Similar expressions hold for the other thresholds (m2 ~ m ~+ x / ~ 2, m~ ~ m2 + ~ The complex two p o i n t function is easily evaluated and a d d e d again.
(13) ).
5. Conclusion A m e t h o d to evaluate the scalar four p o i n t function for arbitrary complex masses has been given, which in almost all cases will give sufficient precision. The exception is a chain decay o f two unstable particles. The m e t h o d has been i m p l e m e n t e d a n d is being used in the calculation o f the electro-weak radiative corrections to W pair p r o d u c t i o n including decay.
References [ 1] G. 't Hooft and M. Veltman, Nucl. Phys. B 153 (1979) 365. [2] G.J. van Oldenborgh and J.A.M. Vermaseren, Z. Phys. C 46 (1990) 425. [ 3 ] G.J. van Oldenborgh, Comput. Phys. Commun. 66 ( 1991 ) 1. [4] W. Beenakker and A. Denner, Nucl. Phys. B 338 (1990) 349. [ 5 ] D.B. Melrose, Nuovo Cimento 40 ( 1965 ) 181. [ 6 ] W.U van Neerven and J.A.M. Vermaseren, Phys. Lett. B 137 ( 1984 ) 241. [7] G.J. van Oldenborgh, Ph.D. thesis, Universiteit van Amsterdam ( 1991 ). [8] A. Denner, U. Nierste and R. Scharf, Nucl. Phys. B 367 ( 1991 ) 637.
189