NO'J lOUUbO
Ntl-NO- - Jl> I
Virtual two-loop corrections to Bhabha scattering
by f Knut Steinar Bjørkevoll
Them pneented at the Uiiircuilj of Beigen for the dt. sdent. degree
pepaxtiMnt of Thyeic* XSBxremiy otBagea Bergen, notway «
' i
rs i' Virtual two-loop corrections to Bhabha scattering
by Knut Steinar Bjørkevoll
Bergen, March 17, 1992 Acknowledgements The work that is presented by this thesis, hu been carried out at the Department of Physics at the University of Bergen during the last three years, with financial support from the Norwegian Research Council for Science and the Humanities. I wish to thank Professor Per Osland, who has been an excellent advisor for my dr. scient. work. I also want to thank him and Docent Goran Faldt for a fruitful collaboration that has been very valuable for me. I have had many interesting and useful discussions about physics and language with Dr. Conrad Newton during his stay at the University of Bergen. I also wish to thank Amanuensis Per Steinar Iversen for being the local computer gum, always ready to help when problems arise. It it impossible to thank my family properly in a few lines. Their support has made this work much easier. To Kjersti and Skjalg Contents
1 Introduction and motivation 1 1.1 Motivation 1 1.2 An overview of our work 2
2 Gauge invariance 6
3 Extraction of infrared-divergent factors 9 3.1 General technique 9 3.2 The two-rung ladder-like diagrams 11 3.3 The three-rung ladder-like diagrams 14 3.4 Explicit expressions for the infrared parts 17
4 Multiple complex contour integrals 19 4.1 Introduction 19 4.2 The factorization formula 20 4.3 Convergence of the complex integrals 21 4.4 Feynman amplitudes 22 4.5 Closing the contours 24 4.6 Algorithms for doing the complex contour integrations 25 4.6.1 Direct integration 27 4.6.2 Algebraic reduction of the integral 29 4.7 Example: A one-loop integral 29
5 The two-loop ladder-like diagrams 32 5.1 The uncrossed diagram 32 5.2 The once-crossed diagram 35 5.3 Results 39
6 Concluding remarks 44
A Proof of two relations 45
i B Momentum integration! and manipuUtion of Dirac matrices with REDUCE 48 B.l The momentum integration! 48 B 2 Simplification of the apinor factor 49 B.3 Contraction of momentum and ipinor factor» 51 B.4 The uncrossed and once-crossed diagrams 51 B.5 Listing of procedures, operators and some other definitions 52 B.6 Output for the uncrossed diagram 56
C The MAPLE-program CINT 60 C.l Calling sequence 60 C.2 Global variables 61 C.3 Procedures 62 C.3.1 Initialization 63 C.3.2 Before the integration starts 63 C.3.3 Doing the integrations 64 C.3.4 Printing results 66 C.4 Input and output for the one-loop example 67 C.5 Listing of procedures 68
D Applications of CINT to two-loop integrals 78 D.l Listing of the main code 79 D.2 Listing of auxiliary routines 63 D.3 Output for the uncrossed diagram 89 D.4 Output for the once-crossed diagram 89
ii Chapter 1 Introduction and motivation
This thesis present the firstpar t of a study of virtual two-loop corrections to Bhabha scattering at small angles and high energies. The three papers [1, 2, 3j that we have finished so far, are enclosed at the end of the thesis. They will be referred to as paper I, II and III respectively, and equations within the three articles will be referred to by adding the prefix I, II or III to their equation number. Bhabha scattering is the reaction
+ e+e- - e e-(n7), (1.1) where e+ is a positron, e~ is an electron, and n is the number of photons in the final state. The present study will be limited to contributions from Feynman diagrams without real photons, and all calculations will be done with
» » |t| > m2, (1.2) where a is the center of mass energy squared, t is the square of the transferred four-momentum, and m is the electron mass. In the next section we will discuss our motivation, and in section 1.2 we will give a short overview over the work that we have completed so far.
1.1 Motivation
I will discuss two motivations for our work: First, higher theoretical precision is needed for the determination of luminosity at e+e~-colliders, and secondly, the study of two-loop effects is interesting from a more theoretical point of view, since it may reveal new insight into the behaviour of perturbation series of gauge theories. The kinematical region defined by (1.2), is used for luminosity measurements at present e+e~ experiments, and will also be used for future experiments. Presently, the full one-loop corrections, and some contributions from higher-order diagrams,
1 are know, see e.g. (4, 5, 8, 7, 8). The experimental and theoretical error* for lumi nosity measurements at LEP, are now around 0.5%, and the experimental errors are expected to decrease to around 0.1% in future experiment» [9|. Two-loop corrections will be suppressed by a factor
^j«0.5-10-' (1.3)
relative to the lowest order amplitude. Extra factors of T appear at the one-loop level, e.g. in terms of the logarithm log(j/u - ie) a -in, and we may similarly get a factor of wa at the two-loop level. Thus a'/»* is replaced by a' wO.5-10"4, (1.4) and an extra factor of 20 is enough to bring the result up to the 0.1% level. Such a factor may appear as e.g.: Iog-^«24 log' i£! « 5» = 25 2irJ » 20 (1.5) at LEP energies with \t\lt « 1/100, which is relevant for luminosity measurements. Two-loop results may also give us some insight into what will happen at higher orders in the perturbation expansion. It is interesting to compare with ^-theory. Here, a study of two- and three-loop effects [10], inspired other studies of higher order effects. Presently, the leading behaviour of ladder diagrams has been summed up to all orders in four and six dimensions [11,12, 13, 14]. The one-loop results are very simple, and do not tell very much about what happens at higher orders, while two- and three-loop results give us some ideas about the structure of amplitudes at higher orders.
1.2 An overview of our work Because QED is a gauge theory, the contribution from a single diagram will depend on an unphysical gauge parameter. We are therefore interested in the contribu tion from a set of gauge invariant diagrams, and we have selected the set of six gauge invariant diagrams shown in fig. 1.1. We call these diagrams ladder-like, since they arise from the ladder diagram (diagram a) by crossing the photon lines in all topologically distinct ways. Gauge invariance of the six diagrams is proven in chapter 2. In chapter 3, we determine the infrared structure of the sum of the six diagrams. This result is in agreement with the results of paper II.
2 a b
X s^. c d
/
Figure 1.1: The three-rung ladder-like diagrams.
Feynman parameterization, integration orer internal momenta, and manipula tion of the Dirac matrices are done in paper I. We managed there to write the Feynman-parameter integrals in a very compact form. The calculations are lengthy, but straightforward, and it is therefore convenient to use an algebraic manipulation program. Appendix B shows how this may be done with the symbolic manipulation program REDUCE. The most complicated part of the calculation is the integration over the Feynman parameters, which we denote by ai ... 0:7. One of the main steps in the calculation, is to use the factorization formula which is described in chapter 4. The idea is to introduce a number of complex integrals, with contours that run parallel to the imaginary axes, in such a way that the Feynman-parameter integrals become easier to do. We have used this formula in two different ways. The first way, which is described in chapter 4, is to introduce a large number of contour integrals, such that the Feynman-parameter integrals may be done ex actly, yielding a product of beta-functions. It is in principle easy to find the leading behaviour of the contour integrals, but since each contour integral may get contribu tions from several pole singularities, we may have to add contributions from many thousands of terms. Thus, I have written a symbolic computer program that finds the leading terms of the integrals. The program needs a few hours to find the leading
3 logarithms for each diagram. By leading logarithms, we mean the terms with the largest number of logarithmic factors. For the two-loop diagrams in question, we find thai each diagram has terms with up to four logarithmic factors. This method is useful for finding terms with three or four logarithmic factors, but the coefficients of terms with only two logarithmic factors appear as sums of multiple contour in tegrals. Il often happens that these sums can be evaluated, but first it is necessary to convert each integral to a standard form. This is possible to do, but we have not yet made a computer program for this purpose. The symbolic program that determines leading terms, is written in MAPLE, and is described in appendix C. The second way of evaluating the Feynman-parameter integrals, is to introduce a smaller number of contour integrals. This is described iu papers II and III, where we have combined the factorization formula with a Mellin transformation. This leads to three and one non-trivial contour integrals respectively, and these can easily be done by hand once the Feynman-parameter integrals have been done. The Feynman- parameter integrals are very complicated in this case, but we managed to find the leading behaviour of the sum of the six diagrams. An important advantage of this method, is that it is possible to sum the contributions from the six diagrams at an early stage of the calculations. Thus, we avoid the calculation of a large number of terms that cancel. Presently, we have calculated all terms with two logarithmic factors, and it is possible to extract the coefficients of linear logarithms in terms of some integrals. The second method seems to work best for the present calculation, but still the first method is useful as a partial check of the results. The first method is also more general, and it will probably be straightforward to apply it to other diagrams. In particular, it will be very useful for setB of diagrams where the leading terms have three of four logarithmic factors. Furthermore, future improvements of the MAPLE program may make it useful also for the calculation of terms with less then three logarithms. We use a small fictitious photon mass A in order to regulate the infrared diver gences. Thus the amplitude depends on four independent invariants, s, t, m and A, and we expect the mathematical structure of the amplitude to be more complicated than that of the amplitude of the corresponding Feynman diagram in ^'-theory. In order to study the mathematical structure, we have done the calculations in two different limits. In paper II, we do the integrations in the limit A -> 0, a and |t| -» oo, (1.6) with ^-ro2>«»|t]>m2. (1.7) We find the result
M(»i) = M^l f-^-log' [ ^) ] -Mower order terms, (1.8)
4 where lower order terms means terms with only one logarithm or terms of relative order |1| /*, nt'/l'l or A'/m1. Here, M1'^ is the matrix element of the lowest order {•channel diagram, and MS*^ is the sum of the matrix elements of the six three- rung ladder-like diagrams in fig. 1.1. We see that the term that is expected from the infrared analysis in chapter 3, is the only term with more than one logarithm that does not cancel when the six diagrams are summed. In paper III we evaluate the integrals in the limit
«-co (1.9) with t, in, and A kept fixed with
|t| > ro'> A*. (1.10)
Although the results are completely different for individual diagrams in the two cases, we find that the leading behaviour of the sum of the six diagrams is given by eq. (1.8) in both cases.
5 Chapter 2 Gauge invariance
We shall here prove that the sum of the six three-rung ladder-like diagrams in fig. 1.1 is gauge invarian t The matrix element of the sum of the six diagrams may be written as
4»' J q\ - A' + te J q\ -X' + ieJ g| -X' + ie x **(
x «Mb7 "J—- *yM T-7a«(pi)V0'«v(2.1) f i + ii - m + it j>\ -ii-m + ie with
T/afl-r _ =(_,\ J _a I fl i -r I -h-ii-m + ie1 -ft + ii-m + ie1
+ 7° 7 -h - i t + »£ -fi + é — m + ie + 7" -*•-* •m + 7-.y"ie -fa + i: — m + ie + 71- js -h-é -m + ie -fa + é — m + ie + r -h-é •m + te7 -#+#, — m + ie + ^ -fc-J,-» + * 7° -* + * — + * 78 } "(P'J)-(22) The tensor A„„ is given by VsM«) = 9^ + «^r, (2.3) where { is a gauge parameter. The momenta of internal and external lines are denned according to fig. 2.1, and
Q = p,-P;=pi-p,. (2.4) Pi Pi - gi Pi + q» pj 7i^ ?a£ft^
9at 9it 9et —(—£ <• -i < 1—t— ~P2 -P2 -9a -P2+ 9e ~Pj
Figure 2.1: Definition of momenta for any of the six three-rung ladder-like diagrams. The indices a, i and c are a permutation of 1, 2 and 3.
In the following, we will suppress ie and the photon mass A, since they would only cause corrections of order e or A* as long as we do not change the sign of any of the denominators. We also introduce the abbrevations u = u(p,), ii' = 0(pi), v' = t-tø), v = v(p,). (2.5)
V*1 is completely symmetric in gi, q-i and ft, and since the labels of the mo menta and indices of the photon propagators are arbitrary, we may symmetrize the integrand of eq. (2.1) by the replacement ^lv"^ (2.6) ti+ii-m • h-éi-™ ' u 3! where
^-'W^ Wl pi — f3 — TTl +y— y— 19
1 1 7 fi + ii-m Pl-«;2-m + V» 1 y 1 y
+ n, T jS3-r^——i + fc-m'V ,« .^,-^-m '
7 _1_ + -r'3rrfrr^v^r=T'}.. (")
We will now prove that all terms that have one or more {acton of ( after the tensors &„>, kgf and h^y have been expanded, are lero. Due to the symmetry of the integrand, it is enough to show that terms in which Aas' it replaced by fti.vW/4? vanish. We perform the ^-integration, which is trivial because of the ^-function, and use relations like
i «(p) = - • ! „ (t - i - "»)«(») = -«(p) (2-8)
and the Feynman identity
_ i i_ (7.9) f - j -m ft- m'
to reduce the factor (qiaqia- £/"•**' V°'*V to
I Jh-fa-n» Pi + fa - m
fj — tn ~~F2 ~ §2 — w*
(2.10) which is zero. It follows that the sum of the six three-rung ladder-like diagrams is gauge invariant. This result was expected from a more intuitive argument. Consider scattering between two particles with charges Z\ and Zj. Now the sum of the matrix ele ments of the six three-rung ladder-like diagrams is proportional to Z\Z\. The only other virtual diagrams that yield this factor, are diagrams with a one-loop vertex correction to each of the fermion lines. But it is easy to see that the latter kind of diagrams form a gauge invariant set, and it follows that six three-rung ladder-like diagrams must be gauge invariant in order to mak» the set of all virtual diagrams gauge invariant.
8 Chapter 3 Extraction of infrared-divergent factors
In this chapter we will show that the structure of the sum of the matrix elements of all one-, two- and three-rung Feynman diagrams is given by
MM = mo, = modi) + rai,
= m° 2 + m'~W {3) where B is infrared divergent, mo, mt and m5 are infrared finite. We have used Af(n7> to denote the sum of the matrix elements of all n-rung Feynman diagrams. We will see that B has the form Mog(A5) + k, (3.2) where A is the photon mass, and 6) and 63 are infrared finite. Note that the functions mo, mi and B are different from the same functions in the famous paper by Yennie, Frautschi and Suura [15], because we only consider virtual photons that connect the two fermion lines. Since M1-'^ is known, we obtain an explicit expression for the infrared-divergent part of Af'3T'. It is straightforward to extend the following analysis to any order, but we will not do this. We will from now on use the Feynman gauge, in which f = 0 in eq. (2.3). 3.1 General technique We will factor out the infrared divergences of Al'"7' by using A technique developed by Grimmer and Yennie [16]. Their basic idea is to split all photon propagators that 9 contribute lo the infrared divergence into one part that contribute» to the infrared divergence and one part that does not, such that the integrals that contribute to the infrared-divergent integrals (actor out. The following description may be used for any order of perturbation theory. The first step is to symmetrize the integrand of Af'"7' in the same way as we did for the integrand of M^ in the previous chapter. Next we would like to split photon propagators that contribute to the infrared divergence into two term» by writing —jf--— -7--—p-—?"• (3-3> where 6 is defined such that only the if-part of the propagator contributes to the infrared divergence. It turns out this is only possible if at least one photon does not contribute to the infrared divergence. For the ladder-like diagrams, however, we may see from power counting that all photon propagators contribute to the infrared divergence in different parts of the momentum space. Therefore we write the integrand of .M'"1' as a sum of terms in which at least one of the photons does not contribute to the infrared divergence. This is obtained by multiplying the integrand with i-gife*)* = ;£«? +7 £•«. (»•«> «3 where Q = p1 — p\ and t is the Mandelstam variable which is equal to Q' by definition. The sum is over all photon momenta. It will now be possible to find a unique expression for b for each photon that contributes to the infrared divergence in each term. Grammer and Yennie find that b depends on whether the photon that we want to split is attached to the initial or finalpar t of each fermion line, where initial and final part is defined relative to a photon that does not contribute to the infrared di vergence. They also give the explicit expression for 6. For a photon with momentum g,-, and our definitions of external momenta, 0 is given by 0 = & = 4>'>.dV'>, r,, = i,f (3.5) where r(«) = »[/) if the photon is attached to the initial [final] part of the electron (positron) Une, and c and d are given by (,) _ -2piM + g,„ (j) 2pl„ + 1i* '" rf + W« Ai) _ 2P»« + 10 Figure 3.1: The two-rung ladder-like diagrams. After splitting photon propagators that contribute to the infrared divergence, we may expand the matrix element into terms in which the original photon propagators are replaced by K- and G-type propagators. In the next two sections we will see that integrals over momenta of A'-type propagators factor out from the rest of the integral. 3.2 The two-rung ladder-like diagrams We could go directly to the three-rung ladder-like diagrams and write all propagators that contribute to the infrared divergence as a sum of G and K propagators. It is, however, possible to express the infrared parts of A4'*'' in terms of the infrared and finite parts of A4'*nr>. We prefer this procedure since the infrared and finite parts of A*'*1' are well known |6). The two Feynman diagrams that contribute to Al''7' are shown in fig. 3.1. After symmetrization, the matrix element of the sum of the two two-rung ladder like diagrams may be written as , ((w _ «? f *9i / *«. 2T' J ql - A» + it i q\ - A* + it X 6*{Q -q,- q1)gaa, g0B, W* V"'»' (3.7) with e , U" = u[-f-D -i -.^ + 10- _i -y*)u (3.8) \ fi + 4i-m + ie' i>\ + h - m + if J and e a V = vL°— -5 _y + 7'_ _i --, )v'. (3-9) The momenta are defined in the same way as for the three-rung diagrams, see fig. 2.1, but now without the internal lines with momenta p\ + 93, q3 and —p'2 + gc, and with a and b either 1 and 2 or vise versa. 11 We «rill alio here suppress it and the photon mass A, since they would only cause correctioni of order e ot A' to the following manipulations at long as we are careful not to change the sign of any of Ibe denominators. Multiplication with eq. (3.4) gives x («. + «,)'ft»'9»'{/""V0'''' = M» + M», (3.10) with «J=°-«l (3.11) after we have used the symmetry of the integrands to combine terms. Now MJI is finite in qt and Mn is finite in both 91 and 41. To get a simpler expression for the infrared part of Mji, we split gaa- into /da' &nd Gm>. Thus we may write M21 as a sum of two terms, Mn = M* + M°, (3.12) where the upper index indicates whether gaa' has been replaced by Kaa> or Go,'. We simplify M([ first by using eqs. (2.8) and (2.9): X 1 t*>' ~1 Now the integral is a scalar quantity which only depends on the masses of the electron and the photon and the Mandelstam variables s, u and t. Thus each term will be invariant under the exchange (pi «-> — p\, pj «-» —pj), and the result simplifies to MK = g iVu6w.| A" (jg) _ jg)j. (3.14) 12 It follows that the matrix dement of the two-rung ladder-like diagrams is given by A^'^'sTnooB + m, (3.15) where . m„ = M™ = ij2 nYuihtv' (316) is the tree-level t-channel matrix dement, and m, = M& + M„. (3.18) Both mt, and mi are infrared finite, and the infrared divergence is given by the function B, which is an important function because it also appears at higher orders. We will calculate B in section 3.4. Next we simplify Jl/jJ by using relations of the form We obtain "ii = JWJI - ™« to2 j 9i «irs 7 q\ ,,, I IT',*) . . M.,7-1 I, which is clearly finite as 91 —• 0. 13 3.3 The three-rung ladder-like diagrams The symmetrized «urn of the matrix elements of the three-rung ladder-like diagrams is given by 4ir» J gf - A3 + ie J q} - A> + it i q\ - A» + « x «4(C - 9i - 9: - ftfe_> W Sr,' tf"*7 V"'*'' (3.21) with f/ and V given by eqs. (2.7) and (2.2), and momenta denned by fig. 2.1. Once more, we suppress ie and the photon mass A, and multiply with eq. (3.4) to obtain J afll v x (9. + 9» + ?3) ft»' fej' Sr,. t/ V°'" = M3i + Mu, (3.22) with (3.23) 9i~Q~1i-ti -io» f^gi /«*(, """IS? X tø-ftWSfWSrr-tr* V"'',V (3.24) s after collection of symmetric terms. M31 is finite in 93 and A/32 > finite in both q2 and 93. Thus the photon with momentum 93 may be used to separate initial and final parts of the fermion lines. After splitting the propagators that contribute to the infrared divergence, we may write a K Af3i = M&* + M* + Mg + Mf,°, (3.25) M„ = M?,° + Mg, (3.26) where the first upper index refers to the —«9oa'/9i propagator and the second to the —ig$P'/ql propagator, the index g indicating that the propagator is not expanded. Due to the symmetry of the integrand we have Af^G = M^f, and we will only need the if-part of the —igaa' /gj-propagator to find the infrared-divergent part of M3l and Afji. The replacement (;„»< -> Kact turns the factor \ 14 into 4'vi^^+c'v = (#>-*!}>-#>+<#) x • {^^^-f-^^^S} V. (3.27) We see that the fermion propagators that depended on both 41 and 42 have cancelled and we are left with a Dirac factor that has the same form as that of Alf'*T' and depends only on 92, times a scalar factor that depends only on qi. Now a simple ex pression is obtained for M^' = Af*G +Af*K by combining eq. (3.27) with eqs. (3.5) and (3.23) and the results of the previous section: = mo£f + M°f. (3.28) The two terms in the final expression are equal to Mj[K and M£G respectively. We may therefore include M^K by multiplying the second term with two. It follows that M31 is given by: M31 = rn„ ^- + M£ aB + Mgf. (3.29) We will later show that M°° is infrared finite. 15 Mjj ir slightly more complicated because of the factor ft • ft _ ii-(Q-li- ft) 3 TrT . ..« • (*•») («-«.-ft)' which prevents direct factorization of the infrared part. This is dealt with by rewrit ing the factor as ft • (Q - ft) . 9i • (g ~ ft ~ ft) ft • {Q ~ ft) ( _ ft • (Q ~ 9a) . 9i • (g ~ 9a) [-g? + 2)' ( M& = Mn aB + M*«, (3.32) where M32° is infrared finite. Summing up all contributions, we have shown that MW = m„ ^- + M° aB + M7, aB + m2 = m0 ^Y~ + m, aB + m2, (3.33) where m0 and mj are given in the previous section, and mj is given by G m, = M3° + Mff + M£>. (3.34) We will now show that M^3 and Af^° are infrared finite. We will do this by proving that G-propagators will not contribute to the infrared divergence. The same proof is given in [16] for the general case. Because of symmetry, it is enough to show that the —iGatx'lq\ propagator will not contribute. By power counting we can immediately see that terms with ?i in the numerator do not contribute to the divergence. We consider first the factor If0"1, which is defined by eq. (2.7). In the first three terms of J/™"7, the photon with momentum "'7° * -f. ) 77 (3-35) Pi + ft - m Pi - f3 - m T «'j£V, I mT , (3-36) 2Pl9i Pi - I» - »» 16 where we have dropped terms linear in qi in the numerator and quadratic in fi in the denominator, since the infrared behaviour is independent of such terms. In the third term of l/"*1, the gj-photon is connected between the ^-photon and the final part of the electron line. In this case, we may see by power counting that the ifi-propagator will not contribute to the infrared divergence unless „2p«» A+m 7 7 (3.38) 2pi,2 2Pl(?. + ft) 2p\q2 2^(91 + M-nn) - (WK-*MIX2M.) -4(ftft) * -4(p»Pi) - (-2pi«i)(2ft«i), (3.39) (-2pi thus we see that the infrared-divergent contributions from the two terms in Ga, cancel. The same happens for the other terms. 3.4 Explicit expressions for the infrared parts The infrared factor given by eqs. (3.17) and (3.5) may be written as A (2pi - «)(2p, + 9) 3 2 2 i ) 2?r J 9 - A + ie {q-2piq + ie)(Lq + 2p2q + ie) (2p,-7)(-2pi + g) (,2 -2p„ + »e)(," -2p',q + ie) 2ft ft (g2 - A2 + te)(g2 - 2pi + ie)(q* + 2p,q + ie) i/* 9 1 1 2 (q - 2pl9 + te)(9 + 2p,q) + ie 1 2 - (ft ~ -P2Y (3.40) (q* - A + ie)(q' + 2p,q + ie) 17 Here, we have re-introduced the photon mau A and the inftniteumal imaginary part ie, «ince they are important for the explicit calculation of B. Scalar integral» with up to four factor! in the denominator have been calculated in general by t'Hooii and Veltman [17). Substitution into their results give»: .-i [„(;-*)„,(£)..v(3-..)-svS) Thus our result is in agreement with that of Bohm, Denner and Hollilc (6), where •M'*"' is given by Milj) = moaB' + m[ (3.42) with *=-NG-*)*(§)• f3-43) The infrared-finite term m[ is of order \t\/i relative to mo. Defining AS = B - B', which is infrared finite, we have m\ =m, + m0oAB, where the infrared finite functions mo and nti are defined by eqs. (3.16) and (3.18). Substituting for B in eq. (3.1), it follows that B AB)]1 M^ = m„ l°( '+2 + m, a(B' + AB) + m2 = ro*>—- — •+• mi åa + m2 , = -^Ylog (g)+mi + 0(iJm0) (3.44) with (m l} {mi) m'a = roo i^l + mi aAg + m, = ' !' ~ ' + m,, (3.45) £ £TflQ and m2 given by eq (3.34). We are not interested in n»i since it is of order \t[ js relative to m0, but it remains to be seen whether this is true also for m». 18 Chapter 4 Multiple complex contour integrals 4.1 Introduction In this chapter, we will discuss the introduction of complex contour integrals in order to simplify the Feynman-parameter integrals. In papers II and III, we show how Feynman-parameter integrals may be simpler to do after the introduction of a small number of complex variables. In this and the following chapter, we show how a larger number of complex variables make it possible to do the Feynman-parameter integrals exactly. It is in principle straightforward to find the leading behaviour of the resulting multiple contour integrals, and we will discuss how this may be done. We will illustrate the method through a simple one-loop example, and do the much more complicated two-loop integrals in the next chapter. We have also used this method in appendix G of paper II and appendix E of paper III. Bergére, Calan and MalbouiBson [18J, have shown that any convergent Euclidian Feynman amplitude may be expressed in terms of multiple contour integrals. More over, this may be done without any approximation. Our approach is less general, but seems to be more suited for the amplitudes that we want to calculate. Similar methods have successfully been used for the ^-calculations that were mentioned in section 1.1. One of these calculations, by Gastmans, Troost and Wu [12], included both leading and non-leading logarithms. Additional applications may be found in [19] and [20]. The calculations in QED are more complicated than the ^-calculations, because: • We have to add up gauge invariant sets of diagrams in order to obtain a physical result. When we do this, we find that the two leading logarithms from each of the two-loop diagrams that we consider, cancel. It is reasonable to expect that the n leading logarithms will cancel for the corresponding n-loop diagrams. 19 • Since we use a non-vanishing photon mass to regulate the infrared divergent, we have two masses in QED, and we are therefore not allowed to use the sime approximations as in ^'-theory. This is important, since a large numb-r of terms in the Fey nman-parameter integrals are proportional to the squ re of one of the masses. 4.2 The factorization formula We will turn Feynman-parameter integrals into complex contour integrals by using the factorization formula: (A. + ... + A,)"' = r£j n [I <«n«)*r] r(p - £ *)An^::;*, (4.i) which is valid for p - Y£=* c» > 0, c; > 0, i = 1,... ,n - 1, and Im(j4j) > 0, t = 1,... ,n. We have here introduced the notation dz = -^-l dz. (4.2) To prove eq. (4.1), we make use of thei relations for Re(p) > 0 and Im(u) > 0, and e'T- JdzTiz)^""^' (4.4) for c > 0 and Im(T) > 0. Proofs of these relations are given in appendix A. Now the proof of eq. (4.1) is straightforward: (A1 + ... + A„)-' rr» Jo r(p) Jo g _-wp/-•»I>/22 .a, n-1 r - 1 dt 1 i iM Vy I f' II [jf --iwp/2 n-1 r . -i X ^P-E^)/»^»^", (4.5) 20 which is equal to the rhs. of eq. (4.1). We may replace Ai in eq. (4.1) with A'1' + it, and perform a Mellin transfor mation with "a fixed,t o obtain: r i* r< {D,** +... + D.J"" + uyr Jo x (D„ + !>)-»+£."'.'".'«• rdtM-<-T.::,'-«- Jo = rL n \I «WMW+")'"} r This is a more general form of eq. (2.3) in [10] and eq. (3.7) in paper II. We have used this relation in paper II, but in the following we will use eq. (4.1) without the Mellin transformation. 4.3 Convergence of the complex integrals We consider the asymptotic form of the integrand of eq. (4.1), as the absolute value of one of the integration variables, Zj, approaches infinity. The integrand may be written as the factor /(«,)sIX«i)r{p-X>i)(^)* («J times a factor that does not depend on Zj. Using Stillings formula, T(z) ~ Æe1'-!'1»1'1-', (4.8) which holds asymptotically for \z\ —> oo with |arg(z)| fixed at a value smaller than T, we find that the absolute value of f(zj) has the asymptotic form |/(ZJ)| ~ 2*exp -irtøjl - arg f—J Vj (Re(p-5>)-l)log|z,-| + log (4.9) Here Xj = Re(zj) and y, = Im(z}). We see that the Zj-integral is absolutely con vergent if |arg(j4„/Aj)| < ir, since the integrand then falls off exponentially as Vj -* ±o°- 21 4.4 Feynman amplitudes The following general considerations are valid Car all the Feynman integrals that we consider. More explicit examples follow-; in section 4.7 and chapter 5. After Feynman parameterization and integration over internal momenta, we are left with a scalar integral that may be written as ;= /' D = D,a + D„u + D,t + D„m' + DXX' + ic, (4.11) where D„ Du, Dt, Dm and D\ are given in paper I and in the next chapter. For all diagrams that we will consider, we will define the polynomials such that D„ DUI and D, are always positive, while Dm and D\ are always negative (the definitions of D., 2 3 A,, Dt and Dm are ambiguous, since s, u, t and m are related by 3 +1 + u = 4m ). It follows that D,s is always positive, while all remaining term6 of D are always negative. It is therefore not. obvious that the complex integrals that we introduce in order to factorize D~"D, are convergent in the limit £->0. We are not aware of a rigorous treatment of this problem, but similar methods have been used e.g. in ip3 calculations. We have also checked that our methods give correct results in a number of one-loop integrals, e.g. in the example in section 4.7. Farther discussion of this problem iB beyond the scope of this thesis, and we will assume that the following procedure gives the correct leading behaviour. We first replace D by ie ! 2 -D.se~ + Z)uu + D,l + Cmm + DXX 2 = -D.ae-*> - A, \u\ - D, \t\ - \Dm\m - |AJ A», (4.12) where all terms are negative for 8 = 0. Now the factorization formula may be used safely if we keep |0| fixed at a value less than IT. After having found the leading behaviour of the integral, we will take the limit 0 —»ir. 22 After factorizing D'"D and A""* into a snffidently large number of factors, the a-integrals will, if they are well defined, only yield products of r-fnnctiou. Thus the function / given by eq. (4.10), is converted into the multiple contour integral where ø,: i = 1 m, and tfs, i — 2,..., m,, are linear combinations of complex integration variables with integer coefficients. The contours must be chosen such that c; > Ofort = l,...,n, Re(V>0 > 0 for >' = 1,... ,m,, Re(6) > 0fort = l,...rm,. (4.14) The constraints ensure that the factorization procedure and the Feynman-parameter integrals are well defined. In the following discussion, we will assume that it is possible to find contours that satisfy eq. (4.14). Dependencies on the invariants a, u, t, m' and A1, and ©, hare been included in the factor a*0, by defining Vto as V-0 = V-. ( 1 - j^T J + 7»^u + 7.V>. + 7mV>m + Wx (4.15) where f log |*| for x = £,u, logW (4.16) 7.= »°g(*') for x = m,A. I log(«) Thus s*° =s*'|u|*'|t|*'(m2)*-(A2)*»e-'e. (4.17) The functions tp,, ^„, ^>m and Vu, are linear combinations of complex variables with coefficients —1, 0 or 1 plus integer constants. We will always have D\ ^ 0, and in this case we may use eq. (4.1) such that i>l = -*1 - • • • - **., $u = -*n.+l — ... - Zn.+n., — V"i = *n,+n.+l — • • • — 2n>+n.+nll V'm — —^ni+n«+n«+l — • • • ~ Znj+fW+nt-t-nf») _n V"A = I) + »i + . . . + 2n,+n»+ni+nI., (4.18) where n,, nu, n, and n. are the number of terms that we have split D„ Du, Dt, Dm and D), into. 23 Expression! of the same form at eq. (4.13), may be obtained in different ways. Bergere, Calan and Malbouiwon [18] have shown that any Euclidean amplitude may be converted to this form, but their approach would give a very large number of complex variables, almost SO for our two-loop integrals. We will therefore use a somewhat different approach, which will be described in the next chapter. 4.5 Closing the contours We will now discuss the possibility of closing the contour of one of the complex integrals in eq. (4.13) at infinity. We consider once more the zj-integral, where z, is one of the complex integration variables. Since the integrand falls off exponentially as Im(zj) -> ±oo, we may dose the contour at infinity around the positive (negative) real axis if the integrand falls off sufficiently fast as Re(zj) -* oo (Re(zj) -« -co). We will therefore discuss the behaviour of the integrand as Re(zj) -• ±co. The arguments of the r-functions in the numerator, are normally sums of some of the arguments of the r-functions in the denominator. In particular, we will obtain integrands where all the r-functions may be combined into beta-functions. This property is important for the asymptotic behaviour of the integrand as |Re(zj)| —> oo. The beta functions will have the form Bl^ + kM + h,)^———^, (4.19) where at and a2 are real numbers of order 1, and fci and ij are complex numbers that do not dependent on Zj. Up to a factor that is proportional to |Re(zj)| raised to a power of order 1, we find, again by using Stirlings formula, that |B(a!«j + 4i, ajZj + frj)| ~ exp[(aiIog|ai| + aalog|aj|-(a, + o,)log|oi +a!|)Re(i3)J (4.20) when Re(z) -» ±oo. Here the coefficient of Re(zj) in the exponent is of order 1. Strictly speaking, (4.20) is not valid for |Re(zy)| —» oo if oiz> < 0 or OJZJ < 0. Still, it may be shown that the total contribution to the Zj-integral will be suppressed as indicated by eq. (4.20). Next, we consider the factor t*n _ eiosW(oo»,+i») (4-21) where we have written V>o as a0Zj + V We see that the behaviour of the integrand of eq (4.13), as Re(z) —> ±oo, is dominated by the factor s*° if \a0\ 3- l/log(s). In this case, we may close the contour at infinity in the half plane where s*° —> 0 as |Re(zj)| —» oo. On the other hand, if |oo| ~ l/log(a) or less, and in particular if ao = 0, it may not be possible to close the contours at infinity without getting a non-zero contribution from the part of the contour at infinity. We will, however, not evaluate such integrals by closing the contour at infinity. 24 4.6 Algorithms for doing the complex contour integrations It is in principle straightforward to find the leading behaviour of eq. (4.13), but the calculation may be very lengthy. I have therefore written a program that can do this calculation, and I will now describe the algorithms t! it the program is based upon. The code is described and listed in appendix C. It is in general not convenient to dote the contours at infinity, since we get contributions from an infinite number of poles for each integration, and it may be difficult to tell which poles contribute to the leading order. Therefore we will make use of considerations by Bergére, Calan and Malbouisson [18, 19). These are very useful for the calculation of the leading terms of eq. (4.13) with © = 0 in V"o- Non-leading terms will either have fewer logarithms than the leading terms, or be suppressed by at least a factor \t\ /«. Thus we may find the leading terms for 3/0, by making the replacement s -. «-ie (4.22) Non-leading terms will still be suppressed. Finally we take the limit 0 -»ir to find the physical result. First we note that Re(i/>o) is independent of the imaginary parts of the integration variables when © = 0, and we may keep «""t*») outside the integrals. Thus the «- dependence of the integrand will be given by ***"<**>, which has absolute value 1. Since the integral is absolutely convergent, it follows that it is bounded when s —» co, and that eq. (4.13) will net increase faster than aRe(*11) as a -» co. Furthermore, defining p as the minimum of Re(V>o) subject to the constraints c; > 0 for t = 1,... , n, Re(^i)>0for»'=l,...,mi, (4.23) it follows that J will increase more slowly than i**' as s —> oo, for any real, positive e. Thus we expect the leading terms of / to have the form p Cs log(o1)...log(in), (4.24) where C is a constant and ai, ..., a* are ratios of invariant quantities, e.g. s/\t\ or s/m*. We define the point Za in the space of the complex integration variables, as a point where V>o = p, eq. (4.23) is fulfilled, and Im(2;) = 0 for t = l,...,n. Z0 will normally be uniquely determined, but this is not important for the following discussion. The problem of finding p and Z0) is known as linear optimalization. Efficient algorithms have been developed for this problem, and we have made use of the SIMPLEX algorithm, which is included in the standard library of MAPLE. 25 We now use the relation T(x) = T{* + l)/x for IMunctions whose argument vanish at 20, such that the singularities at Z0 are removed from the T-functions. This puts eq. (4.13) on the form '=^i/--/j^fij?fe'* (42S) where the IMunctions have been hidden in the function g, which is analytic at Za. The primed products only includes factors that vanish at Z0. The next step is to change variables in eq. (4.25) into z(, ...z'„, such that all new integration variables vanish at Z0- Using factors in the denominator as new integration variables, and dropping the primes, we obtain { I= -^fdzi...[dzng(z) * "V*"* i», (4.26) where all integration variables vanish at Z0. The functions g(z), V'o • • • vW and 4>i • • -4>m', may be different from those in eq. (4.25). The functions Vo • - • V1™; and 4>i... <)>m^ are still linear in the integration variables, but now all except i/>o are homogeneous in the integration variables, while the constant term of V'o is equal to p. The function g{z) is Btill analytic near Zo, and the poles in g(z) will normally only contribute to non-leading terms. A factor * = £«<«, (4.27) where «u,- are rational mun! * Thus we may expand the integrand by using *-r = t«iTr-T> (428) such that we get terms without the factor 26 We repeat the step» above until we obtain a sum of terms that either have no factor 4>i in the numerator, or are suppressed by a factor t/m' to a negative power. Thus we obtain terms of the tonn f du... f dzng(z) -, (4.29) where the functions g(z) and ip0.. .ifij are different from those in eq. (4.26). But still all t/ij's except ipo are linear and homogeneous in the integration variables, V'o is linear with constant term p, and g{z) is analytic near ZB. The contours are still constrained by c, > 0 V,- and Re(i/>j) > 0 V,. Constant factors have been put into g(z). It is straightforward to see that this integral will give terms with at most d logarithmic factors, and we will use the notion "leading logarithms'' about terms with d logarithmic factors. We will discuss two ways to evaluate the complex integrals in eq. (4.29): 1. By direct integration 2. By algebraic reduction of the integral 4.6.1 Direct integration We pick a variable, Zj, which has a nonzero coefficient in Vto- Since g(z) is an alytic near Z0l we may shift the integration contour from Cj to a point ej where Ref^o) < p, without crossing any pole in g(z). Since g(z) falls off exponentially as the imaginary part of one of the complex variables approaches ±oo, we will only pick up contributions from poles in 1/(21...z„ij>\ • ••''I'd), and obtain / dzj = / dzi + contributions from poles in. ; —. (4.30) Jcj /cj Z\ • • . Z„ll>l ...fa The first term on the rhs. will be of order sp, with p' < p. In our case, sT' will be of order \t\js or less relative to sT, and we may discard the first rhs. term. The remaining terms have one less complex integration left. After some integrations, some of the linear forms in the denominator may become equal to each other, e.g. 2j + 22 and 2j — 22 become equal in the contribution from the pole at zj = 0. Thu6 we get higher order poles, and the factor **> will be differentiated when we evaluate the contribution from these. This produce factors of log(a). We carry out the contour integrations successively as described above, until ipo is independent of the remaining integration variables. This gives a sum of terms that are proportional to expressions of the form ^log'W / dzx...\ dzk ^ -, (4.31) 27 where f, k and d are nonnegative integers, and p is defined in section 4.6. Some of the V>>'> may here be equal to each other, or to integration variables. The number of integrals, k, may be zero. In this cue, the result is simply a'log'W (4.32) times a constant. Normally, we will have k = 0 in (4.31) in terms with d logarithms, i.e. no contour integrals will be left in the coefficients of leading logarithms. We may show that this will always happen if the product V>i.. -iij in eq. (4.29) depends on all the integration variables. In other words: If the denominator z\... ZnV>i • • • V'd ** a' least quadratic in all integra tion variables, the coefficients of the leading logarithms will be given as rational numbers, not as contour integrals. The following simple example is typical: 1= f dz, f dz,T(l + Zi)T(l + z,)V{l - z, - z,) f ** Jc, Jc, 2lZ,(*l + Zj)J = / dz, fr(i + *or(i - z,)~ - r(i + *,)r(i _ „)!*jj} Jci [ Z, Zi + terms with no log(«) + O I - ) J = i iog w+ / dz1T(zlyr(-'i) H(») Jci + terms with no log(j) + O (-J . (4.33) We see that no contour integrals are left in the term with two logarithms, while one integral is left in the term with only one logarithm. The one integral was left because the coefficients of z\ and zj in the exponent of s were equal, and thus the exponent of s vanished in the contribution from the pole at Zj = — z\. Such situations may be avoided for one-loop integrals, but not for two-loop integrals. In the latter case, coefficients of non-leading log's may appear as complex contour integrals, but coefficients of leading logarithms will normally emerge directly as numbers. It happens quite often that coefficients that appear as contour integrals, may be reduced to rational numbers. Consider e.g. the following sum of two integrals: ^rd + .^d-^^^rd-»)i"(i + ,) (434) We change variable from z to —z in the second integral, and shift the contour of the same integral through the pole at z — 0. Picking up the contribution from the pole, we obtain j* r(i+*)i-q-») +1 _ jJz rd + ^d-z) = , (4Jfi) 28 In this simple cue, we could u well have replaced the T-functioni by their values at 2 = 0 before we did the integral. For coefficients that are given by multiple contour integrals, however, such replacements may introduce errors. Similar reductions may also be done in the case of multiple contour integrals, but they are more complicated, and we have not yet developed MAPLE-code that does this. 4.6.2 Algebraic reduction of the integral An alternative procedure is to reduce the integrand algebraically before doing any contour integrals. One such procedure is described by Bergere, Calan and Malbouis- son [18, 19|. This procedure turns out to be less convenient than the more direct one, for computer calculations of the leading contribution. It is, however, useful for finding contributions that are suppressed by a factor s/m3 to a positive power relative to the leading contribution. It may be important to find such contributions when the leading contributions cancel exactly. 4.7 Example: A one-loop integral To illustrate the methods described in the previous sections, we will calculate the integral I = s2 I da\ dai das da* 6(1 — V)«)—;- Jo "-1 g2 2 y(l-y)z(l-zf I dxdydz- (4.36) Jo which arises in the calculation of the contribution from the box diagram to e+e" —t p+/j~-8cattering. We have choBen this diagram since it is calculated explicitly by Kraemmer and Lautrup [21] with other methods. We find that our result agrees with theirs. The second form of I in eq. (4.36), is obtained after the change of variables: a — i = 3/(1 *)i a> = (1 — y)(l — z), as = xz and at = (1 — x)z. The denominator is given as 0 g = ctiais + a3att - (aj + 012)^' - (<*3 + 04)03»"' ~~ (°3 + "O *"»* + ie 2 2 2 = S(l - y)(l - z) a + x{l - x)zH - (1 - z)X - xz m\ -(l-x)z2ml + ie, (4.37) where s, t and A are defined as before, while me and m„ are the masses of the electron and the muon respectively. We will now show the details of the calculation of I to leading order in \t\ /«. We will put mc = 1. Defining ^0 by i6 z V"o = -zi - 7t a ~ 7A*3 - 7m,&> + . , ,, (4.38) 29 where the definitions of 71, ix, .. .follow from eq. (4.16), we find , i ) /-«•/ x[(i-*)z'P[*zJ]-'+"+"+J3+%* i=l w«i J i=l X j.-»+«+«+«(l - a)-*-^1 ""(l - j,)1— = ,2n[/^r(zolr(2-t;^) x B(-l f z1 + z3 + z<,l-z2-Z4)B(2-z1,2-z1) x B(-2 + 2zi + 2z3l 4 - 2z, - z3) t*', (4.39) where B is the beta-function, and the Ci's must be chosen such that: Ci > 0 V;, Ci + C2 + C3 + C4 < 2, Ci + C3 + C4 > 1, c2 + e4 < 1, d < 2, c, + c3 > 1, 2c, + c3 < 4. (4.40) Up to this point no approximations have been done, and the result is exact. Now we do the complex integrals as described in section 4.6. We put 0 = 0 in V'o, 2 and assume that 1/A 3> a S- \t\ > m„ 3> 1. The infimum of Re(j/i0) subject to the constraints above is —2, thus the integral will at most be of order s~2 times logarithms. In this simple case, it follows immediately that the leading contribution will come from the poles where Zj = 2 and z2 = z3 = z4 = 0 in eq. (4.39). We extract the poles that contribute to leading order by using the identity T(z) = T(z + l)/z, to obtain 2 I = » I dzi... I dzt g{z,z ,z ,z,)a'hl i 2 3 (4.41) z2z3z4(2 - zi - z2 - z3 - z4)(2 - zi)(4 - 2zj - z3)' where S(«ll«2,Z3,Z4) = 2r(z,)r(*2 + i)r(z3 + i)r(z4 + i)r(3 -*,-*,- *s - z4) r(-i + z1 + z3 + z4)r(i - z2 - z4) r'(3 - zt) r(z!-z2 + z3) r(5-2«i) 30 xT(-2 + 2z,+ 223)r(5 - 2zt - 2,) r(2 + 2j) K ' Contributions from poles in c(2i,2j,23,24), will be suppressed by at least a (actor \t\/s. It is now straightforward to i'nd the leading contributions to the integral, since we know which poles will contribute. But to proceed in a more general way, we make the following considerations. We first shift all the contours, such that Cj is close to 2, and c-,, c3 are close to 0. We do the 2i-integral first, and note that zx has a negative coefficient in i/'o- Thus we may decrease Re(V>o) by increasing cu and we may do this till Re(^>o) is close to —3 without crossing any poles in g. We pick up contributions from the poles Zi — 2 — 22 — 23 — 24, z\ = 2 and 21 = 2 — Z\ — Z3/2, and from the arguments given in section 4.6, we conclude that the remaining 2;-integral will be of order s"3. Proceeding in the same way for the 22-, 23- and 24-integrals, and going to the physical limit, we finally obtain the result / = i[log2 \t\ - 21og \l\ log » + log2 3 + 27rilog |t| - 2?rilog« + 21og |t| - 21og > + 2irt] + O ? = 21og2^ + 2«logy/^+21og^ + ^ + 0^), (4.43) which agrees with eqs. (A.20-22) in [21], where the integral J above is called I3. 31 Chapter 5 The two-loop ladder-like diagrams We will now evaluate the matrix elements of diagram a and diagram 6 in the way that was described in chapter 4. In paper I, we found that the matrix element of each diagram is given by [see eq. (1.6.1)] *[£""'+a5iJV" + 4Z>*]- (51) The numerators TV and the factors A and D, which are polynomials in the ct's, are different for different diagrams, and will be given explicitly for each diagram below. We will first factorize the integrand by introducing a large number of complex integrals. After this, the Feynman-parameter integrals are straightforward, and give a product of beta functions. Finally the complex contour integrals must be carried out. This part is very tedious, but possible to do with a computer program. We will use the notions 'the uncrossed diagram' and 'the once-crossed diagram' to denote diagram a and 6 in the same way as in paper I. 5.1 The uncrossed diagram In this case the numerators are given by Nm = 2s2[aaecW<+) + (aa + bl)(cc + dtf)W<->], <+) Nn = -4s|[(oti + a3 + a6 + a«)aå + (aj + a, + at + aT)cc + 2a«(oc + ac)] JV + [4(ai + a3 + aB + as)(oi + bb) + 4(o2 + on + as + aj)(cc + dd) — at{ai + ad + be + be - ac - ac - bi-bd)]W<->}, AT, = 8[(A + 6a5)lV(+) + 4(4A + 6a3)W<->], (5.2) 32 with o = (oj + O6)(QI + 04 + a, + o7) + 00(04 + aT), b = OS(OJ + a, + ag I- a?) + 040», c = (04 + aT)(ai + a, + as + at) + a»(a3 + as), d = at(ai + aa + as + o«) + a3a«, a = o(ai •-* a3;oj «-• 04), 6 = 4(ai «-» a3;a2 <-» 04), c = c(ai «-» a3; o} <-» 04), d = (±) W = ti(p|)7Aii(pi)5(pj)7Aii(pj) ± u(p',)7»75»(Pj)«(Pj)7J7s«(pi)- (5.4) The polynomial D has the form 2 2 D=* D.3 + D,t + Dmm + DXX + ic, (5.5) with D, = 0103(02 + 04 + 04 + 07) + 0204(01 + o3 + o5 + at) + O1O4O8 + oaa3o«, Dt = 050907, 2 S An = (Ol + 03) (a2 + O4 + 0« + O7) + (02 + 04) (0] + Q3 + Q6 + 0«) + 2(oi + a3)(as + oujoe, D), = -(a6 + at + a7)A, (5.6) while the polynomial A is given by A = (01 + Q3 + o5 + oe)(a2 + a4) + (01 + a3 + o6 + OJJOT + (ai + 013)06 + Q6o«. (5.7) We may write M as a sum of terms that have the form 4 C £° da-,... y°° da7S(l - $>) of • • - o?' A"" XT"" (5.8) where C is a constant and «1, ..., n7, nj and no are nonnegative integers. We note that the integrand of the rhs. of eq. (5.1) is homogeneous of order -7 in the a's, thus T»I + ... + ri7 - 2nA - 3nD = -7. (5.9) 33 After replacing D by (4.12) (with Du = 0), we tue the factorizing formula, eq. (4.1), to write A-"» £>-"«> as r a 0? A T(Sfn[/w^r(z..)] ^-|^ "'- ' ""' J x laiOjfaj + a, + a, + aT)]~ '[a2a4(a, + aj + as + a»)]"" 5 x ja^a»]" [ajasa»]*"|a5a«aT)"* J x [(ai + a3) (a, + 04 + 0(5 + a7)r*[(a3 + a4)'(a, + a3 + as + as)}'" x[2(ai+az)(a, + at)at}— x (a(Aj-'*(a«A]-*",[a7A]-"D+"+-+*"><*", (5.10) where i'o = -(*i + *2 + *3 + *•) f 1 - j-y-r) - 7<*s - 1m(ze + z7 + z8) - 7>(nD - z, - ... - za). (511) The quantities ft, ym and 7A, are defined by eq. (4.16). Using the factorization formula three more times, we also obtain + + A-A-" « -+« = / dzu...[ dz13T(zll)T(zl2)T(z13) r(«D + «A — Z\ — . . . — Za — Zn — Zi2 — Z13) r(7ic + nA - Z] - ... - z») x [(a! + a3 + as + 06)(a2 + 04)]-*" [(ai + a3 + as + ae)a7]-'" ni+ + + + +I + x [(aj +a3)a,]-'"[asa8]-""- " - " «' " *", r(z )r(2Z8 + + Zl3 Zl4) (al + a3)-"«-"-'»= / rfz14- " "' " T(2z, + z, + z,3) xaf'a,"-*14 «-2*' 4 -*t-xn ""•+m• («2 + 04)—— = / ,Zl/(^m2z7 + z. + z,--z,t) Jc,, r(2zT + z, + zn) x apia;2"—-*»+*". (5.12) Combining these results, we may write (5.8) as1 CW° fer*11 • •• jf r(nj — Z] — ... — zip) r(np + TtA — Z\ — ... — zB — zu — Z12 — Z13) r(nD) r(nD + nA - zs - ... - z») r(2z6 + zg + Z13 - Z14) T(2zT + z, + Zu - z16) r(2z« + z, + z13) r(2zT + z» + «ii) - , 0 x a,-* .. .«^(aj + a3 + as + a») "^ + a, + a6 + a7)-"2- "»^ ,(5.13) *We assume that the limit © -* x gives the correct result, see section 4.4. 34 where p, p,0 are linear in «i,..., »u- We will now do t'jt a-integrals, using the same approach as Gastmani, Trooit and Wu (12): Since the integrand of eq. (S.8) is homogeneous in a of order -7, we may replace 6(1 - at - ... - aT) by l{\ - 04) (see also appendix B in paper II), and do the a-integrali by using tbe formula Hdx,...dxn *;••... z?"(l3 + z, +... + *„)-» JO _ «-»-.,-...-.. r(l-a.)...r(l-a.)r(o-n + al-f... + tt.) - p r(*) l ' two times, with 0 = o« = 1. Substituting for p,, ..., p,0, we finally find that eq. (5.8) is equal to C (-1)"" Urn /' do,,... f' da7S( 1 - £ a) fi f / ^(z,)] ®-**rfO «/0 j=I L/c, J r(njj - zi - •. • - »io) r(np + t>A - Zi - • •. - z, - *u - z» - *u) r(nc) r(nD + nA-«,-...-i.) r(2z» + z, + z13 - zu) r(2z7 + z, + zu - zu) r(2z, + za + z13) r(2z7 + z8 + 2ii) 1 r(zj + z7 + zu + zia) r(zi + z«) X T(l - 7lJ> + »7 + *1 + «J + 2» + *4 + 2« + *7 + 2» + 2» + ZlO - «1») X T(l + n, - *j - z3 - z,4) T(l + n3 - Zi - 24 - 2z« - z» - zjj + zI4) x T(l - n0 - n4 + nB + zi + zn + zij + zu + za + z3 + z4 + zt + zj + z3 - z») x T(-3 + nD + n>, - n, - n3 - n6 + zt + zg + z») T(l + m - z, - z, - zlt) x T(l + n4 — z2 — z3 - 2 zi - zt — Zn + zu) x T(-3 + T»D - T»J - n4 - m + z, + zT - zs - Z10 + Zn + 2n)<*°. (5.15) This expression is exact, and we will return to the evaluation of it6 leading terms in 5.3. 5.2 The once-crossed diagram In this case, the numerators are given by eq. (1.6.10): N„j = 2«'[oo(cc+ Nn = 4*{[(a2 + a7)(ad - ac) + 2(a4 + at)(ad - ac) i+) + 4(ot + a3 + a3)aå - (a, + an + a, + a7)(cc + dd)]W + [-{a, + a7)(bc + 2ac) - (a, + a3)(bi + lac) ) + (a, + a3 + as)(au + bb) - 2(as + a„ + o, + a7)cc)W(-), + JV> = 18[3(aj + oT)(a4 + a.) - 2A](W< > + */<-)), (5.16) 35 with a = (a,+ or)(aj + of) + (a, + a, + oT)(o4 + a»), b = (a, + a7)(a3 + Cn) + 0,(04 + a»), c = (oj + at + 07X04 + a») + 07(01 + as + <»«), d = Qjaj - 04(01 + at), å = a(ai *-» as; aj •-» a<; a» *-» 07), 4 = 6(0] «-» QjiO] *» 04; 04 •-« 07), c = c(ai *•» aj; Oj«-* o<; a» *-* 07), rf= il(oi •-» 03:03 «-• a«;a« «-* a?), (5.17) and W^J denned by eq. (5.4). The polynomial D is now givsn by 2 D = D,t + Duu + D,t + Dmm + Dx\* + ie, (5.18) where D. = 0^3(0} + 04 + a» + a7) + 010407 + Q20joe Du = 030405 Dt = aB06Q7 , 2 Dm = (oi + Qs) (oJ + 04 + a, + o7) + (oj + 04) (a, + a, •+ ot) + af(o4 + a») + aJ(o2 + aT) + 2(oi + 03)030» + 2(ai + 03)0407 D* = -(o6 + a, + aT)A, (5.19) and A is given by A = (oi + oa + aB)(oj + a4 + a, + a7) + (o2 + 07X04 + a«). (5.20) The matrix element M may also in this case be written as a sum of terms of the form 1 CJ doi.../, da7ff(l-£a)a7'...a7"A-"'D-" ', (5.21) but with the new definitions of D and A. In this case it is convenient to use an approach that is different from the one we used for the uncrossed diagram. We will first approximate D\ by —A, i.e. replace (05+05+07) by 1 in D\. From the following arguments, we expect this replacement only to give an error of order A/m raised to a positive power. Due to the delta function, (o5 + a» + 07) + (ai + aj + 03 + 04) = 1. Thus, what we want to do, is to add 6A = -(o, +03 + 03 + Q4)A (5.22) 36 to Dy. It is easy to see that \t>x\<2i\Dm\ (S.23) in the parts of the integration space where max(ai, 01,03,04) > max(ai,o«,ar). Since A1 < m*, it follows that \£>xX'\<\D„m'\, (5.24) unless max(as, a», o7) > max(a1, Oj, 03, a4). On the other hand, if max(o(, at, aj) > max(ai, 01,03,04), we have \b>\<\Dx.\. (5.25) It follows that D\\* is always negligible compared to Dmm? + D\\*. Using the same techniques as for the uncrossed diagram, we replace D by (4.12), and write A""' D'"" as tg!n[//^)Jr(nD-|,)ttr....«?-A- , , 5 x [a^o^cij + 04 + a« + a7)J"*[aia4aTJ"*[Q2<»jo«J"* , 1 x [aja4Q5]"*[a5a6a7]"" 2 x [(a, + a3) (o2 + a< + a, + a7)]"«[(aj + a,)*^, + a, + 05)]-" x [alien + a,)]—(afto, + a,)}-" 0 x [2(oi + a3)o,o«]-« [2(ai + 03)0,07]-*" x (Al-""+"+-+"' »*», (5.26) where ^0 = -fa + zi + ts) U ~r~n\) ~y»z*~'i'Zi n - 7m(*« + '7 + 3» + Z» + Z10 + ill) - 7A( f - «1 - . . . - ill). (5.27) Once more, 7,, 7„, 7m and 7* are defined by eq. (4.16). Further factorization yields •/c„ r(nD+»A-*i-"--*n) x [(a, + o3 + a6)(aj + 04 + o« + or)]"*" x [(a, + a7)(o4 + 0,)]-»-»+''+•••+«•», (01 + 04)-'" = / ^^^or»o4-«-», (5.28) and we find that (5.21) becomes C (-1)»" Km jf' do,... jf' do76(l - £ o) II [/. d*At)] 37 r(np - fi - ,.. - tii) r(n0 + n» - «i - ... - x») r(2«T - *u) r(nc) r(no + nA -*,-...-*,,) r(2zT) x af ... of(a, + a,)_»*(a, + a, + «^""(a, + aT)""'^, + a,)-»" p u < x (a, + en + a, + a7)~ "2~' »* '. (5.29) To do the a-integrals, we change coordinate» according to a, = {l~Pl)(l-p,){l-pt) «r = (l ~Pi)(l ~ Pi)P* a* = (1 - ftlftC - Pt) a» = (1-Pi)pap5 <*s = Pi(l-P;i) <*i = PiPs(l - Pe) a3 = pip3p«. (5.30) The Jacobian of the transformation is p|(l - p\)3pi(l - PJ)PS- The p-integrals go from 0 to 1 without a ^-function, and give therefore the following product of six beta-functions: B(3 - pi - pi — Pi - p« — p», 4 - pa - p< - pt — pi - pio - Pn - Pit) x B(2 - p, - pa - pn,2 - p, - p7 - pio) B(2 - pi - ps - p», 1 - p6) xB(l-p7,l-p2)B(l-p6,l-p<)B(l-p3>l-p1). (5.31) Substituting for pj.. .pllt we finally find that (5.21) becomes C(-1)-" Urn I dz,...i dz13T(z1)...rOu) r(np - zi - ... - zu) r(np + nA - z, - ... - z12) T(2z7 - 2ia) r(n0) T(nD + »A - z, - ... - z„) T(2z7) X B(3 + 7»1 + »3 + »S — 2zi - Zj - Z3 - Z4 - Z6 — 2Z6 - Z7 - Zio - Zu — Zi2, 4 - 2nrj - 2n\ + n3 + nt + n« + n7 + Zj + z« - z8 - z» + z«) x B(2 - »x> - n* + »« + n» + zi + z« - z7 - z» + Zn + z«, 2 - nc - n* + z» + z7 - z> + «i + zia + »2 + n7 - J») x B(2 + r>i + na - 2zj — z^ — z3 — 2zs - zio - zii, 1 + nB - z« - z6) X B(l + n7 — zj — Zg — Zn, 1 + xt2 — Z3 — Z4 — 2ZB — Zio — Zja) x B(l + B« - z3 - z5 - zio, 1 + n4 - z2 - z< - 2z7 - 2zj - zn + zu) x B(l + n3 — zx - z3,1 + «i - Zj - zj) x 2-"°-*" a*" (5.32) 38 5.3 Results We will now find the leading contributions from the uncrossed and once-crossed diagrams with s > |i| > ma > X'. (5.33) We start with eq. (5.1), which we express as a sum of terms of the form (5.15) for the uncrossed diagram, and (5.32) for the once-crossed diagram. The calculations are very lengthy due to the large number of complex variables, but with the MAPLE program CINT, terms with four logarithms may be found in a few hours. CINT is described in appendix C, and appendix D describes how to use CINT for the evaluation of eqs. (5.15) and (5.32) We find that the following results hold for both diagrams: Ioda'-Ioda,S{1"Xl--"'7)A^ ;> f' „, , 2s2aacc(Wl+1 + W<">) = / 1 /da1...j[ da7*(l-a1-...-a7)^ = <7Q r*»«-jf'to'«(1-*--H^ = 0g). (S.M) The functions A, D, Ni, Nn, NJU, a, a, c and c, which are different for the two diagrams, are defined in sections 5.1 and 5.2. The integral //// is defined for each diagram according to eq. (II.2.9), as hu = Iiu(»,t) i = fdai...pda7S{l-ai-...-aj)^L\ (5.35) Jo Jo \*D3 lw(+)=H'<-)=i The results for the second and third integrals in eq. (5.34) are in agreement with the results of paper II. Moreover, we findtha t each term in these two integrals is of order 1/s. We have calculated terms with four logarithms in Ifu and l\tlt where indices a and h refers to the uncrossed and once-crossed diagram respectively. We find that these terms depend on the magnitudes of 7„, ft, -fm and fx• Since we neglect terms of order )t|/s, we have 7U = 1, and since we are free to choose units, we may put m = 1 such that fm = 0. Thus the result depends on two independent parameters: 7t and 7a. 39 We have done the calculation! for different valuei of 7» with eq. (5.33) fulfilled, and find that the relation J?„ = -aitf/ + O (log») + O (J, £, ^j) (5.36) holds in the two limits i -• oo (5.37) with t, m, and A fixed, and A -• 0 (5.38) with a, t, and m fixed. Thus, we see from eq. (1.7.10) that terms with four logarithms cancel in Foo for the sum of the six diagrams. This agrees with papers II and III, where we find that only terms with two logarithms do not cancel. We expect eq. (5.36) to be valid in any asymptotic limit where « S> |t| 3> m1 > A9, but we have only checked this in a few cases. In the calculation of /;;/, some of the coefficients appeared as contour integrals, and we used the identities j[d,z3r'(,)r3(-z) = i, (5.39) jdz z6rs(z)r3(-«) = o, (5.40) to put Iflt on the same form as IftI. We also find that the only terms in Njn that give four logarithms in the limit A —» 0, are ^'(ajaj + atø + atø). (5.41) This is in agreement with paper II, where we find that the leading term is .?i(C,'a), defined by eq. (II.4.14), in which the numerator is obtained from Nut by putting <*i = «2 = <*3 — <** — 0. We will now give the asymptotic form of Ifn for different values of it and 7A with eq. (5.33) fulfilled. We use the following notation to make the results more compact and easier to read: *-*•(£). x. = *(3). *>-"••(;£)• ("') We do the calculations with 9 = 0. After making the replacement (4.22), and taking the limit © —> ir, we find that the asymptotic form of l°tI is given by (see also table 5.1) 3 ^L,L t-iL.L\Lx + 4L,LtLl-^L.ll-L* 40 Figure 5.1: Regions where the asymptotic form of IfIt is analytic. We have here used units in which m = 1, such that 7t = 0 and 7A = 0 correspond to |() = m1 and A = m2 respectively. + 3 "t ^* ~~ 2 ^' ^x + 3 ^' ^* 3 toM + 0(H.-*) (5.43) for 0 > log(A2/m2) > log(|t|/a) (region I in fig. 5.1), °M+0(?.^)] <«*> for log(|t|/,) > log(A2/ms) > 21og(|t|/s) (region II), /?„ = -^^^t + l^^ + iLl^-iLlUL, 41 + \iiil-\L.ii + ai.iih-^ + l L* - 1 L* + iLt 48£* for 21og(|<|/j) > log(AJ/»»J) > -21og(j/ma) (region III), and + 2i; XI - | i.i? + 2£. X? £» + I ij for log(A2/m2) < -21og(«/m2) (region IV). We note that Ifjj is continuous on the lines that separate different regions. We also see that the result in region I is in agreement with eq. (III.7.3). As a check of our methods, we have also used eq. (5.15) to recover the results for the three-rung ladder diagram in ^3-theory obtained by Osland and Wu [10]. To do this, we simply interchanged a and t and put A = m in eq. (5.15), and then evaluated eq. (5.1) with JV„, = £A«, «at3 Nt, = Nt = 0. (5.47) The methods used here combined with the computer program CINT, have been very useful for the calculation of leading logarithms. Eqs. (5.15) and (5.32), which are used as input to CINT, may also be used for the calculation of non-leading logarithms and constants, but this will require further development of ;he computer code. 42 Logarithmic Region factor I II III rv i i v. 5 3 4 4 4 L\U 3 3 3 3 4 a 4 L M 3 3 3 L\L\ 2 4 4 L\Ul* -4 -4 -4 L\L\ 2 | 2 8 8 L.L\ 1 3 3 L.L1L* -4 2 2 LaLtL\ 4 L.L\ i -1 ø -1 2 1 1 Li 3 3 3 L3 L 4 t x i 3 1 L]L\ -! 3 L.LI i 1 L\ -s 43 Table 5.1: The coefficients of L*, £j£, etc. in the function tIfrt/2 in each of the four regions shown by fig. 5.1. 43 Chapter 6 Concluding remarks We have developed methods for the calculation of contributions from ladder-like diagrams to Bhabha scattering. We have calculated the leading terms both for separate diagrams, and for the sum of the gauge-invariant set of all six ladder-like diagrams. For separate diagrams, the results depend upon how A2 is related to s, \t\ and m2, whereas the leading term of the sum of the six diagrams is the same in the cases that we have considered. Still, subleading terms, which we have not evaluated yet, may have a more complex structure, and depend on how the above invariants are related. It is not surprising that the result for single diagrams depend on the magnitude of A. In ^'-theory, we see a similar dependence. In this case, the result depends on how |t| compares with s and m2. We see no reason why such dependencies should not be present for the sum of a gauge-invariant set of diagrams. We have evaluated the integrals in two different ways, and we expect both meth ods to be valuable ibr calculations of contributions from other Feynman diagrams. In particular, we have in mind virtual QED corrections to Bhabha scattering or pair production at small angles. In these cases, detailed calculations may be possible even at the three-loop level. 44 Appendix A Proof of two relations We will now prove the two relations that were used in the proof of eq. (4.1). The first relation is p —i*p/2 foo °-°=wrl dur"e'tu ( p , r(P) -^/„ <'" - ^ = T^r(--rr(p) = ^. (A.2) This follows from the definition of the T-function for imaginary u with uji > 0 and Re(p) > 0. The result may be extended to any u with Im(u) > 0 by analytic continuation. The second relation is eiT = fdz T(z) e'"'2 T- (A.3) for c> 0 and Im(T) > 0. Using Stillings formula, eq. (4.8), we find that the absolute value of the integrand is asymptotically Æexp [(as - \) log T - y9 - x - y - x log \T\ + y sng(T)} (A.4) for z —> oo with |arg(z)| < ir. We have U6ed r = \z\, 6 = arg(z), x = Ke{z) and y = Im(z). When y —> ±co with x fixed, the integrand is approximately exp [(x-i)log|!,|-ye-^+yarg(T)] (A.5) times a factor that does not depend on y. This falls off exponentially as |y| —t oo if Im(T) > 0. It follows that the integral is convergent for Im(T') > 0. To prove 45 Ca Ct Figure A.l: The contours Ci, Ci, C3 and CV eq. (A.3), we integrate along the rectangular contour shown in fig. A.l, and then move the sides Ci, Cz and Ct towards infinity away from the origin. T(z) has simple poles for z = —n, n = 0,1,... with residues (—l)n/n!, giving the contribution T £ LiLe-W»r = £ -AiTf = e' (A.6) n=0 "• n=0 n- to the rhs. of eq. (A.3). It follows that eq. (A.3) holds if the contributions from Ct, C3 and C4 vanish as they approach infinity. It follows immediately from eqs. (A.4) and (A.5) that C2 and d do not contribute in this Umit. We consider the contribution from C3 after Cj and Ct have been moved to infinity. By Cauchy's theorem, the contribution from C3 is not changed if we move it along the real axes without crossing any pole. Thus we may choose the intersection with the real axes to be c — n where n is an integer and c is the same as in eq. (A.3). By Cauchy's theorem, we may also choose c such that 0 < c < 1 without loss of generality. With these choices, the contribution from C3 may be written as / dzT(z)é"l2T-'< I dz \T(z)e","T- Je~n Jc-n ' 2 +n rpn = Jdz \V(z-ny"' T-' \ < M (A.7) (»-!)! where M = [dz |r(i)e"'/JT- (A.8) 46 Since M does not depend on n, we have shown that the contribution from Cs vanish as n -» oo. We have thus shown that the contributions from C3, C3 and Ct vanish, and that eq. (A.3) holds. 47 Appendix B Momentum integrations and manipulation of Dirac matrices with REDUCE The main results of paper I have been checked with the symbolic manipulation program REDUCE. In this appendix, I will explain how this was done, and list the REDUCE code. To understand the details of the code, some knowledge of REDUCE is required, see e.g. [22]. Names of REDUCE variables, procedures and operators are printed with bold types. The calculations are mainly done by the procedures and operators listed in sec tion B.5, the most important ones being MQ, MG and MQMGCONTRAC- TIONS. These procedures and operators may be used for any of the six diagrams without modifications. First the procedure MQ performs the Feynman parame terization and momentum integrations, then MG simplifies the spinor factor, and finally MQMGCONTRACTIONS contracts the momentum and spinor factors. B.l The momentum integrations The procedure MQ carries out the momentum integrations according to appendix A of paper I, where we show that Mq, which is defined by eq',. (1.3.5) and (1.3.10) for diagrams a and 6, has the form + T^j LjjT W + aus"**?*; + OMS"**?*' 48 + W*?*; + <»IJ We also write k* = api+bp, + CQA Q *t = cp'I+«jP; + CQB *e = ip1 + op,+ CQC 1 D = D.s + Dtt + Dmm + Dx\* + it. (B.3) Table B.l gives correspondences between REDUCE variables, and variables that are used above and in paper I. In addition, the REDUCE program use the vector quantities PINTERNALt, > = 1,..., 7 to represent the momenta of the internal lines. Here the index i is the same as that of the corresponding Feynman parameter, see figs. 2 and 3 in paper I. B.2 Simplification of the spinpr factor The operator MG (MG is denned as an operator because procedures may not have indices or vectors as parameters) simplifies the spinor factor M1, which is denned by eqs. (1.3.4) and (1.3.9) for diagram a and b. The program calculates the functions FOO, FOS, FSO and F55, which are denned such that (AT)„,„ oc FOO U* Vh + F05 U* V^ + F50 U* V^ + FS5 U^ V*, (B.4) where Ux = «(pi)7««(Pi)i Ox = u(p'i)7a75u(Pi)> V* = «(ftbøf (Pi), V\ = 5(ft)7fl7«"(P2)- (B.5) 49 pi,pip1P2,p*p-.p1,p;>p,,pi S, TT, M, LAM « «, (, m, A Q1.Q2 ~«,,«j Vl, V2, V3, V4 <-> «I,BJ,OJ,04 (in appendix A in paper I) X1,X2,X3,X4<-* Zi,*j,*s,*4 MU,NU,RH,SI~ it,v,p,tr Al,...,A7<-> ai,...,07 LAMB <-> A VXAMB «-» A (explicit value) A12, A1S,..., A34 «-» au, a13,..., on (in appendix A in paper I) VA12, VA13,..., VAS4 «-. au, o13,..., a» (explicit values) A,AT,...,DT <-.o,o,...d VA, VAT VDT «-. o, o,... d (explicit values) STERM.TTERM, MTERM.LAMTERM <-. Dts,Dit KA,KB,KC,KD <-.*„,*!,,*c,Jbj Table B.l: Correspondences between REDUCE variables and variables in paper I. The indices of the REDUCE-variables on the rhs. of eq. (B.4), are such that indices that survive contractions are the same on both sides. In the REDUCE program, we use the variables PHI and PH2 to represent the indices (jfffhst _ jy(AI,iBB) GA,DE,EP,PH)7*. t = 0,5; j = A,S, (B.6) where tVe REDUCE variables AL, BE, GA, DE, EP and PH represent the indices a, p, 7, S, e and 1The notation of vectors and indices in REDUCE may need some explanation. The expression X.Y, is an ordinary dot product if X and Y are vectors. If X (Y) is an index, X.Y means YX (X* ), and if both X and Y are indices, X.Y represents the metric tensor gxr. Upper and lower indices are not distinguished. 50 B.3 Contraction of momentum and spinor fac tors The procedure MQMGCONTRACTIONS contracts W and W to determine the numerators Nm, Nu and Ni in eq. (5.1). The corresponding REDUCE variables are defined by JV„, ot NIIIOO U*t V* + NIII05 U4l V*, + NIII50 V+, V* + NIIISS £/*, VH. (B.7) The explanations after eq. (6.5) apply also here. B.4 The uncrossed and once-crossed diagrams Now the calculations may be done for any diagram, by denning the internal momenta and calling the three procedures. Only three lines of code depend upon which diagram we consider: The definitions of the internal momenta on the positron Une, PINTERNAL3 and PINTERNAL4, and the order of AL, BE and GA in the call to MG. The calculations for the uncrossed diagram, for example, may be done by adding the following lines to the code in section B.5: PI1ITEBHAL1:=P1-Q1$ PMTERHAL2:«P1P+Q2$ PIHTERHAL3: "-P2-Q1$ X Diagram dependent. PIHTERHAL4: =-P2P+Q2$ '/, Diagram dependent. PINTERNALE:-qi$ PIHTERBAL6:-Pl-PlP-Ql-q2* PINTERHAL7:-Q2$ DUMMY:=MqO; DUHHY:=MG(AL,BE,GA)$ 1. Diagram dependent. DUMMY:=MQHGCDHTRACTIOHS(); The output is listed in section B.6. We see that the results of the program agree with those in paper I. In addition, it is easy to include terms with a 75 on only one of the fermion lines. This may be done by replacing FOO with F05 or F50 in the definitions of NIIIOO, NIIOO and NIOO. For the once-crossed diagram, the three diagram-dependent lines should be changed to PISTERNAL3:—P2-Q1$ % Diagram dependent. PIHTERKAL4:—P2-Q1-Q2$ X Diagram dependent. DUMMY:=HG(AL,GA,BE)$ X Diagram dependent. 51 The results of the calculation agree with paper I also in this cue. Clearly, such checks on the algebraic calculations are extremely valuable. Fur thermore, it is easy to modify the REDUCE code to be used for other diagrams. B.5 Listing of procedures, operators and some other definitions OFF ECHO; % To suppress echoing of th» procsdu» definitions, X Faynman paramstsrization and «OBsntum integrations: PROCEDURE MQ; BEGIN VECTOR W,K1,K2,HU,NU,FIH,SI,V1,V2,V3,V4,K1,KB,KC,KD,Q; Vl.W«W.Yl:=V2.W»W.V2:=V3.W»W.V3:«V4.W«W.V4:-0; % PINTERNAL1:«PINTERNAL1+V1; PHrTERH112: -PIHTERML2+V2; PINTERNAL3: «PIHTERNAL3-V3 J PIHTERNAL4: «PINTES» ",4-V4; 1. Dl: =PINTERNAL1. W*M. PINTERNAL1-Ht»2+Il; D2:«PIHTERNAL2. W*W.PIVTERNAL2-H**2+I2; 03: •PIHTERNAL3. W*W. PIN7ERNAL3-M«*2+I3; D4: -PINTERNAL4. W»W. PIHTERHAL4-M**2+I4; DS:=PINTERHALS.W*W .PINTERNAL6-LAM«*2; D6:=PINTERH1L6.W*W.PIHTERNAL6-LAM**2; D7 :=PUfTERSAL7. W*W. PIKTER!(AL7-LAM**2; X D0:=A1*01+A2*D2+A3*D3+A4*D4+A6»D5+A6»D6+A7*D7; CLEAR 01,1)2,03,04,05,06,1)7, PINTERHAL1 ,PINTERNAL2,PINTERNAL3,PIHTERNAL4, PIKTERSAL6, PINTERNAL6, PISTERNAL7 ; VA1:=MYC0EFF(D0,Q1.W); D0:=D0-VAA*qi.W«2; K1.W:=HTCOEFF(DO.Q1.W)/(2«AA); Tl:«REDUCT(D0,qi.W)/lA; CLEAR 00; X Dl :=1A**2*(T1-K1. W*W.Xi> ; VLAMB:«LC0F(Dl,fl2.W); Dl:»01-VLAMB»q2.W**2; AAP:=LC0F(VLAMB,AA); K2.W:«LC0F(Dl,q2.W)/(2»VLAMB); T2:*REDUCT(Dl,q2.W)/VLAMB; CLEAR Dl.Kl.W.Kl.Tl; X AA:«VAA; VLAMB:=VLAHB; 52 D2:«VUMB«*2*(T2-K2.U*W.K2)/ili CV:-HYC0EFFCD2.Y1.W)/(2»A1); V112: —MYCOEFFCCV ,V2 .W)/»2; VA13:— MYC0EFF(CV.V3.V)/A3; VA14:—HYC0EFF(CV.V4.W)/A4; KA.W:-SUB(V2.W-0,V3.W«0.V4.W-0,CV); D2:-SUB(V1.W-0,D2); CV:»MYC0EFF(D2,V2.W)/(2«A2); VA23:—HYCOEFFCCV,V3.H)/A3; VA24:—KYCOEFP(CV.V4.W)/A4; KB.W:«SUB(V3.W»0,Y4.W-0.CV); D2:*SUB(V2.W*0,D2); CV:«MYCOEFF(D2,V3.W)/(2»A3); YA34:«-MYC0EFF(CY.Y4.W)/A4; KC.W:=SUB(V4.W-0,CV);; D2:«SUB(V3.W»0,D2); CLEAR CV; KO.H:«MYCOEFF(D2,V4.W)/(2*14); D2:*SUB(V4.W-0,D2); ( INDEX W i D2-.-D2; STERM:»LTERM(D2,S); TTERH:-LTERM(D2,TT)i MTERM:«LTERM(D2.M); ON BCD; LAMTERM:=LAMB»LTERM(D2,LAM)/VLAMB; OFF CCD; P2P:>P2+Q; P1P:=P1-Q; KA.MU : = KA.W*W.HC; KB.NU := KB.W*W.NU; XC.BI : = KC.W*H.RE; KD.SI := KD.W»W.SI; CLEAE KA.W, KB.H, KC.W. KD.W; VA:«LCOF(KA.MU,Pl.MU); VB:=LC0F(KA.Mn,P2.M0): VC:*LC0F(KB.NU,P1.NU); VD:«LC0F(KB.!IU,P2.NU); VAT:*LC0F(KC.RH,P2.RH); VBT:*LC0FCKC.RH.P1.RH); VCT:«LC0F(H).SI,P2.SI); VDT:«LCOF(KD.SI.Pl.SI>; VCQA:=LCOF(Ki.HU,q.m); VCqB:=LCOF(KB.NU,q.KU) ; VCqC:=LCOF(KC.RH,Q.EH); VCQD:=LCOF(KD.SI,Q.SI); WRITE "Factor is terms oith MU.ND: A12 :« ",VA12 WRITE "Factor in terms »ith MD.RH: A13 := ",VA13 WRITE "Factor in terms with MU.SI: A14 :< ",VA14 WRITE "Factor in terms with NU.RH: A23 :• ",VA23 WRITE "Factor in terms with NU.SI: A24 :- ",VA24 WRITE "Factor in terms with RH.SI: A34 := ",VA34 WRITE "A := ",VA; WRITE "B := ",VBi WRITE "C := ",VC; WRITE "D :«= ",VD; WRITE "AT := ",VATj WRITE "BT :• ",VBT; WRITE "CT :«= ",VCT; WRITE "DT := ",VDT; WRITE "CQA := ".VCqAj WRITE "CQB := ",VCI}B; WRITE "CQC := ".VCQC; WRITE "CQD := ",VCQD; WRITE "Ds := ",STERH/S; WRITE "Dt :- "(TTERM/TT; WRITE "Dm :• ",MTERM/1T2; WRITE "Dlam -.' ",LAMTERM/LAH-2; WRITE "LAMB := ",VLAHB; S3 CLE1R A,AP.D2,X2.H.K2.T2,W; CLEAR KA.MU, XB.NO, KC.BH, KD.SI; CLE1B P1P.P2P; VECTOR P1P.P2P; END! X The Oirac manipulations: VECTOR AL,BE,GA,DE.EP,PE,PH1,PH2; OPERATOR MG,J05,J0A,J6S,J6A; FOR ALL AL,BE.GA,DE,EP,PH LET JOS(AL,BE,GA,DE,EP,PH)* (BE.G1*DE.EP-BE.DE*GA.EP+BE.EP»GA.DE)»AL.PE -AL,EP»Gi,DE*BE.PB+AL.EP*BE.DE»Gi.PB-AL.EP*BE.GA*DE.PH +(AL.BE*GA.DE-AL.GA»BE.DE+AL.DE*BE.GA)*EP.PH; FOR ALL AL,BE,GA,DE,EP,PB LET JOA(AL,BE,GA,DE,EP,PE)< - (AL. GA*DE. EP-AL. DE»GA. EP) «BE. PB + (AL. BE»DE. EP-AL. DE*BE. EP) *GA. PH -(AL.BE«GA.EP-AL.GA*BE.EP)*DE.PH; FOR ALL AL,BE,GA.OE,EP,SI,PH LET JSS(AL,BE,GA,DE,EP,SI,PH>* I»EPS(BE,GA,DE,SI)*(AL.SI»EP.PH+EP.SI*AL.PB-AL.EP*SI.PH); FOR ALL AL,BE,GJI,PE.EP,SI,PH LET J5A(AL,BE,GA,DE,EP,SX,PH> I«(-BE.GA*EPS(AL,DE.EP.SI)+BE.DE*EPS(AL,GA,EP,SI) -GA.DE»EPS(AL,BE,EP,SI))*SI.PB; INDEI AL,BE,GA,U,I2; FOR ALL ALP.BEP.GAP LET MG(ALP,BEP,GAP)= BEGIN F0S0S:=J0S(GA,NU,BE,MU,AL,PH1)*J0S(ALP,RH,BEP,SI,GAP,PH2); FOSOA:=J0S(GA,NU,BE,MU,AL,PBl)*J0A(ALP,RE,BEP,SI,GAP,PH2); F0SSS:=J0S(GA,ND.BE.MU,AL,PH1)*J5S(ALP,RH,BEP,SI,GAP>I2,PH2); FOS5A:=JOS(GA,NO,BE,HU,AL>PE1)>J5A(ALP.RI,':3EP,S1,GAP,I2,PE2); X** F0AOS:=JOA(GA,Ha,BE,MO,AL.PHl)*JOS(ALP.EH,BEP,SI,GAP,PH2); F0A0A:=J0A(GA,NU>BE,mj,AL,PBl)*J0A(iLPÉRB,BEP,SI,GAP,PH2); F0A5S:=J0A(GA,mj,BE,KU,AL,PHl)*J5S(ALP,RH,BEP,SI,GAP,I2,PH2); F0ASA:=J0A(GA,NU,BE,HU,AL,PB1)*J3A(ALP,RB,BEP.SI,GAP,I2,PH2); FS50S:=J5S(GA,NU,BE,MU,AL,I1,PB1)>JOS(ALP,RB,BEP>SI,GAP,PB2); F5S0A:=JSS(GA,MJ,BE>MU,AL,I1,PH1)*J0A(ALP1RB,BEP,SI,GAP,PH2); F5SSS:'JS5(GA,ND,BE,HU,AL,I).,PH1)«JBS(ALP,RE.BEP,SI,GAP,I2,PB2); F5S5A:=J5S(GA,NU,BE.HU,AL,I1.PH1)>JBA(ALP,RE,BEP,SI.GAP,I2,PB2); X** FSA0S:=JSA(GA,NU,BE,ND,AL,I1,PE1)>J0S(ALP,SH,BEP,SI,GAP>PH2); F5A0A:=J5A 54 FS15S:-JS1(G1,NU.BE.HU,ÅL,II,PHl)*JSS(lLP,RB,BEP,SI,6iP,12,PH2); FSiSlr-JSKOi.rø,BE,MU,IL,II,PH1)»J61(1LP,BE,BEP,SI,GIP,12,PH2); X" WRITE FOO: -FOSOS+FOSOA+FOAOS+FOAOi; WRITE FOE: *F0SSS+F0SSA+FOASS+F0A6A; WRITE FSO: «FSS0S+F6S0A+FSA0S+F6A0A; WRITE FEE: =F5S6S+F5SSA+F6A5S+F515A; RETURN NIL; END; X Calculation of NIOO, NISE, NIIOO, .... NIIIBS: PROCEDURE MqMGCONTRlCTIONS; BEGIN M:«0;TT:=0; X Dirac equation vita m=0: P1.PH1:=P1P.PH1:*P2.PH2:=P2P.PH2:=0; P1P.PB2:=P1.PH2; P2P.PHi:=P2.PHl; X Khxiplovich identities Kith |t|/s neglected compared to 1: PI.PH2*P2.PHI:=PH1.PB2«S/2; KA.HU := i*Pl.MU+B«P2.MU+CQA*Q.KU; KB.NU := C*P1.NU+D*P2.NU+C1)B«Q.NU; KC.RH : = BT»Pl.RB+AT*P2.RH+CQC»q.RH; KD.SI := DT*P1.SI+CT*P2.SI+CQD*Q.SI; Q:*P1-P1P; X INDEI MU,JIU,RB,SI; NIIIOO:=FOO*Ki.MU*KB.NU*KC.RB*KD.SI; NIIIEB:=FBB*KA.HU*KB.NU»KC.RH*KD.SI; % NIIOO: =FOO* (A12*HU. NU*KC. RH»KD. SI+A13»KU. RB»KB-. NU«KD. SI +A14»HU.SI*KB.NU»KC.BH+A23*iro.RH»KA.)ro*KD.SI +A24*NU . SI'KA. MU*KC.RH+A34*RH. SI*KA.MU»KB.NU) ; NIISB:=F55*(A12*HU.NU*KC.RH»KD.SI+A13»MU.RB*KB.NU'KD.SI +A14*KU.SI*KB.NU*KC.RH+A23*NU.RH*KA.MU*KD.SI +A24*MJ. SI*KA. MU*KC. RH+A34*RH. SI*KA. HU*KB .NU); X NIOO: =FOO«(A12*A34*MU .NUtRE.SI+A13«A24*MU.RB»NU. SI+Ai4*A23*HU. SI»NU .RH); NIBE: =FSS* (A12*A34*HU .NU*RE. SI+A13»A24*MU. RE*NU. SI+A14*A23*HU. SI*NU .RH) ; X WRITE "NIIIOO := ",NIIIOO; WRITE "NIIISS := ".NIIISS; WRITE "NIIOO := ",NIIOO; WRITE "NIISB := ",NIISS; WRITE "NIOO := ",NIOO;WRITE "NISE := ",NISS; X NIL ENDS 55 PROCEDURE MYCOEFF(F.I); IF FREEOF(F.I) THEN 0 ELSE LCOF(F,I)$ X Definitions of some vactor quantities and vector products between these: LINELEMGTH 85» HISS M>M,P2*M.P1P*H,P2P*M$ HSHELL P1,P1P,P2,P2P* PI.P2:*S/2-M**2» PIP.P2P:«S/2-M**2$ PI .PIP: —TT/2+M**2* P2 .P2P: »-TT/2+M««2* Pl.P2P: = (S+TT)/2-M**2$ P2.PlP:*(S+TT)/2-H**2* % * -U/2+M**2 VECTOR PIHTERN1L1,PINTERNAL2,PINTEBN1L3,PINTERN1L4 , PIHTERN1L5 ,PINTERN1L6, PINTERN1L7. Ql, Q2t ON ECHO: B.6 Output for the uncrossed diagram DDMHY:=MQ(); Factor in terms with MU.NU: 112 : = 16 Factor in terms with MU.RH 113 := - (12 + 14 + 16 + 17) Factor in terms with MU.SI 114 : = - 16 Factor in terms with NU.RH 123 : =- 16 Factor in terms with NU.SI 124 :•= - (11 + 13 + 15 + 16) Factor in terms with RE.SI: 134 := 16 1 := 12*13 + 12*15 + 13*14 + 13*16 + 13*17 + 14*15 + 14*16 + 15*16 + 1S*17 + 16*17 B := 12*13 + 13*14 + 13*16 + 13*17 + 14*16 C : = 11*14 + 11*17 + 13*14 + 13*16 + 13*17 + 14*15 + 14*16 + 15*16 + 15*17 + 16*17 D := 11*14 + 13*14 + 13*16 + 14*15 + 14*16 IT :- 11*12 + 11*14 + 11*16 + 11*17 + 12*15 + 12*16 + 14*15 • 15*16 + 15* 17 + 16*17 BT : = 11*12 + 11*14 + 11*16 + 11*17 + 12*16 CT := 11*12 + 11*16 + 11*17 + 12*13 + 12*15 + 12*16 + 13*17 + 15*16 + 15* 17 * 16*17 ST :- 11*12 + 11*16 * 12*13 + 12*15 + 12*16 CQ1 » - 16*17 CQB • - 17*(11 + 13 + 16 + 16) CQC = 16*17 CQD • 17*(11 • 13 + 15 + 16) 56 DB := 11*12*13 + 11*12*14 • 11*13*14 + 11*13*16 + 11*13*17 * 11*14*16 + 12 *13*14 + 12*13*16 * 12*14*1B + 12*14*16 Dt :* 15*16*17 2 2 2 2 2 Dm :* - (11 «12 + 11 *14 + 11 *16 + 11 «17 * 11*12 + 2*11*12*13 • 2*11* 12*14 + 2*11*12*16 + 2*11*13*14 + 2*11*13*16 + 2*11*13*17 + 11* 2 2 2 2 2 14 + 2*11*14*16 + 12 *13 + 12 *16 + 12 «16 + 12*13 + 2*12*13* 2 2 14 + 2*12*13*16 + 2*12*14*15 • 2*12*14*16 + 13 *14 + 13 *16 + 2 2 2 2 13 «17 t 13*14 + 2*13*14*16 1- 14 *15 + 14 *16) Man ;= - LINE*(15 + 16 + 17) LIMB := 11*12 + 11*14 + 11*16 + 11*17 + 12*13 + 12*15 + 12*16 + 13*14 + 13 *16 + 13*17 + 14*15 + 14*16 + 15*16 + 15*17 + 16*17 DUMMY:=MG(1L,BE,G1)S FOO := 2*(NU.MU*HB.SI*PE1.PB2 - NU.MU*RH.PB1*SI.PH2 - H0.M0*RH.PB2*SI.PE1 + MJ.RH*MU.SI*PB1.FB2 - iTO.RB*MU.PBl*SI.PB2 + 8B.RH*M».PH2*SI. PB1 + 3*NU.SI*HU.RB*PB1.PB2 + NU.SI*MU.PB1*RB.PB2 + NU.SI*HU.PE2 •BE.PHI + mi.PBl*MU.BB*SI.PH2 - NU.PB1*MU.SI*HH.PB2 - MU.PH1*H«. PB2*RB.SI + NU.PB2*MU.RB*SI.PB1 + Nn.PB2*M0.SI*RB.PEl - NU.PB2* MU.PHl'RH.SI) F05 := - 2*I*(HU.MU*EPS(HH,SI,PE1,PH2) + MJ.PB2*EPS(MU,RB,SI.PH1) + HU. PB2*EPSOra,RH,SI,PBl) - BE.SI*EPS(NU,HD,PH1,PB2) + BB.PB1* EPS(11D,MU,SI,PH2) + SI.PHl*EPS(mj,HU,RI,PE2)) F50 := - 2*I*(Hn.mj*EPS(RH,SI,PBl,PH2) + NU.PB2*EPS(HU,HH,SI,PH1) + MO. PH2*EPS(«ra,RH,SI,PHl) - BH.SI*EPSOnj,MU,PHl,PH2) + HH.PH1* EPS(NU,MU,SI,PB2) + SI.PBl*EPS((ra,HO,RB,PB2)> F55 : = 2*(KU.MU*RB.SI*PB1.PB2 - fflJ.MU*HE.PBl*SI.PB2 - NU.MU*RH.FB2*SI.PB1 + NU.RH*MU.SI*PE1.PE2 - NU.RH*Ktf.PBl*SI.PH2 + Kir.RB»MU.PB2*SI. PB1 - 5*NU.SI*MU.BH*PB1.PH2 + HO.SI*MD.PBl*RB.PH2 + HD.SI*MD.PH2 •RB.PB1 + NU.PB1*MU.HE*SI.PE2 - NU.PB1*MJ 5I*BB.PB2 - KD.PB1*MD. PB2*RB.SI + NU.PB2*MU.EH*SI.PB1 + NU.PB2*MU.SI*RB.PB1 - ND.PB2* Mn.PEl*BB.SI) 57 DUKHY.-'MQMGCONTRiCTIOHSO; 2 MIIOO :» 2*PH1.PI2*S *(2*1*C*1T*CT + 1*D«1T*DT + B*C*BT*CT + B*D*BT*DT) 2 NIII5B := - 2*PB1.PE2*S *(1*D*1T*DT + B*C*BT*CT + B*»*BT»»T) SHOO :« 4*PB1.PB2*S*(1»D*134 + B*i*iT*å24 + 3*1*CT*123 + B*C*134 + 4*B*BT «124 + B*DT*123 + 3*C*1T*114 + S*C*CT*113 + D*BT*A14 + 4*D*DT* 113 + BT«CT*U2 + 1T*DT*U2) NII5S := - 4*PH1.PE2*S*(1*D*134 + 3*l*lT*i24 - 1*CT*123 + B*C*134 + 4*B* BT*124 + B»DT*123 - C*1T*114 + 3*C*CT*li3 + D«BT*114 + 4*D*DT« 113 + BT*CT*112 + 1T*DT*112) MOO := 8*PB1.PB2*(6*112*134 + 17*113*124 + 8*114*123) NISS := - 24*PH1.PB2*(112*134 + 5*113*124) y. Check the results of paper I: BEGIN il2:=V112;113:=V113;114:=V114;A23:=V123;124:=Y124;134:=V134; Cgi: =VCQ A; CQB: =VCQB; CDC t *VCQC; CqD: =VCQD; 111: =11+13+15+16; 112: *1?+14+16+17; TT:=0; END; NIII00-2*S**2*( 1*1T*C*CT*PB1.PH2 +(1*1T+B*BT)*(C*CT+D*DT)*PE1.PH2 ); 0 NIIISS-2*S**2*( 1*1T*C*CT*PE1.PH2 -(1*1T+B*BT)*(C*CT+D*DT)*PB1.PH2 ); 0 NII00+4*S*( (111*1*1T+112*C*CT+2*16*(1*CT+1T*C))*PH1.PH2 +(4*111*(1*1T+B*BT)+4*112*(C*CT+D*DT) -16* (i*D+iT*DT+B*C+BT*CT-l*CT-iT*C-B*DT-BT*D) ) *PH1. PH2 ) ; 0 NII55+4*S*( (Ul*l*iT+112*C*CT+2*i6«(1*CT+1T*C))*PE1.PH2 - (4*111* (A*1T+B*BT)+4*1A2* (C*CT+D*DT) -16*(1*D+1T*DT+B*C+BT*CT-1*CT-1T*C-B*DT-BT*D))*PB1.PB2 ); 0 58 NI0O-8*( (iil*U2+5»16"2)*PHl.PE2 +4»(4»Ul«112+2«i6*«2)»PHl.PH2 ); 0 NI66-B»( (AA1»112+S*A6»»2) *PH1.PH2 -4*(4»iil»ii2+2*16»*2)»PB1.PE2 ) ; 0 59 Appendix C The MAPLE-program CINT The program CINT, which is written in the symbolic manipulation language MAP LE [23], calculates the leading contribution to integrals of the form (4.13). Section 4.6 gives a general discussion of the algorithms that are used, and we will now give a more technical description. The code is listed in section C.S. Names of MAPLE variables and procedures will be printed with bold types in the text, while all other variables are defined in section 4.6, if nothing else is said. C.l Calling sequence The program is started with the the call Cint(invars,GAHMAs,c,var); in MAPLE. The arguments of the call are defined as follows: 0 invars = logfs* ) = log(s)i40- We do the calculations with 0 = 0, thus invars should not depend on 0. GAMMAs is a list that represents the product of T-functions in the integrand. An element I which is a linear form in the complex integration variables, represents simply T(2). If / is a list itself, however, it represents the factors (C1) r(/1 + ... + y where lt, ...,ln are the elements of I. c is a constant factor that wili be kept outside the integrals. var is a list of the complex integration variables. In addition, one may influence the calculations by assigning a value to one or more global variables, which will be described in the following section. The program returns the minimum of invars, which is equal to log(a)p. The main results are stored in global variables or written to a file. 60 C.2 Global variables Two kinds of global variables are used in the program. The first kind has names that start with eint, while the second kind has names that start with Cint (MAnLE is case-sensitive). The first kind of global variables may be set by the user to alter the default options of the program. These are: cint.vinvars is a set of equations that give the values of the logarithms log(«), log |u|, The calculations will be valid for these values of the logarithms, and as far as the leading contribution is analytic. Only the ratios between the logarithms are of importance, thus a scaling of the values will not affect the results. The default value is {LB = 25, Lu = 25, Lt = 20, Lm = 0, Llam = -900}, where Ls, Lu, Lt, Lm and Llam represent log(s), Iog|u|, log|(|, log(m3) and log(A2) respectively. This representation is used in order to reduce the size of the results. cint-min-logs is the minimum number of logarithmic factors that we are interested in. Terms with fewer than cint.min Jogs logarithms will be not be calculated. A small value cint.minJogs may exhaust the computer. By default, only leading logarithms are calculated. cint.pole is specified if we are only interested in poles where invars = POLE. The default is to calculate contributions from the leading poles. E.g. POLE = —2 * Ls — Lt asks the program to calculate contributions that are of order c/s2t, where c is defined in section C.l. cint .output is an integer that determines how much output the program writes. If it is 0, nothing is written, if it is 1, only some statistical information is printed, if it is 2, the results are written in a one-dimensional form that later may be used as input to MAPLE, and if cint.output is 3, the results are written both in the one-dimensional form, and in a two-dimensional form that is more readable. Default is 2. cint.nonan may only have the values true or false. If true, possible non-analyti- cities of the result will be calculated. Default is true. cint.file is either 0 (default) or a filename. In the latter case, intermediate results will be stored in the file with this name, and thus free the part of memory normally used for this. The output to cint.file is a sequence of MAPLE statements that adds intermediate results to the variables Cint_sum[i], which are explained below. The file may later be read into MAPLE to sum up the final results. The second kind of global variables, whose names start with Cint (capital C), should normally not be altered by the user. The following list shows their usage. 61 Cint-time The CPU time in seconds from MAPLE is started till CINT is called. Cint-VMr The complex integration variables Cint.c A set of equations of the form z; = a, where the z;'s are the complex integration variables and the c,'s specify contours that satisfy eq. (4.14) for the present integral. Cint-sum An array in which the results are stored. Terms in which ( complex integrations are left (see eq. (4.31) ) when i/>0 becomes independent of the integ.auon variables, are stored in Cint_sum[/]. The leading contribution to the integral will be given as JT f dzi... f dzi Cint_sum[.'], (C.2) 1=0Jct Jc< where n is the number of integration variables and zt,..., z„ are the complex integration variables. Cint-nonanB Returns a set of functions that are linear in log(a), log(u), The resile (Cint-sum) is analytic at the point specified by cint-vinvars, unless one of the functions in Cint.nonans is zero at this point. Furthermore, the result is analytic around this point, in the region where all the functions in Cint.nonans are nonzero. Cint-nzterms Number of nonzero contributions to Cint.sum. Cint.calls Number of calls to Cint.intr at each depth of recursion. In addition, each of the global variables of the first kind, has a capitalized counter part, such that the program can tell which of these variables are changed between two calls to Cint. C.3 Procedures The procedure Cint is merely an interface to the procedures Cint.init, Cint.calc and Citit.end. Its arguments are described above. The variable pole is assigned to the value of invars at the minimum po'nt Zo, which is defined in section 4.6. We will now go through the procedures of the program, more oi less in the same order that they are entered for the first time. We skip some small procedures that may easily be understood from the listing in section C.5. 62 C.3.1 Initialization The arguments of Cint are passed to CintJnit, which initialises a number of global variables. It alio checks that invars does not depend on other variables than the integration variables and those logarithms whose values are given by cint.vinvan. A complete check of the input is not performed, but the program will normally stop at an early stage if illegal input is given. C.3.2 Before the integration starts Before the contour integrations are done recursively by the procedure CintJntr, the integral must be converted to a sum of terms of the form (4.29). This is done by the procedures Cint.calc, Cint.this.pole, Cint.sptit and Cint.int, and procedures that are called within these. We will now briefly go through these procedures. The procedure Cint.calc first puts the arguments of the r-function., into the two lists psis and phis. The elements of the list psis are the I/I'S in eq. (4.13), and the elements of the list phis are the T-functions, now with arguments that vanish at Z0- The next step is to use the relation T(x) = T(z + l)/x for T-functions of ø's that vanish at Zc, and put the T-functions into the variable g. Thus phis will from now on represent factors in the numerator, not T-functions. The same will be done for T-functions oii/i'e, but this is done in Cint.int for reasons of economy. If we would Uke to consider contributions from a non-leading pole, some argu ments of T-functions may have negative values at this pole. In this case we would 63 have to use the relation TIX) - r irr(« + " + i)r(-»-n+i) n>)-(-i) r(-x + i)(« + n) • (C3) if the argument z is -n at Za, instead of T(z) = T(i + 1)/». We see that the two formulas are equivalent for n = 0. This is not necessary for the present calculations, still the more general formula is used in Cint.thu.pole. The procedure Cint.next.pole is called if minimum is larger than Cint.pole. In this case, we are asking for the contributions from poles that are suppressed by a negative power of s/m2 relative to the leading poles. It is possible to find such contributions, but they are not needed for the present calculation. Thus we do not include an implementation of the procedure Cint_next.pole here. The value of minimum may also be smaller than Cint.pole, which means that the present term is of lower order, and is skipped. The procedure Cint.split splits the integrand in eq (4.26) into a sum of terms of the form (4.29). This is done according to the description after eq (4.26). After Cint.split has removed one factor from the numerator, it calls itself recursively to remove the next factor. This is repeated until it arrives at terms that are either of the form (4.29), or suppressed by i/m? raised to a negative power. The procedure Cintjnt does the final preparations before the integrations starts. The functions T(i/>j + 1), where V\ are the V>'s that vanish at Zo, are stored in g, such that the elements of psis now represent factors in the denominator rather than T-functions in the numerator. Now g is identical to the function g(z) in expres sion (4.29). If only leading logarithms are going to be calculated, the complex vari ables that are present in psis, are replaced by their value at Zo in g. Before Cint_int is called, a valid set of contours are determined through a call to Cint.contours. Finally the integrations are started by a call to Cint_intr. C.3.3 Doing the integrations Now the contour integrations will be done recursively by the procedure Cint-intr, which is called with the following parameters: invars = log(a*°) = log(«)V»o. g is the same as g(z) in expression (4.29), with any constants included. h The factors in the denominator. At the first call, h = z,... j„Vi • • • i>d- var The remaining complex integration variables. With these definitions, the multiple contour integral that we are going to evaluate may be written as: / a »>nv«» n Idz§-h~' (C4) 64 where the contours c, were determined by the procedure Cint-contours before Cint.intr was called. The procedure does one integration, and then calls itself recursively to do the next integration, until it arrives at terms of the form (4.31), in which Vo does not depend on the integration variables any more. At the first call, all factors in h will normally be different, but later some of them may coincide to form higher order poles. We will now describe Cint Jntr in some more detail. CintJntr calls Cint.endterm if g vanishes or if V>o " independent of the re maining integration variables. Otherwise it calls Cint_next_z, which determines which of the complex integrations will give the lowest number of terms. The cor responding integration variable is called z. A more random choice of integration variable would give the same result, but could increase the computing time by many orders of magnitude. After a few more definitions, we come to the main loop of Cint.intr. This is a loop over the elements in hi, which is a list of the factors in the denominator h that depend on z. The present factor is called factr within the loop. Within the loop, the quantities lin and ndif are defined such that lin is linear in z, and factr = linndlf+1. (C.5) Now we want to find the contribution from the pole at lin = 0, if this pole is crossed when the contour of z is shifted such that invars decreases. There is one complication at this point: The real part of lin may be 0 at the contours that we use for the calculations. This will not happen while doing the first integration, but it may happen at a later stage. In this case, we may neither say that we croas the pole, nor that we do not cross it. We could shift the contour of this integration by a small amount, but we would have to do this consistently for all terms. This is dealt with as follows: We have calculated a set of numerical values for the intercept of the contours with the real axis. We may, before we start the integration, add an infinitesimal amount to each of these values, i.e. Cj —» Ci + Si, where Si is a small, real number. We choose these such that 0 < (,- < 1, and Si+i/Si J_ fge'-^dz _ 8 (flz + 6)ndif+'ge""r"' (C.7) 2m J h az" 65 The loops ovei the MAPLE-variables i and j are loops over the terms in the derivative in eq. (C.7). The loop index i is the number of times g has been differen tiated, and j the number of times (az + {,)nd"+' /h has been differentiated. Including these loops, rather then differentiating the whole thing in one call to diff, is done for two reasons. First, the number of possible logarithms is reduced with one each time we differentiate g. Thus we want to keep track of the number of times g is differ entiated, to avoid unnecessary calculations of terms with fewer logarithms than we want. The second reason is that we want to reduce the use of computer storage as much as possible. Cint.intr calls itself recursively for each integration variable, and all variable contents must be stored at each depth of recursion. When the number of integration variables is large, it may be important to store as little as possible. Further optimalization of Cint.intr may be possible, but it will to some extent depend on whether we want the program to use less CPU-time or less memory. Optimalization may also make the code more complicated, and less suited for further changes. The procedure Cint_nextjz has much of the same structure as Cint.intr. It has an additional outer loop over the complex integration variables, and the number of terms is calculated by using the formula Number of terms in ^ (AW.../.(*)) = ^f^f. (C.8) The procedure Cint.endterm is called to add a result from CintJntr to the relevant global variables. Some extra attention must be paid to terms in which two or more integrations are left. Consider e.g. the term g(«i,*a)log'(<) , j ' dzi j dz; C9 J ZlZj(zi — Zi) Since here is no s raised to a linear combination of zi and z2 in the integrand, CINT will not do the remaining integrals, but simply return the integrand. But the integral depends on whether Re(z1 — z3) is positive or negative (the real part of each integration variable will always be positive). Thus CINT must tell us what the sign of Re(zi — z2) is. In order to tell this, the procedure Cint-denom replaces the factor zj — za by Cintd(zi — z2) if Re(zt — z2) > 0, and — Cintd(z2 — Z]) if Re(zi — z2) < 0. In general, this is done for any factor in the denominator if the factor has more than one term. C.3.4 Printing results The procedure Cint_print is called at the end of each call to Cint to print the results of the calculation, according to the value of cint.output. 66 C.4 Input and output for the one-loop example The contour integral in eq. (4.39) may be calculated by giving the following com mands in MAPLE. We have stored the program CINT in a file called cint, and we use the variables Ls, Lt, Llam and Lm.mu to represents log(a/mf), log \t\/ml, log(A2/mJ) and log(mJ/mf) respectively. read cint: intarface(QCho-2); _cint_vinvars:*{Ls=25,Lu*26,Lt«20,Llejii»-900,Lin_mu«5}: .cint .min.logs: «0: .cint.output:*3: Cint (-zl »Ls-z2»Lt-z3*Llam-z4»Lm_mu, [[Zl,z2,z3,z4,2-zl-z2-z3-z4],[-1+Z1-Z3-Z4,1-Z2-Z4] , [2-zl,2-zl],[-2+2*zl+2*z3,4-2*zl-z3]], s"2, [Zl,z2,z3.z4]): The call to Cint generates the following output: .cint.vinvars: {Ls = 25, Lu - 25, Lt = 20, Llam = -900, Lm.nu = 5} .cint.min.logs: 0 .eint.pole: I _cint .output: 3 .cint.tost: 0 .cint.sonan: true .cint.file: 0 Final results: .Cint.sumM := l/2*Ls**2-Ls*Lt+l/2*Lt**2-Ls+Lt+l/2»Pi**2 _Cint_sum[0] :« 2 2 2 1/2 Ls - Ls Lt + 1/2 Lt - Ls + Lt + 1/2 Pi Possible points ox non-anonlyticities: {Lt, Ls-Lt, Lt-Ln_nu, Lt-Llam} Numbers of nonzero contributions: _Cint_eum[0]: 3 Number of calls to _Cint_intr: Level 0: 1 Level 1: 1 67 Level 2: 1 Level 3: 6 Level 4: 3 Total: 12 Time used: 10.680 After having made the substitution s —* e~", we obtain the result in eq. (4.43). The output also shows that other values of Ls, Lt, Llam and Lmjnu would give the same expression as long as we do not cross any of the lines Lt = 0, Ls — Lt = 0, Lt—Lm_mu = 0 or Lt — Llani = 0. The present calculation was done with Ls = 25, Lt — 20, Llam = —900 and Lmjnu = 5. The time used for this calculation, was 10.68 seconds. C.5 Listing of procedures Cint:=proc(invars,GlHHls,c,var) local pole; if nargB < 4 then lprintC Usage: Cint(invars,GiMMAs.c,var);'); KETUBNO fi; Cint.init(args); pole:=Cint_calc(invars,GAHHls,c); Cint_print(nops(var)); pole end: Cint_init:=procO local nv; Cint_tima:stine()J Cint _var:=args[4]; 4 Store values of global variables, and print variables that have been changed: Cint.global('cint.vinvars',cint.vinvars,'Cint.vinvars1); Cint.global('eint_min_logs',cint.min_logs,'Cint.min.logs'); Cint_giobal('cint_pole' ,subs(Cint.vinvars(cint.pole) , 'Cint.pole'); Cint.globaK 'cint.output' , ciat .output, 'Cint.output') ; Cint.global (' clnt.nonan', cint.nonan,' Cint_nonan'); Cint.init.file(); Cint.globalCcint.file',cint.filo, 'Cint.file'); # Initialize variables in which results are stored: nv:=nops(Cint_var); if Cint.file«0 then Cint_Bum:stable([0tnv]); Cint.sumCO] :-=0 fi; if Cint.nonan then Cint_nonans:«{} fi; Cint_nzteras:stable([Otnv]); Cint_nztenns[0] :>0; Cint_calls:=table{[Otnv]); Cint.calls[0]:*0; # Check indetarninates in invars: indetsdubt'Cint.vinvars,indets(args[l]))) minus indetB(Cint.var); if " <> O then EML0B('Undefined coefficient(B) in invars',") fi; NULL end: 68 Cint .global: «proe (gl, g2 ,G) t Prints g2 (gl is the name of g2) if it ie not equal to G, and puts G # equal to g2. if eval(G) <> g2 then lprint(cat(gl," : "),eval(gl)); G:»g2 fi; NULL end: Cint_init_file:=proc() local j; * Initializes the file cint.file and the array Cint.sum if t cint.file <> 0, if cint.file=0 or cint_file=Cint_file then RETDRN(NULL) fi; lprint('Results vill be Britten to:*.cint.file); lprint('0ther output will be appended to:', output.file); writeto(cint_file); for j from 0 to nopB(Cint.var) do lprintOCint^sumC .j.']:=0:'); Cint.sumCj] :*0 od; appendto(output_file); NULL end: Cint.calc: =proc (invarB, MHHAs, c) local pBiB, phis, Z0, minimum, invarsp, cp; pBiB:*map(proc(x) if type(x,{list,set}) then op(x) elBe x fi end .GAMMls); phiB:°map(proc(x) if type(x,{list,set}) then convert(x,' + ') else NULL fi end, GlHHis); Cint.cancel(pais,phis,*psis','phis'); f Calculate contours to check that the representation is sell defined: Cint_contourø(iBvars,psis); if Cint_c=NULL then ERRORCThis should not happen') fi; # Minimize Re(invars): ZO:*Cint_min(invars,psiB, 'minimum' .UNRESTRICTED); if Cint_pole-I then Cint.pole:«minimum fi; if Z0=NU1L or minimum < Cint.pole then RETURN(subB(Z0,invars)) fi; * Change variables - use factors that are 0 at Z0 as new variables: invarsp:=Cint.chvar(invars,psis,phis,c,ZO,'psis','phis','cp'); if minimum > Cint.pole then Cint .next.pole (invarBp, 0 ,pBis,phis, cplminimuD> ) else Cint_this_pole(invarsp,pBiB, phis, cp, minimum) fi; subs(Z0,invars) end: 69 Cint_cancel:*proc(a,b,ap,bp) local app, bpp, j, n; # RemoveB equal elements in the two lists a and b, and returns them as # ap and bp. app:=a; bpp:*b; j:»ij do if j > nops(app) then break fi; if member (app f j] ,bpp, *n') then app:»BubBop(j-NULL,app) > bpp:«subsop(n*NULL,bpp) else j:=j+l fi od; ap:*app; bp:=bpp; NULL end: Cint.contours :=proc (invars .factors) local expsp, j , eps, enst , i; expsp:-subs(Cint_vinvars,invars); for j to 4 do eps:=l/10_j; cnBt:={(factors[i] >= eps)$i=l..nops(factors)}; Cint_c: =simplex[minimize] (expsp,enst) ; if Cint.c <> O then break fi od; if Cint_c*0 then ERROR ('Cannot find contours') fi; NULL end: Cint_min: «proe (invars ,f actors ,minimum) local argm, ZO: argm:"subs(Cint.vinvars,invars) , map(proc(x) x >= 0 end, factors); if nargs > 3 then argm:»argm,args[4] fi; ZO:«simplex[minimize] (argm); jf Z0*O then ERROBCNo feasible solution') fi; if Z0*NULL then minimum: «-infinity else minimum:=subs(ZO,op(l,[argm])) fi; ZO end: Cint.chvar: «proc (invars ,psis ,phis ,c ,Z0 .psisp ,phisp, cp) local invarsp, psiBl, pais2, I, phispp, cpp, nf, z, k, coeffz, zrepl; invarsp: 'invars; psisl:"[]; pBis2:xD ; for f in psis do if snbs(Z0,f)'0 then pBiBl:*[op(psisl),f] else psis2:>[op(psis2),f] fi; od; 70 phispp:-phis; cpp:-c; nf:*nops(psisl); for z in Cint.vax do k.—0; for k to nf do if hat(pBisl[k],z) then break fi od; if k > nf then cpp:«cpp*z/GAMMA(z+l); lprintC WARNING (Cint.chvar):','No factor in psisl depends on'.z); next fif coeffz:-coeff(psiBl[k],z); zrepl:•(z-8ubs(z=0,psis1[k]))/coeffz; psial:=subBop(k=[fULL,psisl); invarsp:s subs (z=zrepl, invaxsp); pBiBl:=8ubs(z=zrepl,psisl); psis2:=snbg(z*zrepl,psis2); phispp:=8ubs(z-zrepl,phispp); cpp:=subs(z=zrepl,cpp)/abs(coeffz); nf:=nf-l od; psiBp:<[psiBi,psis2]; phisp^hispp; cp: snormal(cpp); expand(invarsp) end: Cint_this_pole:=proc(invarB,pB is,phis,c,minimum) local Z0, nlogs, g, psi. psiO, phisp; Z0:=map(proc(x) x-0 end.Cint.var); nlogs:*nops(psis[11); g:-c; phisp : = []; for psi in phis do psiO:*subs(Z0,pBi); if type(psiO,integer) and psiO <= 0 then nlogs:Bnlogs-l; g:*g*(-l)~pBiO*GAMMA(-psi fl> p8i0: BubB(Z0linvars); g:'g*Bnb8(exp(Ls)*B,exp(Lu)su,exp(Lt)t,expand(exp(psi0))); 71 il aubs(Cint_vinvar8,Ls-Lu)*0 then g:«BUbs(u»-a,g) fi; 11 Cint.min.logs < 0 then Cint.nin.logs:•slogs fi; il nlogs < Cint.min_logs then AETURN(NULL) li; il nops(phisp) B 0 then Cint_int(invar8-paiO,pBiB[l3 ,g) else Cint.split(inTars-psiO.psisCl]iphisp.g) li; NULL end: Cint_next_pole:=proc() ERROR('Not incladsd in this version') end: Cint_split:-proc(invarB,psis,phis,g) local ni, phisp, phispp, nn, invarsp, pBisp, gp, k, z, coellz, zrepl, nev_min; nf:=nops(p6is); phisp:smap(Cint_n.o_t,phiB); k:=min(op(phiBp)); for nn to ni do il phisp[nn]=k then break li od; phisp:s8Ub8op(nnsNULL,phis); lor z in Cint.var do il not has(phisCns],z) then next li; lor k to nf do il has(pBis[k] ,z) then break li od; il k > ni then * z is the only lactor in the denominator that depends on z: coeffz:*coeff(subs(Cint_vinvarB,invars) ,z); il coellz > 0 then next li; il coellz < 0 then ERRORCThis should not happen') fi; lprint('Cint_split: Denominator becomes independent of, z); invar Bp:s invars; gp:=g*coel1(phis[nn],z)*z; psisp:=psis; phispp:s phisp; else gp:*g*GAMMA(z+l); coeffz:*coeff(psis[k],z); zrepl:•(z-subs(z-0a psis[k]})/coeffz; invarsp: =expand(subs (z=zrepl,invars)); gp:-subs(zszrepl,gp)«coef1CphiBthnj,z)/abs(coeffz); psisp: =snbs (z'zrepl, snbsop(k=NUn.,psis)); phispp:ssub8(z*zrepl,phisp); Cint_min(inversp,psisp, 'new_min" .NONNEGATTVE); il newjnin <> -infinity and nev.min > 0 then ERRORCThis should not happen: new_min:' ,new_min) li; if nev_min=-infinity or neo_min < 0 then next fi fi; 72 il nops(phispp) > 0 then Cint_split(invarBp,psisp,phiBpp,gp) else Cint_int(invarsp,psisp,gp) fi od; NULL end: Cint.int: =proc(invars,psis,g) local gp, psi, nlogB, ZO, varp; gp--*g; for psi in pais do gp:=gp*GiMMA(psi+l) od; for pBi in Cint.var do gp:*gp*GlMMA(psi+l) od; varp:={op(Cint_var)} minus indets(psis); nlogB:=nops(psis); if nlogB = Cint_min_logs then ZO:=m«p(proc(x) x=0 end,indets(psis)); gp:=normal(sub6(Z0,gp)); gp:=eval(gp); * To get rid of GIHHKl) f.ex. f i; # Call the routine that does the integration: psi:=[op(psis),op(Cint_var)]; if traperror(Cint_contours(invars,psi)) = lasterror then ERROROThis should not happen:',lasterror) fi; if Cint.cNDLL then RETURNO fi; Cint.intr (expand(invare) .normal(convert (psi,'•')) ,gp,Cint.var ,0.nlogs); HULL end: Cint.intr: 73 hnz:Bnorm&lCconvert(hnz ,'•')); hz:-normal(convert(hi,'*')); t Loop over the factors: for 1 actr in hi do if typeCiactr,'"') then lin:=op(l,factr); ndif:=op(2,factr)-l; elBe lin:=factr; ndif:*Q; fi; lin:=lin/coeff (lin.z); c_lin:=Eubs(Cint_c,lin); if c.lin = 0 then for varp in Cint.var do if has(lin,varp) then break fi od; c.lin:= costf(lin,varp)j fi; if path*c_lin < 0 then next fi; zO:=z-lin; if ndif=0 then Cint.intr (expand (subs (z=z0,invars) ) ,normal(sub8(z=z0, normal (h/lin))) ,path*subs(z=zO,g),BubB(zsNULL,var),nlogs,max_logB); next fi; invarsp:-expand(snbs(z-zO,invars)); varp:=6ubs(z=tnjLL,var); f :«normal(lin"(ndif+l)/hz); gp: =path*g/ndif!; for i from 0 to ndif do t g is differentiated i times fp:=f; for j from 0 to ndif-i do * fp is differentiated j times c: =binomial (ndif-i,j)*coef f (invars,z) "(ndif-i-j); if type(fp,,+") then tox term in fp do nterm:=normal(subs(z=z0,term/hnz)); s Cint_intr(invarsp,denosi(ntenn)f c*subs(z zO,gp)*nnmer(nterm) ,varp, nlogs+ndif-i-j,max_logs-i) od; else ntenn:=normal(subs(z=zO,lp/hnz)); Cint_intr(invar8p,denom(nterm),c*subs(z=zO,gp)*mimer(nterm),varp, nlogs+ndif-i-j ,max_logs-i) fi; if j < ndif-i then fp:=diff(i'p.z) fi od; if max.logs-i <= Cint_min_logs then break fi; gp:=diff(gp,z)*(ndif-i)/(i+l) 74 od od; NULL end: Cint.next.z: *proc (invars,hl) local iavarsp, z, path, txms, nf, factr, lin, ndif, cz, c.lin, zp; invarsp:=subs(Cint_vinvar6,invars); for z in Cint.var do if has(invarsp,z) then break fi od; for z in Cint.var do cz:=coeff(invarsp,z); if cz=0 then next fi; if Cint.nonan then Cint.nonans:*Cint_nonanB union {coeff (invars,z)} fi; path:*signum(cz); tnns[z]:=0; nf :*convert(map(proc(x,z) if has(x,z) then 1 else 0 fi end,hl,z),' + '); for factr in hl do if not has(factr,z) then next fi; if type(factr,'"') then lin:=op(l,factr); ndif:=op(2,factr)-l; eise lin:-factr; ndif:-0; f i; cz:=coeff(lin.z); lin:=lin/cz; c.lin:=subs(Cint_c.lin); if c.lin = 0 then for zp in Cint.var do if has(lin,zp) then break fi od; c.lin:= coeff(lin,zp); f i; if path*c_lin < 0 then next f i; trms[z] :-trms[z3+binomial(ndif+nf-l,ndif); od; if trms[z]=0 then RETURN(0) fi; od; zp:=0; for z in Cint.var do if has(invarsp,z) then if zp=0 then zp:»z elif trmetz] < trms[zp] then zp:=z f i fi od; zp end: Cint.endterm:«proe(nv,g,h,nlogs,var) local gp, i, newvar; if g=0 or nlogs < Cint.min.log8 then BETOHK(NULL) f i; Cint_nzterm8[nv]:=Cint_nzterms[nv]+l; if nv=0 then 75 gp:»g/°: «lii nv«l than gp:«sub8(var[0*Cint_var[l] ,g/h) else nauvar : = • ; for i to nv do newvar^CopCnewvar^varCU^Cint.vartt]] od; gp: =sube (newar, g/Cint .denom(i)) fi; Cint_sum[nv] :=Cint_sum[nY3+cint_siinp_eum(expand(gp)); if Cint_file <> 0 and nv > 0 and l«nf;th(Cint..Bum[nv]) > 5000 tben Cint_store(nv) fi; NULL end: cint_aimp_Bum:sproc(x) x and: Cint.store:=proc(nv) * Stores Cint.sumCnv] in the file Cint.file, and resets it to 0. appendto(Cint.file); lprintC'Cint^sumC .nv.'] :=Cint_sum[* .nv.']+('); lprint(Cint_Bum[nvJ); lprint('):'); appendto(outpnt.f ile); Cint.sumtnv] :=0; end: Cintd: = 'Cintd': Cint_denom:=proc(d) local v; if tvpetd,'*') then map(Cint_denom,d) elif type(d,'"') than Cint_denom(op(i,d))"op(2,d) elif type (d .numeric) or nopa(d)sl then d elif subs(Cint_c,d) < 0 then -Cintd(-d) elif aubc(Cint_c,d) - 0 then for v in Cint„var do if haB(d,v) then break fi od; if coeff(d,v) < 0 then -Cintd(-d) else Cirtd(d) fi else Cintd(d) fi end: Cint.print:«proc(nv) local 1, flag; if 0 >= Cint.output then RETURN (NULL) fi; flag:«0; for 1 from 0 to nv do if Cint.nzterm&flJ 0 0 then flag:=l fi od; 76 if flag-0 than RETUMdULL) fi; if 1 < Cint.output than lprintO; lprintCFinal reralta:'); lprintO fi; for 1 from 0 to nv do if Cint.aum[l] <> 0 than if Cint.output < 2 than naxt fi; lprintC Cint_sum['.l.*] :•'.Cint.eumCl]); if Cint.output < 3 than naxt fi; lprintC Cint.aumf.l.'] :«'); print(Cint.iuaCl]) fi od; if Cintjaonan and 1 < Cint.output than lprintO; lprintC'Poiiible point» of non-ananlyticitiaa:'); lprint(Cint.nonans) fi; lprintC); lprintC Numbers of nonzaro contributiona:') ; for 1 from 0 to nv do if Cint.nzterns[1] <> 0 than lprintC Cint.aunC .1.'] :' ,Cint_nztarnB[l]) ; fi; od; lprintC Number of calls to Cint.intr:'); for 1 from 0 to nv do if Cint_calls[nv-1]»0 than braak fi; lprintC Laval '.l.':',Cint_call»[nv-l]) od; IprintC Total:' ,aumCCint.calls[,l,1.,'l'»0..nv)); lprintC'Tim» osed:'.tiaeO-Cint.time); ITJIL and: Cint_n.o_t:«proc(x) local y; # Raturna the ncnbar of tarma in x. y:*expand(x); if typeCy»'+') than nopaCy) also 1 fi and: cint_vinvaxs:- 77 Appendix D Applications of CINT to two-loop integrals We will now show how the contour integrals in chapter 5 may be evaluated with MAPLE and the program CINT described in the previous appendix. The first part of the listing in section D.l, shows how the steps leading from eq. (5.8) to eqs. (5.15) may be programmed in MAPLE. This code is written such that it may easily be modified to be used for other diagrams. The calculations are described in chapter 5, and in the comments within the code. The second part of the code in section D.l, defines the numerator of the integral that we consider. In the present example, we have chosen A/// for the uncrossed di agram. Since we are only interested in terms of order «~3t_1, we have put cint.pole equal to — 2 Ls — Lt. Otherwise, the program would spend a lot of time calculating terms that are of order »~3. Next, the program CINT is used to determine the leading contribution from each term in the numerator. For this part, we use the auxiliary procedures listed in section D.2. We will now describe these briefly. First, the procedure colLsym collects terms in the numerator that give the same contributions because of symmetries in the denominator. It loops through the four symmetries of the uncrossed and once-crossed diagram, and applies those that are present in the present denominator. Next, the procedure init.SUM initializes the variables SUM and NON ANS, in which the total results will be stored. SUM is only initialized when cintjile = 0, which is the default value. If cintJHe is different from 0, the intermediate results will be written to the file cint.file, and nothing will be stored in SUM. The procedure N-term is called for each term in the numerator. It prepares input for CINT, calls CINT, and finally stores the results for the present term. Before the results are stored, they are simplified through calls to simp_SUM and simpJVONANS. Is simp.SUM, the identities (5.39) and (5.40) are applied to SUM[1]. In simp-NON ANS, Lu is replaced by Ls if the calculation has been done with Ls = Lu (i.e. Ls = Lu in cint.vinvarB). Next the linear forms in NONANS ere divided by Ls, and the new variables gamma.u = Lu/Ls, gamma.t = Lt/Ls 78 and gammaJaui = Llam/Ls are introduced. Finally, the procedure end.SUM is called to print the total result». If cint.fHe is a filename, the file with this name is terminated, such that it contains a complete MAPLE program which may be read into MAPLE later to sum up the results. The variable NONANS is now a set of linear forms in gamma.t and gamma Jam if the calculation has been done with Ls = Lu. The solution is now valid at the point specified by cint.vinvars, and as far as the linear forms in gamma.t are non zero. When the number of linear forms is large, the procedure region is very useful. It determines the corners of the region in which the solution is valid. The corners do not specify the region uniquely unless we know that it is bounded, therefore the four extra linear forms gamma.t ± 99999 and gamma-lam ± 99999 are added to NONANS before the call to region. As long as 99999 is much larger than any number that enters the problem, we may conclude that a region that reach one of the lines on which one of the four extra linear forms is zero, was not bounded before we added the extra linear forms. The final output is shown for the uncrossed and the once-crossed diagram in sections D.3 and D.4. In both cases the calculation is done with Ls = 25, Lu = 25, Lt = 20, Lm = 0 (m = 1) and Llam = —900, and it turns out that the results - are valid for arbitrarily small values of Llam. A factor 2/t is excluded from the numerator, thus the final result must be multiplied by this factor. D.l Listing of the main code read(cint): read(cint.nux): vits(student): intertac«(echo=2); DB := al*a3*12+a2»a4»Al+al*a4*a6+a2*a3»a6: Du := 0: Dt :•= a&*a6»a7: Dm := -al3_2*A2-a24_2»il-2«al3*a24*a6: Dlam := -Lam*(a5+a6+a7): LamV := Al«(a24+a7)+(al3+a5)»a6: sub.*: =Lam=LsmV, Al=al+a3+a6+n6, al3*al*a3, A2*a2+a4+a6+a7 ,a24 Ds:=expand(Ss): ns:=n_o_t(Ds): Dt:=expand(Dt): nt:=n_o_t(Dt): nu:*0: Da:>sxpand(Dm): nm:=n_o_tCDm): Dlam: "expand(Dlam) : alam:*n_o_t(Dlam}: LamV:*expand(LamV): 79 # With Lm-O: invaxs:*exp&nd(-Ls*Bum('z.i' ,i»i. .ns) -Lu*Bum( 'z. i', i»ns+l. .n»+nu) -Lt*Bum('z.i *,i»ns+nu+l..ns+nu+nt) -Llam*sum('z.i',i*ns*ati+nt+nin+l..ns+nn+nt+nm+nlan)): t Pat the terns in Ds, £>u, ... into an array called term: ntrms:*0: for d in [Ds,Du,Dt,Dm,Dlam] do if d=0 then next fi; if type(d,'+') then for trm in d do ntrms."=atnas+l; term[ntrms] :»trm od; else ntnDB:sntrms41; termCntrms] :«d; fi; od: ntrms J): =ntrzB8: # Gamma~fmiction6 that come from factorizing: GiHHts:«[[z.(l..ntrms.D)]]: # Constant coefficients in 0: const:«1: for j to strms do if type(term[j] ,'»') then conBt:«const*(map(proc(x) if typed,numeric) then abs(x) else 1 fi end, term[j]))-(-z.j) elif type(term[j].numeric) then const:=const*term[j]"(-z.j) fi; od: i Idd the terms in lambda to the array: for d in [LamV] do if d*0 then next fi; if type(d,'+') then for trm in d do ntrmB:*tLtrm8+l; termCntrms]:*trm od; else ntrms:«ntrms+l; term[ntrms]:*d; fi; od: ntrms_L: *ntrsis: t Gamma-functions from factorizing Lam: GiHHls:*[op(GlMMis), [z. (ntrns.D+l. .ntrms.L)]] : 80 for j to ntrms do lprint(*term['.j.'] «*,term[j]): od: * Calculate the povers of al, a2, ... after factorizing. * p[...] is no« the «negative» poller of its argument: for r in [al,a2,a3,a4,a5,a6,a7,al3,Al,a24,A2,Lam] do p[r]:=0; for j to ntrms do if has(term[j],r) then p[r]:=p[r]+degree(term[j],r)»z.j fi od; od: a:*'»': _n:='_n': for j to 7 do p[a.j] .••p[a. j]-_n. j od: for r in [al,a2,a3,a4,aS,a6,a7,al3,Al,a24,12,Lam] do lprintCpf.r.'] =',pW): od: t Split al+a3 into al and a3: ntrms: =ntrms+1: GAMMAs: = [op(GAHMAs), [z.ntrms ,p [al3]-z.ntrms]] : p[al] :-p[al]+z.ntrms: p[a3] :«p[e3]+p[al3]-z.ntrms: • Split a2+a4 into a2 and a4: ntrms:*ntrms+l: GAMMAs:•= [op(GAHHAs), [z.ntrmB ,p [a24] -z .ntrms]] : p [a2] : # Do the alpha integrals: GAMMAs:=[op(GAMMAs), [l-p[al] ,i-p[a3] ,l-p[a6] ,+p[Al]-3+p[ai]+p[a3]+p[aS]] , [l-p[a2] ,l-p[a4] ,l-p[a7] .+P[A2]-3+p[a2]+p[a4]+p[a7]]] : lprint{GAMMAs); var:«[z.(l. .ntrms_D-l),z. (ntrms.D+l. .ntrms_L-l) ,z. (ntrms.L+l. .ntrms)] : i Impose complex delta functions: z.ntrms_D:=_deg_D-convert([z. (1. .ntrms.D-D] ,'+'): z.ntrmB.L:•.deg.lam+p[Lam] -convert([z. (ntrms_D+l. .ntræs.L-1)] ,' + '): invarB:"norjnal(invaxs): GAMMAs :*normal(GAMMAs): const:•normal(const) : var:«var: 81 * Print input to cint in general, before apecializing to a specific term: lprint(invars); lprint(GlMMls); Iprint(const); lprint(var); lprintC Number of variables:' ,nops(var)); a := a2«a3+a2»aS+a3*a4+a3*a6+a3*a7+a4»i.-<-a4*a6+a5*a6+aS*a7+a6*a7: b := a2*a3+a3»a4+a3*a6+a3*a7+a4*a6: c :» al»a4+al*a7+a3*a4+a3*a6+a?«a7+a4»aB+a4»a6+a5*a6+a5«a7+a6*a7: d :- al*a4+a3»a4*a3*a6+a4*a5+a4*a6: at := al*a2+al*a4+al*a6+al*a7+a2*a5+a2«a6+a4*a5+aS*a6+a5*a7+a6*a7: bt := al*a2+al«a4+al*a6+al*a7+a2»a6: ct :• al*a2+al*a6+al«a7+a2*a3+a2»aE+a2»a6+a3*a7+aS*a6+aS»a7+a6*a7: dt : = al»a2+al*a6+a2*a3+a2*a6+a2*a6: * A lactor 2/t is not included in Niii: Niii : = expand(2«s"2*t*c*a*at*ct): * W"+ « 1, • V- = 0. * Niii := eitpand(s"2*t»b»c*bt»ct+b»d*bt*dt+d«a«at»dt): # Remaining tarns Niii:»coll.8ym(Niii,BUbB(sub.l,DB»B+Dt*t+Iji«m"2*Dlam«lajn"2)): * Reset some variables in order to free some memory: Ds:*'Ds': Du:='Du': Dm:='Dm': Dlam:='Dlam': a567V:»'aS67V': lamV:='LanV: snb.l:*'anb.l': term^'term1: p: = 'p': z.ntrm8_D:*evaln(z.ntniiB_D): z.ntrms2:*evaln(z.ntrms2): z.ntrms:=evaln(z.ntnns): interface(quiet-true); cint_min_logs:B 4: cint.pole:=-2*lB-Lt: cint_output:-0: init.SDMO; if not tjrpe(Niii,'+') then Niii: = [Hiii] ii: niii_tQrms:Ktable(): 82 for trm in Niii do minimum:«N_tarm(trm,3,3); if assigned (niii.terms [minimum]) than niii.terms [minimum] : *niii_terms [minimum] +trm else niii.termB[minimum]:=trm fi od: for i in indices (niii.terniB) do lprintCTaras in niii.terms [',op(i)F'] :' ,n_o_t(niii_terms[op(i)])) od: ond.SDMO; lprint('Total time used:',time{)); D.2 Listing of auxiliary routines coll.sym: «proc (f ,D) local old.quiet, fp, sym, tzm, trap, i, j, n; old.quiet:'interface(quiet); interface(quiet=true); fp:*expand(f); if type(fp,'+') then fp:=convert(fp,list) else interface(quiet=old_quiet); RETUBN(f) ii; lprint('••• Collecting symmetrical terms:'); lprint('Number of terms (1):',nops(fp)); n:=l; for sym in [ [al*a2,a2*al,a3*a4,a4«a3,a5*a7,a7=aE], [al=e3,a3-al,a2*a4,a4=a2], [al-a4,a4=al,a2=a3,a3=a2,aS=a7,a7=a5], [al 83 fp:»map(proc(x) if x«0 then NULL else x fi end, fp); lprint(* Number of termB ('.n.'):',nops(fp)); fi; od; interlace (quiet=old.quiet) ; expand(convert(fp,'+')) end: init_SUM:=proc() if cint.file = 0 then SUM:=table([Olnops(var)]); SUM[0]:=0 fi; N0NANS:=O; if eint.file <> 0 then cint.simp.sun:=simp_SUH fi; NULL end: end.SUM: =proc 0 local j, vgamma; if cint.file <> 0 then for j from 0 to nops(var) do if Cint.sumCj] <> 0 then Cint.Btore(j) fi od; appendto(c.int_f ile); lprint('read cint.aux:'); lprint (' cint.file: «0:') ; lprint('cint.end.file():'); appendto(output.file) else for j from 0 to nopo("ar) do if SUM[j] <> 0 then lprint('SUM['.j.'] :=',SUM[j3) f i od; f i; lprintCNONANS : = ',N0N1NS); if cint.f ile*0 then for j from 0 to nope(var) do if SUM[j] <> 0 then lprintCSUMC.j.'] :='); print(SUM[j]) f i od; if cint_min_logB = 4 and SUH[1] <> 0 then lprint('SUM[0] + SUM[1] :*'); print(snbs(Lu=La,SUH[0]+SUM[l])) fi f i; N0NlNS:*{op(N0NlNS), gamma. t+99999,gamma.t-99999,gamma.lan+99999,gamma.lam-99999}: N0N1NS:«regionCNONINS, op ( subs (cint.vinvars, [gamma.fLt/Ls, gamma_lam*LlaBi/Ls] ) ) ); lprintCCornera of the region of analyticity, Lganma_t>gainma.lam] : *,N0N1NS); 84 vgamma: «subs (cint.vinvare, -{gamnia_uBLu/Ls, ga&ma_t*Lt/L8 ,gamma_la&]*Llai_/Ls}); lprint('Calculated at:'.vgamma); NULL end: N.term: =proc (numer.term ,n_D ,n_lam) local nterm, xx, gg, ccv j, nj; if indetB(numer_term) minus -Cal,a2,a3,a-,a5,a6,a7,8,t} <> {} then ERROR('numer.term:',Mimer_term) fi; nterm: sexpandCaumer.term); if tvpetaterm,'+') then map(N_term,convert(nterm,list)); RETURN (HULL) fi; xx: =degree (numer_tenn.-Cal.a2, a3 ,a4.,a5,a6t a7» -2*n_lam-3*n_D; if xx <> -7 then ERRORCTotal degree of integrand is not -7, but'.xx) fi; cc:=lcoeff(nterm,{al,a2,a3,a4,a5,a6,a7})*const; xx:ssubs(_deg_D=n_D,_deg_lamsn_lam,invars); gg: *subs (.deg.O^n.D, _deg_lamsn_lam, Gi_l^_a); cc: =subs (_deg_D*n_D, _deg_lam=n_lam, cc) ; for j to 7 do nj:»degree(nterm,a.j); gg:*subs(„n.j*nj,gg) od; xx:'Cint(xx,gg,cc,var); lrøtUNS:=iI01UNS union -imp_NOKiNS(Cint.nonans) ; gg:=0; for 1 from 0 to nops(var) do if Cint_n9!*arms[l] <> 0 then gg:-l fi od; if Cint_calls_nopB(var)] <> 0 then if gg=0 and cint.file = 0 then lprintO; lprintC 'Numerator:' ,numar_term,'':'',' (No contribution)') else lprint () ; lprint ('Numerator:' ,numer_term,'': ") fi fi; if gg'O or cint.file <> 0 then RETURN (xx) fi; for j from 0 to nops(var) do Cint.«urn [j] :*aimp_SUM(Cint_Bum[j_ ); SUHCj] :=SUM[j]+Cint_Bum[j] od; Cint.output:«100; Cint_nonan:«false; Cint_print(nops(var)); lprintO; Cint_output: •cint.ontput; Cint.nonan: Bcint_nonan; xx end: 85 simp_SDM:»proc(R) local Rp, z, c; # Simplifies the answer from Cint: Rp:'expand (A); if typsCRp,'*') then RETUAN(map(Bimp_SUM,Rp)) fi; z:=Cint.var[1]; if not has(Rp,z} then RETURN(Ap) fi; if has(Rp,GAMMA(l/2*z)) then Rp:«normal(2»subs(z=2«z,Rp)) elif has(Ap,GAMMAC2*z)) then Rp:«normal(subs(z=z/2,Rp)/2) fi; if type(Rp.'*') then c:=map(proc(f ,z) if not has(f,z) then f else 1 fi end, Ap, z); Ap:=normal(Rp/c); it Rp=z-B*GiHMl(-z)-2*GaMHl(z)-2 then 0 elif Rp=z-3*GlHHlC-z)-2*GAMm 86 if cint_min_logB = 4 then lprint('Cint_sum[0] + Cint_sum[l] : = '); print (subs (Lu-Ls,Cint_fium£0)+Ci2it_snmIl])) fi end: region: *proc (lin_fon»B ,xxO ,yyO) # lin_forms is a list of linear forms t xxO and yyO are equations that define a point in the x,y plane, local x,y,fs,online,f0,cps,i,al,bl,cl,pl,j,&2,b2.c2,cp,fsp; x:=lhs(xxO); y:=lls(yyO)j if indets(lin.forms) <> {x,y} then RETURN(NULL) fi; if member(0,lin_fonuB) then lprintCWARNING (region): 0 found in the list of linear forms'); fi; fs: = [op(map(proc(f) if type(f .numeric) then NULL else f fi end, lin_forms))3; online: *0; * Reverse Bign on linear forms that are negative at (xxO.yyO), and # check whether (xxO.yyO) lies on one or more lines: i:=i; do if i > nops(fs) then break fi; fO:*subs(xxO,yyO,fs[i]); if to < 0 then fs:»snbsop(i«-fs[i],fs) elif f0«0 then if online«0 then online:*fs[i] ; fs:*subaop(isNULL,f8) else RETURN({[rhs(xxO).rhB(yyO)]}) fi elBe i:=i+l fi od; • (xxO.yyO) lioE on a line: cps :=•(}; if online <> 0 then al:=coeff(online,x); bl:>coeff(online,y); cl:*snbs(x*0,y»0,online); for j from 1 to nopa(fa) do a2:=coeff(fs[j],x); b2:>coeff(fs[j],y); c2:-subs(x*0,y>0,fs[j]); ep:aal*b2-a2*bl; if cp <> 0 then cp:»C(bi*c2-b2»cl)/cp,-(al*c2-a2»cl)/ep]; fsp:>subs(x*cp[l],y"cp[2],f»); if nin(op(fsp)) >> 0 then cp»:-{op(cp»),cp> f i fi od; 87 RETURN(cpB); H; i Remove one of each pair ox parallell lines that are on the same side of * (xiO.yyO): It pi: A point on ls[i]s0, cp: A point on ls[j]=0. do if i > nops(ls)-l then break ii; al:=coell(lsti3.x); bl:=eoelf(ls[i] ,y) ; cl:=subs(x*0,y=0,ls[i:i) ; if al=0 then pl:= do if j > nops(ls) then break fi; a2:=coeff(fs[j],x); b2:=coeff(fs[j],y); c2:=subs(x=0,y=0,fs[j]); if al=0 and a2=0 or al <> 0 and a2 <> 0 and M/al*b2/a2 then if subB(pl,fG[j]) < 0 then f£:=EubBop(i=NUlL,f6); ai:=coeff(fs[i],x); bl:=coefl(lsti],y)i cl:=Bubs(x*0,y*0,fs[i]); if al=0 then pl:={x=0,y=-cl/bl} else pl:={x=-cl/al,y=0} fi; j:»i+l; next fi; if a2=0 then cp:= if subs(cp,fs[i]) < 0 then fs:=8ubsop(j=NULI,)fB}; next li li; od; i:»i+l od; # Find corners of the convex hull around xO: for i to nops(fB)-l do al:=coell(ls[i],x); bl:»coefl(fs[i],y); cl:=8ubs(x=0.y=0,fs[i]); lor j from i+1 to sops(Is) do a2:-coell(ls[j],x); b2:«coell(fB[j],y); c2;«8ubB(x»0,y«0.fB[J]); cp:»ai«b2-a2*bl; il cp <> 0 then cp:»[(bite2-b2*cl)/cp,-(ai*r?-a2»cl)/ep]; lsp;=aubs(x=cp[l],y=cp[2],10; il minCopCfsp)} >• 0 then cps:> 88 D.3 Output for the uncrossed diagram SUM[03 := -4/3*LB**3*Lt+4*Ls**2*it**2-4*Llam*Ls**2*it+2*Llam*Ls*Lt**2+4/3*Ls **3*Llam+2*L8**2*Llam**2+i/3*L8**4-8/3*L8*Lt**3+l/3*Lt**4 N0N1NS ;* {1, gamma.lam, -gamma»t+gamma_lam+l, -2*gamma_t-2+gamma_lam, gannoa.t , gamma.t-4, 2+gamDia_t, ~2+gannna._'t > "•l/2+gainnia..t, l+ganniia_t , -l+gannna_t , 2+gamma_lam, ganma_lam-l/2J, i+gamma_lam, -l+gamma.lam, -2+gamma_laai, _ gaBBtta..lam-3/2*gamma_t, gww* i ww-pmrnin 1.1 -l/3*gMniia_'t+gamina..lam., l/2*gannBa_"t+ gomma^lam, -2*gamma_t+gainma_lam, -2/3*gannQa-t+gaiania_lam1 -l+gamma,..t+gainnia..laiii, -gaBnna_t+2+gannna_laBi, l/2-l/2*gaimiLa..t+gammanlam, l-2*gannnav't+ganffla_lam1 2-2* na | n , gwTmi»^_t.-fg^T[ff _1^r• -1 •»ffWTnmn_i mn-g^iimiwt t -2/3*gainnia_t~2/3+gamina_Xam}' SBMtO] : = 3 2 2 2 2 3 - 4/3 LB Lt + 4 LB Lt - 4 Llam LB Lt + 2 Llam LB Lt + 4/3 LB Llam 2 2 4 3 4 + 2 LB Llam + 1/3 Ls - 8/3 LB Lt + 1/3 Lt Coiners of the region of analyticity, [gammb_t,gamma_lam] : {[1/2, -2], [1/2, -999991, [1, -2], [1, -99999]} D.4 Output for the once-crossed diagram SUM[0] := -Ls»LlBffl*Lt**2+L8**2»Lu»Llsm-7/12»Lt**4+2»Lu*LB«Llam*Lt-l/3*Lt»*3* Lu-l/6*Lu*»3»Lt+6/6»Ln*»3«LB+B/4*Ln**2»Lt**2-l/8*Lu**4+l/2«Lu»Lt*L«*»2-3/2»Lu«« 2»Lt*Lg-l/2«Lu*Lt**2*L8+l/4*Lu**2*Ls**2+3/2«Lt«L8**3-5/6*Lu*LB**3-13/4»Lt**2*Ls **2-Lu*Ls»Llain«'»2-ll/6*Ls»*3*Llam+2/3*Lu*»3«Llam+l/8»LB»«4+8/3«Ls*Lt*»3 SimCl] := LB*»2*Lu*Llam+6/12*Lt»*4+l/3*Lt»«3*Lu+l/3*Ln**3»Lt-10/3*Ln*»3»LB+l/ 2*La**2*Lt**2+3/4*Lu**4 89 , -2»t—».t«g—I.IM t. 3*gu»i.1u 3«giui.t. -Hg—i.lu'gu»i.t. 9-3/3* gU»».t«g—1.1—, gU»l.1U'g—l.t+4. (HH.lM-3«a*|»BM.t, g—l.lU'6-a* IUH.C, -a/3. -2/3*guni.t. Hg—l.lu-3»glUl.t. glul.t*3/2. glul.t«l/2, 3 ')IMI.lM 3»g—I.t. -l«|atJM.lW(MB«.t, g—M.t-2. gUM.t-1. -l*gUM.lU , 2*gu»i.lu-3'giu».t. -i—i.t «gun i.1u*l, -gn—_t«g—. 1 u. l*gua«.lw- a>guM.t, g—i.t-3/a. -a»g—.t«g—I.IM, g—11.1—113. g—.in a. -i/a* guu.t. a/a*gf i.T-a«g—i.t. |m.tn, gi—i.t«a. g—juii/a+g—i.t. gi—i.lwi«a«giin.t. guaa_lu«0. -3/2*giui.lu. guu_lu*l/2. l/3»g—!.)•• guM_t. gi—i.lm-4/3. -3*g«i.t. -l/2»g—.t«g—• ,1M. -l/a*g—.1». 3* gaau.li*. 2*gUM.lu-l/2*guw.t, gi—,.lu*3-g—i.t, gu—.lu*4-«»gu«_t . -3>guu.t»swi.la4<, (UM.lu«l-l/l>pw.t, -4/S»|«u.t, -6*g— i.t) 3UMC0] :• a a 4 3 - Li Ilia Lt • Li Ln Liu • 7/12 Lt Olali Liu Lt - 1/3 Lt Ln 3 3 2 3 4 2 - 1/6 Ln Lt + 6/6 Ln Li • 6/4 Ltt Lt - 1/8 Ln * 1/2 Ln Lt L a a a a a 3 - 3/2 Lu Lt Li - 1/2 Ln Lt Li • 1/4 Ln L» • 3/2 Lt L» - 6/6 Ln L» 2 2 2 3 - 4 - 13/4 Lt Li - Lu Li lie* - 11/6 Li Liu • 2/3 Ltt Liu * 1/8 Li 3 • 8/3 L« Lt SUM[13 :« 2 4 3 3 3 2 2 I, Ln Liu + 6/12 Lt + 1/3 Lt Lu + 1/3 Lu Lt - 10/3 Ln Li + 1/2 Ln Lt 4 2 2 2 2 2 3 + 3/4 Lu + 4 Lu Lt Ls - 2 Lu Lt Ls - 2 Lu Lt Li • 6 Lu Ls - 2 Lt Ls 3 2 2 3 3 4 - 4 Lu Ls + 2 Lt Lu - 1/6 Ls Liu + 2/3 Lu Liu + 7/6 Ls 3 2 - 4/3 Ls Lt - 2 Lu Ls Liu 90 mn[ø] • son[]} ••• 3 3 4 2 3 4 - La Ua Lt - a/3 U Llaa - 1/6 Lt • 3 Li Uaa Lt « 3/3 It U • 1/6 U a a a a s - 3 It La - La Uaa * 4/3 La Lt Coroaxa of taa raglon of aoalytidty, [gaaaa.t.giaaa.lari: <(a/3, -98899], [1. -99999], [3/3. -6], [1, -6]) Calculated at: {gaBaa.ii • 1, gaaaa.t • 4/6, gaaaa_laa • -36} 91 Bibliography [1] K. S. Bjørkevoll, G. Fildt, and P. OsUnd. Two-loop Udder-diagram con tribution* to Bhabha scattering. I. Structure of the matrix elementi. Scien tific/Technical Report no. 1991-01, University of Bergen. Submitted to Nuclear Physics B., 1991. |2] K. S. Bjørkevoll, G. Faldt, and P. Osland. Two-loop ladder-diagram contri butions to Bhabha scattering. II. Asymptotic results for high energies. Scien tific/Technical Report no. 199S-01 University of Bergen. Submitted to Nuclear Physics B., 1992. [3] K. S. Bjørkevoll, G. Faldt, and P. Osland. Two-loop ladder-diagram contribu tions to Bhabha scattering. III. The 4>' limit of QED. In preparation, 1992. [4] F. A. Berends and R. Kleiss. Distributions in the process e+e" —> e+e" (7). AW. Phys., B228:537-551, 1983. [5] M. Greco. Bhabha scattering near the Z„. Phys. Lett., B177:97-105, 1986. [6] M. Bohm, A. Denner, and W. Hollik. Radiative corrections to Bhabha scat tering at high energies. I: Virtual and soft photon corrections. AW. Phys., B304:687-711, 1988. [7] F. A. Berends, R. Kleiss, and W. Hollik. Radiative corrections to Bhabha scat- term at high energies. II: Hard photon corrections and Monte Carlo treatment. AW. Phys., B304.712-748, 1988. [8] S. Jadach and B. F. L. Ward. Muitiphoton Moate Carlo event generator for Bhabha scattering at small angles. Phys. Rev., D40:3582-3589, 1989. [9] Alex Read, CERN (private communication, February 1992). He tells that both OPAL, ALEPH and DELPHI may improve the presisjon of luminosity measure ments to around 0.1%. He gives the following reference: "Proposal for upgrading the OPAL luminosity detector", CERN/LEPC 91-8. [10] P. Osland and T. T. Wu. High-energy, large-momentum-transfer processes: Ladder diagrams in tj? theory (I and II). AW. Phys., B288:77-130, 1987. 92 (ll| R Gajlmani and W. Troost. The asymptotics of ladder diagrams involving maulets scalar*. Pkyt. Lttt, BJ49:523-527, 1990. [12| R. Gastmans, W. Troost, and T. T. Wu. The asymptotic» of «V* ladder diagrams in six dimensions: High energy and large momentum transfer. Nucl. Phys., B36S:404-430, 1991. [13) M. C. Bergere and L. Siymanowski. Scattering processes at the high-energy and large-transfer-momentum limit: The asymptotic» of «V* ladder diagrams. Nucl. Phys., B350:82-110, 1991. [14] C. Newton and T. T. Wu. High energy, large-momentum-transfer processes: The n-rung ladder diagram of 4? theory. J. Math. Phys., 32:1619-1637, 1990. [15] D. R. Yennie, S. C. Frautichi, and H. Suura. The infrared divergence phenom ena and high-energy processes. Annals of Physics, 13:379-452, 1961. 116] G. Grimmer and D. R. Yennie. Improved treatment of the infrared-divergence problem in quantum electrodynamics. Phys. Rev., D8:4332-4344, 1973. [17] G. 't Hooft and M. Veltman. Scalar one-loop integrals. Nucl. Phys., B153:365- 401, 1979. [18] M. C. Bergere, C. de Calan, and A. P. C. Malbouisson. A theorem on asymptotic expansion of Feynman amplitudes. Commun. Math. Phys, 62:137-158, 1976. [19] C. de Calan and A. P. C. Malbouiason. Complete Mellin representation and asymptotic behaviours of Feynman amplitudes. Ann. Inst. Henri Poincare, 32:91-107, 1980. [20] L. Lukaszuk and L. Szymanowski. A new technique for the evaluation of Feyn man diagrams in the high energy, large momentum transfer limit. Z. Phys., 043:133-139, 1989. [21] A. B. Kraemmer and B. Lautrup. Radiative asymmetry around the resonance in e+e" -> fi+[i-. AW. Pays., B95:380-396, 1975. [22] Anthony C. Beam. REDUCE User's Manual Version S.S. Rand PubUcation, 1987. [23] Bruce W. Char, Keith 0. Geddes, Gaston H. Gonnet, Michael B. Monagan, and Stephen M. Watt. MAPLE Reference Manual. Fifth edition. Waterloo Maple Publishing, 1988. 93 Paper I Daivarsity of Berg**, Dapaxtaaat of Physics Selratlflc/ToekalMl laport lo. 1M1-07 ISSI 0603-MM TWO-LOOP LADDER-DIAGRAM CONTRIBUTIONS TO BHABHA SCATTERING I. STRUCTURE OF THE MATRIX ELEMENTS Knut Steinar BjerkevoU Department of Physics, University of Bergen, AlUgt. 55 N-5007 Bergen, Norway Goran Faldt Gustaf Werner» Jmtitut, Box 535 S-7S1 tl Uppsala, Sweden and Per Osland Department of Phytic», University of Bergen, AUigt. 55 N-5007 Bergen, Norway ABSTRACT We discuss contributions to Bhabha scattering from the gauge-invariant set of six t- channel, photon-exchange, two-loop, ladder-like diagrams. The contributions from these diagrams to the unpolarized cross section can be expressed in terms of four scalar functions. These functions are given in terms of Feynman-parameter integrals in the limit 3 3> |t| > m*. 1 1. Introduction One of the most important tasks at LEP is to perform precision measurements, to test the consistency of the standard model. AD Important input in such cheeks is the absolute luminosity, which at LEP is obtained from a measurement of the Bbabba scattering cross section, e*e~ - e+e". At high energies the radiative corrections become large, due to the large logarithms log-fj, (1.1) associated with the cottinear emission of soft photons. At small angles there are also enhancements due to logarithms of the kind *i- (1* For these reasons, it is believed that the presently available one-loop-corrected cross section might not be sufficiently accurate [1], In fact, the theoretical uncertainty is in the angular range down to ~ 40 mrad estimated to be of the order of 1%, which is one of the largest uncertainties in the analysis of the data [2]. For annihilation processes, the QED corrections of relative order a7, associated with the initial-state radiation, have already been calculated [3]. They are important for the determination of the £°-properties, and taken into account in realistic Monte Carlo calculations [4]. For small-angle Bhabha scattering, on the other hand, it is the lowest-order (-channel photon-exchange diagram that dominates. Initial-state corrections are thus largely irrele vant. We have initiated the evaluation of QED corrections to the Bhabha scattering cross section, of relative order a2. The first part of this investigation is a study of the r-channe! ladder-like diagrams. There are six of these (see fig. 1), forming a gauge-invariant set. In addition to the experimental importance of these calculations (when completed), th;:re is also a theoretical motivation for studying ladder-like diagrams. It was found in the context of ^-theory, that in the kinematical region m2«|t|«j, (1.3) the asymptotic form of the amplitudes for the ladder diagrams are given by different formulas, depending upon how log \t\ compares with various rational fractions of logs [5J, 2 [6], [7|. In «t'-theory, thii phenomenon fint occur* for the four-rungdiagra m (i.e., at the level of three loopa) [5|. For QED, being a gauge theory, there are substantial cancellation! among the leading termi. The amplitude thus become* sensitive to term* beyond those that for an individual diagram are the leading one*, and which correspond to the «t'-results. The question then arises whether or not the iub-leading terms are analytic at the two-loop order. The article is organised as follows. In sects. 2 and 3 we discuss general properties of the Bbabha crou section and the six ladder-like diagrams that contribute to 0(<*'). The contributions from two of these six diagrams are decomposed into momentum integrals and spinor factors. Sect. 4 gives the results of the momentum integrations, while sect. 5 presents the spinor algebra. In sect. 6 these results are combined to obtain the numerators in the Feynman-parameter representation of the amplitude». Sect. 7 contains a discussion of the remaining four ladder-like diagrams, and in sect. 8 we summarize our results. 2. The Bhabha cross section At small scattering angles the Bhabha scattering cross section [8] is dominated by the one-photon exchange diagram. We shall investigate contributions to this cross section coming from a gauge-invariant set of three-photon exchange diagrams. In the cross section th»y first enter as interference terms with the one-photon exchange term. The amputated matrix element that we consider is M = M{1->> + Mii-,) + Ml,y\ (2.1) where .M'1'"' is the one-photon exchange term M™ = ««H-feVMi*) =%r *(W)(-"7>(P;) with Q = Pi-p'i=P2-P2, (2.3) and Al'2'*' the two-photon exchange term as given in refs. [9], [10], and [11]. The Bixth-order diagrams we shall consider are displayed in fig. 1. We denote their sum M^L The tensor structure of M^'^ is quite complicated, but simplifies considerably when contributions proportional to the electron mass are neglected. In this limit the general 3 structure become» + <«Wh»«(B»«(wh.i»»W) + •••flV«Wh.7»«(j»i)fl(p»h»f(ri)]- (2.4) Since we do not include the effect* of /"-exchange, the term* Fj£ and F£ do not con tribute to the unpolarixed differential cross section. The fact that terms proportional to m are neglected, doe* not imply that the invariant function* Fn and Fu are independent of tht electron mass. Logarithmic dependences Uke log */m* and log \t\/m7 are still possible, and do in fact appear. The unpolarized cm. differential cross section is afc-asi;^!:^ (2-5) ipln where m is the electron mass and a the square of the total cm. energy. The contributions from (-channel diagrams alone give ^-^{l+ = *<-..)+ $*(..*>}. (2-6, We are interested in the contribution to fofat) due to the two-loop ladder-like diagrams. In terms of the functions defined in eq. (2.4) it is •?f7) ('. «)•=!*• [fi« + 5^7^'] = s Re [F°° + : F»] • <2-7) where in the second step we have neglected terms multiplied by relative factors tja. In the above, the kinematical variables are given by * £ (pi + ft)' = (pi + pi)2 = 2p, • pj = 2p; • pi, 2 < s (pi - P',) = (P2 - Pif = -2pi • p'i = -2pj • p'2, 2 2 u = (p, - pi) = (pi - p2) ^ -2p, • pi = -2P; • P2. (2.8) 3. General properties of the six diagrams We denote the six diagrams a, b, c, d, e, and /, and the corresponding Feynman amplitudes Ma, ..., Mj. It turns out that they can all be expressed in terms of the functions that define the amplitudes Ma and Mb. We therefore start out with a more detailed discussion of the amplitudes for the uncrossed (MJ and once-crossed {Mb) three- rung amplitudes. 4 3.1 THE UNCROSSED THREE-RUNG LADDER AMPLITUDE We use the Icinematica] variable* defined in fig. 2. Our choice of Fejmman parameter* i» given in tig. 3; it coincide* with thai used by Cheng and Wu |12] in their discussion of the corresponding ^'-diapram. The Feynman amplitude can then be written as ' Vi+»)*-">'+»« (Pi -9i)a-m2 + "v ' ' •«' a (3.1) «j-A' + u ((J-^-ftJJ-A' + te 9|-A -t-«>* Here, we extract a coefficient, and sort terms according to powers of m: 11 2 Ma = M™ + mMi + m M^ + m'M^ + mtM™. (3.2) We neglect amplitudes containing multiplicative factors of m. The amplitude of interest Ma is then factorized as M™ = -ie'M? • Ml = -«'(AC WeO""". (3.3) with the spinor factor (^)M^ = |fi(pi)7",7^7fl7(.7a"(pi)][«(P2)7a7p7fl7»7-rv(pi)l, (3.4) and the momentum-space factor + 92) 1 2 2 2 "' 7 0)« y (2*-)< (n - «)• - m + >e tø + «,s); — m + »e ( (P2+9l)' (Pi ~ 9»)° (P2 + 9i)2 - m2 + te (pi - 92)' - m2 + te 1 1 1 (3.5) fjJ-A'+« (Q-g, - g,)2 - A2 + ie gJ-A' + te' 5 3.2 THE ONCE-CROSSED THREE-RUNG LADDER AMPLITUDE The notation for the Fernman parameter* U given in fig. 3. Again, it coincide* with that used in |12|. The Feynman amplitude U (Pi + Xtf- A»+u lO-ji-fljJ'-A'+.e «J-A»+«' (3'6J Again, we extract a coefficient, and «ort terms according to powers of m: M» = M1? + mMi" + rn'Mj" + msM + m4M(,". (3.7) Only Mh is retained and factorized as M[0) = -«"WW^Atf )•""*, (3.8) with the spinor factor WW = [«(pibWWMPi )l[«(P2)7»7p7,7<,7^f (Pi)]- (3-9) The ordering of the 7-matrices in the v-factor differs from that of M2, and with the æcmentum-space factor (M1Y"" - [ É3l- f *** (Pi-9i)* (Pi+«)" V 2 * ' 7 (2*)* y (2TT)« (Pl - gi) - m» + ie (p', + q2)* - m> + it (P2 + 9i)p (P2 + 91 + gi)J (P2 + 9i)2 -m2 + it (p2 + 91 + 92 )2 - m5 + ie 1 1 1 (3.10) gf - A* +« (Q - 91 ~ 92)2 - A2 + te «f - A2 + ic' 4. The momentum integrations The momentum integrations are straightforward. The approach outlined in Ap pendix A provides a convenient method for handling the momentum dependence of the numerators. 6 4.1 THE UNCROSSED THREE-RUNG LADDER AMPLITUDE In order to facilitate comparison with the ^'-results, (12|, (5|, we introduce A = A(a) = (QI + a» + Q» + a«)(aj + at + and 3 D(a) = D,t + D,t + D„m* + Dx\ + it, (4.2) with D, = a,aj(a} + a4 + a« + QT) + aja«(ai + o» + a» + at) + a«(aia, + a,a3), Dt = QjajaT, D„ = -(ai + a3 + a, + a«)A + (ai + aj)(aj + a«)(a» + OT) + (ai + aj + QJ + Q4)(ajaT + asa( + a ten), Dx = -(a» + a, + a,)A. (4.3) In fact, in the limit when the masses are the same, A = m, the relevant quantities are D„ Dt, and Dm + Dx, with Dm + Dx identical to the Dm of refs. [12] and [5]. The integrations in eq. (3.5) yield the following result: 1 2 -( + a, + a + aWkZk' - (a + a+a,+ aWkfå 2£(a) Q] b 2 t + O.UT«£*? - g^kik't - g<"Kk> + iT*X)] (ai + a + a + a)(ai + a + a + aiWg™ 40(c) 3 5 t 4 6 + a\{gl"g>' + g"'g"ty. (4.4) Here, i„ = opi + 6p2 - Q8Q7Q = op', + 6p'2 + as(ot2 + 014 + a6 + a7)Q, kb = ep', + op, + a5aeQ » cpi + dpi - 07(01 + o» + o» + a,)Q, i, » i* •»• in + o,oT • ipi + *Pi - °»(°» + 04 + o« + ot)Q. fc< - ip\ + cp, - a,a,Q * ip, +ipi + OT(O) + o» + o» + o«)Q, (4.5) with a(o) = (OI + os)(aj + 04 + ot + 07) + 04(04 + OT), 4(e) = a» (a j + 04 + at + or) + 0404, e(a) = (a4 + OT)(OI + o» + a» + a«) + 04(0.1 + o8), d{a) = 04(01 + Oj + a» + a«) + Q»Q4. a(o) = a(ai *•» as;ai <-» 04), 6(0) = k(ai *•• OJ;QJ •-» 04), i(a) = c(oi «-> Qjjoa «-» a4), d(a) = d(oi «-• QJJQJ «-» 04). (4.6) Furthermore, these quantities satisfy c(a) = 0(01 <-»ojjos «-• 04; a» <-» 07), d(a) = 4(ai <-» oj;a3 «-» 04:05 «-» 07). (4.7) 4.2 THE ONCE-CROSSED THRES-RUNG LADDER AMPLITUDE The momentum integrations are again straightforward. We express the result in terms of the same functions as those used by Cheng and Wu [12J, A = A(Q) = (oi + 03 + os)(o2 + 04 + oj + 07) + (a* + as)(a: + 07) 2 = (01 +03 + a4 + a5 + ae)(o2 + 04 + Q« + 07) - (<*4 + o») , (4.8) and 2 7 D(o) = D.s + D„u + Dti + Dmm + D\\ + ie, (4.9) with D, = O] 03(02 + o« + o« + a7) + 010407 + 02030s, Z?« = 02040s, 8 D, m atatai, Dm • -(oi + aj + 0| + 04) A + (ai + a» + a» + o4)(aia« + o»oT + O»OT) + at(ai + ai)(ai + a<) + 0401(01 +01 + 01) + OjOr(ai + OI + o,) + 0,04(0, + a,), Dx * -(o, + a* + aT)A. (4.10) Obierve that the A- and /Munition* for the once-eroHed ladder diagram differ from the correiponding one* for the uncrossed ladder diagram. Again the function* of the 4* theory are obtained (or A = m, in which case the retarant nun function become* Dm + Dx- The integration yield* *{DW WW + 2DW[(0< + a,)(s'"'*'*J - S"'W) + (a* + «TKJT W - j"*f *.') - (a, + a, + a, + 07)0"'***? + (01 + °» + a»)»"'*:*? 1 |(a, + Q )(O + a,)(y V" + J*V) + 40(a) I' T 4 - (<*i + as + OB)(<»3 + a4 + Qe + OT)S" V |, (4.11) where fc« = opi + 6p2 + (0504 - a(a7)Q = op'j + bp2 + a5(£»2 + a4 + a» + 07)*?, kb = cp\ + dp'2 + a5(a4 + a,)Q = cpi + dp2 - [{on + at){ai + a3 + as) + 07(04 + at))Q, kc = bpi + 0P2 - (a2at - a«aT)Q = bp\ + opj - a5(aj + a4 + a< + a7)Q, *d = dp\ +cp2- a5{as + a7)Q = dPl +cp2 + [(aj + as)(ai + a, + as) + a«(as + a7)]Q, (4.12) 9 with a(o) • (o» + o7)(oj + a») + (o» + o» + aT)(o4 + a»), Mo) • (oj + oi)(ai + a4) + cti(o4 + »s). c(a) = (a. + as -*• OT)(O4 + o») + o7(a, + o» + a4), <<(Q) = aja* - a<(oi + <>•), 5(a) = o(ai •-* at;«i -•* ~\-,a* — aj), 6(a) = b(ai —> aj;aj -• o4;o« «-«Oi), é(a) = e(oi •-• ujiaj «-• a4;as «-• at), d(a) = rf(oi «-« OjjQj •-« 04;0| •-• 07). (4.13) The last four relations in (4.13) seem to differ from the corresponding ones in (4.6). A closer inspection shows that this is actually not the case, because the latter are invariant under a« <-• 07. 5. The spinor algebra We use the conventions [13] e.,a, = l, e01" = -l, (5.1) 7s -- frV-rV. tl = i- (5-2) Then Tr[7aW-As] = 4se"**. (5.3) We need to reduce a product of five 7 matrices [see eqs. (3.4) and (3.9)). There is no unique way to do this. For our purpose it is convenient to do it in a way that exhibits clearly the symmetries under interchange of various indices. We start out from the identity a T 7°7V = <,« V - S^V + ff°V - if ^ 7r7s , (5-4) which we use first to reduce the product of the three 7 matrices in the middle, and next to the resulting triple products, obtaining thus 7 WlV = 7*W - 9*V + ff"V - ^n"ir7r76]7' f = (J0)°^«' + (J5)-^ ', (5.5) 10 where Jt are the terms linear in fT and A those linear la l»7i- Some calculations are considerably simplified if we in addition divide A and A into parts which are symmetric (5) and antisymmetric (A) under exchange of the outer indices, A - Ji + A*, A - J? + J.x, (5«) where (j/)-*--" = 7VV - rV + /V'l - TVV 1 +7vy -7 w+vis-V - s"V+«"Vi. (J*)-™' = .V*"[ff,V + »J7° - Sa'7rl7», (J*)*™' = i[-/V" + /*«•"' - o^t^^hrTs. (5.7) The symmetries of these expressions are as follows, (jSr»yt< = (jS)./»*. = (JS)*!-,/). = (J-»)««^«, (5.8) (J^)"^1' = -{Jf)'0''10 = -{Jf)"™' = (Jo-4)"7***, (5.9) 1 S < , 5 0 (j/)«»-»<« = (J*)* '" = _(JS )" •'' • = -(J, )""»" , (5.10) (J*)»™' = -(Jf)">-'6a = [J*)alie' = -(Jj*)''^". (5.11) In particular we observe that 7™ ' is unchanged when the order of the indices is reversed, whereas J°fiyl' changes sign. For the crossed diagram, it is mere useful to split up A and A according to their symmetries with respect to the first and third indices, Jo = Jo + A i A = # + Ji", (5.12) where 4 (jsTw< = [_7«s«, + 7y. _ 7./*]fl«^ + [7"S^ + 7 Vis ' - 7 W" + »V*] + ^"V* + S^V], (#)«** = .yy» _ 7V%0« + [yyrf _ 7 V V + TVV-S1Vl, (jM)=^'=i[^^(/,.7°-j°'7r) - (e°s'V + 11 The first three quantities satisfy the symmetry relations, <#)•»*• - (tfr*-*. (j*)'™' = -(#)•"••*. (5.14) whereas JB*' is a remainder with mixed symmetries. Substituting (5.7) and (5.13) into (3.4)and (3.9), one obtains a large number of terms. In order to write the result in a compact form, we introduce some further notation: tf» = fi(p',b*«(P.). #A = a(j>i)7A7»»(P))> VA = %ahxt>(pi), VA S C(w)7V»»(pi). (5.15) and X„0 = UaVff + U*Vp, Xa0 = VaVp - UaVp, Yaff = UaV0 + VaV0, Yafi = UaVff - Ua%- (5-16) Then, for the uncrossed diagram [see eq. (3.4)) we obtain a (Ml)»,,?, = 2[(g/1„g(K, - g^fg,a + g»o9vp)X a — 9pu\Apa + Xtrp) + Siip\X,,ø + X.att) + 9p.o\™pv ~ ^vp) + 9vp\Xtr^ — -*p,a) + Ji»cf\Xn/> + -ApM) ~ ffpff(-*p». "*" -*****) a + 4g»PgvvX a} C + €MpffaJ ^TC^p^oJ ^ CpKpo ^ (7 pi/ffO*p J, W'*'J whereas for the once-crossed diagram [see eq. (3.9)] we obtain (Ml )prp, = 2 [(SnaSi-p + Sp.pS.-i7 - 9pv9pa )X"a — + ^pc(Ap£r Xap) — 9pp\Xvo — Xav) + Sp^C^^P*. ~ ^vp) + 9vp\XatL — -Xfiff) + 9vo{Xpp — Xpit) ~ Spø{Xp.y ~" Xvlt) + 49*<,Xflt] 12 -»»«•*•+ f*«w« Kl"*-'-) - 9**t„* + M'^W" + ?•*)]. (5.18) 8. Numerators When combining the results of eqs. (3.3), (4.4), and (5.17), and using also a = e*/4ir, we obtain -SI-J.'-'-^^-S^AeF 2 1 1 a N 3 (6.1) [ZJ(a) "l' " 4 ori22?(a), „ Nil" + TK7-4.0(aT) #/ with the numerators A" to be given below. Similarly, combining eqs. (3.8), (4.11), and (5.18), we get an identical expression for the amplitude Mt, but with other numerators N, as well as the corresponding functions A(a) and D(a). In order to determine the numerators Nm, JV/j, and JVj, we first note that p?tf„ = 0(m), p[aVa = 0{m), tfVp = 0(m), pfVp = 0(m), and similarly for the Ua and Vø. Furthermore, 0aJ7<» = 0, CfVø = 0, a e Q Va = 0(m), Q Vg = 0{m). Defining now U2=p?Ua, Vl=p*Vg, Us=pfUa, Vi=pfVi, (6.2) we find, to leading order in t/s, k5Ua^cU2, kgVe^W, (6.3) 13 and similarly for 0„ and i'a- 3 i x Since |l| > m , the quantities l/jV| ± f/,Vi and UVx ± O Vj> may be related, using the Khriplovich identities (11] (there is a misprint in |U|): l«(p'i)>MPi)ll*(Pa)P>(pi)l + |fi(p'i)l»i1»«(Pi)|[«(w)j»iT.«'(pi)l = (PI Pr){|fl(p;hx«(pj))I«(p»hAf(pi)) + r»tthS«tø)ll«tahvh»(Fi)l}, (6.4) [«tø W»«(PI)1WI».»I»CI4)I - [c(p'l)M5»(p1)IWpJ)i»i7.v(ri)l x = to •p;){|a(pih »(p1)]|«(BhA»(pi)] - [fi(p',bSi«(p.)P(wbx7»«>(pi)l}. («•«) valid for m2 <£ |1|. These identities require pi + Pa = p'i + Pi- In the above notation, and to leading order in t/a, these results may be written as = isW<->, (6.6) with KJH« = r/*v» ± «7*VA. (6.7) Furtnermore, to leading order in t/a, we have ka • ki, = %(ad + bc)t, K-kj = \(ac + M)J, tk • kc = j(oc + bd)s, kf kd — j(cc + rftf)«, kc-kj=^ad + bc)s. (6.8) 6.1 THE UNCROSSED THREE-RUNG LADDER AMPLITUDE The numerators of eq. (6.1) for the uncrossed diagram are found to be Nm = 2j2[aaccW(+) + (aa + bb)(cc + dd)Wi~'>], +) Nu = -4«{[(QI + a3 + as + a»)ao + (a2 + o4 + a, + a7)cc + 2a,(ac + åc)}W<- + [4(a! + 03 + a5 + o«)(oa + bib) + 4(02 + a4 + a» + a?)(cc + dd) - - aB(od + id + be + be - ac - åc - bd - id)]^ '}, JVj = 8[(A + 6al)WM + 4(4A + 6a?)W(->], (6.9) 14 with A, a, b, e, d, å, i, e, and d given by eqs. (4.1) and (4.6). We have bere omilled the Y terms in eq. (5.17), since they do not contribute to the unpoUriwd cross section. 6.2 THE ONCE-CROSSED THREE-RUNG LADDER AMPLITUDE For the once-crossed diagram we similarly find 1 l+) 1 Nn, = 2a [aå(ec + ay*)W + floeeM' -'), Nn = 4J{[(QJ + aj)(ad- åe) + (at +at)(åå-ac) l+) + 4(Q, + O3 + as)a5 - (oj + at + a, + a7){cc + dd)]W + |-(QJ + aT)(6c + 2oc) - (a4 + a8)(6c + 2oc) 1 + (a, + a, + a5)(aa + 66) - 2(a2 + at + ae + a^^W^' }, ( + ) <_) N, = 16[(a2 + a7)(a4 + a«) - 2(a, + a, + a6)(aj + at + a, + a7)](W + W ) <+) ( = 16|3(o2 + aT)(a« + a.) - 2A](W + W ->). (6.10) It should be noted, however, that here the abbreviations A, a, 6, c, d, a, 6, c, and d are given by eqs. (4.8) and (4.13). We have here omitted the Y and Y terms in eq. (5.18), since they do not contribute to the unpolarized cross section. 7. Summing the contributions from the six diagrams The sum of the six amplitudes corresponding to the diagrams in fig. 1 can be expressed in terms of the amplitudes for diagrams la and lb. We shall demonstrate how it is done. The sum is M^>= £ Mm-ie* £ (AC)„„„(Af?)'»""', (7.1) •=«,...,/ >=o,...,/ where in the last step terms multiplied by m are neglected. In this approximation, the general structure of Al'3""' is given i>y eq. (2.4), i.e., M<3 ) = Foofi p 7 u pi w ' '' ^r"[ ( i) * ( ) '(ftfrfXpi) + f55«(pi)7 75«(pi) S(p2)7„75D(p'2) + iKi^p'i hd«(pi) «(P2 hrfMti) + »'*s!r»(pi )T^75«(PI ) »(PI h*v{P2)] • (7.2) For the contributions from the individual diagrams la to If we have identical decomposi tions but with F-fnnctions with suffixes a to f. 15 Feynman diagram Id goes over into diagram la if we twist the positron line around. The contributions from these two diagrams are therefore related. The spin factor of dia gram la is 1 > (M2)**„ = |S(p;h T,,'Ai.7' »(j>i)]|0(Pjba7#7lfl7»7-»«'(P»)) = [Knuppa) + K^i-rvØpaW^apØcn) + K^(apØ where K(0) denotes the part corresponding to the J(0) part of eq. (5,5) and similarly for K(t). The momentum-space factor of diagram la, (4.4), we write symbolically as (M.THPi.PaiPi.Pi) = jVq\(Vx ~ «.)"(pi + ft)"(P» + iiViPi ~ «»)']• (7-4) Let us now inspect diagram Id. If we interchange p and a, as compared with diagram la, we get for the spin factor, (MJ W = [tf&(7"/3/*a-) + K^uØ^WK^^pa) + K^yaØpa)}. (7.5) The order of the indices in the u-factor is the same as for diagram la. The positron line has been twisted around, however, and therefore the indices in the v-factor are in the reverse order as compared with (7.3). The order of the 7-matrices can be reversed, employing the symmetries of eqs. (5.8)-(5.11). This results ia WW = [K^vØpa) + K^uØ^a^K^apØcy) - Kfa(ap0*-r)], (7.6) where the orderings of the indices are now identical to those in eq. (7.3). For the momen tum-space factor we have (MSrHPi^Pi.Pi) = /P?[(PI - 9I)"(P'I + 92)"(P= - 9i)'(P2 + »)'] ^(MSr^bu-fcp^-p,), (7.7) as it should, since diagram d is obtained from diagram a by twisting around the positron Une. Indices in eqs. (7.7) and (7.4) are identically ordered. Under the transformation P2 <-• —P2, we have » <-» 11, with t unchanged (see eq. (2.8)]. It follows that with the decompositions of Ma and Mi as in eq. (7.2), ;ft W) = fft W), Fl?(s,t) = -F£\u,t). (7.8) 16 Diagrams le, le and If are all related to diagram lb by twisting around the positron (or electron) line or both electron and positron lines. The discussion is similar to the one given above for the diagram Id, and we do not repeat it. The results are fi?(>, t) - **•"<*. 0, rtfl; ') = fii\., t), riPl:t) = r£\*,t), *&"(*,t) = -fj*»(«, i). (7.9) The Foo and F55 for the sum of the six diagrams, eq. (7.2), become Foo = [F&\.,t) + 1#W)] + 2 [*£>(.,«) + F&\u,i)] F» = [*•¥(*.*) - tftW)] + 2[*ff('.«) - *ilW)]. (7.10) We are not interested in the contributions .Fos and Fso since, in the absence of contributions from Z°-exchange, they cancel in the unpolarized cross section. 8. Summary and outlook We have discussed a set of sixth-order QED Feynman diagrams contributing to Bhabha scattering. We are interested in the limit a 3> |i| » m2 and have therefore neglected contri butions of relative size |t|/s and m!/«. Furthermore, we have focussed on the unpolarized cross section. In this situation there are only four analytic functions that need to be cal culated. The reduction in the number of functions is a consequence of the QED crossing symmetry. The four functions are all of the form F(,,t)=^..jrV-.<*M(i-£«ox^ a For two of the functions, F0 0(s,i) and Fgs(s,t), the numerators Mm, Mn, and Mi are obtained from eq. (6.9) by putting W(+) = W(_) = 1 for the function F^(s,t) and H"( + > = —W'(~) = 1 for the function F^s(t,t). The denominator functions A(a) and D(a) are given 6 by eqs. (4.1) and (4.2). For the other two functions, Fj0(a,i) and F5 5(a,t), the numerators Mm, Mn, and Mi are obtained from eq. (6.10) by putting W*+ ' = W'-' = 1 for the - 6 function F0'0(s,<) and W<+> = -PT< ' = 1 for the function F5 5(*,i). The denominator functions A(a) and D(a) are given by eqs. (4.8) and (4.9). The physically interesting quantity is the dominant 4r(loga/m2)* contribution. Un fortunately, it is not obtained by simply determining the dominant term for each of the 17 diagram*. We expect substantial cancellations between the various contributions. This is already seen for the fourth order QED Feynman diagrams. In this order there are two ladder-like diagrams, the box diagram and the crossed box diagram. The dominant summed contributions from these two diagrams is, relative to the one-photon exchange term, ^\H(J + «j log JI\ {a(pihMpi)s(ft)7xv(p',) +«(p[hSa(Pi)''(P»hi7»»(pi)}- (8.2) If we look at each term separately the picture is very different. Each of them contains a contribution proportional to log^-t/m2), which however cancels in the sum. Furthermore, one diagram has a term log(-m2/»+ie)log(-(/A') while the other one has a corresponding term log(-u/m2)Iog(-t/As). These two add up to the result oi (8.2). Thus, it follows that it is not sufficient to determine the dominant contribution from a single Feynman diagram. The cancellation encountered in fourth order QED has been found by Cheng and Wu [12] also for the set of sixth-order Feynman diagrams that we are interested in, but in the ^'-theory. Consequently, we expect that a physically relevant determination of the contribution of our set of sixth-order Feynman diagrams to Bhabha scattering must neces sarily involve the determination also of the subdominant contributions for each individual diagram. These results will be presented elsewhere [14). Acknowledgments A major part of this research was done while G. F. was a visiting scientist supported by the University of Bergen. This research has also been supported by the Norwegian Research Council for Science and the Humanities. Appendix A The integrations over the loop momenta in Mq may conveniently be performed using a trick similar to one used by Karplus and Kroll [15]: -* LiT^fi. ._ L (A.l) -l>-dv? (p ± v)2 - m2 + x + it This allows us to express fermion propagators in terms of scalar propagators. With U being linear combinations of pi, pa, pi, pi, 9i, and qi, we write, Miy*-= / *«» ** '? 3 l' It 2 1 ; J {»« (2ir)* I2 - m2 + it ll~m2 + it ll-m2+ it I2 - mm + it 18 1 1 1 Il - A» + it Pt - A» + it J? - A' + w a a a a dxidxtdxi if*». Øvip Bvtv Bvzp Bvtø is;f .d*9 i ' __! I I 1 (A.2) I| - A» + « IJ - A* + « I? - A* + iei.4_0 The integrations over q< are now reduced to scalar integrals. After Feynman parameteri zation and integration over the g< we obtain, 2 i f. . . . r a e a a (M")" dvtc (A.3) where 2 2 I> = D.s + C„u + D,t + Dmm + D\\ + 2[ai(fc„ •t)i) + 02(fcf«2) + aj(fcc -usj + a*^ -vt) 3 4 4 -^2 51 <*i<*i<^iiVi-Vj]+\'J2 + 2^i- |«i!ffM"fcc'*J + «us"'*»*? + W^KK , + 02Ss "fc;*;+ww + OMS"*;*»" ']}' (A.5) The &*, fcj, if, fcj and Oj,- depend on the topology of the diagram under consideration. 19 Reference» (li see, e.g., M. Caffo and E. Remiddi, in Z Posies at LBPl, CERN 89-08 (1989), p. 17) [2] see, e.g., "The DELPHI detector at LEP", DELPHI Collaboration, P. Aarnio et al., CERN/EP 90-5, Nucl. Instr. and Methods, to be published (3| F. A. Berends, W. L. van Neerven and G. J. H. Burgers, Nud. Phys. B297 (1988) 429; E. A. Kuraev and V. S. Fadin, Sov. J. Nucl. Phys. 41 (1985) 466; O. Nicrosini and L. TrenUdue, Phys. Lett. B196 (1987) 551 [4] see, e.g., R. Kleiss, in Proc. of the Ringberg Workshop on "Electroweak Radiative Corrections", April 3-7,1989, Ringberg Castle, Germany (also CERN-TH.5439/89); S. Jadach and B. F. L. Ward, Phys. Rev. D40 (1989) 3582 [5] P. Osland and T. T. Wu, Nucl. Phys. B288 (1987) 77, 95 [6| M. C. Bergrre and L. Szymanowski, Phys. Lett. B237 (1990) 503 [7| C. Newton, Ph. D. thesis, Harvard University, 1990, and Journal of Math. Physics, in press [8] H. J. Bhabha, Proe. Roy. Soc. A154 (1936) 195 [9] R. W. Brown, V. K. Cung, K. 0. Mikaelian and E. A. Paschos, Phys. Lett. 43B (1973) 403 [10] A. B. Kraemmer and B. Lautrup, Nucl. Phys. B95 (1975) 380 (11] I. B. Khriplovich, Sov. J. Nucl. Phys. 17 (1973) 298 [12] H. Cheng and T. T. Wu, Expanding Protons: Scattering at High Energies, (The MIT Press, Cambridge, Massachusetts, 1987) [13] J. D. Bjorken and S. D. Drell, Relativistie Quantum Fields, (McGraw-Hill, New York, 1965) [14] K. S. Bjørkevoll, G. Faldt and P. Osland, to be published [15] R. Karplus and N. M. Kroll, Phys. Rev. 77 (1950) 536 20 Figure captions Fig. 1. The six three-rung ladder-like Feynman diagrams considered. Fig. 2. Choice of momentum variables for the uncrossed (a) and once-crossed (b) three- rung ladder Feynman diagrams. Fig. 3. Choice of Feynman parameters for the uncrossed (a) and once-crossed (A) three rung ladder Feynman diagrams. 21 x x Fig. 1 22 Pl Pi-9i Pi + 92 Pi »— > i *- 9i , Pl - Pi x, 92 -9l - 92 W -P2 -P2 - 9i -pi + 92 -Pa Pl + 92 Pi * z>—*- (») Pi ~ 91 -P2 -91-92 -P2 Fig. 2 23 1 Pl <*1 <*3 P ! «5 a» on (a) aj -P2 on -Å Pi «i a2 as W a an ~P2 3 -ri Fig. 3 24 Paper II Onivaralty of Borgos, Doportaoat of Pkyiica Seioatific/Tochaicol Roport lo. 1M3-01 I8SI 0803-3690 TWO-LOOP LADDER-DIAGRAM CONTRIBUTIONS TO BHABHA SCATTERING II. ASYMPTOTIC RESULTS FOR HIGH ENERGIES Knut Steinar Bjørkevoll Department of Physics, University of Bergen, AlUgt. 55 N-5007 Bergen, Norway Goran Faldt Gustaf Werners Institut, Box 535 S-751 tl Uppsala, Sweden and Per Osland Department of Physics, University of Bergen, Allcgt. 55 N-5007 Bergen, Norway ABSTRACT We determine the dominant contribution to the Bhabha-scattering amplitude arising from the six two-loop ladder-like diagrams in the limit s 2> \t\ S> ro2 S> A2, where A is the photon mass. A method is developed that allows for the simultaneous presence of these four different scales. While each individual diagram contributes four logarithmic factor!; to the amplitude, the two leading orders cancel, leaving only two such factors in the sum. They are precisely the logarithmic terms associated with two infrared photons. Integrals are also given for the terms that could possibly give single logarithms, but their sum is expected to vanish. 1 1. Introduction The Feynman amplitudes presented in ref. [1|, determine a part of the amplitude for Bhabha scattering at small angles, that is of O(a') as compared with the lowest order ones. Here, we present an evaluation of these amplitudes. On the one hand, it is known from 4s theory |2j that the leading terms from each diagram contain /our logarithmic factors, but the arguments of these logarithms could a priori be any dimen "onless function of the kinematical variables t, t, mJ, and A1, where A is the photon mass that we introduce as an infrared regulator. On the other hand, we know that in QED (like in other gauge theories) there are cancellations between the contributions from different diagrams. For example, at the level of one-loop ladder-like diagrams, each diagram contributes two logarithmic factors, whereas the sum has only one such factor, namely the one containing the infrared regulator, m(^Vl'l). We expect the leading logarithms to cancel also for the two-loop contributions. This turns out to be the case. Furthermore, all term» involving time logarithm» eanctl. We are not aware of any argument explaining this fact. Thus, for the sum over all six diagr- TUB, we are left with only two logarithmic factors. They are the logarithms associated with two infrared photons. If v>e expand the two-loop ladder amplitude as in eq. (I.Y.2) of paper I (we refer to equations from ref. [1] by the prefix 'I.') l r M = — [i ooii(p'1)7'''»(Pi) »(P2)7««(pi) + ftsttfølfr^uføi) "(Pab^svføi) + iKsHp'ihMpi) 5(P2h*7s»(p!>) + »-Fs7s(pib*75«(pi) *(pihMti)]> (1.1) then, for A2 ^C m2 -C \t\ \^j+ single logs + For comparison, the lowest-order term in the amplitude is the one-photon exchange amplitude ^(17) = -iE^tøJy^) 6(K)7„„(pi). (1.3) Thus, ignoring terms involving 75, we have •ML^ = M(l7){-y [W (f)+ singlelogs] +o(;)}- (1-4) 3. Structure of the amplitude* A* indicated by eq. (1.7.10), the full amplitude corresponding to the sum of the six diagrams, can be constructed from just the two amplitude* -M, and Aft, with suitable substitutions among the kinematical variable* t, t and u. For the sum of the contribution* from all six diagrams, we have from (1.7.10), Foo = (fftWj + fftW)] +2[j4'W) + /i(o'W)]- (2.1) Thus, it suffices to consider the two amplitudes Ma and Mt. For the uncrossed ladder diagram, we found the amplitude to be given by the following integral [1), where 2 A(a) = A" = (ai + as + a5 + a8)(a2 +at+a,+ a7) - a ,, (2.3) 2 2 D(a) = D*(a) = D,i + Dtt + Dmm + £>AA + te, (2.4) with D, = aia3(a2 + a4 + at + a7) + «2a4(ai + aj + as + a«) + a«(aia4 + a3aj), Dt = Q5asa7, Dm = -(a! + a2 + as + a4)A + (ai + as)(ai + a4)(a5 + a7) + (ai + a2 + as + a4)(a507 + asa« + 0(0:7), Dx = -(a5 + a8 + a7)A. (2.5) The numerators Niu, Nu, and Ni are given by equation (1.6.9). Similarly, the amplitude Mb for the once-crossed diagram is given by an expression identical to (2.2), but with numerators given by eq. (1.6.10), and A1 and Db(a) given by eqs. (1.4.8) and (1.4.9), respectively, as" A* = (ai + as + on)(ai +cn + ae + a7) + (a4 + at)(ai + a7) 2 = (ai + as + a4 + a5 + a«)(a2 + a4 + a, + a7) - (a, + a,) , (2.6) * Except where confusion might arise, we shall leave out the indices a and 6 used here to distinguish quantities referring to the two different diagrams. and 3 />*(«>) = D,M + D.u + 0,1 + Dnm + D\X* + ie, (2.7) with D„ D«, i>„ />„,, and D\ given by eq. (1.4.10). In order to extract the amplitude Fon from the integral* appearing in eq. (2.2), we (+) (_) substitute W = W = 1 in the numerators JV///, Njt, and Nj [see (1.6.9) and (1.6.10)], and write *lo=[2/«/+J J//+$//], (2.8) with 1 , w..«) = / .../ -.,...^*(i-|;ay)x^[-5fI /^-jf-"jffc*"-^«1-g^A(ifej- (29) Here JV/// = iVm|Hr(+)=H,(.)=], with e. g., JV/// = 2a2[oocc + (oo + bb){cc + dd)}, (2.11) and the a, 2, etc. bilinear expressions in the a's, independent of the kinematical variables. Among the three terms in eq. (2.8), the first one will dominate. A major part of the present paper is therefore devoted to a study of ////(a,t) and ijjj(a,t). In sect. 9 we shall briefly discuss the other integrals Iji(s}t) and Ii(s,i). The amplitudes (2.9) depend on four kinematical variables, », t, m2, and V, and thus potentially have a very rich structure. We are interested in the combined limits Ai » 4 - °°- (2-12) and - - 0. (2.13) However, we do not know a priori whether these two limits commute. It is not difficult to imagine functions that have different limits, depending on which limit is taken first. Consider, for example log(j/mJ) for * —• oo, (,m, A fixed, (2.14) *(*•» log(l«l/A') for A-0, »,t,m fixed. We therefore have to be very careful with the order of these limits, and shall evaluate the integrals using a Mellin transform that yields a correct result even when the relative magnitudes of the different logarithms are comparable. 3. General structure of//// In order not to overburden the notation, we extract a factors2 from Nm [cf. eq. (2.11)] J and write the first equation in (2.9) as ////(s,t) = s2/(*,*), (3.1) where for the a-diagram, with N- = AT?///*2 = 2[aacc + (aa + bb)(cc + dd)]. (3.3) As A —» 0, we expect the dominant terms in I(s,t) to behave like (l/«22) multiplied by four logarithms. (Among these logarithms, the dependence on A is at most like log2 A.) These dominant terms can be extracted by a Mellin transform [3] (see Appendix A), /(o = rii-<+i i(M,t). (3.4) Jo More specifically, we rescale the kinematical variables as |i| = l, m2=a*-, A2=s"\ (3.5) 5 with nji < Vm < 0, (3.6) and perform the «-integration as follows |2] (see also the factorization formula given in Appendix A), f°° ds*-<+1 - x f(2 - < - z, - nmzs - »;A(3 - zi - zt - zs)) s+ x (D. +ie)—(-D, + U)—(Dm +ic)-"(Dx + te)- "—, (3.7) where all contours of integration run parallel to the imaginary axis such that Re(z()>0, » = 1.2,3, Re (3 - zj - z2 - Zj) > 0. (3.8) Since the dominant amplitude behaves as l/(s*t), it is convenient to introduce new variables of integration, defined by *i = 2-f»i, *2 = 1 - Cjfe, zs = Cya. (3.9) Furthermore, with 3/4 =3/i + y2-»s- (3.10) we have RejfaX), Re jfc > 0. (3.11) Invoking the result (3.7), and using the variables (3.9), we write HO = jT'... jf2*n...*»T«(l - X>)^'(<,«)• (3.12) with x r(i + C»j)r[i + C(yi + »2 - »)]«[i -si+ «?mys + I?A(»I + y2 - »)] 2+ 1+ x (D, +«)- <»'(-I>, + ie)- <*>(Dm +i€)-<**{Dx +j£)-«iri+»»-».). (3.13) 6 The integration over jij yields ,+t x (£>, + «)-»+<»•(-D, + U)- "(Dm + it)-l*'(Dx + »€)-<»', (3.14) with J/j and j/4 given in terms of yt and J/J as y» = [-1 +(-TA + l)jfi -IMft], 3/4 = (l + tøm - Itøl + VmSfe]- (3.15) Vm ~ »?A Since we are only interested in the structure at small f, we expand the product of T- functions, r(2-cjfi)r(i-cs2)r(i+tø,)r(i+cift) = (i-tø) i + ^Hri+s/f+yJ+^+otO (3.16) Substituting back into eq. (3.12), neglecting terms of third order in £, we obtain with Ac.-».»)-/l...jf-«....^Ki-i:^^ 2+ +(v -4 x (U, + i£)- ^(--D« + U)-' '(Dm + ie)-<*'(Dx + it) *. (3.18) If we substitute for N(a) following (2.11) and (1.4.6) the appropriate expression for the a-diagram, or the appropriate one for the 6-diagram, we see that the integrand is homogeneous in the a, of degree -7, and may thus use a theorem of Cheng and Wu [3] (see Appendix B) to transfer the ^-function constraint to a Bubset of the variables of integration, ^((;Vi>Va) = / ••• / dot!... da I ... I da da,da 6(1 - o - a, - a ) Jo Jo t Jo Jo s 7 6 7 2+ 1+ x J^(D. + U)- <»(-Dt +it)- <»(Dm +ic)-<»{Dx +ie)-<". (3.19) It follow* immediately from eq. (2.5) that Dx =-[at + a, + a,)A(a) = -A(o). (3.20) At this point we extract a few phase factors. Since Dt > 0 and Dx = -A < 0 for all six diagrams, whereas Dm < 0 for diagram a, we write 1+ 1 C,, i = _ e-"<». (/>, _ it)-+("(-Dm - it)"'» A-<»\ (3.21) Next, inspired by the procedure of ref. [3], we rescale the Feynman parameters asso ciated with the fermion lines, Qi = pi, at = p(l - x), as = p'y, o4 = p'(l - y), (3.22) with 0 We repeat this rescaling for p and p', p = TZ, p' = r(l - z), (3.25) with 0<*<1, 0 < T < oo, (3.26) and pp'dpdp' = r*z(l - z)drdz. (3.27) 2 2 Substituting into D,(a) and Dm(a), we extract the factors r z(l - z) and -r present in D,(a) and Dm(a) for all six diagrams (for more details, see appendices D and E), D.(a)s r'z(l-z)Q.(a), ! Dm(a) = -r Sm(a), (3.28) 8 where for the a-diagram, e. g-, C;(o) = 0,5(1 - x)(l -y) + a»+ 07IV + r(«(l - x) + (1 - x)y(l - y)|, (3.29) and 1 Q^c) = a,|z(l - *) + (1 - z)(l - »)]" + a, + a7[zx + (1 - z)y} + T\ZX + (1 - z)y\[z{l - x) + (1 - z)[l - y)}. (3.30) Hence, eq. (3.19) can be expressed as ix Ittiyuyi) = -e- <" f°°dr I ix f dy f dz f ... f daida,da7S(X - os - a, - a7) Jo JO JO JO JO JO s x r-i+2«vi-»3) [2(! _ z)]-i+<*.J\r(o)A(o)- -<»* i+t 1+ _CW x (C. + U)- »(Dt - «)- «"(Gm - M) , (3-31) where we have also used the result (3.20). We aim for a determination of the leading powers of logarithms, of any argument. These are associated with the terms that are most singular, as £ —» 0. An inspection of eq. (3.31) reveals that the integrations over r and z both yield factors of l/£. Similarly, the integrations over as, a«, and at yield at most two powers of l/£. There will be at most a power 1/C from the integrations over x and y. Thus, if we imagine an expansion, /(Ci»,») = ^^ + ^f^1 + ^f^1 + less singular terms, (3.32) then ^5(3/1,j/2), .44(2/1,312) and As(yi,yi) are the coefficients that are required. They are related to four, three and two powers of logarithms. The terms .42(1(1,1/2) and .4i (2/1,2/2) cannot, å priori, be determined since we have neglected terms of 0(£s) in the the expansion (3.16). However, we shall later see that when we sum the contributions from all six diagrams, the leading terms .45(2/1,2/2) and JU(y 1,112) cancel. This cancellation takes place before 2/1 and 2/2 are integrated over. Hence, eq. (3.17) can indeed be used to determine also ^2(2/112/2) an<* -4i(Vli!/2)- We shall refrain from doing so. In the present investigation we shall only determine the dominant contribution and that will, as explained, turn out to be At(yi ,2/2). 4. Integrating over r and z We rewrite eq. (3.31), for an arbitrary diagram, as /(<;»,») = /"* / dzr-1+2«»'-w) [z(l - z)]-1+<»i *(,,»), (4.1) Jo Jo 9 with E{r,t) = -e-'"**' f dx f dy f ... f da, x JV(a)A(a)-s-c»«(e, + u)-a+f»'(2J, - »e)"1+C,*(Qm - te)-4"». (4-2) In order for the integrals over r and z to be convergent, we must require Re yi > 0, Re (jn - w) > 0. (4.3) Invoking the expression (3.15) for gft, the second inequality can be written as Re3/i(|i7m| + l) + Reyi|ijAl^l. (4.4) Furthermore, in order for the integrations over QJ, as, and 07 to converge, we must require Re J/2 > 0. (4.5) The integrations over r and 2 are discussed in appendix C. Combining those results, we obtain, * + 7T f T^1) ~ -fft0'1)] + 7-1°° T^'l + 'l'-1» + less singular terms. (4.6) This expression can be simplified. We note that the factors in the original integrand eq. (3.31), JV, A, C, Dt and Qm, are all symmetric under the simultaneous substitutions oti «-» as, c<2 +~* 0J4, and as +-+ 017, or equivalently z «-* 1 — z, x «-» y, Q8 •-» a7. (4.7) Hence, 10 The function H{T,I), eq. (4.2), simplifies considerably when one or both of itt argu ments vanish. From the formulas in appendix D and E it follows that Dt(r = 0 and/or * = 0) = 040407, (4.9) for all diagrams. Furthermore, N{a)\ ' 4A.(o), (4.10) for all six diagrams. Here, AO(Q) = a5Qj + a«07 + OT<>S- (4.11) When z = 0, on the other hand, N(a) TQN{T;Z=0) (4.12) Ma) A(r;z = 0) \(ay where QN{Z = 0) and A(z = 0) depend on the diagram considered. Since the second term on the right-hand side is multiplied by r, the two integrals in eq. (4.8) involving QN(Z = 0), can be added together to an integral over r from 0 to infinity. We therefore rewrite eq. (4.6) as a sum over five terms HCyuVi) =-t-"'*1 f ••• / rfa da,dQ *(l-a -a, -a-,) Ja Jo 6 7 5 5 1+ x (a5a,a7)- <»' £*»(C;aj, (4.13) where the different .?*(£; a) are given by x [fi.(r = 0;z = 0) + «]-2+Cl"[Qm(r = 0;z = O)]"**» (4.14) x {[C.(r = 0;z)]-^<"[Qm{r = 0;z)]-<" - [Q.(r = 0;z = 0)]-,+<»»[a«(r = 0;* = 0)]"'»}, (4.15) , , ^ci-).^A,(.)jr*jr*jf * 11 ,+ x {|0.(r;i = 0)]- <»MGm(r;J = 0)]-<»[A.(r;, = 0)|-«»« 7 - |Q.(r = 0;* = 0))- *<*>\Qmlr = 0;J = 0)]"<»»|A,(r = 0;* = 0)J-<"}, (4-16) u<''a)=^A<,(o)/1 */ */" £ (c'(r:*=or,+<'" x [em(r;» = 0)l-<»[A(rj2 = 0)]"<»\ (4.17) CVi Jo Jo Jo A(r;« =0) x |Q.(r;i = 0)]-J+<»>[Qm(ri* = 0)]-<»[A(r;* = 0))"<»«. (4.18) We are interested in the terms which are most singular in the limit ( —» 0. As suggested by the prefactors in the equations above, Fi((;a) is in this limit the dominant term. It will be evaluated in the next section. We shall see that when we add the contributions from all six diagrams, the dominant terms in Ti((; a) cancel. Consequently it is important to check also the remaining Fi((\ a) to T${(,\a). There are also cancellations in those, as will be seen in sect. 6. S. Summing dominant terms and integrating over x and y We consider first the contribution from the first term in (4.8), referred to as Ti, and defined in eq. (4.14). Since cancellations occur when we sum over the six diagrams, we are primarily interested in the sum. Anticipating the result after integrating over z, we define •FiK; •*)=,,,, , w jAolal'-'-j-, (5.1) Cs(i- Here, the contributions from the individual diagrams are labelled by the the corresponding superscripts. We start with diagrams a and d, where (cf. appendices D and E) e;(r = 0;z = 0) = -Q;(r = 0;z = 0) = x[aTy - «s(l - y)] + a, + a6(l - y), (S.3) eS,(r = 0;z = 0) = ei(r = 0;* = 0) 2 = a5(l-y) +ae+a7y*. (5.4) 12 It follows from the definition of ?i((;a), eq. (4.14), that T* = «'"<»> *?. (5.5) The integration over z in eq. (4.14) it straightforward. We encounter x*= f MQ"i(r = 0; z = 0) + «]-,+<»' i - C»i «T» - 015(1 - y) +»« x {(a, + OTy]-I+<*' - [a, + ai(l - »)l-l+<*«}. (5.6) The remaining factors in the integrands of eqs. (4.13) and (4.14) are unchanged under the simultaneous interchange 05 <-» aj and y *•» (1 - y). We may therefore combine the two terms in eq. (5.6), whereby the imaginary part drops out, giving I+ Ia, + a7y]- <»'. (5.7) ,ymm 1-C»i aTV-Ml-!/) Invoking eq. (5.5), the terms Ja and Jd of eqs. (5.1) and (5.2) become J° = / *y t-n—> l°» + aiy]~1+ 7o <»7y - o5(l - y) Jd = ei»C„ j. (g g) For diagrams 6 and e we have (cf. appendices D and E) Qj(r = 0;z = 0) = -e;(r = 0,z = 0) = z{ai + a5(l - y)] + Q«y - a5(l - y), (5.10) Qj,(r = 0;z = 0) = QJ,(r = 0i» = 0) 2 = as(l-y)*+aty + a,. (5.11) Since 1-Ql + »e]-2+Cl" = e"<»> [Si - ie]-2+Cvi, (5.12) it follows from the definition of Fi(ba), eq. (4.14), that jrc = ei 13 The integration over x i» again straightforward, I* = J 1 - Cvi OT + <*s(l - y) ,+ + x {[a.y + a7)- <»' - |<»eV - a,(l - y) + tt)-' <»>}. (5.14) We now change integration variables, interchanging a« and a7. As a result, Qj^fr = 0;z = 0) becomes CJ,(r = 0;z = 0), while no other factor in the integrands of equations (4.13) and (4.14) is affected. Adding the identical contributions from diagram 4 and c, and bringing the .'«suit into the form of eqs. (5.1) and (5.2), we get h s+ J + J< = f dy ' {[a, + a7y]- <»' J0 a, + as(l - y) I+ 2 2 - [a7y - as(l - y) + te]- <»'}[a5(l - y) + a, + a7y ]-<». (5.15) By eqs. (5.13) and (E.6), we get J' + J' = e'^y [J" + J%^_t. (5.16) Next, we calculate the sums JV> and J<2> of eq. (5.2). Noting that 1 | * ai+aw (517) a7y-«s(l-y) a«+a5(l-y) [a7y - a5(l-y)]|ae + a5(l - y)]' we obtain J(1) = /' *lr h w j. n « ["• + a™l°" Jo [ [<*7y ~ 05(1 - y)][c [a7y - a6(l - y) + ie][ae + a^l - y)\ J 2 x [a5(l - yf + «« + «ry ]"^, (5.18) J ftl 1) jt > = e" j( |,__f. (5.19) When f —» 0, the sum of jl1' and jW vanishes identically. This means that when the contributions from the six diagrams are added together, the dominant term vanishes identically. Hence, J is proportional to C and ^i(C;a) to l/£. Before continuing, it is convenient to rearrange sUghtly the expressions for j'1' and ji'). We observe that in the first term of J^ we have a principal value factor while in 14 the second term the tame factor appear! with the it prescription. We would like to iiuert the if prescription alto into the first tenn, bui mutt then add a correction term V tj J ) I [a J|-a»(l - y) a y-ai -I'M: T 7 t (1-y)+ « <,l : x [a, + aTv] [a,(l - y)' + a, + aTy*]-<» A,(cQ ]«'•-»> (5.20) AO(Q) aj + ai As a result of this procedure, we may write (5.21) jV> = ,."<». (j-0>)| ^ _«j), (5.22) where dy-, Jo cny - a»(l - y) + ie][ac + as(l - »)] 2 2 xlas(l-y) +all+ajy ]-<*>. (5.23) For small values of (, we may expand the two terms inside the curly bracket of eq. (5.23), and also replace the last factor by unity. Hence, to leading order in £, we obtain Jm = (yifd 1 y any - asfl -y)+»e][a, + a (l - y)} Jo 6 x {log(a6 + any) - log[aTy - as(l - y) + »e]}. (5.24) The integration over y is discussed in appendix F. With 1 1 ' = Re f dy Jo <*7jr - 05(1 - y) + « ««+ 05(1 - y) x {log(a« + a7y) - log[aTt/ - a5(l - y) + ie}}, (5.25) we obtain, when symmetrizing in the variables as and 07, -2 *--d5{-T+'K2S)-(-w)] + u,(, tp \ / tp u V (a$ + at)(at + a7) J \(a5 + a,)(at + a7) J } (5.26) 15 The subscript 'symrn' indicate* that we have neglected terms that are odd under inter change of as and en- Such term» disappear when integrating over those variables. We rewrite this expression as where , -Ii1/ 1_J M^l iVW, *P iV (5-28) 3 \ (a5+a,)(a,+aT)/ \(»i + a,){at + a7) J ' ' This function appears also in the 3 limit [4J. Thus, adding the contributions from SJ, eq. (5.20), and J^"\ eq. (5.24), the result of the x and y integration is simply j=-£fe[fl(a)-T]+o«2>- <«•> Substituting this into eq. (5.1), we have (5.30) C(i-C»i)(»i -w) «(«)• 6. Summing subdominant terms In the previous section we found that the leading l/£2 singularity in .?i(£;c<) cancelled when summed over the six diagrams. This implies that the order of ^i(f; a) is the same as the apparent order of Ft(£; a), • • •, ^j(C;<*)• In this section we shall show that a similar cancellation takes place for the latter amplitudes too, so that they remain of lower order. 6.1 Tt The expression for Fi((; a) is given by eq. (4.15). Since Q,(T = 0; z) = Q,(T = 0; z = 0), and independent of z, this factor can be taken outside the ^-integration, **( x/ —{[Qm(r = 0;JS)]-«»*-[a,.(r = 0;z = 0)]-«w}. (6.1) Jo z 16 For diagrams a and d we have (cf. appendices D and E) G:(r = 0;z = 0) = -Qi(r = 0;* = 0) = at(l - i)(l -y) + a,+ ajxy, 1 Qm{r = 0;z) = a6|*(l - »)+(1 - *)(1 - v)} +<*, + a,[tx + (1 - z)y\', Ci(r = 0;.) = a,tøl -*)-(!- z)(l - y)]2 + a,[l - 4*(1 - *)] + a7[« - (1 - *)!/]'• (6.2) For diagrams b and e we have Q;(r = O;* = 0) = -Q;(r = 0;z = 0) = -<*s(l - x)(l - !/) + a»!/ + a7x, 1 2 Q^r = 0; 2) = a5[z(l - x) - (1 - z)(l - y)} + c,[z + (1 - z)y] 2 + a7[zx + (l-*)] , C^(r = 0; z) = a»[z(l - x) + (1 - z)(l - j/)]2 + a,[z - (1 - z)y]2 2 + a7[zx-(l-z)] . (6.3) An inspection of the functions Qm and Q, suggests that there will be no further simpli fication when the six diagrams are added together. The lead'ng term in an expansion in powers of £ becomes ^(C;a) = -=^-[Ao(a)]1-^« t dx f1dy[Q.(r = Q;z = 0)}-' Vi - J/3 Jo Jo x „ f dz Qm(r = 0;z) X log M /o T Qm(r = 0;, = 0)' < > which is to be summed for the six diagrams. Comparing this expression with ^Fi((;a), eq. (5.30), we note that the 1/C factor is absent, making J^CCi**) of lower order. 6.2 F3 According to eq. (4.16) the term F*(C; a) is given by *i(C;") = ^-A,(a) /' dx /' dy f - kVi Jo Jo Jo r 2+ x {[Q.(T;Z = 0T ^[Qm(r;z = 0)]-<»[Ao(i-;* = 0)]"<»« - [Q.(r = 0;* = 0)]-2+<»'[Sm(r = Q;z = 0)]-<»MAo(i- = 0;* = 0)]"C»«}. (6.5) 17 For diagram» a and d we have (cf. appendices D and E), e:(r;i = o) = -c;(r;* = o) = as(l - x)(\ - y) + a, + a7*y + ry(l - y), Q^(r;z = 0) = Cj,(r;z=0) 7 2 = a4(l - y) + a« + a7y + ry(l - y), d A*(rlz = 0) = A {riz =0) = A0(a) + r[a6(l - y) + at + a7y] + r'y(l - y), (6.6) whereas for diagrams A and e e;(r.;z = 0) = -e:(r;* = 0) = -a5(l - x)(l -y) + aty + a7x + rxy(l - y), Ql(r;z = 0) = Q'm(r;z = 0) 7 = a6(l - yf + asy + a7+ry(l- y), A\T;Z = 0) = A'(I",Z = 0) s = A0(a) + r[a5(l - y) + a,y + a7] + r y(l - y). (6.7) We observe that each of the functions Qm(r;z = 0) and A(r; 2=0) becom-s identical for all diagrams if we make the substitutions as «-* a7 in diagrams b and e. Furthermore, x appears only in Q,. We shall now show that due to these symmetries the leading terms cancel when the six diagrams are added. Consider the integral r=/ -1 1 - tvi cny - a5(l -y) + ie 1+( 1 1 x [{a, + a7y + ry(l - y)}- " - {a5(l - y) + a, + ry(l - i/)}- ^» ]. (6.8) For diagram d we set ilr( Wd = j* In the expression for diagram 6 we Srst relabel the integration variables, interchanging a« and ct7. The integration over x yields -1 1 W = 1 - (Vl "s(l - y) + a, + ry(l - y) 1+c i+< 1 x [{at + a7y + Ty(l-y)}- ^-{-as(l-y) + a7 + ie}- '' ]. (6.10) 18 We now concentrate on the leading term in an expansion in power» of (. Summing over the six diagrams we obtain y wim 2 f I + L 1. i=fl. ajy - at(l - y) [at + ajy + ry[l ~ y) o5(l - y) + o« + rjj(l - y)\ (6.11) The remaining parts of the integrands in eqs. (4.13) and (4.16) are symmetric under the simultaneous replacements as <-» a?, y «-» (1 - y). The sum £j=«,...,e W"> however, is antisymmetric under this replacement. Consequently, we can effectively set £ w=o. t=o,...,e The upshot of this analysis is that to get a non-vanishing contribution we must expand the integrand in .?s(C;a) in £yi. As a result the leading contribution to ,?3(f;a) comes from the term ^3(C;a) = 8A0(a)/ dx f dy f - Jo Jo Jo T ! 2 x {[Q.(T;Z = 0)]- loge.(r;z = 0) - [S.(r = 0;z = 0)]- logC.(r = 0;z = 0)}, (6.12) summed over the six diagrams. In this order there i6 no dependence on Qm or A. Compar ing with T\(£;a), eq. (5.30), we again note that the l/£ factor is absent, making ^j(^;a) of lower order. 6.3 Ft According to eq. (4.17), *(C;«0 = fa A.(a) f dx £ dy J°° t [fi.(rj * = 0)1 ~2+<^ x [Qm(r;z = 0)]-<»'[A(ri« = O)]""". (6.13) This contribution is treated in the same way as the previous one, Æ((;a). When summed over the six diagrams there remains to leading order ?4C\*) = Uo(a)J dxj dy Ja°^-{Q.(r;z = 0)}-2log G.(r;* = 0). (6.14) Again we conclude that ;7\t(C;a) is of lower order than Ti {(,•,&)• 19 6.4 T* According to eq. (4.18), V C»l Jo Jo Jo A(r;2 = 0) + ( x [G.(r;z = 0)]-' «"|fim(r; J = 0)]- »'(A(r;« = 0)]"<»«. (6.15) The expressions for S,(r;z = 0), Qm(r;z = 0), and A(r;z = 0) for the six diagrams are given by eqs. (6.6) and (6.7). Furthermore, for diagrams a and d a Q N(r;* = 0) = Qtir;* = 0) = aB[ry(l - j)(a5 + aT) + A„(a)], (6.16) and for diagrams b and e = a7[r»(l - y)(o5 + a,) + A0(a)j. (6.17) b s It follows that Q N = Q%(at <-> en) and that Sjy i unchanged under the substitution as «-» cti, y <-> (1 — y). The arguments that were applied to ^j(C; a) thus apply also here, implying that the dominant term in ?>{(; a) vanishes when summed over the six diagrams. The leading non-vanishing contribution to Ts(C;a) becomes X Q.(r; * = 0)"2 iog Q.(r; z = 0), (6.18) summed over the six diagrams. Further simplifications can be made. We h?ve <25v(r;« = 0) = a«(a5 + a7)[ry(l -y)+ a,} + asa,a7. (6.19) The term a6a8a7 can be ignored. The additional factor asaaOr makes the integration over as, as, and a? regular, i. e., there is no singularity at £ = 0. The first term in (6.19) contains the factor a8(a5 + a7) = aa(l-a«), (6.20) which has the effect of softening the singularities in the as, a» and a7 integrations. As a consequence, the order of .?s(£;a) is lower than that of J^f^ja) to .^((^a). It only contributes to the constant in Joo- 20 6.S T\ revisited The expression for T\((,\a) is given by eq. (4.14). In sect. 5 we first performed the x and y integrations and then summed over the six diagrams. We can also apply the method used in sect. 6.2 for Ts((;a). There we showed that the leading contribution vanished when summed over the six diagrams. In the present case we have both r = 0 and 2 = 0, which means that the analysis in sect. 6.2 applies also to this case. The conclusion is that after summing the six diagrams the dominant non-vanishing part of .Fi(C;a) becomes 4 *«;«)= „„ t,/.M'-ft" fdx fdy ClS/i-Ss) Jo Jo x [fi,(r = 0; z = 0)] -J log Q,(T = 0; z = 0), (6.21) a formula that could also have been used as a starting point. 7. Integrating over as, a., and 07 In the previous two sections we have shown that the leading contribution to the function /(£;th,tb) comes from the term yri((;a). Substituting the expression (5.30) for .^(C; a) into (4.13), we get J(CiWi.») = -^^""TT; 7—w si ••• I dasda,da7 «(1 - as - a. - 017) C(i-Cyi)(si-»s)y0 Jo 1+ X (asata7)- ^ A.(a)-<" ]jt(a) - y ], (7.1) where, according to eq. (5.28), «••-T-'KSHM'-W)] ~ LU ((«s + «.Ma. + *,)) + Ut V ~ («. + a.K« + «T)J ' (7'2) When £ is small, the dominant contributions in the integration over as, a., and 07 clearly come from the regions where two of them are close to zero. The expression for R(a) can be simplified considerably. We observe that the function A0(a) and the product asatcn are unchanged under permutations of as, a., and an. Therefore, we can symmetrize the expression for R(a), giving 21 i fu, ( S| -) _ Li, f, _ . $ ) +Li,(- S| r) - LiJl - -. 2| -) +Li,( $ )_Li2fa_ ^ \1 \(a7 + OS)(OIT + at)J \ (ai + as)[a7 + a, /} (7.3) This expression is manifestly symmetric in as, a« and a7. It looks more complicated than the expression in eq. (7.2). The amazing fact is, it is much simpler. It is independent of as, 04 and aT. This is most easily demonstrated by differentiating with respect to as. Once we know that the function fi,ymm(a) is a constant we evaluate eq. (7.3) at as = 0, and get JW>(a) = y. (7.4) With the simple integrand that results, the integral for /(£;jrii]/a) is easily evaluated in the limit £ —> 0. Details of the integration are given in appendix G. The dominant terms 2TT2 8 /«;»,») = «-*"*' 2 ((!-(»)(»-») < !fj(2y2-!/4) •T('^5»=)]3 - <"> where 3/4 is expressed in terms of yi and j/2 in eq. (3.15). The last term in the above equation will not be kept since it leads to a constant in Foo, terms which are not considered in the present study. We remark that the integral (7.1) is convergent only if Re tø - W2) > 0. (7.6) Substituting for /(C'>3/ijtft) in eq. (3.17), keeping only the leading term l/£a, we obtain j(() = ^1 / *£ / *L I . (7.7) 8. Integrating over j/i and y; We substitute in eq. (7.7) for yj and y< according to eq. (3.15), Sv'iVm-Vx)* fdyi [ dyi *' C J 2m J 2iri 1 + tø* - ltø+VXV2 22 1 + (fm - l)yi + ImSJ 1 + (lm - Itøl + VXVl 1 + (lm - 1)»1 + (2»JA - Vm)Vt Vi' (8.1) By the inequalities (3.11), (4.4), and (7.6), along the contour of integration, the real parts of the first and fourth denominators are negative, while those of the second and third are positive. Integrating first over j/i, we close the contour in the right-hand half-plane. The contributions then comes from the poles in the second and third denominator. Their positions are at l/i = r-^— {1 + VmV»), (8.2) and 1 -Vm The corresponding residues yield m = ~?~Jw[2T+Sg"(i + n»»)i?|- (8'4) We also here close the contour in the right-hand half-plane. Since the pole of the first term is in the left-hand half-plane, only the second term contributes. It has a pole at the position 1/2 = > 0. Inserting this value in (8.3), we observe that the contribution from Fi(C; oi) to /(£) comes from a residue at yi ~ 0. Our final result for the Mellin transform is thus hO = ^rvl. (8.5) Substituting into eq. (3.4), and using the results of appendix A, we get JW = V^l0ga' (8-6) where by eq. (3.5) r)x = log A2/logs. We have previously fixed the scale by setting \t{ = 1. If we now reintroduce the t- variable, it follows that w-SMw)- (8-7) 23 Invoking c-qs. (2.8) and (2.1), we obtain for the amplitude 0. The integrals /// and // We shall now show that at high energies the integrals /// and 7/ of eq. (2.9) tend to zero more rapidly that ////. Consequently, they can be neglected. To demonstrate this statement it is sufficient to consider the contributions from diagram a. Furthermore, since we here only want to determine the power behaviour in a, one single Mellin transform suffices. We start with the integral Ifj and define According to eq. (1.6.9) Nu/a is independent of a. We assume that J(a) behaves as 1/a2 for large values of a, aside from powers of log(j). This is the naive «-dependence since D(a) is linear in t. (In contrast, recall that the factor 1/D(a)s appearing in the integrand for Ifj, yields l/(j'|(|).) We define the Mellin transform /(C) = f" dsa~<+1 J{a). (9.2) Jo Owing to our assumption about the behaviour of J(a), the transform /(C) is well-defined in the strip 0 < Re ( < 2. Define D(a) = >Q(a) + <*(<*) + ie. (9.3) Then the integration over a can be performed, with the result «,.jm^±flJf...Jf*,..**-g„^^^. ,», At C — 0 there are singularities with a leading term l/(5> originating from the r and z integrations, [cf. eq. (3.28)] at r ~ 0, and z ~ 0 or z ~ 1. This implies that for large values of a, the dominant term of J(a) behaves as (l/«2)log2(«). There will also be terms (l/j2)log(a) and 1/J2. Some of these terms may vanish when the contributions from all 24 the six diagram» are added together. For /*/(<) this implies a power behaviour of 1/J at large t to be compared with ////(*) which behave? as 1/|(|. Is our assumption about J{§) correct or could it behave as 1/j for large values of s? Such a behaviour would generate a singularity in /(() at ( = 1. Setting ( = 1 in eq. (9.4) yields a factor Q''d'1 in the integrand. The subsequent integrations over the a-variables are convergent. This is easily understood by comparing with the corresponding integrand for IJU which at ( ~ 0 has the factor Q-i*(d~1~(. The only singularities in this case are the end-point singularities at r ~ 0 and z ~ 0 or z ~ I. With the power Q'1 there is no end-point singularity and the integral /(( = 1) must be finite. We conclude that J (a) indeed behaves as 1/s3 for large values of a. We can repeat the argument for I}. Define (cf. eq. (2.9)) '«-fl-jf-i^-^-g^dW (9-5) According to eq. (1.6.9) Nj is independent of a. We assume that J(a) behaves as 1/a for large values of a, i. e., naive «-dependence arising from the linearity of D(d) in s, eq. (9.3). Define the Mdlin transform J(Q = f daa-<-r> J(a) Jo = ir(l - CMI + 0 £ ... jf' da,... da, *(1 - £ a,) J^Q-»+ negligible in comparison with Ifn, 10. Concluding remarks As expected, our calculation has demonstrated that when we sum the contributions from all six ladder-like diagrams, the terms with four logarithmic factors cancel. Also the terms with three logarithmic factors cancel. This result, it seems to us, is not obvious. Even though the amplitude can contain at most two infrared logarithmic factors, i. e., only two factors of log A2, we are not aware of any reason why it could not contain, e. g., a term 25 log'(|t|/m'). From the corresponding diagrams in the 4>* theory, it U known that there are no logarithms with argument! involving * |3J. However, because of the presence of an additional scale (A / m), it is not clear whether or not that particular result applies here. Furthermore, as suggested above, the argument of such a cubic logarithm might have been «-independent. A calculation of the one-loop ladder-like diagrams gives [Sj M£*L = -w'11' [-y lo«J (jjj)] +lower order term6- (10-2) The coefficients of the infrared logarithmic powers can also be calculated with other meth ods. Applying those developed by Yennie, Frautschi and Suura [6], the above results may be confirmed. Furthermore, one can show that, when summed to all orders, the infrared logarithms from ladder-like diagrams exponentiate, yielding an expression of the form ail E MJSL = «P [~ °S (H) + O(JJ)] (m, + m, + m2 + ...). (10.3) Here m< is of order a'+1, and is the infrared finite part of A4j^Jj when the latter is written as a polynomial in log(AJ/|<(). The two first m-factors are known, mo = Al'17' and mj = moC?(|t|/a). The contribution from the ladder-like diagrams above to the cross section becomes M\ ("7) = |(m +m +m + ...)|s 1 + 0 Mi! . (10.4) 0 1 i -(f)] Thus, the infrared divergent terms are at most of relative order \t\/s. In order or4 the infrared terms receive contributions from 2Re M^M^ and [M^l . From the results (10.1) and (10.2) we conclude that they indeed do cancel, leaving a rest of at most relative order |<|/s, in agreement with (10.4). In conclusion, we have shown that there is no term with two logarithmic factors, other than the infrared ones. It is straightforward to extract from our expressions the integrals that yield single logarithms, but that is beyond the scope of the present investigation. Moreover, there are strong indications [4] that they cancel. We have not studied any of the non-ladder-like contributions to the virtual corrections at the level of two loops. It is quite possible, however, that the present methods will turn out to be useful also for some of those diagrams. Acknowledgments It is a pleasure to thank professor Tai Tsun Wu for helpful discussions. This research has been supported by the Norwegian Research Council for Science and the Humanities and by the Swedish National Research Council. 27 Appendix A. Mellin traiuform and factorisation formula Consider a function /(') = 9('-l)^(l°g')'- (A.1) Then its Mellin transform is defined by /«)= f°°d.,-<+,I(3) Jo m (-g)V(.Ji-T (A-2) for Re C > ' + 1 - n. The choice l = n-l, (A.3) gives Another useful relation is the factorization formula (see refs. |7J and [8]) [A1+... + A„r^^)j^^jdz1T(z1)...JdznT(zn)Ar'...A-'' xl(p-zi-...-z„), (A.5) where all contours of integration run parallel to the imaginary axis for Re (z;) > 0 and Im (Ai) > 0. Combining these results, one obtains »•;. (3.7). Appendix B. Theorem of Cheng and Wu Consider /=/,TT*»i«(l-I>)*'(a), (B.1) where F(a) is a homogenous function in the a, of dimension —n. One may then write this as [3] /= / n'dai f n"dai6(l-Z"ai)F(a), (B.2) Jo Jo where the a's have been split into two sets, , {a} = {a1,...,ah}, (B.3) {a"} = {at+1,...,an}, (B.4) and the primed and double-primed products and sums in (B.2) refer to the sets (B.3) and (B.4). 28 Appendix C. Integration! over r and z Consider the integral appearing in eq. (3.31), f\B\* /°°drr-1+« /' d*[«(l - z)|-I+< H{v,z), (C.l) Jo Jo with € = 2C(yi-»s), * = fa. (C.2) The integration over r is of the form JF; = / drr-1+'F(r) Jo = f drr-i+,F(0)+ f dTT-1+'[F(r)-F(0)}+ f°°drT-1+'F{r) Jo Jo Ji i+ 1+ = i F(0) + f drT- ' [F(r) - F(0)} + j°° drr- 'F(r). (C.3) Up to terms of O(e), this can be written as TT = \ F(0) + f ^IF(r) - F(0)] + /°° % F(r) + 0(e). (C.4) e Jo T Ji T Similarly, the integration over z yields F,= fdz[z{l~z)]-i+sG(z) Jo = f dz{z-i+sG{0) + (1 - z)-1+«G(l)} Jo + f dz{[z(\ - z)]-1+eG(z) - z~1+tG{0) - (1 - z)-'+eG(l)} Jo = i[O(0) + 0(1)] + jf dz{[z(l - ,)]->+« G(.) - z"1+s G(0) - (1 - z)"1+{ G(l)} = ±[G(0) + G(1)] + J' dz{z->+s{(l - zfG(z) - 0(0)] + (l-z)-I+«[/G(z)-G(l)]} = I[O(0) + 0(1)] + jT ^\G(z) - 0(0)] + jf j^[C(«) - G(l)l + 0(«). (C5) Having thus established the two leading orders in 1/e and l/j, we may combine them, to give the result quoted in section 4. 29 Appendix D. Diagrams a and b In general, the denominator function will be written in the form (2.7). Substituting for u in terms of », t, and m2, we define the functions Q„ Qt, and Cm by the following equations, 2 1 D(a) = D.i + Duu + Dtt + Dmm + DxX + it 7 = T'Z(\ - z)Q,, + Qtt - r Qmm' - Qx\* + ie, (D.I) where we have used the variables r and z of eqs. (3.22) and (3-25). These functions depend on the diagram considered. The functions D,, ...,£> depend on the variables oj,... ,07. The functions Ql^. •• ,Q\, on the other hand, depend on the variables r, z, z, y, and as,..., 0:7, with as + at + 017 = 1. We also define the function A0(o) sos«i + 0,107 + 0705, (D.2) which is symmetric under any interchange among 05, a», and 07. For diagram o, we have in terms of the variables of eqs. (3.22) and (3.25) A°(o) = Ao(o) + r{o5[z(l - x) + (1 - z)(l - y)} + ae + o7[zx + (1 - 2)»]} + r2[zz + (1 - z)y][*(l - *)+(1 - 2)(1 - »)], (D.3) where A"(o) is defined by eq. (2.3). Furthermore, Q\ = o6(l - a)(l - jr) + o« + a7xy + r\zx(l - x) + (1 - z)t/(l - »)], Cf = 050,07, ! QJ, = o5[2(l - as) + (1 - 2)(1 - y)f + a, + o7[zx + (1 - z)y] + r[zx + (1 - 2)3,][2(1 - x)+(1 - z)(l - »)], CS = A'(o). (D.4) The expression for the numerator function Na(a), eq. (3.3), is in general very compli cated. Fortunately, only two special cases are needed for the present investigation, namely those where either r = 0, (or equivalently Oi = 02 = Qj = 04 = 0) or 2 = 0 (or equivalently Oi = 02 = 0). For those two cases we have AT-(a) ) 4A°(a)' for r = °' F(o? = 1,L,,, rgft<.-0), ,_„ (D"5) 30 with Q'N(z = 0) = aa[Ao(a) + rj,(l - y)(a5 + a7)\. (D.6) For the once-crossed diagram 6, we get b \ {a) = Ao(a) + r{a5[z(l - *) + (1 - z)(l - y)} + a,[z + (1 - z)y] + a7[zx + (1 - z)}} + T'{Z3X(1 -*) + (!- z)2y(l - y) + «(I - *)(1 - xy)}, (D.7) and 6 h b 2 b 2 Z) (Q) = Djj + D uu + D\t + D mm + D xX + ie = (Uj - /?>)* + (/?«' - Dl)t + (Dl + 4DbJm2 + Z>jA2 + ie. (D.8) With the definitions of eq. (D.l), Qj = -a5(l - *)(1 -y) + asy + a7x + rxy[z(l - x) + (1 - z)(l - j»)], (D.9) Q\ = a5ae<»7 - T*Z{1 - z)(l - x)(l - y)as, 2 2 2 Ql = a5[z(l - x) - (1 - *)(1 - J/)] + a,[z + (1 - *)y] + a7[Za; + (1 - z)] 2 2 + r{z x{l - x)\z + (1 - z)y] + (1 - z) j/(l - v)[zz + (1 - z)] + z(l - z)(l - xy)}, Qi = A6(a). (D.10) For the ^-diagram, the numerator function is given by eq. (1.6.10). Substituting W<+) = W*-' = 1, and taking out the factor s2, we obtain Nb{a) = 2[ao(cc + dd) + aåcc], (D.11) corresponding to eq. (3.3) for the uncrossed diagram. However, the quantities a, a, etc. are different for the two diagrams. For the two cases where the numerator function Nb(a) is needed, we have in analogy with (D.S) N>) 4A,(«), for, = 0, with b Q N(z = 0) = a7[Ao(o) + ry(l - y)(o5 + a,)]- (D.13) 31 Appendix E. Remaining diagrams The denominators of the remaining four diagrams c, d, e, and / are related to those of a and 6 by simple substitutions. Consider first diagram d, for which Dd{a) = D"(3 <- u) = D> + D?i + D^m? + ØJA2 + it a 2 J = -D°,s + (Z>? - I»;)* + {D"m + 4D ,)m + Z)JA + ie, (E.l) since i + i + u = 4m2. (E.2) The decomposition (D.l) yields Qd. = ~Q°, 2 d Q m=Q°m-4z{l-z)Q°, Qt = QJ. (E.3) Consider next diagram e, for which D'{a) = Db{> <-> u) b b J h 2 = {Dl - D t)s + {D\ - D\)t + {D m + 4Dj)m + D x> + it. (E.4) It follows from (D.l) and (D.9) that s: = -Qb„ 2 Q' = asata7 - r z(l - z)\rxy{z{l - x) + (1 - z)(l - j/)} + j/oe + x«T], 2 2 2 CS, = as[z(l - x) + (1 - *)(1 - j,)] + ae[z - (1 - z)jr] + a7[zx - (1 - z)] 2 2 + r{z x(l - x)[z + (1 - z)S] + (1 - z) jr(l - j/)[zx + (1 - z)} + z{l - z)(l - xy) - 4z(l - z)sy[z(l - x) + (1 - z)(l - „)]}, Ql = Ql (E.5) Finally, b D'{a) = D {a), D'{a) = D'{a). (E.6) 32 Appendix F. The integral T This appendix is devoted to the evaluation of the integral /"' 1 1 TIE Re / dy -j- r ^ r—- Jo at + ai{l - y) a,y - ai(l - y) + u x {log(a6 + a7y) - log[aTt/ - Q5(1 - y) + t'e]} (F.l) of eq. (5.2S). We note that 1 1 at + 05(1 - y) "ry - o5(l - y) + te 1 r 1 , 1 1 Ao(a) l y- (t»s + a«)/a y- «5/(05 +07) 5 + *] Hence, with T = Ti +T2, (F.2) we get T • * f\\ 1 1 V ] A0(a)yo "[ y-{as+at)/as 3) - a5/{a5 + a7). 1 „ Z"1 , f 1 1 Ao(or) Jo ~*[y-(as + at)/as y - a5/(a5 + 07) + »e. x log[a7j/ - as(l - ») + te]. (F.4) Exploiting eq. (3.13.1) of ref. [9] and familiar properties of the dilogarithm, we arrive at + Li,(,_. Mi .)_„,(. *£! 1 V (05 +at)(a, + a1)J \{as + a,)(ae + a?) J + T (F.5) R }• with the "remainder" TR = logf -*£>-) logf S.) + I ^(-—Mf)—-) \a5+a7/ \a5/ 2 ° \(a5+aT)(o;6+aT)/ X i 2/ A0(a) \ f a«Q7 *\ .. /«5°7\ - 2l0 B V(a,+«.)(«.+ «0,/ + *"' \MS)) ~ L" [MS)) (F.6) Since the remaining factors in the integrations over <*j, a«, and aj are symmetric under interchanges of these variables, the remainder will not contribute and may be ignored. Appendix G. Integrations over as, a», and at The integration over ai, as, and 0:7 reduces to the evaluation of S(t,6)= I da da,da S(l - a - a„-a )(a.sa6a )~1+'[Ao(as,06,0.7)]"*. (G.l) Ja s T s 7 T In our application t = (ylt S = £3/4 and we are interested in the limit £ —> 0, Since the function S(e, 6) diverges when e —» 0, our aim is to isolate the leading contributions when C-0. We dispose of the ^-function by introducing new variables <*5 = P1P2, ae -I-pi, a7=pi(l-p2). (G.2) In terms of pi and P2, HpuPz) = />i[(l - pi) + PiPt(l - pi)\- (G.3) It is convenient to transform Ao(/>],P2)-{ into a product of factors with the aid of the formula (see appendix A) 6 dz [M1 IMPuPi)]' = éijc ^^i^ - "iM'~V«(l - Pi)]", (G.4) where the contour C is a straight line parallel to the imaginary axis with Re z > 0 and Re (6 — z) >). The integrations over pi and pz each gives rise to a beta-function. As a result / dz T{z)r(s-z) r(2c-s-z)r(-s + t + z) [r(*-z)]2 ( 5) ''~j02*i V(S) r(3£-2«) r(2e-2*)- ^ ' We close the path of integration in the left half-plane. There are poles at 2 = - n, n = 0,1,... and at z = — n + 6 — e, n = 0,1,..., giving *M) = £^(-1)f- T(6 + n) T(2e - S + n)T(-S + e - n) [r(e + n)f 0 T(6) r(3e-2S) r(2* + 2n) T{6 - i - n)T(n + 34 Here it is necessary to point out that we must require 2e - S > 0, since otherwise the integral S(e,i) will diverge. We first observe that the contributions to the sum in (G.6) which arise from terms with n > 1 vanish as f2, when £ —> 0. They are therefore ignored. The contributions from the two terms with n = 0, add up to i u s\ = 2&-2S) r(i + 2c-<)r(i-< + «) [r(i + e)]a K ° '' £(2 _ M r(i + g-c)r(i + e) [r(i + 2C-g)p e(e-f)(2t-f) r(l+f) r(l+4e-2«)" l ' We are only interested in the values of the function S(e, S) for small values of S and e. Therefore, we can insert the expansion r(i + a)r(i + 6) *» r(l + a + i) -1--J--4-T-. (G-8) into eq. (G.7). After some rearrangements, the final result, valid for small values of e and 6, such that 2« — 6 > 0, becomes s^ = ^bj - *211 - 7+ &h]+ °<^'d>- (G-9) 35 References [1] K. S. Bjørkevoll, G. Faldt and P. Osland, to be published, referred to as "Paper I" [2] P. Osland and T. T. Wu, Nucl. Phys. B288 (1987) 77, 95 [3] H. Cheng and T. T. Wu, Expanding Proton»: Scattering at High Energies, (The MIT Press, Cambridge, Massachusetts, 1987), see, in particular, Appendix C [4] K. S. Bjørkevoll, G. Fåldt and P. Osland, to be published [5] M. Bohm, A. Denner and W. HoUik, Nucl. Phys., B304 (1988) 687 [6] D. R. Yennie, S. C. Frautschi and H. Suura, Annals of Physics, 13 (1961) 379 [7] M. C. Bergére, C. de Calan and A. P. C. Malbouisson, Commun. math. Phys. 62 (1978) 137 [8] R. Gastmans, W. Troost, and T. T. Wu, Nucl. Phys. B365 (1991) 404 [9J A. Devoto and D. W. Duke, La Rivista del Nuovo Cimento 7 (1984) 1 36 Paper III TWO-LOOP LADDER-DIAGRAM CONTRIBUTIONS TO BHABHA SCATTERING III. THE <£'-LIMIT OF QED Knut Steinar Bjørkevoll Department of Physics, University of Bergen, Allcgt. 55 N-5007 Bergen, Norway Goran Fåldt Gustaf Werners Institut, Box 5S5 S-751 SI Uppsala, Sweden and Per Osland Department of Physics, University of Bergen, Allégt. 55 N-5007 Bergen, Norway ABSTRACT We evaluate, in the high-energy limit, the sum of the Feynman amplitudes correspond ing to the six two-loop ladder-like diagrams in Bhabha scattering. This is the limit where s —> oo, while t, tne electron mass m and the photon mass A are all being held fixed. In this limit the sum of the six Feynman amplitudes does not depend on the electron mass. When specialized to the region s S> \t\ 3> m2 S> A2 this result complements the one previously obtained. The connection with 4>s theory is also investigated. 1 1. Introduction In two-body scattering in QED, with the photon mass A as an infrared regulator, there are four independent dimensionful quantities, a, t, m7, and A1. Thus, apart from an over-all scale, the scattering amplitude is in general a function of three dimensionless quantities. Clearly, such an amplitude may have a very rich structure. In two previous papers [1], [2], we discussed the contribution to Bhabha scattering from the six two-loop ladder-like diagrams that together constitute a gauge-invariant set. The limit considered in [2], is A1 -» 0, with a > \t\ > m1 being held fixed. In the present paper, we discuss the amplitudes arising from the same set of diagrams but in the limit where A, m and t are kept fixed, while a —» oo. This limit is easier to handle since only a single Mellin transform is needed. Simultaneously, we get the corresponding results for (6s theory, basically by taking A = m and removing the complications arising from the spinor couplings of QED. For the uncrossed ladder diagram, we found the QED amplitude in the large-j limit to be given by the following integral [lj, where A" = Ala) = (aj +a,+as+ aj)(o2 + 04 + a, + a7) - a\, (1.2) 3 D"(o) = D.a + Dtt + Dmm* + DXX + U, (1.3) with D, = a1a3(a!2 + a* + oi« + 017) + a2<*t(<*i + Qj + a5 + a«) + a8(aia4 + ajotj), Dt = aiatCiT, A„ = -(ai + o2 + as + at)A + (ai + a3)(ai + at)(at + a7) + (ai + a2 + os + 014X015017 + ajcij -r atctj), D\ = -(os + a, + a7)A. (1.4) The numerators NftI, Nfj, and Nf are given by equation (1.6.9) of paper I (we refer to equations from ref. [1] by the prefix 'I.'). 2 Similarly, the amplitude Mi for the once-crosied diagram i« given by an expression identical to (1.1), but with numerators given by eq. (1.6.10), and A* and D'(«0 given by eqs. (1.4.8) and (1.4.9), respectively, as A* = A(a) = (a\ + aj + a5)(aj + on + a» + 07) + (0.4 + at){ai + 0:7) ! = (<*i + Qj + a, + a5 + Q»)(aj + a4 + a» + 07) - (0:4 + oe) , (1.S) and 2 2 £*(«) = D'» + Ai" + D,t + Dmm + Dx\ + te, (1.6) with D„ Z)», Dt, Dm, and Dx given by eq. (1.4.10). Unless confusion may arise, we shall leave out the indices a and 6 used here to distinguish quantities referring to the two different diagrams. As indicated by eq. (1.7.10), the full amplitude corresponding to the sum of the six diagrams, can be constructed from just these two amplitudes Ma and Mb, with suitable substitutions among the kinematical variables s, t and v.. In *-(isW -L ^•••^1-J>>A^H^ (17) where N' = [A»]4, (1.8) D'(*) = D°(a)\ . (1.9) Hence, the 2. The integrals ////, //;, and 7/ We shall not evaluate the complete amplitude M„ for the uncrossed diagram, but only that part which contributes to the unpolarized differential cross section. In our previous papers I and II this part was denoted Fga(i,t), and defined in eq. (1.2.4). Its decomposition is identical to that of A4„, *» = 2/?//(M) + ii?/<»,<) + \m>,t), (2.1) 3 with ^(M)-jf-..jf*n-..--.*(i-±-i)3?jSJfc(Sy. (2.2) The functions JV are obtained by setting W'+' = W<_) = 1 in the functions N entering eq. (1.1), and defined in eq. (1.6.9), = 2s2[aacc + (oo + bb)(cc + dd)], N ftr h = • //|vr(+)=wc-)=i' *? = ^lw«-m-)-.- (2-3) Similar expressions hold for the amplitude Fg0 arising from the once-crossed diagram. Among the three terms in eq. (2.1), the first one will dominate. A major part of the present paper is therefore devoted to a study of ////(«,<) and ////(»,<). The integrals Jjj(ai*) and //(s,t) have been shown [2] to contain additional powers of 1/a and are therefore neglected. 3. General structure of//// As i —' oo, the dominant terms in ////(*,t) behave like (l/aJ)loga and 1/V. They can be determined by a Mellin transform [3j (see also Appendix A of [2]), Im(s,t) = ssJ(s,t), (3.1) H(,t)= [°° dss-<+1 H,,t). (3.2) Jo Observe that we here deviate from the procedure followed in paper II, in that i is kept as an independent variable, which is not Mellin transformed. We write the denominator function D(a) as D(a) = D.{a)é+V(a), (3.3) 4 where, for the a-diagram [see eq. (1-3)], a 2 , •D ^a) = Dtt + Dmm + Dx\+it. (3.4) Furthermore, we extract two powers of s by denning 3 N{a) = Nm/s . (3.5) Performing the integration over a in eq. (3.2) the MeDin transform becomes Ht,t) =* (i - c) jf ••• jf^. ...*»T«(I -X»XJ§* [-D.(«)r2+C m.*)]-1-* -»-<#+t) AQ AI - AO . . = £2" + —£—. (3-6) where we have neglected terms that are finite when £ -» 0. Inverting the Meliin transform (see Appendix A of [2]), we get the asymptotic form /(*,*) = i [A, log* + (4,-A„)], (3.7) in terms of the coefficients Ao and A\ defined by eq. (3.6) above. Substituting for 7Va(a) according to eqs. (2.3) and (1.4.6) for the a-diagram, or the appropriate equations for the fr-diagram,w e see that the integrand is homogeneous in the o-variables, and of degree -7. We may thus use a theorem of Cheng and Wu [3] (see Appendix B of [2]) to transfer the {-function constraint to a subset of the variables of integration, I((,t) ~ (1 - () f ... j da1...dai f ... f dasdaodenS^-as-at-ari Jo Jo Jo Jo The only factor here that depends on the kinematics, is V(a). We continue to follow the procedure of ref. [3], and rescale the Feynman parameters associated with the fermion lines, a\ = px, en = />(1 - *), a, = p'y, o4 = />'(! - y). (3.9) 5 Substituting into D.(a), we obtain D.{a) = pp'Q( where for the a-diagram Q°(a) = 0.5(1 - x)(l - y) + at + aTxy + px(l -x) + p'y(l - y). (3.11) In an effort to keep the notation as simple as possible, we have denoted the argument of Q° by a. In the following, we shall also use a to refer to a subset of the a-variables. Thus, a may refer to the full set, ai,... ,07, to p,p',z,y,as,ae,aT, or simply to 05,05,07. We now return to eq. (3.6), which can be expressed as -i + —• ~ I ... I da5datda76(1 - 05 - a6 - 0:7) / dx idy C C Jo Jo Jo Jo The important point is now that the singularities in £ can only arise from the regions where p —> 0 and/or p' —> 0. In order to extract the coefficients Ao and Ai, we split the integrations over p and p' as follows, fOO POO fl f\ fOO [1 fOO tOO / dpi dp'= I dp dp' + 2 dp dp'+j dpi dp'. (3.13) Jo Jo Jo Jo Ji Jo J\ J\ Since the original integrand is symmetric under the simultaneous interchanges, p «-» p\ x «-» y, there are two identical contributions. This symmetry accounts for the factor of two in front of the second term. It follows that -å +-y - I • •• I dasdaidaT$(l - as - a« -a7) / dx I dy C i Jo Jo Jo Jo x/y dpj^ dp'+ 2 J" dp J dp' + J°° dp J°° dp'^pp')-'^ x^lCWr2^!^)!-1^. (3-14) The third term of eq. (3.14) does not contribute to A0 nor to A\. 6 3.1 THE DOMINANT TERM As £ —• 0 the dominant term of eq. (3.14) comes from the first integral and from the region where both p and p' are small. Furthermore, we can there replace £ by zero everywhere except in the powers of p and p', yielding dasdaodct7 Æ(l — aj — 04 — 07) Jo Jo dx dy z{a) (3.15) -t [ Wr p=/>'=0 with ^ ' A(a)sD(a) v 3.2 THE SUBDOMINANT TERM There are three separate contributions to the subdominant term, A\. The first one, A\~ , is due to the p- and p'-dependences of the factors Z(a) and 2(a) in the integrand of the first integral of eq. (3.14). The second one, A\ ', is due to the fact that the exponents of Q and V deviate from -2 and -1, respectively. Finally, the third one, A] , comes from the second integral of eq. (3.14) and from the region where p' is small. With A^A^ + Ay+A™, (3.17) we have Ay'= / ... / dasdaidctT6{1 - a5 -cti - a7) I dx I dy Jo Jo Jo Jo <-» J» Jo y ' lew2 Q(«)2U^=o/ 4I = / •••/ dasdaiddT6(1 — as — at - 0:7) I dx l dy Jo Jo Jo Jo x Urn 4" = 2/ .../ do da«do «(l-a -c<(i-07) / J4, = / ... / da5datdaT6(1 - as - at - ai) I dx I dy Jo Jo Jo Jo f*£.llQ\ fW x P I S(«)2 l,_. 2(a)' . }. p'=0) ^i = / ... / dot5dafida7 £(1 - a5 - a6 - CKT) / dx j dy Jo Jo JO JO A} = 2 / ... / dasdaflda? 5(1 — as — as - a?) j dx j dy Jo Jo Jo Jo XL P eJ(*)U„- (319) Since all integrands above are to be evaluated at p' = 0, it follows from eq. (3.9) that the function Z(a)\p'-o is independent of y. However, as seen from eq. (3.11), Q(a)lP'-0 still depends on y. Let [cf. eq. (3.16)) where, for brevity, we have exhibited explicitly only the dependence on p. As already mentioned, Zo{p) is independent of y. Then Z1 fl fl do f1 Aj = / ... / dcLidatdaTS(\ - a& — cto — cti) t — I dx Jo Jo Jo P Jo A^ = I ... / dasda6da76(l-as-<*t~a7) J dxZ0(p = Q) Jo Jo Jo s) A[ = 2 [ ... f dasda,dcn6(1 - a6 - a, - o7) f ^ f dx Zn(p) Jo Jo Ji P Jo X/'TOU (3-21) 4. Cancellation of dominant terms For both diagram a and b we have (see appendix B) N(a) -2A0(a) Z0(p = 0)= 2 (4.1) \(a)'V{a) p= with Ao(a) = asae + aja7 + 0:705, (4.2) i.e., Z0(p = 0) is independent of the electron mass. We shall find it useful to introduce the following three functions, v{a^i?x[dyW? (4.3) (4.4) (4.5) where on the left-hand sides the argument a now refers to the set (as, o«, 07). The first of these integrals is evaluated below, while the other two are discussed in the following section. 4.1 THE INTEGRAL V For A V'{a) = / dx f dyj- n-, X ' Jo Jo {iv + Vo+^Y (4.7) Jo fotøo + 0' with i = a7x - a5(l - x), a (4.8) ijo = V\P=O = 5(l - x) + at > 0. The denominator in eq. (4.7) is always non-negative, so the integral is real. We get V'(a) = —-r log — + log (4.9) Ao(a) [ o« at Since the remaining integrand in AQ, eq. (3.15), is symmetric in as, as, and a-j, we may replace V*(a) by the more compact expression V>{a)\ = 2 j »L±^. (4.10) k K ""«"» A0(a) a, ' For the h diagram, the quantities ( and TJO in Qb(a) are interchanged, as compared with Q°(a). Since the denominator may vanish, some care is required in carrying out the integration. The ^-integration yields Vb{a)= f dx(~^-—^~)~. (4.11) Bere, £ may pass through zero, so the first integral is complex. For the part which is symmetric in the variables a5, a«, and aj, we obtain the result as + as V"la)\ = -7-7-7 log 1- ix -*""«--w (4-12) 4.2 THE CANCELLATION The dominant term, as » —» co, arising bom one particular diagram, is given by eq. (3.15), where the only dependence on x and y is through 2(a). Using the abbreviations (4.1) and (4.3), we find a Ao = I ••• I dasdatdaT 6(1 - o« - e ~ en)Zo[p = 0) V(a), (4.13) Jo Jo where the result of the integrations over x and y are given by the integral V{a) of eq. (4.3). Invoking the results (4.10) and (4.12), we can write the contributions to A% and Aj as Al = -4C31(t), AS = 2[C„(i)+iC,o(0], (4.14) 10 where we have introduced the functions [3]: 1 1 „ ,,-. f f dasda,da7 8(1 - o5 - a» - a7) CS (<) ° -/.-/. a5a,aT|i| + A„(«)A> *' 2 Jo Jo asa,a7\t\ + Ao(a)X on Furthermore, by the substitutions of section 7 of paper I [1], F«\s,t) = iftW), i^M) = *#W). (4-16) we get AS=2[(7,i(l) + iC,o(«)I, 4. = -4Cal(<), ^5 = 2[Csi(t) - iC,o(0]. ili* = 2[C,1(<)-iC,0(0]. (4.17) Thus, the sum of the contributions from all six diagrams vanishes, Ao=Al+Al + A% + At + Al + A{ = 0. (4.18) This implies that in eq. (3.7) the terras proportional to log s cancel, when summed over all six diagrams. 5. Integrating over x and y in the subdominant terms In the integrals (3.21), defining the subdominant terms, the dependence on the kine- matical variables enters through Zo(p). It follows from eq. (4.1) that the term A*\ which only depends on Za(p = 0), is independent of the electron mass, m. The terms A\ and A) , on the other hand, do depend on the electron mass. With the functions V, W and X defined by eqs. (4.3), (4.4), and (4.5), the three terms of eq. (3.21) can be written as Ai = I ... I dasdadcn 6(1 - as - a, - aj) Jo Jo t 11 x [jJ 3) A\ = I ... I dasdatda76{1 - as - a, -a7)Z0(p = O) Jc Jo x [x(a) - V(a)logV(p = 0)}, (5.2) 45) = 2 / ... / dasdaida, 6(1 - a - a, - a ) / — Jo ./o 5 T /i ^ x / dx Z0(p)W(p). (5.3) JO As several times before, we use a compact notation where W(p) refers to the function W that depends on all three argumentsp, x and a = (05,08,07), while W(p = 0) refers to the function W with p = 0 (and thus x irrelevant) and similarly for the function Zo(p)- The function V(a) was evaluated in the previous section. We shall first evaluate the remaining two functions W(p) of eq. (4.4) and X(a) of eq. (4.5). 5.1 THE INTEGRAL W Consider the integral (remember that the arguments x and a are not written out) (5.4) •w-M*) 2 •=o first introduced in eq. (4.4). For the a diagram, taking Qa(a) from Appendix B, we get v t+*\v v+ij 1 Q7X - 05(1 - x) +ie [05(1 — x) + at + />x(l — z) 07* + o6 + px(l - x)J (5.5) Here, since ( can vanish, the ie prescription must be retained. However, the remaining factors in the integrands of eqs. (5.1) and (5.3) are symmetric under the simultaneous interchanges at <-> 02, as <-> 07. Thus, after integration over as, a« and 07, the two terms in the square bracket of eq. (5.5) give identical contributions. In particular, the {-function part from the te prescripts 'tops out, leaving only tl principal value part. Thus, we may replace W"*(p) by V 1 = 2 (56) ^"MLymm o7x - a5(l - x) a5(l - x) + a„ + px(l - x)' 12 For the b diagram, taking Qb(a) from Appendix B, we get w V\{ + i* V + ZJ (5.7) Q7« - e»5(l - x) - ie 0:5(1 - x) + a8 + pi(l - x)' which leads to V 1 W*(P)I •ymm ~~ aiX _ a,.(l _ j.) a6(l _ j.) 4. „, + px{\ _ x) (5.8) 5.2 THE INTEGRAL X For j4, we shall also need the integral X(a) of eq. (4.5). For the a diagram, this integral takes the form X*(a) = f dx [ dy ,,.*•» logtø + * + te), (5.9) Jo Jo («» + ^o + «r with £ and i/o as defined in eq. (4.8). Integrating by parts, ive obtain 1 log(fy + i7o)- ^"/"l &+>» fø + wJo 1 = / <& Ti \"T~H 1 [log(a« + a7z) + 1 1 J„ a7x -a5(l -x)+»e I an + a7a! + „ -LA ,|Nl + "5(1 -»)) + !])• (5.10) a8 + a5(l-x) J Since the remaining factors of the integrand of eq. (5.2), denning A\ , are symmetric uudcr the interchange 05 «-> 07, we may replace Xa(a) by *•<<_-/*[- 071 — 05(1 - x) +— U + 0:5(1 - x) - [log(a« + aTx) + l] a« + ajx 13 i dx —r- ^ f- [log(<*6 + a7x) + l] a (l -x)-a-ix ai + ajx L ' •r s ihfd<-i^-x-^^)H' + -)+^ Ct7 + 1 (5.11) In order to proceed, we need the dilogarithmic function, for which we use the notation [4], [5] hU(z) = -J *log(l-*0 (5.12) = -/>g(i-t). The second term above may then be integrated to give P di-^— log(c + x) = log(c + o) log ^ Li f ^— j + Li ( —^- J - (5.13) J: s 2 Again, exploiting the symmetry of the remaining part of the integrand of eq. (5.2) under the interchange as *•* <*?, we arrive at Finally, for the 6 diagram, we have 1°«(£ + >?o) Jo W É + iJo log«+«)- (5.15) £ + te f + i/o f + «J This function is evaluated in Appendix C. We only need that part which is symmetric under interchange of Q5 «-» 07. It is given by vb, \\ ! / /, , 1 M «5 + OT 1, 2as+a7 "•(-Btj)**(35)+*» (as + arfl) •iir 1 Iog^sA^rJj (as + 07)' = -iJT"(a) 1-log 16) liymm ^ a A (a) }, (S. fe)N- 5 0 14 with ^-gSH'teS)} + hJ *£) T)-i*(l-7 ^ TY (5.1T) 2 v V(a5 +<«»)(«» +a?)/ V (a6 + a,)(a8 + a7)/ ' However, X(a) enters the integrand of eq. (5.2) with factors which are completely sym metric in 015, aj and 07. Therefore, R(a) can be further symmetrized, replacing it by -i-)U.-T-!hte)-*(1-®)+u'(ra) J kJ- fl ) _ uJi - , 4 -) 3 I V(«s+ae)(a5+a7)/ V («5 + a«)(o6 + a-,)) M< v IT^-M1-, + v + ,) V(<*« + tts)(<»J+a7)/ \ (Q3 + Q5)(a6 + 0!7)/ +lJ f! rV^fl-T i )]. \(ar + as)(ocj + at) j \ (a7 + o.5)(c<7 + a6/J (5.18) It was pointed out in paper II that this expression is in fact independent of the variables 0:5, at and a7 and that furthermore *Wl,y,»» = Y- (5-19) 6. Summing contributions to the subdominant term Aj The subdominant term Ai consists of three parts, A\ ', J4J and A\s , which are evaluated for the a and b diagrams in the following three subsections. 6.1 THE COEFFICIENT A^ We define the two functions r1 r1 t1 do Dso{i) = -- I ••• / dasdatda76(1 - as - at - on) I — * Jo Jo Jo P 1 *N .. . ;-» , -<•«•» -Zo(P = 0)- 15 _ D3i(0= s/ ••• / x \UP) __„^ ' _.„_^ - *b» = °): * as(l - x) + a» + ps(l - *) o5(l-i) + ae (6.1) It then follows from eqs. (5.1), (5.6), and (5.8) that A\1)m = -4IW0, 1)i A\ = 2lD3,(t) + iDu(t))- (6.2) 6.2 THE COEFFICIENT Af We first note that (see appendix B) 2 log 0(0)1^^ = log(asa,a7|t| + A0(a)A ) + IT. (6.3) By eqs. (4.1), (5.2) and (6.3), it follows that A™ = -.Til» - 2 / ... / da5da,da76(1 - a6 - a« - aT) . ' . . ... 2 Jo Jo 'aso«ci7| It is convenient to express the coefficient A\ ' in terms of the functions •Eso(t) = / ••• / dasdatda76(l - a5 - a< - a7) 2 o ot5a«a7|t| + A0(a)A (a + a )21 x 1-log 5 7 a5A0(a) J' B»i(t)= / •••/ dasdatda7 «(1 - a5 - a» - a7) . . 2 2 7o ./o QjQ«o7|t| + A0(a)A (6.5) F»o(0 = / • • • / dalda,da16(1 - a5 - a, - a7) 2 Jo Jo a6a«a7| •FaiC) = / ••• / daidatda-, 6{1 - a* - at - a7) ...,.• .., 3 Jo Jo a6a,a7\t\ + A0(a)A 2 x log(a5Q,a,|.| + A0(a)A ) log ft±£l. (6.6) Invoking now the results (4.10), (4.12), (5.14), and (5.16), as well as the definitions (6.5) and (6.6), we find for the a and b diagrams, respectively, J) 4< * = ~inA"0(t) - 4£,1(t) +«"„(«), A?» = -WilS(i) + 2[£„(J) + i£„(«)] - y C3„(.) - 2[F„(0 + %F„{t)]. (6.7) The Cso-term comes from the iJ(a)-part of eq. (5.16), where R(a) is replaced by its value eq. (5.19). 6.3 THE COEFFICIENT A\3) Next, in order to write out A\ , we define the functions •Dao(t) =-"• / ... / da5datda7 6(1 - a$ - ae - a7) I — Ja Jo Ji P x Z„( J * On(t) = — I ... I dasdatdaTS(l — as — ag — ar) I — I dx Jo Jo Ji P Jo V 1 X ZM (6 8) a7X-as{l-x) o.(l-») + ««•+ /«(!-»)' ' Invoking now eqs. (5.3), (5.6), and (5.8), we find for the a and 6 diagrams, respectively, A')a = -4ft, (I), s)t 4 = 2[D51(t) + iDS0(0]- (6.9) 6.4 ADDING THE CONTRIBUTIONS FROM ALL SIX DIAGRAMS Before adding the contributions from all six diagrams, we add the three terms that make up A\, eq. (3.17). We obtain for the a and b diagrams, A\ = -ivAlit) - 4D,1(<) - 4£S1(.) + 4F„(t) - 4Dsl(i), A\ = -trr^JW + 2[D,i(t) + iDM(t)} + 2[£ji(t) + i-Ejorø] - YCjo(i) - 2[F,i(t) + .Fs„(t)] + 2{Dii(t) + W,o(t)]. (6.10) 17 With the substitutions of section 7 of paper I, we find for the other diagrams (see eq. (4.16) and appendix D of paper II) C = 2", Qd = -C' = -Q", S* = Q/=-Cb*. (6.11) Thus, log(Q'i + tc) = logC,' + ijr, \og(Q' + «) = (log Q')* + «r, (6.12) and the coefficients for the remaining four diagrams become Ac — Ah A\ = -4JD,I(<) - 4Etl(i) + 4F51(t) - 4Dal(t), A\ = 2[0si(<) - >A«(')1 + 2[£si(<) - «J5»o(«)] - y CM(<) - 2[*ii(«) - iFjo(i)] + 2[D„(t) - i2>s0(«)l. 4f = A\. (6.13) Summing the contributions from all six diagrams, we obtain b Al = ~in[A'0(t) + 2A 0(t)]-^Catt{t) = jCS0(t). (6.14) Thus, Ai is completely determined by the function Cj0(t). 7. Summary of high-energy results Combining <-qs. (6.14) and (3.7) with eqs. (2.1) and (3.1), we get in the high-energy limit for the sum of the six ladder-like diagrams, fio(M) = y C,.(t). (7.1) The function CJO(<) depends on \t\ and A!, but not on m2. This result complements the one obtained in paper II [2]. 18 In Bhabha scattering at LEP the interesting limit is s » |t| » m2 > A2. The 2 2 asymptotic value of the function CM(i, A ) in the region |t| ~S> A is easily determined. The mathematical details are given in appendices D and E, where two different methods of evaluation are presented. They give the result *.(,,*) = 5£ log» ($). (7.2) for s » |(| » A2. There are no linear logarithms nor constants. The expression quoted in eq. (7.2) is identical to the one given in paper II for leading logarithms. In that paper, though, no attempt was made to determine the subleading terms. In spite of the agreement, there is an important difference between the two calcu lations. The present one has been performed in the limit J-IOO with t, m2 and A2 fixed. It is not a. priori obvious that this result will hold also in the QED limit A2 —> 0, with s, i and m2 fixed. Our guess is that this will be the case, but to be certain one must also carry out the calculation of the subleading terms along the lines of paper II. There are many instances in which the two limits do not commute [6]. The coefficients Ao for each of the six diagrams has previously been calculated by Cheng and Wu [3] in oS3 theory [cf. eq. (1.7)]. Our result agrees with theirs. This is an important check, since for the determination of Ao we can set p = p' = 0 in the numerator function, and since Nm{p = p' = 0) = 4[A(p = />' = 0)]4, it follows that the coefficients Ao must be the same in QED and ^* theory. Cheng and Wu [3] have also shown that, for the sum of the six diagrams, part of the term independent of logs, is simply related by analytic continuation to the logs terms of the individual diagrams. We have calculated all terms, also those that are not related in this way. Consequently, our result for Foo, eq. (7.1), being more complete, differs from theirs by a factor of §. It might also be of some interest to display the leading log a contribution to each of the six diagrams. In section 4.2 they are shown to be determined by the functions Cjo(i) and Cai(z). The asymptotic values of both functions in the region |i| ^ A2 are determined in Appendices D and E. The results obtained there give *" ~m -(B) MS)+**"«ø0 -H+-' <"» (7.4) 19 where the dots denote terms independent of logs. They remain to be determined. To do so, all the functions of section 6 must be calculated. Some of them might depend on the electron mass m. Acknowledgments It is a pleasure to thank professor Taj Tsun Wu for helpful discussions. This research has been supported by the Norwegian Research Council for Science and the Humanities and by the Swedish National Research Council. 20 Appendix A. Integrations over p and p' Consider first an integral appearing in eq. (3.16), Mistone/1p-i^F(p)dp. (A.l) Integrating by parts, we find m = ^{P = P(0). (A.2) Next, consider h[F] = UmC2 l\ f dp-(pp')-,+(F(p,p') <—° Jo Jo = Km C f dpp->+< \p«F[p,A - f dp'^^-F^p1)} i—0 Jo I y=o Jo aP J -F(1 '1) - J! dp'i>F^ - J! dpiF^+jf * jf *^w> = F(0,0). (A.3) Both results (A.2) and (A.3) are needed for the evaluation of eq. (3.18). Appendix B. Diagrams a and b for p' = 0 All the functions pertaining to the uncrossed diagram a are defined in section 4.1 of paper I. With the abbreviation Ao(o) = cuae + aear + ajas, (B.l) 21 which is symmetric under any interchange among as, ag, and 07, we find A"(a)| = A0(a) + Oi(o8 +aT) + a2(a6 + a8) + a:a2. (B.2) Furthermore, according to eq. (3.11), C(a) = as(l - x)(l -y) + a, + a7xy + px(l - x) + it l<>'=0 S iy + V + «. (B-3) with f = arx - 05(1 - x), T) = as(l - x) + at + px(l - x). i,BA) We note that Q°(a) is non-negative. With p' = 0, or equivalently 03 = a4 = 0, we have the following equalities [for definition of the variables, see eq. (1.4.6)] 4 = d = 0, a — Ao(tt) 4- 020:5, å = Ao(a) -f ai(a« + a7) + «2(05 + <*«) + «jQ2 = A(O)|,,'=D, c = Ao(a) +0^0:7, c = a. (B.5) Thus, the numerator function of eq. (2.3) becomes **(«0|,., = *** 2 = 4[A0(oi) + Qi(c«s + a7) + a2(a5 + a«) + aia2] x (A0(a) + or2as)(A0(a) + a, a7). (B.6) Finally, for D°(a)L_0 we need Di = a5a»o7, D Q a a _ a • "»lp'=o ~ - i 2( i + "2) ( i + ocifat - a|a5 - a\a7, a Q Q 0»Uo = -( 5 + « + 0A(a)|(J,=0. (B.7) For the special case where also p = 0, we have 2 Z>°(<*)| =aso«a7i-(a6 + asJ--iT)Ao(a)A . (B.8) 22 All functions pertaining to diagram 6 are defined in section 4.2 of paper I. With Ao(a) as defined by eq. (B.l), 6 A (a)l = A0(a) +01(06+07)+ a2(a5 + a6) +ai«2. (B.9) Furthermore, b Q {a)\ = -os(l - x)(l - y) + aey + a7x + px(l - x)y + ic = W + f + w, (B.10) with r) and £ as defined in eq. (B.4). We note that this function changes sign in the domain of integration. For the particular case p' — 0, or equivalently 03 = 04 = 0, the variables 6, d, a, a, c and c become the same for diagram b and diagram a, eq. (B.5). This is also the case for the numerator function, and for the remainder of the denominator function, Nb{a)\ , =4oocc, (B.ll) lp'=0 ^Hl„.=o = ^(a)L,=0. (B.12) Appendix C. The integral Xb(a) The integral Xb(a) is defined by eq. (5.15). We split it into three terms, Fi, fj and F„ b X(a) = f dx — I°B(£ + '?o) Jo V» I Z + vo 1 , /* • ^ 1 1 t + it v t + Vo i + "J = Fi + F2 + F„ (C.l) with FlS-/^0)1Oga + "0)' (a2) f^i'^)l0g(£ +i£) ' (C3) FiE/(^kj' (C4) 23 The first integral Fi, can by partial fractioning be written as A0(oi) [ aB 2 Q7 1 2as + ae /o6a7\ /(as + ae)a7\l 2 as \Ao(a)/ \ A0(o) }\ For the second integral Ft, we obtain upon substitution for T)o and £, 5 x [log(x - °j_ + «) + log(as + o7)l. (C.6) We here invoke eq. (5.13), and project out the part that is symmetric under a5 «-» a7, 1 T . . , T2 , Ao(a) . a« ''l.ymn. A0(a) | -i7rlog a5 + — - log log - 05 05 + a6 W'-OTM"!*-*)]- (C.7) The dilogarithms can be rearranged making use of the formulas (see ref. [4]) 1+ i: L (q5 + at)[as + a7) aj A (a) °6 A (a) H é)~ )- <-é))-^ 0 0 ("5 +Q«)(«5 +« ) + — - MT log 7 Ao(a) . /", a5« \ . / o a \ a a , (05 + 07)' T 7 T 5 7 5 7 a'+^, (C8) L" i1 - A^J J = -Ll2 tø J - l0g ATR l08 -MOT giving (a + a»)(a!i +0:7) ir trr log 5 + — -loga log- «« '•J™"» A„(a)l ° a Ao(a) 5 5 i5 as + oe A 7) + lovleg °(°> +Li;(°«(Y; )-fLia(--^1)1.(C9) of o8(os + Q7) \ A0(a) / \ A0(o)/J For the last integral Ft, we get 1 at+07 , . F,= log + »x (CIO) Ao(o)L The symmetrized Bum, Fi + F2 + Ft, iB given by eq. (5.16). 24 2 2 Appendix D. Evaluation of £73o(<,A ) and C?JI(<,A ) In order to evaluate Cjo(<, A2) of eq. (4.15), we first homogenize the integrand in the or variables and then apply the theorem of Cheng and Wu [3], to obtain Cfc A») = 1 r da, f da, f da, '£/T ? ~v° -u V 1*1 Jo Jo Jo asa,a7 + P(l + ae)(a6 + asa7) where '-w*1- (D.2) Next, we write the integral as C o(t,A2) = ^ [°°dz ( dx-=^- (D.3) S 1*1 Jo Jo D(z,») ' with D(z,x) = i(l - x)[z + «2(1 + 2)] + *2z(l + z) = 6*{z - zx){z - *2). (D.4) We are interested in the structure of Cjo(t,A2) when 62 becomes small, but we are not interested in terms that vanish in this limit. For this application we may approximate the roots zj and Z2 by •(!-•) z ~-±(x + 62)(l + 6*-x). (D.5) (x + P^l+f-x) 2 Since furthermore, in the limit of small 62, 1 2 2 f {z1-z1)^(x + i )(l + S -x), (D.6) we can write 1 1 1 1 I (D.7) D(z,x) [x+S + (l-x); + 6 }[z-z1 z-zj The integration over z is straightforward, and for small S2 we get the integral representation 2ar /•' dx (»+f)2(i+f2-»)2' (D.8) ^V'TtLlS** S*x\l - x) 25 The integration over x is done by the formula +P+1 - ^~l '°«" (i) + "lPK-irC„,p+l, (D.9) where in the last step we have neglected terms that vanish when S2 ~* 0. The constants Cn,p+i are defined by the formula c"*+l = lår f TT7[log"'" log"(1+1)] log'(1 +1)' (D' 10) and identical to the corresponding constants in ref. [5]. The first few values of CntP+i are = = Co,] = 0^0,2 = OotS ^2,1 0 Ci,i = C(2), Ci,2 = C(3), (D.ll) where £(n) is the Riemann zeta function. Neglecting contributions which vanish when £2 —* 0, we get C(i,A») = ^log»*» The integrand of CSI(J, A2) has an additional factor log[(os + aj)/cts]. In the inte gration variables of eq. (D.3) this is simply logs. Going through the same steps as for Cie(t, A2) we get for small S2, the representation (x + 6*)2(1 + 61 - x)> log|*(l - x)]. (D.13) S*x(l - x) Applying formulas (D.9) and retaining only terms that survive in the limit 6 —» 0, we have C„M«) - iJi [§ *$)+¥*(B) - H- (D-14) If we are only interested in the leading logarithms, the detailed expansions given above are not required. An alternative, and often simpler method is given in appendix E. However, that method may possibly fail for some of the other integrals of section 6. 26 Appendix E. Evaluation of Cjo(t,A2) and Cn(t,X2) by contour integration We introduce variables similar to those used in appendix G of paper II, «5 = 1 ~ Pi, ot« = pi(l-P2), <*T = PiP2- (E.l) The function Cjo(t, A2) of eq. (4.15) can then be written as C<,(i,As) = ^jT d jT dp Pi s Pl 2 PW1 -/>i)(l-ft) + /»i[(l -P^+Piwtl-Pz)]*2' (E.2) 2 2 2 with J = \ /\t\. Using (II.A.5), we transform Cs„(t, A ) into 2 J Cso(<, A) = |7j j[ to / to (2^ £ to /, to r(z,)r(z2)r(i - *, - *2) X Pl[rf(ft(l - Pl)(l - P2)]-I+ "+"[pi(l - ft)]"" Wft(l - ft)]"" x(jS)-*l—1 x^.^lfllz,,»,)^)--'- T = 5 (5S? /e, ** /«, *» l«W»W ~ •» ~ *) „ i-frEW (f3)-„-,3- (E3) r(zi +z2)r(2ij)' We do the Z2-integration first. Since S2 = AJ/|<| 27 With the same variables, we have Pi l°gP2 PiPaf1 - Pi)(l - Pi) + M(l - Pi) + P1P2U - PiW ' PWI - pi)(i - PZ) + pi[(i - PI) + pip2(i - PiW x JB^.zjJBfz! +e,2,)(«2)-11-". (E.5) We proceed in the same way as for CJO(<, A2), arriving at (D.14). For further applications of this technique, which is well suited for computer treatment, we refer to ref. [6]. 28 References [1] K. S. Bjørkevoll, G. Faldt and P. Osland, Bergen Scientific/Technical Report No. 1991-07, to be published, referred to as "Paper I" [2] K. S. Bjørkevoll, G. Faldt and P. Osland, Bergen Scientific/Technical Report No. 1992-01, to be published, referred to as "Paper II" [3] H. Cheng and T. T. Wu, Expanding Protons: Scattering at High Energies, (The MIT Press, Cambridge, Massachusetts, 1987), see, in particular, Appendix C [4] K. S. Kolbig, J. A. Mignaco, and E. Reraiddi, BIT 10 (1970) 38 [5] A. Devoto and D. W. Duke, La Rivista del Nuovo Cimento 7 (1984) 1 [6] K. S. Bjørkevoll, thesis for the dr. scient. degree, University of Bergen, 1992 93- U. I Z 29
(Pi) (pj+9i)' -m' + w1 '(pi+^+flaJ'-m' + u* "" vr"