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Simple rules for the signature and the singularity structure of the multiparticle amplitudes
Nuclear Physics BI02 (1976) 447-460 © North-Holland Publishing Company
SIMPLE RULES FOR THE SIGNATURE AND THE SINGULARITY STRUCTURE OF THE MULTIPARTICLE AMPLITUDES A.J. BURAS * The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark Received 20 August 1975 We show that the representation of the 2 ~ N amplitude in the multiregge limit which exhibits signature factors and singularity structure allowed by the Steinmann relations can be reproduced by means of a few simple rules. We illustrate the rules with several examples and point out their importance for the reggeoncalculus.
1. Introduction Some time ago Drummond, Landshoff and Zakrzewski [1 ] found a representation of the 2 ~ 3 production amplitude in the multiregge limit which exhibits signature factors and singularity structure allowed by the Steinmann relations [2]. This representation has been subsequently generalized to the 2 ~ 4 case by Weis [3] and recently to the 2 ~ N case by Bartels [4]. The representation has a structure of a decomposition of the amplitude into terms each of which corresponds to a certain set of simultaneous singularities in the energy variables. Furthermore each term in the decomposition is characterized by certain signature factors and real analytic coefficient functions. Typical examples are given in sect. 2. Such a representation of the amplitude (hereafter called the proper decomposition o f the amplitude) was originally discussed for amplitudes with pure Regge pole exchange only [1,3]. Recently it has been shown by Bartels [5], on the basis of hybrid Feynman diagrams for the production amplitude, that the proper decomposition remains valid in the presence of cuts and consequently is one of the ingredients of the reggeon calculus scheme for the multiparticle processes [6]. Indeed the 2 ~ N amplitude in the reggeon calculus can be written in the form of the proper decomposition with the coefficient functions receiving contributions from both poles and cuts. We shall discuss this in sect. 5. Furthermore the validity of the proper decomposition for the particular case of the 2 ~ 3 amplitude has been proved by partial wave analysis of the five-point function [7]. * Present address: CERN, Geneva, Switzerland. 447
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All this indicates that in any scheme which takes into account both the signature and the analytic structure of the multiparticle amplitudes, the proper decomposition must be an important ingredient. Therefore it is of interest to develop a systematic method for writing down the proper decomposition of any 2 ~ N amplitude. As will be clear from sect. 2, the number of terms in the proper decompositon grows rapidly with N and each term becomes increasingly more complicated. It is therefore desirable to have a simple and efficient technique for the calculation of each term of the decomposition. Recently Bartels [4] demonstrated how each term in the decomposition can be understood in terms of the 2 ~ 2 amplitudes. He has also shown how each term can be constructed once the set of simultaneous singularities corresponding to it is given. Unfortunately Bartels construction, when applied to processes with N > 3 is quite laborious and it is desirable to find a simpler method. In this paper we present such a method. It is based on three simple rules, which we have found by analyzing Barrels' construction. The paper is organized as follows: in sect. 2 we review briefly the arguments for the proper decomposition by discussing 2 ~ 3 and 2 ~ 4 amplitudes. In sect. 3 we present the general idea behind Bartels' construction and illustrate it using the 2 -* 6 amplitude. In sect. 4 we formulate our rules and show that they are equivalent to the method of ref. [4]. The effectiveness of our method is illustrated for a few examples. Finally in sect. 5 we summarize our results and comment how our rules should be applied in the reggeon calculus scheme.
2. Proper decomposition of the 2 ~ 3 and 2 ~ 4 amplitudes; comments on the 2 ~ N case In this section we review some of the results of refs. [1,3]. For derivations and detailed treatment we refer to the original papers as well as to ref. [8] and sect. 2 of ref. [4].
2.1. Preliminaries Consider the 2 ~ N process (fig. 1) in the multiregge limit:
s, si/~oo,
(2.1)
L j = I ..... N ,
with 7/k-
S k - l , k Sk, k+l Sk-l,k+l
,
k = 2 .... , N -
1,
(2.2)
and the momentum transfers tl, ..., t N_ 1 ftxed. Here s = (Pa + Pb) 2 = S l N ,
(2.3)
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(2.4)
Si/= (Pi +Pi+I + "'" +p/)2 ,
where p~ denote the four-momenta of the external particles. As usual we define two channels to be overlapping when they have particles in common but are not subchannels of each other. For instance s14 and s45 correspond to the overlapping channels, while s23 corresponds to the subchannel of s14. We also recall that according to the Steinmann relations [2] the amplitudes are not allowed to have simultaneous discontinuities in the energy variables corresponding to the overlapping channels. This means for instance that the product s ~ wer- s ~ wet must not appear in the 2 ~ 5 amplitude or in any other amplitude withN > 5. 2.Z Proper decomposition defined The multiregge amplitudes with pure pole exchange for the 2 ~ N process depend on N - 1 large energy variables chosen from the set (2. I), certain signature factors, particle-particle- reggeon vertex functions g (t) and reggeon-particle-reggeon coupling V(~k) * If the amplitude is written in a representation in which (i) large energy variables are combined in accordance with Steinmann relations; (ii) all signature factors are explicitly exhibited; (iii) the functions V(rlk) are real and analytic in their arguments, then this representation will be called the proper decomposition of the amplitude.
2. 3. Examples The proper decompositions of the 2 ~ 2, 2 ~ 3 and 2 ~ 4 amplitudes involving only pure Regge-pole exchange are the following:
(2.5)
T2--, 2 = g(t) Sa(t) ~ g(t) , T2--'3 =g(tl)g(t2)[sals23
~al ~a2-at VL(r~2)
+s~2s~-~2~ 12 ~¢~2~ ~ot I - ct2 VR(~2)] T2--, 4 = g ( t l ) g ( t 3 ) [ s
~I ¢~2--0q ~3--0~2
"t"S~3S°t2--°t3s°tl--°t2~ 13
+S
12
s24
s34
(2.6) ~Otl~Ot2--Otl~t~3--O~2 V L ( r / 2 ) V L ( r / 3 )
~j ~ --a2 YR(Y/2 ) VR('03 ) a3 oc2-¢x3 ~1
o~3 ,~,1-o~3 oc2-o~1~ ~ ~ Tr . \ S13 S23 ~Jot3~tl_~3/;a2_cxl VLI, r / 2 ) V R ( r / 3 )
* The dependence of V(rlk) on tk_ l and t k is tacitly understood here.
450
A.Z Buras / Multiparticle amplitudes + SalSa3-alsa2-a3~24 23 al ~a3-al ~a2- a3 VL(r/Z) VR (r/3) "t" ¢ ~ 2 e 0:1 -- or2 eOt3-- or2
°12
o34
h a 2 ' ~ i -- or2 ~ot3-- ~2
VR(r/2) VL(t/3)] •
(2.7)
He re
~a=
e - ina "1"7 sinrrt~ '
(2.8)
e - i~r(ai - a/) + ri r~
~a i - a / -
sin rr(a i - t~/)
(2.9)
Here, r -- +-1 is the signature. Eq. (2.5) is the usual Regge pole formula. Eqs. (2.6) and (2.7) have been derived by Drummond et al. [1] and Weis [3], respectively. The meaning of the indices L and R as well as the structure of eqs. (2.6) and (2.7) will be explained below. For N > 4 the number of terms * in the proper decompositions grows rapidly with N and each term becomes more and more complicated. Therefore it would be convenient to have simple rules for the construction of expressions like (2.7) for any N. Before we come to this we want to make a few comments about eqs. (2.6) and (2.7) which will turn out to be helpful in finding such rules. 2. 4. The structure o f the proper decomposition First notice that each term in the decomposition corresponds to a certain set of simultaneous discontinuities in the si/variables which are allowed by the Steinmann relations (see subsect. 2.1). Thus we can associate [4] with each term a diagram with a certain topology of discontinuity lines (see dashed lines in figs. 2, 3 and 5), each line corresponding to one variable sip A given topology is allowed by the Steinmann relations if (i) the lines do not intersect and (ii) no further lines can be drawn without violating (i). In order to obtain the amplitude we have to sum over all allowed topologies. In this way the amplitudes T2~ 3 and T2_ 4 as given by (2.6) and (2.7) can be represented by sets of diagrams in figs. 2 and 3, respectively. These diagrams differ from those discussed in ref. [4] in that they exhibit reggeons on the boundary of the diagram and also involve open circles at certain crossings of the reggeons with the discontinuity lines. Crossings so denoted will be termed relevant. The definition of a relevant crossing is given in sect. 4. This method of drawing diagrams, with reggeons and relevant crossings exhibited, will turn out to be very powerful in constructing the corresponding term in the proper decomposition of the amplitude. Next notice that for each r/i there are two kinds of reggeon-reggeon-particle * If k N denotes the number of terms in the proper decomposition for the 2 ~ N case, then as can be deduced from subsect. 2.4 k 3 = 2, k4 = 5, k s = 14, k 6 = 42, etc.
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451
~S~i
'S ~____
JQ
I' or-2
~i-I
I'
~i . . . .
(:zj-I
r' 'N
OLj
I~N-~
~N-I - 1
S"
Fig. l. Multiregge limit of the 2 -- N amplitude with Regge pole exchange.
2
i
I
,3
(a)
2
(b) Fig. 2. The diagrams representing the proper decomposition of the 2 --* 3 amplitude.
1
,
2,
3
,4
1
Ca) 1
2
2
4
(b) 3
4
1
2
3
4
(d)
(c) !
2
3
4
Fig. 3. The diagrams representing the proper decomposition of the 2 ~ 4 amplitude.
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452
1
2
3
4
5
'-- ..... _o"--:../___
(:I.2
6
/
~b- C......."~'" A
(b) Fig. 4. Reduction of fig. 2a to 2 ~ 2 amplitudes according to the method of ref. [4 ]. Fig. 5. One of the diagrams in the proper decomposition of the 2 --, 6 amplitude.
couplings distinguished by the indices R and L. The meaning of these indices will become clear for Bartels' construction, which we present in the next section. Here we would like to remark that for each diagram in the decomposition there is a certain combination of the couplings V with various indices L and R and one of the tasks is to find these combinations from the topology of the diagram. Our task can be summarized as follows: find simple rules, which for a' given diagram in the proper decomposition would give the powers of si/, the signature factors, as well as the labels L or R of the couplings V. Bartels' construction [4] is of great help in finding such rules.
3. Bartels' construction [4] By analyzing the arguments which led the authors of refs. [1,3] ,to expressions (2.6) and (2.7), Bartels has proposed a method which allows to construct, for any diagram with a given set of discontinuity lines, the corresponding term in the proper decomposition.
3.1. General idea Let us illustrate the idea of the method in question with a simple example. Consider the diagram of fig. 2a and denote its contribution to the 2 ~ 3 amplitude by T~1_2)3which is given by the first term of eq. (2.6). The way of reproducing T~1_2)3 from the diagram of fig. 2a is as follows. First the particles 2 and 3 are grouped into a cluster of mass squared s23 and the whole process is considered as the two-body process: a + b ~ 1 + cluster (2.3) with exchange of the reggeon ct 1 ( see fig. 4a). The
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453
corresponding amplitude is given by
with G(a 1, (2, 3), b) denoting the reggeon a 1 - cluster (2, 3) - particle b vertex. Next G(Otl, (2, 3),b) is considered as the amplitude for the process b + a 1 ~ 2 + 3 with exchange of the reggeon a 2 (see fig. 4b). Usually the signature factor for the process in question would be ~a2" However, because one of the external particles has non-integer spin e 1 ,/Ja2 is generalized to ~a2-a~ *" Thus we write G(ot 1, (2, 3), b) = VL(r/2) s ~ / J a 2 - a l g(t2) '
(3.2)
where the reggeon-reggeon-particle vertex is denoted by VL(rt2). The label L refers to the fact that the non-physical "external particle" (reggeon ¢x2) appears on the left-hand side of the vertex V. The final step is to insert (3.2) into (3.1). The resultant expression is equal to the first term in eq. (2.6). Similarly the second term of eq. (2.6) can be reproduced from the diagram of fig. 2b. This time the subprocess relevant for the construction are a + b ~ 3 + cluster (1,2) with exchange of ~2 and a + a2 ~ 1 + 2 with exchange o f o q . Now the role of the non-physical "external particle" is played by the reggeon a 2. This explains the factors/L,~-a2 and VR (rt2) **. The remaining factors in the second term of eq. (2.6) are also easily reproduced. This procedure can be easily generalized to more complicated processes. For later purposes we enumerate the essential steps in the construction. 3.2. Essential steps in the construction
In the initial step the amplitude is decomposed into terms allowed by the Steinmann relations (see subsect. 2.4). Each term in the decomposition corresponds to a diagram with a certain topology of discontinuity lines. The expression corresponding to a given diagram can be calculated by means of the following procedure: (i) The particles in the final state are grouped into clusters according to the topology of the discontinuity lines in a given diagram. For instance in the diagram of fig. 5 the particles 5 and 6 constitute the cluster B, the particles 2, 3 and 4 constitute the cluster D and so on. (ii) The diagram is decomposed into two-body processes by the subsequent "resolution" of large clusters into smaller clusters and particles until all clusters are resolved into particles. For instance the diagram of fig. 5 is decomposed into five two-body * That this interpretation makes sence has been shown in ref. [4]. In particular for integer al, eq. (3.2) reduces to a 2 -* 2 scattering amplitude with physical particles. ** Notice that the non-physical particle (reggeon a2) appears on the right-hand side of the vertex V(rl2). Therefore the label R. For the general properties of VL(n) and VR(n) see refs. [1,4].
A.J. Buras / Multiparticleamplitudes
454
processes with clusters and particles in the final state and particles as well as reggeons in the initial state (see fig. 6). This particular example will be calculated in subsect. 3.3. (iii) Because of the appearance of the unphysical particles (reggeons) in the initial state of the two-body subprocess, the usual signature factor ~ai is generalized to ~ i _ a/_..Here c~i is the exchanged reggeon, while ~i denotes the reggeon which plays the role of the external particle. For instance, the signature factor corresponding to the diagram of fig. 6b is ~as-a4' in the cases where two reggeons appear in the initial state (e.g. fig. 6d) one should, as discussed in ref. [4], choose as a~ the reggeon which has been exchanged in the preceding step of the construction. For instance, the signature factor corresponding to the diagram of fig. 6d is/Ja2-a~ and not ~a.2-a4' (iv) For each reggeon-particle-reggeon vertex the label L or R is chosen according to on which side of the vertex the unphysical particle (the reggeon) appears. For instance the diagram of fig. 6a gives VL(r/3) and VR(r/4). Following this procedure on can obtain all the terms in eq. (2.7) from the diagrams of fig. 3. This time there are three two-body processes for each diagram of fig. 3. Generally for the 2 ~ N process one has to decompose each contributing diagram into N - 1 two-body processes. Consequently, although very interesting and straightforward, the above way of obtaining the proper decomposition from the set of allowed diagrams is not very economical. Fortunately, as we shall show in sect. 4 Barrels' construction is equivalent to three simple diagrammatic rules by means of which one can immediately (without the need of decomposing the diagram into twobody processes) write a mathematical expression corresponding to a given set of discontinuity lines. The effectiveness of our rules and their equivalence to the method just discussed is nicely illustrated in the diagram of fig. 5 to which we now turn still in the context of Bartels' construction.
3. 3. The 2 ~ 6 amplitude We calculate the contribution of the diagram in fig. 5 to the 2 ~ 6 amplitude (denote this contribution by T2--,6) by decomposing the diagram into two-body processes according to the rules (i) and (ii) of the preceding subsection (see fig. 6) and then applying the rules (iii) and (iv) to each of the diagrams in fig. 6. We obtain for the diagrams (a)-(e) in fig. 6 the following expressions:
T2-,6=G( a, C , a 4 "[) ~ s) \¢q ~a4G(°t4'B,b) '
(3.3a)
G(a 4, B, b) = VL(B5) s;~ ~c~s_'~4g(t51,
(3.3b)
. . / s 1 4 ) al G(a,C, a4)=gttl)~-~24 ~ a l - a , G ( a l , D, a4),
(3.3c)
A.Z Bums / Multiparticle amplitudes,
455
~
~b (a)
(b)
a2
~/4
4
(c)
3
(d)
a3
4
(e)
Fig. 6. Reduction of fig. 5 to 2 ~ 2 amplitudes according to the method of ref. [4].
[s24)°2
G(°q, D, a4) = VL(r/2) \ s34
(3.3d)
/Ja2- al G(¢2, E, or4),
(3.3e)
G(t~2, E, o~4) = V L (r/3 ) (s34)a3 ~a 3- a2 VR (114),
where the capital letters B, C, D and E denote the clusters as shown in fig. 5 and the functions G have been defined in subsect. 3.1 (see eq. (3.1)). Putting (3.3a)-(3.3e) together we obtain after some algebra the expression corresponding to the diagram in fig. 5, ~
e e l 4 ¢rO~ 1 - - & 4 eO~2 - - ~ 1 ¢CX3 - - CX2 ¢0~5 - - Cv'4
T2-'6 =g(tl)g(t5)°
°14
°24
~34
°56
X ~0t4 ~ o t l _ ot4 ~ot2 - ot I ~0t3 - ot2 ~ot5 - ot4 V L ( r I 2 ) V L ( r l 3 ) V R ( ~ g ) V L ( ' r I 5
).
(3.4)
This is already a complicated expression and the way in which it was obtained was quite long. A natural question arises. Could we find a method by means of which expression (3.4) can be immediately read from fig. 5? An affirmative answer is given in sect. 4.
4. Simple rules for the proper decomposition Each diagram in the proper decomposition is characterized by a set of discontinuity lines, which cross the reggeons in a certain way. In order to state our rules we have to characterize the crossings of the discontinuity lines with the reggeons as well as the discontinuity lines itself.
A.J. Buras / Multiparticle amplitudes
456
4.1. Characteristics of crossings (i) A crossing is said to be of the type L (R) if the discontinuity line leaves the diagram (crossing a reggeon) in the clock-wise (counter clock-wise) direction. For instance the crossing between the line D and the reggeon % in fig. 5 is of the type R, whereas the crossing between the line E and the reggeon ~t2 is of the type L. (ii) The crossing of the type L (R) is called relevant if there is no other crossing (with the same reggeon) of the same type to the left (right) of it. The relevant crossings are marked by open circles.
4. 2. Characteristics of discontinuity lines (i) Among the reggeons, which are surrounded (not crossed) by a given discontinuity line there is one which is not separated from the line in question by other discontinuity lines. Such a reggeon we shall denote by o z. For instance for the line C in fig. 5 OZ = % while for the line D oZ = a 2. (ii) To each discontinuity line there corresponds one relevant crossing *. We denote the reggeon involved in this crossing by t~x. For instance, for the line B in fig. 5 a x = t~4 while for the line E, '~x = ~2" The only exception is the discontinuity line corresponding to the energy variable s (line A in fig. 5). In this case there is no crossing and in our formulae below we put ctx = 0. In summary each discontinuity line is characterized by the pair (oZ, etx). For instance the line E in fig. 5 is characterized by the pair (t~3, t~2).
4.3. The rules As in the method of ref. [4] in the initial step the amplitude is decomposed into terms allowed by the Steinmann relations. Each term is then represented by a diagram which exhibits a certain topology of discontinuity lines and reggeons as well as relevant crossings. Our rules for writing down a mathematical expression corresponding to a given diagram are as follows: (a) With every discontinuity line (characterized by the pair (OZ, t~x)) associate the factor ~ z - ax
s~
~az-ax'
(4.1)
where sii is the subenergy corresponding to the given line and ~az_axiS given by (2.9). For instance, the factor associated with the line D in fig. 5 is equal to
(4.2) * That there is only one relevant crossing for each discontinuity line follows from the fact that the discontinuity lines are not allowed to cross each other.
A.J. Buras /Multiparticle amplitudes
457
(b) With each reggeon-reggeon-particle k vertex associate the factor VL(r/k) or VR(r/k). Here, L or R is chosen according to on which side of the vertex there is a reggeon (attached to this vertex) which is crossed by at least one of the discontinuity lines surrounding the vertex in question *. For instance, in fig. 5 we have VL(r~2) and VR (r/4). (c) With each reggeon-reggeon-particle vertex associate g(t). Before we give arguments for the equivalence of these rules with Bartels' construction we would like to illustrate their effectiveness with a few examples.
4.4. Examples 2 -~ 6 amplitude (fig. 5j. Applying the rules to the diagram of fig. 5 we immediately obtain t~56
1 r~al-a4L 1 [ea2-al~a2_al" ' a 3 - a 2 ~ • ~as- ct4JB t°14 s a l - a4JC t~24 ID IS34 ~;a3-allE
X g(tl) g (ts) VL(r/2) VL(r/3) VR(r/4) FL(r/5),
(4.3)
which is equal to (3.4). The indices A, B, C, D and E have been put for convenience. 2 -~ 4 amplitude (fig. 3c). Our rules lead to the following expression for the diagram 3c: [sa3~a3] [Sl~-a3~al_a3] [s2~-al/ja2_al] g(tl)g(t3)VL(~2)VR(713) ,
(4.4)
which is equal to the third term in eq. (2.7) as it should be. 2 -* 4 amplitude (f~g. 3e). The corresponding expression is found to be
r al-a2~~al-c~2J l[S~-a2~a3_a2]g(tl)g(t3)ZR(r12)VL(n3), [SaZ~r,2] tSl2
(4.5)
which is just the last term in eq. (2.7). Analogously one can obtain the decomposition (2.6) from the diagrams of fig. 2 and the decomposition (2.7) from the diagrams of fig. 3. Also the example of the 2 ~ 5 amplitude discussed in ref. [4] is easily reproduced.
4. 5. Simple rules and Barrels' construction It is not difficult to show the equivalence between our rules and the method of ref. [4]. In eq " (4 " 1) sa..z~a is just a reflection of the fact that the diagram can be decomtl Z posed into two-body subprocesses dominated by the reggeons % . Because most of these processes do have a reggeon in the initial state, ~a, is modified to ~az_ax. Ac*
Because each discontinuity line surrounds at least two particles and the discontinuity lines are not allowed to cross each other there is only one reggeonwith this p r o p e r t y f o r each vertex.
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A.J. Buras / Multiparticle amplitudes
cording to rule (iii) of subsect. 3.2 c~x is equal to the reggeon which has been exchanged in the preceding step of the construction. It is obvious that this is just the reggeon which has the relevant crossing with the discontinuity line corresponding'to the subenergy si]. The appearence of the factor sit c~x in eq. (4.1) is now also clearly understood. In the preceding step of the construction, where ctx was exchanged, si] was considered as the mass of the cluster and thus appeared (for kinematic reasons) in the denominator (see for instance S34 in (3.3d)). Our rule (b) is directly related to rule (iv) of subsect. 3.2. It helps in finding immediately the right label for a given vertex from the topology of the diagram. For instance, in the case of the vertex V(~2) in fig. 5, it is difficult to decide without our rule (b) or without going back to Bartels' construction which label we should choose. Indeed since the reggeons on both sides of the vertex V(r/2) are crossed, they play the role of the initial particles in certain subprocesses of the construction of ref. [4] and consequently decide about the labels of the vertices V. Rule (b) in this case helps in finding which of the two reggeons (cq and a2) is responsible for the label of the vertex V(r/2), It chooses the reggeon O~1 (i.e. the label L) in accordance with the method of ref. [4]. Rule (c) is obvious. In summary, our simple rules are equivalent to Bartels' construction.
5. Summary and final remarks In this paper we have developed a simple method for the construction of the proper decomposition of the multiparticle amplitude. The method consists of two steps. In the first step one decomposes the amplitude into terms allowed by the Steinmann relations and associates a diagram with each term (see subsect. 2.4). For a 2 --' N amplitude only diagrams with N - 1 non-intersecting discontinuity lines are allowed. In this context the concept of so called relevant crossings (see subsect. 4.1) turns out to be useful. In the second step the mathematical expression corresponding to a given diagram is found by means of the rules of subsect. 4.3. This method, while being equivalent to Bartels' construction [4], has the advantage over the latter in that even for higher values of N the expression correspond ing to each term in the decomposition can be found directly from the topology of the corresponding diagram. For completeness, our final remarks concern the application of the rules just developed in the framework of the reggeon calculus [6,9]. Until now the discussion was entirely based on the amplitudes with pure Regge pole exchange. Fortunately, Barrels has shown [5] that the general structure of the proper decomposition remains unaffected when Regge cuts are included. The only difference is that to each term in the decomposition there are contributions from many reggeon diagrams.
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For.instance, the 2 ~ 3 amplitudes in the reggeon calculus can be cast [5] in the form of the proper decomposition as follows *: 1
T 2 - (2hi) 2
f f d Jl d J2 (sJ' sJ~-J' ~J1~J2- JI FL(r/2 ;J1, J2)
J2 JI-J2
s S12
~J2~Jt-J2FR(~12;J1, J2)}"
(5.1)
Here the amplitude is written as a double Mellin transform with F L and F R being the partial wave amplitudes, which contian the whole information about the/'-plane structure. They receive contributions from both poles and cuts. They can be calculated by means of reggeon rules which are a straightforward extension [5] of the original Gribov rules for the elastic amplitude [9]. The index L (R) indicates that in calculating F L (FR) one should use the vertex VL (VR). I f F L and F R contain only pole contributions, eq. (5.1) reduces to (2.6) ** Similar formulae can be written for more complicated processes. They are always straightforward generalizations of the expressions which we have discussed extensively in this paper. For instance the third term of eq. (2.7) is generalized to
(21i)3yff dJl dJ2dJ3 sJaSl3JI-J3 s23J2-Jl~J3~Jl_J3 X ~J2- Ji FLR (r/2' r/3 ; J l, J2, J3 )"
(5.2)
Here again, FLR is calculated by means of the reggeon rules. This time the vertices VL(r/2) and VR(r/3) enter the calculus. We refer to ref. [5] for other examples. This discussion makes it clear that the evaluation of any amplitude in the reggeon calculus scheme naturally proceeds in two steps. In the first step the amplitude is written in the form of the proper decomposition. Here the simple rules of subsect. 4.3 can be utilized. In the second step the partial wave amplitudes (e.g. FLR in eq. (5.2)) are calculated by means of the reggeon rules [5,9]. From what we have just said it is evident that the proper decomposition is an important ingredient in the reggeon calculus for multiparticle processes. We believe that this is the case of any scheme of multiparticle amplitudes. If so, our simple rules are of general usefulness. I am grateful to S. Chadha for interesting discussions and a careful reading of the manuscript. I thank H.B. Nielsen, A.B. Kraemmer and P. Olesen for helpful comments.
* We suppress the m o m e n t u m transfers. ** Notice that the v e r t i c e s g ( t ) are included in the functions F.
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A.J. Bums / Multiparticle amplitudes
References [ 1] i.T. Drummond, P.V. Landshoff and W.J. Zakrzewski, Phys. Letters 28B (1969) 676; Nucl. Phys. Bll (1969) 383. [2] O. Steinmann, Heir. Phys. Acta 33 (1960) 257,347. [3] J.H. Weis, Phys. Rev. D4 (1971) 1777; D5 (1972) 1043. [4] J. Bartels, Phys. Rev. D11 (1975) 2977. [5] J. Bartels, Phys. Rev. D l l (1975) 2989. [6] H.D.l. Abarbanel, J.B. Bronzan, R.L. Sugar and A.R. White, Phys. Reports 21 (1975) 119. [7] A.R. White, Nucl. Phys. B67 (1973) 189. [8] R.C. Brower, C.E. De Tar and J.H. Weis, Phys. Reports 14 (1974) 257. [9] V.N. Gribov, JETP (Soy. Phys.) 26 (1968) 414.
SIMPLE RULES FOR THE SIGNATURE AND THE SINGULARITY STRUCTURE OF THE MULTIPARTICLE AMPLITUDES A.J. BURAS * The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark Received 20 August 1975 We show that the representation of the 2 ~ N amplitude in the multiregge limit which exhibits signature factors and singularity structure allowed by the Steinmann relations can be reproduced by means of a few simple rules. We illustrate the rules with several examples and point out their importance for the reggeoncalculus.
1. Introduction Some time ago Drummond, Landshoff and Zakrzewski [1 ] found a representation of the 2 ~ 3 production amplitude in the multiregge limit which exhibits signature factors and singularity structure allowed by the Steinmann relations [2]. This representation has been subsequently generalized to the 2 ~ 4 case by Weis [3] and recently to the 2 ~ N case by Bartels [4]. The representation has a structure of a decomposition of the amplitude into terms each of which corresponds to a certain set of simultaneous singularities in the energy variables. Furthermore each term in the decomposition is characterized by certain signature factors and real analytic coefficient functions. Typical examples are given in sect. 2. Such a representation of the amplitude (hereafter called the proper decomposition o f the amplitude) was originally discussed for amplitudes with pure Regge pole exchange only [1,3]. Recently it has been shown by Bartels [5], on the basis of hybrid Feynman diagrams for the production amplitude, that the proper decomposition remains valid in the presence of cuts and consequently is one of the ingredients of the reggeon calculus scheme for the multiparticle processes [6]. Indeed the 2 ~ N amplitude in the reggeon calculus can be written in the form of the proper decomposition with the coefficient functions receiving contributions from both poles and cuts. We shall discuss this in sect. 5. Furthermore the validity of the proper decomposition for the particular case of the 2 ~ 3 amplitude has been proved by partial wave analysis of the five-point function [7]. * Present address: CERN, Geneva, Switzerland. 447
A.J. Bums / Multiparticle amplitudes
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All this indicates that in any scheme which takes into account both the signature and the analytic structure of the multiparticle amplitudes, the proper decomposition must be an important ingredient. Therefore it is of interest to develop a systematic method for writing down the proper decomposition of any 2 ~ N amplitude. As will be clear from sect. 2, the number of terms in the proper decompositon grows rapidly with N and each term becomes increasingly more complicated. It is therefore desirable to have a simple and efficient technique for the calculation of each term of the decomposition. Recently Bartels [4] demonstrated how each term in the decomposition can be understood in terms of the 2 ~ 2 amplitudes. He has also shown how each term can be constructed once the set of simultaneous singularities corresponding to it is given. Unfortunately Bartels construction, when applied to processes with N > 3 is quite laborious and it is desirable to find a simpler method. In this paper we present such a method. It is based on three simple rules, which we have found by analyzing Barrels' construction. The paper is organized as follows: in sect. 2 we review briefly the arguments for the proper decomposition by discussing 2 ~ 3 and 2 ~ 4 amplitudes. In sect. 3 we present the general idea behind Bartels' construction and illustrate it using the 2 -* 6 amplitude. In sect. 4 we formulate our rules and show that they are equivalent to the method of ref. [4]. The effectiveness of our method is illustrated for a few examples. Finally in sect. 5 we summarize our results and comment how our rules should be applied in the reggeon calculus scheme.
2. Proper decomposition of the 2 ~ 3 and 2 ~ 4 amplitudes; comments on the 2 ~ N case In this section we review some of the results of refs. [1,3]. For derivations and detailed treatment we refer to the original papers as well as to ref. [8] and sect. 2 of ref. [4].
2.1. Preliminaries Consider the 2 ~ N process (fig. 1) in the multiregge limit:
s, si/~oo,
(2.1)
L j = I ..... N ,
with 7/k-
S k - l , k Sk, k+l Sk-l,k+l
,
k = 2 .... , N -
1,
(2.2)
and the momentum transfers tl, ..., t N_ 1 ftxed. Here s = (Pa + Pb) 2 = S l N ,
(2.3)
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(2.4)
Si/= (Pi +Pi+I + "'" +p/)2 ,
where p~ denote the four-momenta of the external particles. As usual we define two channels to be overlapping when they have particles in common but are not subchannels of each other. For instance s14 and s45 correspond to the overlapping channels, while s23 corresponds to the subchannel of s14. We also recall that according to the Steinmann relations [2] the amplitudes are not allowed to have simultaneous discontinuities in the energy variables corresponding to the overlapping channels. This means for instance that the product s ~ wer- s ~ wet must not appear in the 2 ~ 5 amplitude or in any other amplitude withN > 5. 2.Z Proper decomposition defined The multiregge amplitudes with pure pole exchange for the 2 ~ N process depend on N - 1 large energy variables chosen from the set (2. I), certain signature factors, particle-particle- reggeon vertex functions g (t) and reggeon-particle-reggeon coupling V(~k) * If the amplitude is written in a representation in which (i) large energy variables are combined in accordance with Steinmann relations; (ii) all signature factors are explicitly exhibited; (iii) the functions V(rlk) are real and analytic in their arguments, then this representation will be called the proper decomposition of the amplitude.
2. 3. Examples The proper decompositions of the 2 ~ 2, 2 ~ 3 and 2 ~ 4 amplitudes involving only pure Regge-pole exchange are the following:
(2.5)
T2--, 2 = g(t) Sa(t) ~ g(t) , T2--'3 =g(tl)g(t2)[sals23
~al ~a2-at VL(r~2)
+s~2s~-~2~ 12 ~¢~2~ ~ot I - ct2 VR(~2)] T2--, 4 = g ( t l ) g ( t 3 ) [ s
~I ¢~2--0q ~3--0~2
"t"S~3S°t2--°t3s°tl--°t2~ 13
+S
12
s24
s34
(2.6) ~Otl~Ot2--Otl~t~3--O~2 V L ( r / 2 ) V L ( r / 3 )
~j ~ --a2 YR(Y/2 ) VR('03 ) a3 oc2-¢x3 ~1
o~3 ,~,1-o~3 oc2-o~1~ ~ ~ Tr . \ S13 S23 ~Jot3~tl_~3/;a2_cxl VLI, r / 2 ) V R ( r / 3 )
* The dependence of V(rlk) on tk_ l and t k is tacitly understood here.
450
A.Z Buras / Multiparticle amplitudes + SalSa3-alsa2-a3~24 23 al ~a3-al ~a2- a3 VL(r/Z) VR (r/3) "t" ¢ ~ 2 e 0:1 -- or2 eOt3-- or2
°12
o34
h a 2 ' ~ i -- or2 ~ot3-- ~2
VR(r/2) VL(t/3)] •
(2.7)
He re
~a=
e - ina "1"7 sinrrt~ '
(2.8)
e - i~r(ai - a/) + ri r~
~a i - a / -
sin rr(a i - t~/)
(2.9)
Here, r -- +-1 is the signature. Eq. (2.5) is the usual Regge pole formula. Eqs. (2.6) and (2.7) have been derived by Drummond et al. [1] and Weis [3], respectively. The meaning of the indices L and R as well as the structure of eqs. (2.6) and (2.7) will be explained below. For N > 4 the number of terms * in the proper decompositions grows rapidly with N and each term becomes more and more complicated. Therefore it would be convenient to have simple rules for the construction of expressions like (2.7) for any N. Before we come to this we want to make a few comments about eqs. (2.6) and (2.7) which will turn out to be helpful in finding such rules. 2. 4. The structure o f the proper decomposition First notice that each term in the decomposition corresponds to a certain set of simultaneous discontinuities in the si/variables which are allowed by the Steinmann relations (see subsect. 2.1). Thus we can associate [4] with each term a diagram with a certain topology of discontinuity lines (see dashed lines in figs. 2, 3 and 5), each line corresponding to one variable sip A given topology is allowed by the Steinmann relations if (i) the lines do not intersect and (ii) no further lines can be drawn without violating (i). In order to obtain the amplitude we have to sum over all allowed topologies. In this way the amplitudes T2~ 3 and T2_ 4 as given by (2.6) and (2.7) can be represented by sets of diagrams in figs. 2 and 3, respectively. These diagrams differ from those discussed in ref. [4] in that they exhibit reggeons on the boundary of the diagram and also involve open circles at certain crossings of the reggeons with the discontinuity lines. Crossings so denoted will be termed relevant. The definition of a relevant crossing is given in sect. 4. This method of drawing diagrams, with reggeons and relevant crossings exhibited, will turn out to be very powerful in constructing the corresponding term in the proper decomposition of the amplitude. Next notice that for each r/i there are two kinds of reggeon-reggeon-particle * If k N denotes the number of terms in the proper decomposition for the 2 ~ N case, then as can be deduced from subsect. 2.4 k 3 = 2, k4 = 5, k s = 14, k 6 = 42, etc.
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451
~S~i
'S ~____
JQ
I' or-2
~i-I
I'
~i . . . .
(:zj-I
r' 'N
OLj
I~N-~
~N-I - 1
S"
Fig. l. Multiregge limit of the 2 -- N amplitude with Regge pole exchange.
2
i
I
,3
(a)
2
(b) Fig. 2. The diagrams representing the proper decomposition of the 2 --* 3 amplitude.
1
,
2,
3
,4
1
Ca) 1
2
2
4
(b) 3
4
1
2
3
4
(d)
(c) !
2
3
4
Fig. 3. The diagrams representing the proper decomposition of the 2 ~ 4 amplitude.
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452
1
2
3
4
5
'-- ..... _o"--:../___
(:I.2
6
/
~b- C......."~'" A
(b) Fig. 4. Reduction of fig. 2a to 2 ~ 2 amplitudes according to the method of ref. [4 ]. Fig. 5. One of the diagrams in the proper decomposition of the 2 --, 6 amplitude.
couplings distinguished by the indices R and L. The meaning of these indices will become clear for Bartels' construction, which we present in the next section. Here we would like to remark that for each diagram in the decomposition there is a certain combination of the couplings V with various indices L and R and one of the tasks is to find these combinations from the topology of the diagram. Our task can be summarized as follows: find simple rules, which for a' given diagram in the proper decomposition would give the powers of si/, the signature factors, as well as the labels L or R of the couplings V. Bartels' construction [4] is of great help in finding such rules.
3. Bartels' construction [4] By analyzing the arguments which led the authors of refs. [1,3] ,to expressions (2.6) and (2.7), Bartels has proposed a method which allows to construct, for any diagram with a given set of discontinuity lines, the corresponding term in the proper decomposition.
3.1. General idea Let us illustrate the idea of the method in question with a simple example. Consider the diagram of fig. 2a and denote its contribution to the 2 ~ 3 amplitude by T~1_2)3which is given by the first term of eq. (2.6). The way of reproducing T~1_2)3 from the diagram of fig. 2a is as follows. First the particles 2 and 3 are grouped into a cluster of mass squared s23 and the whole process is considered as the two-body process: a + b ~ 1 + cluster (2.3) with exchange of the reggeon ct 1 ( see fig. 4a). The
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453
corresponding amplitude is given by
with G(a 1, (2, 3), b) denoting the reggeon a 1 - cluster (2, 3) - particle b vertex. Next G(Otl, (2, 3),b) is considered as the amplitude for the process b + a 1 ~ 2 + 3 with exchange of the reggeon a 2 (see fig. 4b). Usually the signature factor for the process in question would be ~a2" However, because one of the external particles has non-integer spin e 1 ,/Ja2 is generalized to ~a2-a~ *" Thus we write G(ot 1, (2, 3), b) = VL(r/2) s ~ / J a 2 - a l g(t2) '
(3.2)
where the reggeon-reggeon-particle vertex is denoted by VL(rt2). The label L refers to the fact that the non-physical "external particle" (reggeon ¢x2) appears on the left-hand side of the vertex V. The final step is to insert (3.2) into (3.1). The resultant expression is equal to the first term in eq. (2.6). Similarly the second term of eq. (2.6) can be reproduced from the diagram of fig. 2b. This time the subprocess relevant for the construction are a + b ~ 3 + cluster (1,2) with exchange of ~2 and a + a2 ~ 1 + 2 with exchange o f o q . Now the role of the non-physical "external particle" is played by the reggeon a 2. This explains the factors/L,~-a2 and VR (rt2) **. The remaining factors in the second term of eq. (2.6) are also easily reproduced. This procedure can be easily generalized to more complicated processes. For later purposes we enumerate the essential steps in the construction. 3.2. Essential steps in the construction
In the initial step the amplitude is decomposed into terms allowed by the Steinmann relations (see subsect. 2.4). Each term in the decomposition corresponds to a diagram with a certain topology of discontinuity lines. The expression corresponding to a given diagram can be calculated by means of the following procedure: (i) The particles in the final state are grouped into clusters according to the topology of the discontinuity lines in a given diagram. For instance in the diagram of fig. 5 the particles 5 and 6 constitute the cluster B, the particles 2, 3 and 4 constitute the cluster D and so on. (ii) The diagram is decomposed into two-body processes by the subsequent "resolution" of large clusters into smaller clusters and particles until all clusters are resolved into particles. For instance the diagram of fig. 5 is decomposed into five two-body * That this interpretation makes sence has been shown in ref. [4]. In particular for integer al, eq. (3.2) reduces to a 2 -* 2 scattering amplitude with physical particles. ** Notice that the non-physical particle (reggeon a2) appears on the right-hand side of the vertex V(rl2). Therefore the label R. For the general properties of VL(n) and VR(n) see refs. [1,4].
A.J. Buras / Multiparticleamplitudes
454
processes with clusters and particles in the final state and particles as well as reggeons in the initial state (see fig. 6). This particular example will be calculated in subsect. 3.3. (iii) Because of the appearance of the unphysical particles (reggeons) in the initial state of the two-body subprocess, the usual signature factor ~ai is generalized to ~ i _ a/_..Here c~i is the exchanged reggeon, while ~i denotes the reggeon which plays the role of the external particle. For instance, the signature factor corresponding to the diagram of fig. 6b is ~as-a4' in the cases where two reggeons appear in the initial state (e.g. fig. 6d) one should, as discussed in ref. [4], choose as a~ the reggeon which has been exchanged in the preceding step of the construction. For instance, the signature factor corresponding to the diagram of fig. 6d is/Ja2-a~ and not ~a.2-a4' (iv) For each reggeon-particle-reggeon vertex the label L or R is chosen according to on which side of the vertex the unphysical particle (the reggeon) appears. For instance the diagram of fig. 6a gives VL(r/3) and VR(r/4). Following this procedure on can obtain all the terms in eq. (2.7) from the diagrams of fig. 3. This time there are three two-body processes for each diagram of fig. 3. Generally for the 2 ~ N process one has to decompose each contributing diagram into N - 1 two-body processes. Consequently, although very interesting and straightforward, the above way of obtaining the proper decomposition from the set of allowed diagrams is not very economical. Fortunately, as we shall show in sect. 4 Barrels' construction is equivalent to three simple diagrammatic rules by means of which one can immediately (without the need of decomposing the diagram into twobody processes) write a mathematical expression corresponding to a given set of discontinuity lines. The effectiveness of our rules and their equivalence to the method just discussed is nicely illustrated in the diagram of fig. 5 to which we now turn still in the context of Bartels' construction.
3. 3. The 2 ~ 6 amplitude We calculate the contribution of the diagram in fig. 5 to the 2 ~ 6 amplitude (denote this contribution by T2--,6) by decomposing the diagram into two-body processes according to the rules (i) and (ii) of the preceding subsection (see fig. 6) and then applying the rules (iii) and (iv) to each of the diagrams in fig. 6. We obtain for the diagrams (a)-(e) in fig. 6 the following expressions:
T2-,6=G( a, C , a 4 "[) ~ s) \¢q ~a4G(°t4'B,b) '
(3.3a)
G(a 4, B, b) = VL(B5) s;~ ~c~s_'~4g(t51,
(3.3b)
. . / s 1 4 ) al G(a,C, a4)=gttl)~-~24 ~ a l - a , G ( a l , D, a4),
(3.3c)
A.Z Bums / Multiparticle amplitudes,
455
~
~b (a)
(b)
a2
~/4
4
(c)
3
(d)
a3
4
(e)
Fig. 6. Reduction of fig. 5 to 2 ~ 2 amplitudes according to the method of ref. [4].
[s24)°2
G(°q, D, a4) = VL(r/2) \ s34
(3.3d)
/Ja2- al G(¢2, E, or4),
(3.3e)
G(t~2, E, o~4) = V L (r/3 ) (s34)a3 ~a 3- a2 VR (114),
where the capital letters B, C, D and E denote the clusters as shown in fig. 5 and the functions G have been defined in subsect. 3.1 (see eq. (3.1)). Putting (3.3a)-(3.3e) together we obtain after some algebra the expression corresponding to the diagram in fig. 5, ~
e e l 4 ¢rO~ 1 - - & 4 eO~2 - - ~ 1 ¢CX3 - - CX2 ¢0~5 - - Cv'4
T2-'6 =g(tl)g(t5)°
°14
°24
~34
°56
X ~0t4 ~ o t l _ ot4 ~ot2 - ot I ~0t3 - ot2 ~ot5 - ot4 V L ( r I 2 ) V L ( r l 3 ) V R ( ~ g ) V L ( ' r I 5
).
(3.4)
This is already a complicated expression and the way in which it was obtained was quite long. A natural question arises. Could we find a method by means of which expression (3.4) can be immediately read from fig. 5? An affirmative answer is given in sect. 4.
4. Simple rules for the proper decomposition Each diagram in the proper decomposition is characterized by a set of discontinuity lines, which cross the reggeons in a certain way. In order to state our rules we have to characterize the crossings of the discontinuity lines with the reggeons as well as the discontinuity lines itself.
A.J. Buras / Multiparticle amplitudes
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4.1. Characteristics of crossings (i) A crossing is said to be of the type L (R) if the discontinuity line leaves the diagram (crossing a reggeon) in the clock-wise (counter clock-wise) direction. For instance the crossing between the line D and the reggeon % in fig. 5 is of the type R, whereas the crossing between the line E and the reggeon ~t2 is of the type L. (ii) The crossing of the type L (R) is called relevant if there is no other crossing (with the same reggeon) of the same type to the left (right) of it. The relevant crossings are marked by open circles.
4. 2. Characteristics of discontinuity lines (i) Among the reggeons, which are surrounded (not crossed) by a given discontinuity line there is one which is not separated from the line in question by other discontinuity lines. Such a reggeon we shall denote by o z. For instance for the line C in fig. 5 OZ = % while for the line D oZ = a 2. (ii) To each discontinuity line there corresponds one relevant crossing *. We denote the reggeon involved in this crossing by t~x. For instance, for the line B in fig. 5 a x = t~4 while for the line E, '~x = ~2" The only exception is the discontinuity line corresponding to the energy variable s (line A in fig. 5). In this case there is no crossing and in our formulae below we put ctx = 0. In summary each discontinuity line is characterized by the pair (oZ, etx). For instance the line E in fig. 5 is characterized by the pair (t~3, t~2).
4.3. The rules As in the method of ref. [4] in the initial step the amplitude is decomposed into terms allowed by the Steinmann relations. Each term is then represented by a diagram which exhibits a certain topology of discontinuity lines and reggeons as well as relevant crossings. Our rules for writing down a mathematical expression corresponding to a given diagram are as follows: (a) With every discontinuity line (characterized by the pair (OZ, t~x)) associate the factor ~ z - ax
s~
~az-ax'
(4.1)
where sii is the subenergy corresponding to the given line and ~az_axiS given by (2.9). For instance, the factor associated with the line D in fig. 5 is equal to
(4.2) * That there is only one relevant crossing for each discontinuity line follows from the fact that the discontinuity lines are not allowed to cross each other.
A.J. Buras /Multiparticle amplitudes
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(b) With each reggeon-reggeon-particle k vertex associate the factor VL(r/k) or VR(r/k). Here, L or R is chosen according to on which side of the vertex there is a reggeon (attached to this vertex) which is crossed by at least one of the discontinuity lines surrounding the vertex in question *. For instance, in fig. 5 we have VL(r~2) and VR (r/4). (c) With each reggeon-reggeon-particle vertex associate g(t). Before we give arguments for the equivalence of these rules with Bartels' construction we would like to illustrate their effectiveness with a few examples.
4.4. Examples 2 -~ 6 amplitude (fig. 5j. Applying the rules to the diagram of fig. 5 we immediately obtain t~56
1 r~al-a4L 1 [ea2-al~a2_al" ' a 3 - a 2 ~ • ~as- ct4JB t°14 s a l - a4JC t~24 ID IS34 ~;a3-allE
X g(tl) g (ts) VL(r/2) VL(r/3) VR(r/4) FL(r/5),
(4.3)
which is equal to (3.4). The indices A, B, C, D and E have been put for convenience. 2 -~ 4 amplitude (fig. 3c). Our rules lead to the following expression for the diagram 3c: [sa3~a3] [Sl~-a3~al_a3] [s2~-al/ja2_al] g(tl)g(t3)VL(~2)VR(713) ,
(4.4)
which is equal to the third term in eq. (2.7) as it should be. 2 -* 4 amplitude (f~g. 3e). The corresponding expression is found to be
r al-a2~~al-c~2J l[S~-a2~a3_a2]g(tl)g(t3)ZR(r12)VL(n3), [SaZ~r,2] tSl2
(4.5)
which is just the last term in eq. (2.7). Analogously one can obtain the decomposition (2.6) from the diagrams of fig. 2 and the decomposition (2.7) from the diagrams of fig. 3. Also the example of the 2 ~ 5 amplitude discussed in ref. [4] is easily reproduced.
4. 5. Simple rules and Barrels' construction It is not difficult to show the equivalence between our rules and the method of ref. [4]. In eq " (4 " 1) sa..z~a is just a reflection of the fact that the diagram can be decomtl Z posed into two-body subprocesses dominated by the reggeons % . Because most of these processes do have a reggeon in the initial state, ~a, is modified to ~az_ax. Ac*
Because each discontinuity line surrounds at least two particles and the discontinuity lines are not allowed to cross each other there is only one reggeonwith this p r o p e r t y f o r each vertex.
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A.J. Buras / Multiparticle amplitudes
cording to rule (iii) of subsect. 3.2 c~x is equal to the reggeon which has been exchanged in the preceding step of the construction. It is obvious that this is just the reggeon which has the relevant crossing with the discontinuity line corresponding'to the subenergy si]. The appearence of the factor sit c~x in eq. (4.1) is now also clearly understood. In the preceding step of the construction, where ctx was exchanged, si] was considered as the mass of the cluster and thus appeared (for kinematic reasons) in the denominator (see for instance S34 in (3.3d)). Our rule (b) is directly related to rule (iv) of subsect. 3.2. It helps in finding immediately the right label for a given vertex from the topology of the diagram. For instance, in the case of the vertex V(~2) in fig. 5, it is difficult to decide without our rule (b) or without going back to Bartels' construction which label we should choose. Indeed since the reggeons on both sides of the vertex V(r/2) are crossed, they play the role of the initial particles in certain subprocesses of the construction of ref. [4] and consequently decide about the labels of the vertices V. Rule (b) in this case helps in finding which of the two reggeons (cq and a2) is responsible for the label of the vertex V(r/2), It chooses the reggeon O~1 (i.e. the label L) in accordance with the method of ref. [4]. Rule (c) is obvious. In summary, our simple rules are equivalent to Bartels' construction.
5. Summary and final remarks In this paper we have developed a simple method for the construction of the proper decomposition of the multiparticle amplitude. The method consists of two steps. In the first step one decomposes the amplitude into terms allowed by the Steinmann relations and associates a diagram with each term (see subsect. 2.4). For a 2 --' N amplitude only diagrams with N - 1 non-intersecting discontinuity lines are allowed. In this context the concept of so called relevant crossings (see subsect. 4.1) turns out to be useful. In the second step the mathematical expression corresponding to a given diagram is found by means of the rules of subsect. 4.3. This method, while being equivalent to Bartels' construction [4], has the advantage over the latter in that even for higher values of N the expression correspond ing to each term in the decomposition can be found directly from the topology of the corresponding diagram. For completeness, our final remarks concern the application of the rules just developed in the framework of the reggeon calculus [6,9]. Until now the discussion was entirely based on the amplitudes with pure Regge pole exchange. Fortunately, Barrels has shown [5] that the general structure of the proper decomposition remains unaffected when Regge cuts are included. The only difference is that to each term in the decomposition there are contributions from many reggeon diagrams.
A.Z Buras/ Multiparticleamplitudes
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For.instance, the 2 ~ 3 amplitudes in the reggeon calculus can be cast [5] in the form of the proper decomposition as follows *: 1
T 2 - (2hi) 2
f f d Jl d J2 (sJ' sJ~-J' ~J1~J2- JI FL(r/2 ;J1, J2)
J2 JI-J2
s S12
~J2~Jt-J2FR(~12;J1, J2)}"
(5.1)
Here the amplitude is written as a double Mellin transform with F L and F R being the partial wave amplitudes, which contian the whole information about the/'-plane structure. They receive contributions from both poles and cuts. They can be calculated by means of reggeon rules which are a straightforward extension [5] of the original Gribov rules for the elastic amplitude [9]. The index L (R) indicates that in calculating F L (FR) one should use the vertex VL (VR). I f F L and F R contain only pole contributions, eq. (5.1) reduces to (2.6) ** Similar formulae can be written for more complicated processes. They are always straightforward generalizations of the expressions which we have discussed extensively in this paper. For instance the third term of eq. (2.7) is generalized to
(21i)3yff dJl dJ2dJ3 sJaSl3JI-J3 s23J2-Jl~J3~Jl_J3 X ~J2- Ji FLR (r/2' r/3 ; J l, J2, J3 )"
(5.2)
Here again, FLR is calculated by means of the reggeon rules. This time the vertices VL(r/2) and VR(r/3) enter the calculus. We refer to ref. [5] for other examples. This discussion makes it clear that the evaluation of any amplitude in the reggeon calculus scheme naturally proceeds in two steps. In the first step the amplitude is written in the form of the proper decomposition. Here the simple rules of subsect. 4.3 can be utilized. In the second step the partial wave amplitudes (e.g. FLR in eq. (5.2)) are calculated by means of the reggeon rules [5,9]. From what we have just said it is evident that the proper decomposition is an important ingredient in the reggeon calculus for multiparticle processes. We believe that this is the case of any scheme of multiparticle amplitudes. If so, our simple rules are of general usefulness. I am grateful to S. Chadha for interesting discussions and a careful reading of the manuscript. I thank H.B. Nielsen, A.B. Kraemmer and P. Olesen for helpful comments.
* We suppress the m o m e n t u m transfers. ** Notice that the v e r t i c e s g ( t ) are included in the functions F.
460
A.J. Bums / Multiparticle amplitudes
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