Symmetry properties, tests, and reduction of the crossing matrix

ANNALS

OF PHYSICS:

Symmetry

62, 492-543 (1971)

Properties,

Tests, and Reduction JAMIL

Department

of Physics,

Tetttple

of the Crossing

Matrix

DABOUL Universiry.

Philadelphia,

Pennsylvania

19122

Received May 6, 1970

W/e derive symmetry properties of the crossing arguments. These properties can be used

matrix

from

general symmetry

(a) To test the known crossing matrices. We find that the crossing matrix of C-TMN [l] satisfies these conditions only if its overall phase factor is modified. We also show that the crossing matrix of Trueman and Wick [2], as we interpret it, is equivalent to that of C-TMN if one takes into account the different continuation paths. (b) To reduce the crossing matrix into two or more submatrices, such that one submatrix connects “symmetry-conserving” amplitudes of the s and i channels with each other, whereas the other submatrix connects only the “symmetry-breaking” amplitudes of the two channels with each other. This fact may be useful for bootstrap calculation. In addition, we derive in a simple way the crossing matrices for the crossing of any two particles in terms of that for particles 1 and 4. Furthermore, we derive in the appendices the exact symmetry relations of the c.m. helicity amplitudes under T, CPT, E,, , E:,, , and E for general reactions using consistent intrinsic phases, as these relations are not available in the literature. Our method is simpler than that of Jacob and Wick [3], since it doesn’t involve any partial wave amplitudes.

1. INTRODUCTION

The crossing matrix [l, 2, 41 relates the cm. helicity amplitudes for the s channel

R = (1 + 2 -+ 3 + 4) to the analytic continuation of the amplitudes for the t channel XR = (4 + 2 ---f 3 + i), where X = XI, denotes the crossing operation. The crossing matrix is useful in Regge pole calculations, if one is interested in calculating individual s-channel amplitudes from Reggeized t-channel amplitudes. However, its explicit form is not necessary for calculating differential cross sections -for this it suffices to know that the crossing matrix is unitary. The crossing matrix has also been used for deriving kinematic singularities and constraints of c.m. helicity amplitudes [l, 5, 61. The history of the crossing matrix is quite interesting. Apparently, three Russian physicists [7] who were quoted by TW were the first to attempt to derive the crossing matrix. The two quotations referred to preprints, so we were unable to see the 492

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original papers. But according to TW, these authors derived expressions with wrong phases, and also they did not dicuss the continuation paths. In their wellknown paper TW used the general helicity amplitudes as an intermediate step to derive the crossing matrix. Shortly afterwards, Muzinich [4] clarified this derivation by explaining it in terms of the crossing of the M functions. A few years later, C-TMN rederived the crossing matrix using a different continuation path from the one of TW. C-TMN worked with the M functions directly without using general helicity amplitudes. This saved them the trouble of analytically continuing these amplitudes to the crossed momenta XK = (k, -- k, ; -k&J. C-TMN also derived the overall phase factor of the crossing matrix, which TW had neglected. The crossing matrix of C-TMN looks formally like the one of TW except for nonoverall factors. However, see Eq. (I. 1) below. The present paper is technical in nature, as all the previous papers on the crossing matrix. It deals with four essentially different but interrelated topics. The reader who may be interested in only one of these topics should be able to read through the sections of interest without reading the others. The first and the main topic is concerned with the derivation and the application of several symmetry properties of the crossing matrix (see Eqs. (3.1))(3.7)). These properties are derived using relations among products of symmetry operations. The multiplication of symmetry operations was defined and discussed in an earlier paper [8], and will be reviewed partly in Sections 2 and 3. The above symmetry relations of the crossing matrix can be used as conditions to test the known expressions of the crossing matrix, and in particular to test its overall phase factor. This factor is important for determining the sign of residues in Regge pole calculations. We find that the crossing matrix of C-TMN can satisfy all the above conditions if we modify its overall phase factors. The discrepancy of the phase comes about since C-TMN use the order YQ$,$,4,&} for the timeordered product instead of T{&$1} for defining the amplitudes. Some of the above symmetry conditions have another application besides testing the crossing matrix: They can be used to reduce the crossing matrix into two or more orthogonal submatrices. This reduction could be of importance in bootstrap calculations. It has a simple geometrical interpretation which is discussed in Section 5. The second topic deals with simple relations between the crossing matrices for the operations Xi? and that for X,, , where Xij is the crossing of the particles in positions i and j. These relations provide a simple method of evaluating the crossing matrices for Xij, which includes the crossing matrix from the s to the u channel. The third topic deals with the discrepancy between the crossing matrices derived by TW and C-TMN using different continuation paths. At first sight, the two matrices seem to differ by nonoverall phases. However, a careful look shows that the two crossing matrices are indeed different but not by nonoverall phases: There

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is no simple relationship between the same elements of the two matrices. Instead there is a simple relationship between different matrix elements, viz., in ST, in T-l,

(1.1)

with h = &A, ; X,X,)

and

where the crossing matrices are defined in Section 2, 5 and 5’ are overall constants, and S+ and T+ refer to the physical regions of the s and t channels. Apparently, when C-TMN compared their results with those of TW, they did not take into account that the quantities used by TW were defined differently from theirs due to the different continuation paths. In fact, if we assume that the two paths lead to the same analytic continuations of the M functions, then we can account for the relations (1.1) and show that the two crossing matrices are equivalent to each other. The disagreement arises because the analytically continued f-channel amplitudes as defined by TW and C-TMN lie, roughly speaking, on two different sheets, which makes them differ by more than phases. The fourth topic deals with the exact symmetry relations among the cm. amplitudes for T, CPT, E,, , and Es4 for general reactions, where Eij are the exchange of particles in positions i and j. These phases are needed for evaluating the above symmetry conditions, and they are not available in the literature. For example, we were unable to find in the literature the general symmetry relation of the c.m. amplitudes under T; when Jacob and Wick [3] discussed time reversal, they mentioned two conventions for defining qr, the intrinsic phase for T, viz., the Wigner-Eisenbud and the Coester conventions [9]. But when they derived the symmetry of the c.m. amplitudes under T, they neglected these vT completely, arguing that one could always redefine the amplitudes such that the 7T would drop out from the final symmetry relation. This argument does not help us here, since what we are looking for are precisely the exact and consistent relations. Anyway, we adopt a consistent definition of qT that is different from the above two conventions. In deriving the symmetry relations for T and also for P and Ejj we use a method that is simpler and more direct than that of JW since it does not involve any partial wave amplitudes, as the method of JW does, The reason why JW could not have used our method is due to the fact that they did not work with general helicity amplitudes as we do in this paper. These remarks will become clear in Appendix B. In this appendix we also review the derivation of the crossing matrix in such a way that we can immediately obtain both crossing matrices depending on the path we choose. In addition, we describe a direct method for finding the overall phase factor of the crossing matrix.

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The plan of this paper: In Section 2 we define the general helicity amplitudes, the M functions, and the c.m. helicity amplitudes with the “E prescription.” We also discuss the ambiguities of the analytic continuation. In Section 3 we derive the symmetry properties of the crossing matrix, and in Section 4 we verify that the slightly modified crossing matrix of C-TMN does satisfy these conditions. In Section 5 we discuss the reduction of the crossing matrix. In Section 6 we derive the crossing matrices for Xij . In Section 7 we discuss the continuation paths of C-TMN and TW and show the equivalence of their crossing matrices. In Appendix A we relate the symmetry properties of the general helicity amplitudes to the symmetry of fields, and in Appendix B we derive the symmetries of the c.m. amplitudes from those of the H amplitudes. Finally, in Appendix C we evaluate some Lorentz transformations and Wigner rotations and in Appendix D we derive equalities involving the “communutation factors” (sij .

2. DEFINITIONS

AND NOTATIONS

In this section we only give some basic definitions and notations. In general, we shall follow our notations given in earlier papers [6, 81. 2. I. Processes and Operations As in Ref. [S] we define a process v = (R, K, 11)by a reaction R = (1 + 2 --f 3 + 4), its momenta K = (k9k4 ; k,k,), and its helicities h = (X,h, ; h,h,). We use 0 to denote the operation that maps a process ‘$J= (R, K, II) into another process O$ = (OR, OK, O/l). For example, the crossing operation X I: X,, is defined by X!@=(XR=(~$2+3+i),XK=(k,, -k,; -k,, k,), XII = (A, - A,; -h,X,)). The definitions of the standard discrete symmetry operations are given in Table I. We also define operations for the three Mandelstam variables s, t, and u (See Eq. (2.W O(s, I, 11)= (x, 4’. z). (2.1) where x, y, and z are permutations of the s, t, and U, and denote energy, momentum transfer, and mementum-exchange variables for the reaction OR, respectively. The symmetry 0 between the cm. helicity amplitudes (see Section 2.6) can be written in the form

fodOR, O(s,tN = 2 V%@,

R, J’,t).h(R, s, th

(2.2)

h’

where the matrix Vcm is called the “symmetry matrix.” Thus, for example, the crossing matrix is nothing but the symmetry matrix for the crossing operation

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Definition 0

TABLE I of the Usual Discrete Operations

OR

1 P

C T

b,-CPT El, E X,4 I St

OK

R-(1 +2+3+4) (1+2+3+4) (T$2-3+q (3+4-1+2) 13+4-T+2) (2+1-3+4) t.2 + J -4 + 3) (4+2-3+i)

A’ = (k,k,

; k,k,)

(I&

; L,R,)

(knk4

: k,k,)

(I;*& ; La.&) (k,k, ; k&d (k:,k, (k4k:: (k, ~ (p~k,

(1+2-3+4)

; k,k,) ; k,k,) k, ; -k,k,) ~ k, : -k,

-- k,)

n Here ff = (k, , -k). b The definition of Oh in case of crossing and Zt is not unique, but depends on the analytic continuation. See text.

0 = x,, . Since the crossing matrix is used quite often in this paper, we use a special notation for it, (2.3) h’

with (2.4)

The product 0301 of two symmetry operations 0, and OI is defined by [8] (O,O,YP = O,(GW

(2.5)

The corresponding product relation among the symmetry matrices is [8] (in matrix notation) Vc”(0201,

R, s, t) = Vcm(O,

, OIR, O,(s, t)) Vcm(Ol

, R, s, t).

(2.6)

The positions of particles a, b, c, and d in R = (a + b + c + d) are called positions 1, 2, 3, and 4, respectively. We shall also name the particles in position i in R as particle i. Then, e.g., for time reversal TR = (3 + 4 - 1 + 2) we have particle 3 in position 1, particle 4 in position 2, and so on. For all the discrete symmetries defined in Table I the particles in OR are the permutation of the particles or the antiparticles of R. Similarly the helicities in Oh are also permutations of the original helicities in h = (&I4 ; X,X,) with possible

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minus sign factors. Therefore, we find it very useful in the following to introduce two more notations: To every operation 0, we define a set of four numbers (01, 02, 03, 04) which are the permutations of the number 1, 2, 3, and 4: the Oi are defined by OR = (6: + ol+

63 + 6;)

(2.7)

where -? indicates a possible bar “-” for denoting an antiparticle. In other words, Oi is the number of the particle, or its antiparticle, in position i of the reaction OR. For example, the crossed reaction is XR = (4 + 2 -* 3 + I), so that (Xl, x2, X3, X4) = (4,2, 3, 1). We also define sign factors ~~(0) by

Oh = (E03(O)hO3 2Eo4(Wo4 ; l Ol(WOl3~02(OPo,). For example, X/I = (A, - A, ; -A& (4X),

(2.8)

so that

‘z(X), %?(‘a EI(X)) = (- 1,

-t 1, +

1, -1).

These ci(X) will occur quite often in this paper, so we shall often use simply Ei instead of ci(X). From the above definitions one can deduce that O,(O1i) = (O,O,)i = O,O&

(2.9)

and %,i(0,01)

= pi

%,i(Ol).

(2.10)

We define 0-l i using Eq. (2.9), O(O-Ii) = i.

(2.11)

The number 0-l i can be interpreted as the position of particle i in the reaction OR. 2.2. Kitleinatics The three channels of a two-body reaction are defined as schannel

=(l

+2+3{-4).

t channel = (4 + 2 + 3 + i), tl channel = (3 + 2 + T + 4). The Mandelstam

(2.12)

variables are

s = (k, + k,)“,

t = (kl - ,k$,

II = (k, ~~ k,)“.

(2.13)

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To simplify many of the following formulas we define i, as the number denoting the energy partner of particle i in the x channel. Thus (1,2?3,4)=(2,,1,,4,,3,)=(3t,4,,1t,2t)=(4,,,3,,2,,l,,).

(2.14)

With the help of Table I it is easy to verify that i, = Ei,

it = 8i,

i,, = BEi = (i,s)t = (it), .

These relations will be useful in Section 4. The c.m. energy and momenta belonging to particle i (or its antiparticle the x channel (x = s, t, or U) are given by

(2.15)

T) in

(2.16) (2.17) where Xii,

z

(x

-

(n7,

+

“7i,)y

(x

-

(n7,

-

I17i,)2)1’“‘.

(2.18)

The Kibble function is defined by &R, s, t) = so

- s - t) - sA,,A,, - tA12A34 + A(7n1~n1~~ - n12%z~), (2.19)

where

Aii = nli3 A

=

A,,

2

=

.&?Zj”.

+

1~7~~, A,,

>

(2.20)

We denote the physical region of the s channel by X+. We define d#(R, s, t) to be positive in the physical region S+ oft he reaction R (s channel). This is enough to determine z/+(R, s, t) everywhere else, given the path of analytic continuation. Thus if we continue z/$(R, s, t) along the continuation paths rC-TMN and rTW of C-TMN and TW we find that d+(R, s, t) in T+ has positive and negative values, respectively (see Fig. 2). Therefore, it is important to distinguish, between dD= and ~#J(R, s, t), since 1/$(XR, t, s) is positive in T +, by definition. Apparently, this distinction or at least its significance has not been recognized; in Appendix B we shall see how serious errors would result if we overlook this distinction.

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In terms of the previously defined quantities, the scattering angles in the c.m. system of the s and f channels are defined by

(2.21)

and 0 -< 0, < 7~in 9. Similarly, we have

(2.22)

FIG. 1. plane. The

The real part of the paths r c -- TMN(-.-.-)andI’TW(-----)inthe(Res,Ret) numbers 1-14 indicate different points on these paths for

convenient

reference.

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FIG. 2. The image of rCMTMN and YTw in the complex $ plane. The numbers l-14 are the image of the corresponding points in the (s, I) diagram in Fig. 1. The cut from 0 to x is the square root cut of A$. 2.3. General

Helicity

amplitudes

The general helicity amplitudes H,(R, K) I H,\,,,,:Aln, ((1 -’ 2 -

3 -t 4), (W-4 ; h&N

are scattering amplitudes that are directly related to cross sections in all Lorentz frames. These amplitudes transform under Lorentz transformations fl E L+(C) as follows: (2.23) H,(R, flK) = 1 &,dfl, R, K) H,,(R, K), h’ with

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where Dyy’[ WQl,

k)] :: D;,,,+-lWH(il,

k) El

with

E

io, .

(2.25)

Here CV”(A, k-) are Wigner rotations defined by WH(A, k) = k-l(Ak)

(2.26)

All(k),

(2.27)

h(k) = L(k) R(k), -.

(2.28) R(k) = R,,(O) = cos $8 = &cd) Me)

in . (r sin i0

(2.29)

h-4),

where

(2.30)

Ri(cx) -5 cos 4~ - iat sin 4~.

Here L(k) is a boost which takes a four-vector from its rest frame I, = (nr, 0) to k = (IQ,, k). In u ’ k =. q,ko + ts * k, the u< are the usual Pauli matrices and a, is the unit matrix. R(k) is a rotation which rotates eZ , the unit vector in the z direction, around the normal n = (eZ x Q/sin 0 = (-sin

4, cos c,&0)

(2.31)

into the direction

I$ == k/II k i/ = (sin 19cos 4, sin 6 sin 8, cos e),

(2.32)

/, k

(2.33)

,: _

(k,"

$

k,l

-t

li,")l~"

_

~~ko2

_

111‘).

The reason for emphasizing the definition off; will become apparent later when we discuss its analytic continuation. We now note a well-known but unavoidable source of trouble and headache [3, lo]: The rotation R(k) is discontinuous at Ai = -e, ; R(k) has different signs, as I$ approaches the negative z axis from opposite sides. This in turn makes the H amplitudes discontinuous at Ri = -e, , if the corresponding particle has a half-integral spin & . Therefore, our definition of H(R, K) with the R(k) as specified in (2.29) can be regarded as defining only “the first sheet” of the H amplitudes. From now on, unless otherwise specified, we shall always consider H amplitudes on the first sheet. So, if in applying a Lorentz transformation we encounter the negative z axis, then we “jump” over it to remain on the first sheet; otherwise if we keep going continuously we end up on the second sheet, where R(k) would

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have the opposite sign from that R(k) defined in (2.29). Only in some cases, such as when we consider the cm. amplitudes defined below? will we actually analytically continue the N amplitudes, and only in such cases will we need to worry about how R(k) exactly behaves along the path in order to determine its sign ambiguity.

2.4. Analytic Continuation Equation (2.27) uniquely defines the 2 x 2 matrix representation of /r(k) for a four-vector k in the physical region. However, if we continue h(k) as a function of k, we can end up with four different determinations of h(k) for the same vector k, depending on the path of continuation. Two possibilities are due to sign ambiguities in determining L(k) and R(k): The sign ambiguity of L(k) is due to the ambiguity in the square root t/k,, + m (see Eq. (2.2X)), whereas the sign ambiguity in defining R(k) depends on the number of times the (-eZ) axis is crossed during the process of analytic continuation. The sign ambiguity of h(k) can only contribute an overall phase difference of (- 1)2s in defining the general helicity amplitudes. The other two determinations of h(k) are caused by the ambiguity in defining R(k): Since R(k) depends on the unit vector & = k/Ii k I/, there are two determinations R(fi) depending on the sign ambiguity in defining the 11k /I = v/k0 - m -\/k,, + m (see Eq. (2.33)). This ambiguity is more serious than just the minus sign ambiguity, since R(h) and R(-&) differ by a rotation R,(r), where n = (e, x &sin 19,and since DfA,(R,(n)) cc 6,,-,, . The meaning of these opposite helicities is intuitively clear, since helicities are the spin projections with respect to the direction of motion A, so that helicities change sign as // k I/ changes its sign.

2.5. The M functions [ll] The A4 functions are linear combinations

of general helicity amplitudes,

Unlike the general helicity amplitudes, the M functions have no direct physical interpretation, but they are useful because of their simple transformation properties,

(2.35) Because of these simple transformation properties, it is usually postulated that the M functions are free of kinematic singularities, i.e., they have only dynamical singularities which are caused by unitarity.

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The absence of kinematic singularities makes, the A4 functions very useful objects to work with in problems involving crossing, since their values are not so sensitive to the choice of the continuation path as the general helicity amplitudes are. The reason is that the helicity amplitudes have kinematic singularities due to the square root factors contained in h(k) in addition to the dynamical singularities, which the A4 functions have. Therefore, in analytically continuing the A4 functions we have to avoid only the dynamical branchpoints, while in continuing the H amplitudes we have to worry about the kinematic branch points as well. 2.6. The c.m. helicity amplitudes The c.m. helicity amplitudes&(R, s, t) are functions of only the invariant energy and momentum transfer variables s = (k, -t k$ and t = (k, - k,)?. They are equal to the general helicity amplitudes for special c.m. momenta, (2.36)

f,(R, s, t) = H,(R, K&n),

where the momenta K& = K&R, S, t) are well-defined functions of s and t and the masses of the particles indicated by R. We shall often drop R or just write I&,, when no confusion can arise. These c.m. momenta are defined as follows: K&R,

s, t) := (ksS,kds; kls, kzS)

where ks = (qs, kl+, 0, km), kzs = (wqs, -k,,E, 0, -k& k,S +e (c+s, k&sin 0, + E), 0, kS4cos e,), k,” L (was, -k,,,(sm

0, + c), 0, -k,,

(2.37)

cos es),

where wiS, kiis , and 8, are the c.m. energies, momenta, and scattering angle defined in (2.16), (2.17), and (2.21). The E is an infinitesimal positive quantity, which indicates the E prescription described below. Note that according to this definition, the space components of all c.m. momenta lie in the x-z plane. The norms of the c.m. momenta are defined by j] kIS I] = I/ k,” jl = k,, and II k,” II = II bS II = k,, > which leads, using (2.32), to the following unit vectors: &J = (E, 0, l), x,s = (-E., 0, -I),

(2.38)

is8 = (sin 8, + E, 0, cos e,), fGIS= (-(sin

6, + E), 0, -cos

0,).

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Note that the directions of the incoming particles are always in the positive and negative z directions and that the azimuthal angle $ for R, and A3 is always equal to zero, whereas the r$for & and R, is always equal to n. This insures that the normal to the scattering plane A, x &/sin 8, is alwu~.~in the positive y direction. THE E PRESCRIPTION. The E in (2.37) is to help us calculate the rotations R(k), whenever the definition (2.29) becomes ambiguous, viz., when the momenta are at the negative z axis. We need this convention for calculating R(k,“), and also for calculating R(k,“) and R(k,“) for backwards and forwards scattering, respectively. In these cases, we treat E at first as a small positive number, which we let go to zero after we have calculated R(k). With this “E prescription” and definition (2.38) of the unit vectors of the c.m. momenta, Eq. (2.29) gives

R(kzS) = R&n) and

(2.39) R(k,“) = R,(--sr) R(k,“)

for all scattering angles 0 < 8 < n.

These expressions for R(k,“) and R(k,“) differ by R,(2n) from those of JW [3], which were also adopted by C-TMN. In addition, following C-TMN, we shall not multiply the amplitudes by factors (- l)S-A for particles 2 and 4 as JW do. This leads to the following relations between our amplitudes and those of C-TMN and JW:

The advantage of our phase convention is that it allows us to define the cm. amplitudes as special cases of general helicity amplitudes for momenta Kern, without introducing any additional phase factors (see Eq. (2.36)). In this paper we never use C-TMN’s phase convention. We shall use a superscript JW to denote quantities given in the JW phase convention; quantities without such superscript are given in our phase convention. In general we shall give important final results in both JW and our conventions, but we shall carry out all intermediate steps with our convention only. The definitions of the cm. momenta and amplitudes in the t channel are exactly similar to those in the s channel: We only replace the reaction R by XR and (s, t) by (t, 4.

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2.7. The crossing matrix The crossing matrix of C-TMN follows [6]:

can be written in the JW phase convention as

where ci = ci(X) = -1 for i = 1, 4, and 1 otherwise. The crossing angles wi are defined completely in S+ by COS

Wi =

E& +

+ dii,) - 2I?P&l

&)(t

9

&i,Tii*

sin wi = 2m,[~(XR, t, ‘)ll” Sti8Tiil

~ mi Ti’i’ sinBt, z/t

(2.42)

sii,

and the conditions -2n < wi < 0

for

i-1,4,

-?T
for

i = 2, 3.

(2.43)

We shall show below that these inequalities also hold in T+ and U+. The “crossing phase” vcr , which is the phase one obtains in the crossing of the M functions (see Eq. (A.24)) is given by C-TMN as 7;~‘~~ = (-1)“11, where g14 = 1 if both particles 1 and 4 are fermions and 0 otherwise (see Appendix D). This phase follows if the scattering amplitudes f(1 + 2 + 3 + 4) are defined in terms of the time-ordered product T{ ~3~4~1~2}. H owever, we find it more consistent to use different ordering for the fields, viz., T{~,~,~,~,} or T{~4q53$1~2}. For these orderings we get

where F is the number of fermions in the scattering process. We shall use the first alternative for 7Cr in (2.41), throughout this paper. We show in Section 4 that the crossing matrix can satisfy all our conditions (3.1)-(3.6) only if qCr is given by (2.44) or (2.45).

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From (2.41) it is easy to obtain the crossing matrix in our phase conventions, (2.46) with

where qcr = P-

1)2s3and where wi are defined as before in Eqs. (2.42) and (2.43).

Conditions on wi irt Tf and lJ+ The crossing matrix is completely defined in S+ (where the s-channel amplitudes are in their physical region and the t-channel amplitudes are continued) by Eqs. (2.42) and the inequalities (2.43). However, sometimes, as, for example, in evaluating conditions (3.6) and (3.7), we want to use the crossing matrix for calculating the physical t-channel amplitudes in terms of the analytic continuation of the s-channel amplitudes. To do this we have to analytically continue the whole crossing relation from S+ to T+ along the path I’ c ~TMN. Although cos wi and sin wi become real again in T+, it is not clear whether the wi would still obey the inequalities (2.43) in T+ as they do in Si. A difference of 2~ would lead to a (-1)2s factor, so such considerations are important in testing the overall phase factor of the crossing matrix using Eqs. (3.6) and (3.7). We now prove that wi still obey the inequalities (2.43), modulo 47r, in Ty and I/+; the 4n ambiguities do not affect the values of ds(w). We prove this by showing that the values of sin +wi = ((I - cos wJ/2)‘/’ and cos ~Wi = ((1 + cos CIJ~)/~)~/~ lie in the same intervals in T+ and U+ as they do in S+. For this we trace the images of PpTMN in the complex cos wi planes and observe that they do not make full circles around the branch points cos wi = fl. This is illustrated in Fig. 4, where

I

13,14 . II I \ \\ , N-m 4” FIG. 3. The image of rc-TMN and rTW in the complex t plane. The cut along the real axis belongs to d\/T. Note that point 6 is the “crossing point” of rTW .

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1COS 9

507 plane 1

FIG. 4. The image of rC-=MN in th e cos w2 plane. Point 14 belongs to S+, where w2 is welldefined by the inequalities (2.43).

the image of rcTMN is plotted in the cos w2 plane; for simplicity, cos wQfor equal mass reactions were used, where it is given by cos co* =

i

1- ~

41112 -l 1 ~. /E-l. s 1 f t

(2.47)

This conclusion is not true for the crossing angles UT” of TW if continued along the continuation path Fw of TW, as we shall see in Section 7 (See Eqs. (7.17) and (7.18)).

3. DERIVATION

OF CONDITIONS

ON THE CROSSING MATRIX

In this section we derive the following arguments: x-,I,,-,,(R,

relations by using general symmetry

s, t) = (- l)z(Ac-AJ X,,,,(R,

s, t),

x .gXh,m,(ER, s, t) = (-1)s1-s2+“3--““X,,,,,(R,

s, t),

x Exn,Oh,(BR,s, t) = (- l)s1+sz-s3~~s4 X,,,,,(R, x mum@ER,

s, t> = (-l)2s1-zs4 Xx,,,@,

x ,.,m4XR, f, s) = X&dR,

s, t),

s, t),

s, f>>

2 XF~~X~,--Y~~~“(X~SR, 21,t) X.YE,,I~“.E,,~,(E~~R, s, u> = (- 1)““” X,m,@,

(3.1) (3.2) (3.3) (3.4) (3.5) s, t). (3.6)

h”

Equation (3.1) relates different elements of the same crossing matrix, whereas Eqs. (3.2)-(3.5) relate crossing matrices for d@erent reactions R and OR (0 = E,

508

DABOUL

0, BE, or X,,). However, Eqs. (3.2))(3.5) become symmetry relations for the same crossing matrix for special reactions R, for which the particles in OR have the same masses and spins as the corresponding particles in R. For these cases Eqs. (3.2)(3.5) become

These equations follow from (3.2)-(3.5), because the crossing matrix depends only on the masses and spins of the particles and does not depend on any other quantum number such as electric charge or baryon number, etc. Note that except for (3.4), which follows from (3.2) and (3.3), all the other conditions (3.1)-(3.6) are independent, in the sense that none of them follows from the others. To derive the above conditions we recall the multiplication rule (2.6) for the c.m. symmetry matrices,

VC"(O,O1,R,s,t)=

VCm(O~,OIR,Ol(s,t))

Vcm(O,,R,s,t).

(2.6)

We further note the following relations among products of operations, which can be easily verified using Table I:

(4

x = PXP,

(b)

I,,X = 8XE,

(cl

Is,X = EX0,

(4

x=

(e)

I = xx,

(3.7)

BEXBE,

PE,,X = XPE,,XE,,.

(0

Our conditions (3.1)-(3.6) then follow by using (2.6) and (3.7). This application is greatly simplified by observing that

V %,td4

R, s, t> = $?'(O, RI S,,,,

for

0 = P, Eij, 8.

(3.9)

PROPERTlES OF THE CROSSING MATRIX

509

Therefore, using the short notation X(R, s, t) z Vcm(X, R, s, t), we get ‘YXh,h, CR, s. t) = $%(O,

X04

~~,m,m~(OR,

s, t) $?‘(O,

for 6 /‘,I“

= ;

Xh,Xh 4XR,

t. .y> Xx,-,,L,(R,

R)

(a-d)

0 = P, E, 8, dE, s, t>,

(e> (3.10)

where in (f) the relation X1, = Es4XEs4 (6.1) was used. Equations (3.10) lead to (3.1)-(3.6) by substituting for 7 Cn1their explicit expressions given in Table II.

4. VERIFICATION

OF CONDITIONS

ON THE CROSSING MATRIX

In this section we verify that the crossing matrix given in (2.46) satisfies all the conditions (3.1)-(3.6) that we derived in the previous section. The expression (2.46) is valid for arzy two-body scattering. Therefore, it allows us to write down immediately the crossing matrices for the reactions OR, by simply replacing the spins Si , masses mj , helicities hi , and the energy and momentumtransfer variables (,s,t) for R = (1 + 2 --j 3 + 4) by the corresponding quantities for OR = (01 + 02 -+ 03 + 04), viz., Soi , ~~~~~ , E~~(O)A,~, and (x, ~1)=: O(s, t). More explicitly,

(4.1) with S’,yoh,ori,(OR) = iF(- 1)=03 e-i~(Eo3(0)~03+f~(X)E~~(o)~~~+f~~(o)~~~+~~~(o),~~~) y (4.2) where we made use of Eq. (2.10). The indices Oi and O-Ii are defined in Section 2. We recall their interpretation: Oi = the /lame of the particle in position i in OR; O-Ii = the position of particle i in OR. It is easy to verify condition wi(PR) = q(R).

(3.1), using d-,,-,(w)

(4.3) = (-l)“-A’

d,,,,(w)

and

510

DABOUL

TABLE

11

Symmetry Phases and Related Quantities” Symmetry operation 0 -.

A,“,

MO.

(-

&A4

P

c

,)?s1+Is3

1

T

R&n

CPT

R,(e + W&(x)

R)

1

+ 0 + 2~)R,(a)

(-I fA*+2S1+2S2 (~,)m‘+“s”+2.sj

El,

(-1f4’2S’

E34

(- p’+esl

E

(F-1)

%S1+*.S3

(-1p

(-

,)Sl+%+h+~a

(-1)“34 iF(- 1)3.s’+2s2

(F-1) (F-1)

s3+s4+AB+A4 as, 4A,)

T%, RI

T?%AN

T:“CO,R)

d”CO,N

for general R

for OR = R

for general R

for OR = R

same as for general R same

$(-1)

s,+s,-s,-S,+A-P 4

same as for general R same (1 for allowed reactions)

0 ‘Here &m, SF+,qH, tJw qcm are defined by Eqs. (B.4), (B.7), (B.2), (2.40), (B.9), respectively. ([;~(OR)/f~w(R)) x vCm, where rlrw corresponds to vcm in the JW phase convention.

71w =

511

PROPERTIES OF THE CROSSING MATRIX

To verify Eqs. (3.2)-(3.6) we need Eqs. (4.8) which can be derived from the following general result: E~-#‘)(s

cos w. l,(OR, x, y) = _

+ &iv) - 2~1~d(OR)

+ &,)(y

,

(4.4)

xii,yii,

(4.5)

Proof. By replacing every m, , s, and t by YIIoi , X, and y in expression (2.42) for cos wi(Rp S, t), we get 4X)(x

+ do,.oi)(y

cos w,(OR, s, y) = -~-~~~

+ iloi.oi,)

- 2nzoi 4OR)

xoi,oi, yoi,oi,

.

(4.6)

This will lead to (4.4), by using O(i,) = (Oi),

and

O(i,) = (Oi),

(4.7)

We verify (4.7) for a specific i, say i = 1. Since particles 01 and 02 are the energy partners in the x channel OR = (01 + 02 + 03 + 04), it follows that (Ol), = 02 = O(1,). Similarly, 01 and 03 are the momentum transfer partners in OR, so that (Ol), = 03 = O(l,), and Eq. (4.7) is proved. Expression (4.5) follows similarly. COROLLARY.

Equations (4.2) and (4.3) lead to the following identities: - Wi(R, S, t),

w,,(ER, S, t) = -T

w,,(%R, s, t) = -TT - wi(R, s, t), (4.8)

wm(%ER, 3, t) = wi(R, s, t>,

w,$(XR, f, s) = w,(R, s, t). Proof. We observe that ci = Q(X) and the quantity d(R) = d,, + A,, (see Eq. (2.20)) and the Kibble function cj(R, s, t) have the following symmetry properties: Ei = -Eki = -E8i = c&Gi= EXi

d(OR) = F-d(R) 2

d&W

O(s, t>> = d$tR,

(i = 1, 2, 3, 4),

(0 = E, %, %E, X), s, t>

(0 = E, 8, %E, X).

(4.9) (4.10) (4.11)

512

DAESOUL

Note that z/4(XR, t, s) = -\/$(R, s, t) does not hold if we use TW’s continuation path (see (7.13)). Substituting (4.9)-(4.11) in Eqs. (4.4) and (4.5) gives cos w&OR,

O(s, t)) = ?

cos mi(R, s, r), 2 sin woi(OR, O(s, t)) = sin q(R, s, t).

(4.12)

These two equations lead to (4.8), modulo 2n. However, Eq. (4.8) must hold exactly, because of the inequalities -2r

< wAOR,

O(s, t)) < 0

and

- m- < u&OR,

O(s, t>) < r,

which hold if O(s, t) belongs to any of the three physical regions S+, T+, or Us’(see Section 2.7). Conditions (3.2)-(3.5) can now be verified easily by using Eqs. (4.8) and the well-known identities df&fJ)

= (- l)s+A’ d?,,, ,(-37 -

w) = (-l)A-G&--w).

(4.13)

Condition (3.6) can also be proved using similar techniques, but we shall not give this proof here, since it is somewhat complicated. We have thus verified that the crossing matrix (2.46), with qcr are given in (2.44) or (2.45), satisfies all our six conditions (3.1)-(3.6). It follows that the original crossing matrix of C-TMN with vcr = (-l)“,, does not satisfy (3.2) and (3.3), although it does satisfy (3.5) and (3.6). In fact, (3.5) was used by C-TMN to determine the overall phase factor of the crossing matrix [3]. This factor can also be determined directly (see Appendix B). 5. REDUCTION OF THE CROSSINGMATRIX The symmetry properties of the crossing matrix (3.1) (3.2’)-(3.4’) allow us to reduce the crossing matrix into two or more submatrices. These submatrices relate “symmetry-conserving” amplitudes in the s and t channels among themselves, and similarly they relate the “symmetry-breaking” amplitudes among themselves; for an exact definition of the “symmetry-conserving” and the “symmetry-breaking” amplitudes, we need some special notations. 5.1. Decomposition of Helicity Indices In this section we are interested only in operations 0 and reactions R, for which OR = R and O(s, t) = (s, t). For these cases the set of helicity indices H = {h} is mapped into itself by the operation 0, i.e., if h E H then Oh E H.

PROPERTIES OF

THE

CROSSING MATRIX

513

We can decompose H into three subsets as follows: First we define the subset Ho which contains all indices h with Oh = h. The remaining indices H - Ho can be decomposed into two disjoint subsets H+ and H- such that if h E H+, then Oh E Hand vice versa. Thus we have H :==H+ + Ho -t H-- with Ho = {/I j h E H with Oh = h), H+ ={hIOhfhandifhEH+thenOh$H+}, H- = (Oh 1h E H+}.

(5.1)

We shall sometimes also use HoO, Ho+ to indicate the symmetry 0 relative to which the decomposition is given. We use dim(H) to denote the number of elements belonging to a set H. We also use the following abbreviations: t? -= dim(H) := the number of amplitudes, n + = dim(H+), 12~= dim(HO). Since dim(H+) = dim(H-),

(5.2)

it follows that 12= no + 2n+ .

(5.3)

EXAMPLES. (1) Parity. Since Ph z P(;\& ; h,h,) = (--h, - h, ; --Xl - X,), then HP0 can have at most one element h = (00; 00), which exists only for reactions involving no fermions. El+ and H- can be chosen arbitrarily using some convention.

(2) Exchange Symmetry.

This operation

EH = E(&l,

; &I,)

is defined by

= (h4& ; h,h,).

Let us consider the specific reaction R = ( p + p -+ p + p), for which ER = R. Then HE0 = ((++, ++), (++, --), (--, ++-), (--, ---)I. The HE+ may be chosen as follows: HE+ = ((+ +, + -), (6-y + -), (:+ -2 + +), (+ -1 --), (+ -, + -1, (+ -, - +)>, so that HE- = i(++,

-+>, (--, -+j,

(-+,

ii-),

(-+.-->,

(-+,

5.2. Symmetry-Conserving and Symmetry-Breaking

-+>, c-+,

+-13.

Amplitudes

Let OR = R and O(s, t) = (s, t), so that the corresponding symmetry is defined by fodR

3, t) = r)hOfh@>s, t>,

(5.4)

514

DABOUL

where QO z= $jLm(O, R) are phases. Then we define for for

gt = \fh” = (.fi, 5 %OJOh)i D I !ih

II E Hs, h E HO.

(5.5)

If the symmetry (5.4) holds, then fi,+ = ~‘2 f,& for /z E H+ and fiL- = 0 for h E H. Therefore, we shall call the combinations &+ “synznzetr!l-consen:i/zg” amplitudes and the combinations fh- “symmetry-breaking” amplitudes. The fj with h E Ho will belong to the symmetry-conserving or to the symmetry-breaking amplitudes depending on whether ~7ho == + 1 or - 1, sinceftEvanishes if vbo = - I and the symmetry (5.4) holds. Therefore, the number of symmetry-conserving amplitudes is equal to n, + $(l + ~0) no where h E Ho. LEMMA.

Let 0’ and 0 be two operations related by (examples are the relations

(3.7a-d)) 0’ = X0X (modulo Ist).

(54

If we decompe the helicity indices of the s and t chanaels relative to 0 and 0’ as foliows: H = Hoi- + Ho0 + Ho-, XH = (XH);,

+ (XH)o,, + (XH),,

,

(5.7)

then dim(H,O) = dim((XH)t,). Proof.

Since I,,h

(5.8)

= h, Eq. (5.6) says O’(Xh) = X(0/7),

so that O’(Xh) = Xh

for

Oh = h.

Hence Xh E (XII):, for all h E H o”. From this Eq. (5.8) follows by noting that Xh f Xh’ if h f h’. COROLLARY. The number of symmetry-conserving amplitudes in the s channel (relative to 0) is equal to the corresponding number in the t channel (relative to 0’ = X0X).

ProojY

This corollary follows from the previous lemma if Tg;=?g=

fl

for

Oh, h’ E Hoe.

(5.9)

PROPERTIES

OF

THE

CROSSING

515

MATRIX

This equality follows in turn from the relation ~ 0’ 0 .u,h,h~ ~ rlXh77,~‘~OXh,O,,= $G&YYn,~t’

for

h, h’ E Ho”

(5.10)

since XX,,,,, do not vanish identically; in (5.10) the first equality follows from (3.1Oa-d), and the second equality follows since O’XZI = XZr and Oh’ = h’ for h and 11’E HoO. 5.3. Simple Reduction of the Crossing Matrix For the following it is useful to write (5.5) in matrix form. Let fh and g,, be written as vectorsfand g, ordered such that the components fh and g,L with h E Hf come first, next come those with h E H”, and finally come the components with h E H-. Then Eq. (5.5) becomes (5.11)

s = C(O).L where C(0) is an IZ x II orthogonal

matrix of the form

Gt,,~(O) = NL@?d + 44 %“b,h’)

(5.12)

with (5.13) and (5.14) Equation (5.11) allows us to rewrite the crossing relationfr

(5.15)

gt = rgg”, where gt = C(O’)j”,

gS = C(O)f”,

Xf” in the form

and (5.16)

X’ = C(0’) XC-l(O). We now prove that x’ is a direct sum of two matrices.

LEMMA. Zf OR = R, O(s, t) = (s, t), and 0’ := XOX(modulo Zst), then the crossing matrix is reduced by the matrices C(0) and C(O’), such that q

X’ = C(C)‘) XC-l(O)

:= (,’

i),

(5.17)

516

DABOUL

where Y and Z are orthogonal matrices. Y has the dimension on whether ~0 = + 1 or ~ 1 ,for h E Hoe. Proof.

The matrix C in (5.12) is orthogonal. (C-%h, = WTh

Hence

= Ct,,t, = Ntd%,,

:= (S,,, + 6 t,.oli4’)

tz+ + n, or n, depetldittg

+ 4h’)

&%ott,,tJ

(5.18)

$4 N,, .

Therefore

where we used Xo,,,L,o,, = q’&~, Ox Xh,h~in the final step. From this expression and noting (5.13) and (5.14) it follows that 11’ E Ho’

xx,,,,

+ rli%rh,oh~

1 X’ = 2/2 (1 + $4 XXh,h~

h’ E Ho0

$2

(1 + $>

x Xh,h’

h’ E Ho0

X,Yh,h,

Xh E (XH);,

1 72

(1

-

&)

xx,,,*

1

X/z E (X-H);,

0

Clearly this matrix has the form (5.17); for 7: = + 1 or - 1 the submatrix Y has the dimension II+ + tie or II+ , respectively. Equations (5.15) and (5.17) say that the crossing matrix relates symmetrybreaking terms in the s and t channels only among themselves. This is an interesting result: It means that in a bootstrap calculation the symmetry-breaking terms will “bootstrap” into themselves and are affected by the symmetry-conserving terms only via unitarity. Therefore, by introducing small parity violating “forces” (or equivalently, small parity-breaking amplitudes in the t channel), we can only end up with small parity-breaking amplitudes in the s channel.

PROPERTIES

5.4. Multiple

OF

THE

CROSSING

MATRIX

517

Reduction of the Crossing Matrix

The crossing matrix can always be reduced once for anI> reaction, because of the parity symmetry (3.1). In addition it can also be reduced for special reactions R, if ER = R, l3R = R. or BER = R, because of Eqs. (3.2’)-(3.4’). Let us consider a reaction R such that ER = R. We can first construct parity-conserving (PC) and parity-breaking (PB) amplitudes and then take linear combinations of these to construct exchange-conserving (EC) and exchange-breaking (EB) amplitudes. In this way we get four types of amplitudes, viz., (PC, EC), (PC, EB), (PB, EC). and (PB, EB) amplitudes. Since the crossing matrix relates (EC) amplitudes to CPTconserving (65) amplitudes. etc., each of these above four types in the s channel is related to only its corresponding type in the t channel, viz., to (PC, KY), (PC, BB), (PB, KY), and (PB, 6B) amplitudes, respectively. Therefore, the crossing matrix must be reducible into four square submatrices connecting the above types with each other. This reduction can be carried out further for reactions with ER = OR = R. Since the crossing matrices for TR and OR are the same, the same considerations are also applicable to reactions with ER = TR == R, and so on. For example, the crossing matrix for R = (p +- p + p + p) can be decomposed into eight submatrices, two with dimension five and six with dimension one. They correspond to the eight (8 = 2”) possible conserving-breaking combinations with respect to the three symmetries P, E, and T. If all these three symmetries are satisfied, then only one type of these combinations survives, viz., the (PC, EC. TC) amplitudes. This type has five independent combinations, which correspond to the usual five linearly independent amplitudes in elastic proton scattering. 5.5. Relationship Between Reducing the Crossing Matrix Using Symmetries and Using Transversity Amplitudes [ 121 The symmetry properties of the crossing matrix allow us to reduce it into submatrices for certain reactions. On the other hand, using transversity amplitudes one can, in fact, always reduce the crossing matrix completely into submatrices of dimension one. Thus there must be some simple relationships between the transversity amplitudes and our symmetry-conserving and -breaking combinations. It may be interesting to study these relationships for special reactions, but we shall not do this here. We only note the following example for parity: We know that about half of the transversity amplitudes vanish identically if parity is satisfied. Hence, these vanishing transversity amplitudes must be linear combinations of our parity-breaking amplitudes. Therefore, the vanishing transversity amplitudes in the s and t channels must be related by the crossing matrix only among themselves, which is true, as can be verified explicitly.

518

DABOUL

5.6. Geometrical

Irlterpretation

of the Reductiotl

The reduction of the crossing matrix can be understood geometrically by picturing the scattering amplitudes as coordinates of the vector f = (fi ,...,fn) in an fz-dimensional space A. Then if the amplitudes obey a certain self-symmetry 0 of the form (5.4), the vectorf become restricted to lie in an m-dimensional subspace C of A (nz < n). When the symmetry 0 is broken,fcan be written as a direct sum of two vectors (5.21)

f(R) = f=(R) GfW

where f c E C and f B E B = A - C. Similarly the corresponding vector f (XR) for the t-channel amplitudes can be decomposed relative to the symmetry 0’ = X0X such that f(XR)

= fC’(XR)

@f”‘(XR),

(5.22)

wherefC’(XR) E C’ andfB’(XR) E B’ with A’ = XA = C’ @ B’. Now the relation 0’ = X0X tells us that the subspaces C’ and B’ must be images of C and B by the mapping X, i.e., C’ = XC and B’ = XB. Hence, C’ and C have the same dimension m, since X is orthogonal; this result was obtained by different arguments in the corollary of Section 5.2. This makes it clear why X must be reducible, since it is mapping subspaces of one vector space into subspaces of the same dimension of another vector space. The details for carrying out such a reduction were given explicitly in Section 5.3. If the scattering amplitudes obey two or three symmetries, the vector space A becomes decomposed into four or eight subspaces and the same arguments would tell us that the crossing matrix can be reduced accordingly. This multiple reduction was discussed in detail in Section 5.4 with an example.

6. THE CROSSING MATRIX

FOR A GENERAL

CROSSING OPERATION

Xii

For backwards scattering one sometimes needs the crossing matrix for XI,, which connects the s channel to the u channel, X,,R = (3 + 2 + i t 4). Also the crossing matrices for X,, and X,, could be useful. These matrices follow simply by applying Eq. (2.6) to the relations Xii = EliE4jXl4EliE4j 7 This gives

Eti = 1 (i = 1, 2;,j = 3, 4).

(6.1)

PROPERTIES

OF

THE

CROSSINCi

MATRIX

519

Substituting for the phases 7 their values as given in Table I, we get

=

(-

1 )s1-s”is+4

Xx,,&

.y, t),

V.Y13h,h,(X13 , R, s, t) = (- 1)o?i+zsLi-p(A~+AL’) .k’xE,,n,E:,,h$E34R,s, u),

X

j”(-1)

s1-sz-s3+s4VX,3h,h+Y13, R, s, t),

(6.3) (6.4)

(6.5)

where we used condition (3.2) to relate X(ER, s, t) to X(R, S, t) in (6.3) and X(E,,R, S, U) to X(&R, S, u) in (6.5). Note the particularly simple relationships between the crossing matrices of X,, and A’,, and also between those of X,, and A’?, ; the first two take the s channel to the t channel and the last two take the s channel to the u channel. These simple relations allow us to write down immediately the explicit expressions for V(Xij , R, S, t) using (4.1). For example, (6.4) gives

with L?;2,= coE3J,5&R, s, u), so that using (4.4) and (4.5) we get

(6.7)

520

DABOUL

In deriving (6.6) we made use of the following identities: (F-1)

u13+Qs = i”( _ 1p+2s4

%vLJ %,,iW) = 4xbl)

(see (D.9)),

= 1, (see (2.10)).

= %(X1,)

In the JW phase convention (6.6) becomes

7. EQUIVALENCE

OF THE CROSSING MATRICES

OF C-TMN

AND TW

In this section we show that the crossing matrices of C-TMN and TW are essentially equivalent, if the crossing matrix of TW is interpreted as in Section 7.2. But before doing this we must first discuss their different continuation paths. 7.1. The continuation paths of C-TMN

and TW

The path I’ = rcTMN of C-TMN is made up of two pieces. The first piece r, connects T+ with U- and the second piece r, connect U- to S,

r,: s = -2p, rz :S = ~(1-

t= 3e8’),

p(1 + 3ei’);

t = -2p,

(7.1)

where p is a large number and 0 < + < rr. The real part of this path is plotted in Fig. 1. The path rTw is defined by pw .

s=

Res+ic,

t = Re t - k,

(7.2)

where the real parts of the variables are illustrated in Fig. 1, and where E is a small constant number which is positive along the path between S+ and the “crossing point” (point 6 in Fig. l), and negative between the crossing point and T+. This path was given by TW for TN scattering only, and TW did not explain how to generalize this path for other reactions. Neither is it straightforward to find a satisfactory generalization: For example, in the TN case, the t variable crosses the real axis between 0 and 4p2 and s crosses the real axis between (m - ,u)” and (nz f P)~. Therefore, one might try to generalize Pw by requiring that the variable

PROPERTIES

OF

THE

CROSSING;

521

MATRIX

x cross its real axis between the pseudothreshold and the threshold of the x channel. This would lead to the same analytic continuation of the c.m. momenta as rCeTMN. However, there are many reactions, such as rrn + rrp, which have two pseudothresholds and two thresholds in the s channel, and where one pseudothreshold is larger than one of the thresholds; in our example (1~~- m,)% > (m, + nz,)‘. The above generalization of r * does not work also for reactions involving massless particles, since, e.g., in yp --f np the threshold and the pseudothreshold for y + p are identical. Thus, it is not clear how to generalize rTW in the above cases. The path 7 Tw is not even suitable without modification for reactions in which all masses are equal. For these reactions the Kibble function is given by C#J(S, t) = stu, so that C$would vanish completely at Re u = 0 if II is kept real all the time, as in (7.2). This would make the analytic continuation of d$(R, s, t) ambiguous since the image of Pw in the complex C#J plane would pass exactly through the square root branch point of d$(R, s, t). Therefore, also for the equal mass case rTW has to be modified: We must give zda.n imaginary part when we cross Re II = 0. For all these reasons we shall confine ourselves to the nN scattering when we compare the two crossing matrices. Tn view of the above difficulties to generalize rTW, one might ask whether it is worth it at all to go into all the trouble of explaining the origin of the discrepancy between the crossing matrices. There are two justifications: (1) It is instructive to learn where mistakes and confusion were made. This should clarify the ideas and the techniques involved. (2) We hope to save other people time and effort, who might also be curious as about this discrepancy as the author was. The analytic continuation of the various kinematic quantities along r and rTW from the physical regions S+(T+) leads to different determinations of these quantities in T+(S); these are related as follows (see Figs. (l-6)):

(7.3) in St,

(7.4) in T+.

522

DABOUL

Kcm

= (k3 k4,kl

K’

k2 )

=‘k’3k&

k’2)=PK=(&;4.kU,i;2)

*

IhI

(a)

k;=-i3 K’=TK=

,..--..a (k, k2,k3

K”=

k4)

R3(“)TK (d)

(c1

FIG. 5 (a) The space components of the cm. momenta KCm in the x-z plane. (b-d) The image of Kcm under the operations P, r, and R&r)T. The small angles Eare described by the Eprescription (Section 2.6). These figures are helpful for determining A& (see Appendix B).

h,x,jiar

crossed

v! -m

I,+

..

1 0

R2 (70

i?

ic-

m

: i- r TW

FIG. 6. The image of rc-TMN and r TW in the complex energy planes of the four particles. (-4&)-l takes the continued cm. energies w, ‘into the crossed energies E~oJ,~(c~= -1 for i = 1, 4 and Q = 1 for i = 2, 3).

PROPERTIES

OF

THE

CROSSING

MATRIX

523

From these relations and using the relevant definitions in Section 7.2 we get (cos o,yw = cos 9, ) (sin 0,5)-rw= sin 19,) (Wiq-rw = +!q

(7.5) in T+.

Similar relations hold for cos Bt ., sin 8,) and wit in St, where wiv are the cm. energies (2.16).

7.2. What is the Crossing Matrix of TW? The crossing matrix of TW is given in Eqs. (41)-(43) of Ref. [2]. In our notation, this matrix becomes

or in our phase convention, (7.7) with (7.8) where cos c!JT” = (

4s + &,>(t + Oii,) - 2mi2Ll TW 1 Sic,Tii,

(7.9)

and (7.10) where the superscripts JTW and TW indicate that the amplitudes are continued along Pv and given in the JW and our phase conventions, respectively; for the JW phase convention we have taken particles “2” in the t channel XR = @+2+3+ i) to be particles 2 and i. We also use superscript TW on the right side of Eqs. (7.9) and (7.10) to indicate that the quantities involved are continued along Pw. For example, in T-;- the quantity S;y = 2 &‘W kzy = -Sii (see Eq. (7.4)). This notation will be used throughout this section. The factor $“is an overall factor, which was neglected by TW. 5’ is also an overall factor. We shall not be concerned in the following with these factors. Although the expression for cos c$” looks exactly the same as that for cos wi

524

DABOUL

given in (2.42), cos UT” and cos wi differ by a minus sign (see (7.17)) since (Sq Tqw = -Sii,Tiit in both regions Simand T--. For sin wz” the situation is more complicated. In (7.10) sin wi is defined in terms of sin 8, but it is not clear from TW’s paper what the substitute is for sin 0: On the one hand, 0 was clearly meant to be the c.m. scattering angle in the t channel, which we are calling 8, in this paper. On the other hand, TW use the following relation (which is given in Eq. (A.6) in TW):

However, using dmt,s)TW

= -\/+(R,

s, t)‘“‘,

(7.12)

sin Bs)‘\I.

(7.13)

we get

sin fj%w = -

i

25~ 13

2.1

Therefore sin 0 = -

sin 0:“.

(7.14)

So, either the relation (7.14) is wrong or 0 is not to be interpreted as the c.m. scattering angle, as we understand it. If we try to substitute ~9= tit in (7.10) we get a wrong crossing matrix. But we get a correct crossing matrix (as shown below) if we formally substitute (7.14) in (7.10) and forget about the interpretation of 0 as being the c.m. angle of the t channel. Apparently the mathematical manipulations of TW which led to the crossing matrix (7.6) were correct, and probably the wrong interpretation of 0 has resulted from not distinguishing between z/&R, s, t) and d+(XR, t, s). One might argue that one could define the scattering angle in the t channel to take values between -r and 0 instead of 0 and rr, as in the s channel. But there is no point in using different conventions for the s and t channels, since a priori there is no distinction between these two channels in their own physical regions. Anyway, one fact should be clear: If we substitute formally for sin 8 in (7.10) its value as given in (7.14), then we shall show below that the resulting crossing matrix is correct provided that the amplitudes for both channels are defined in their physical regions as in the present paper, with the normals to the scattering plane pointing in the positive y axis for both channels. Hence, from now on we use sin

wTW

1

I

-

2rqkii, (

sii,

TW

sin Bt 1

= -

(

2m,(#GR t, s)Y” TW 1 Sii,Tiit

(7 15>

PROPERTIES

OF

THE

CROSSING

525

MATRIX

and

However, we shall see that sincefTW andfare continued along different continuation paths, they will not be equal in the “crossed physical region” (see 7.22)). By comparing cos wi and sin wi given in (2.42) with cos WY and sin wTw given in (7.9) and (7.15), we get cos coy = - cos wi

in T+ and S+,

sin wT” = E sin wi ,

E = -tl

(7.17)

in T+~ and

E = -1 inSt,

(7.18)

so that TW

wi

= c:(n -

ml)

(module 2~).

(7.19)

Substituting

in the expression (7.8) for XTw, and comparing this with X given in Eq. (2.46), we finally get the exact relationships between the crossing matrices of TW and C-TMN, (7.21) where q and q’ are overall phases. Clearly, in this equation XTw and X are continued along PW and rcTMN in going from S+ to T+. 7.3. Proof

of the Equir~a~erzce

In this section we prove that if Xrw and X are related as in (7.21), then they are equivalent to each other. To do this we show that the analytically continued c.m. s-channel (t-channel) amplitudes along rTW and r are related by .h?‘(R, s, t> = TLFLR, s, t> f;~~(xR,

t, s> = rl’f-x&W

s, t)

in T+,

(7.22)

in S+,

where 71 and 7’ are phases. This follows, since by definition fTW (R, s, t) = H(R, Klr) and fh(R, s, t) = H,(R, K&J and since the momenta KiT and K&, ,

526

DABOUL

which are by definition equal to each other in S+, take on different values as they reach T+ via rTW and r. In fact, Kz and Kcm are related by the Lorentz transformation I,tRz(r), with the continuation path indicated in Fig. 6, K”TW cm

=

LR,i4

in T-l-,

K&n

K;=mw= Z,,R,(n-) K&,

in

(7.23)

S+.

This relation follows from the definition (2.37) of K,& and the relations (7.3)(7.5). Equation (7.23) immediately leads to (7.22) as follows: f,:tv(XR.

t, s) = H&CR,

KAw)

= D xn,xdL,R,i~)~

XR, K&II) H,,,(XR,

= zkf--XhiXR I, s)

Kim)

(7.24)

in S+,

where we used Eqs. (2.23) and (2.36) and the following identity: D mxdZstM4,

XR, K&n) = i

&,-,j

(7.25)

,

which follows from f+‘H(L&i4,

4 = WH(Jst , R*(n) k) W’(R,(n),

= R,(+r)

k)

(7.26)

Rz(n27r) = R,(f(-l)?),

where we used Eqs. (C.5) and (C. 7b), and the fact that 24(k) = 27~ or 0 for k in the s - z plane. We used (C. 7b), since li:p = &it in S+ (see Eq. (7.3)), which means that the norm /I k I/ doesn’t change its sign under Jqt (see also Fig. 6). Our proof is thus complete, so that the crossing matrix of TW, as we interpreted it, is equivalent to that of C-TMN.

APPENDIX

A.

SYMMETRY

PROPERTIES

OF GENERAL

HELICITY

AMPLITUDES

The phases qH for the H amplitudes given in Table I are consistent with Lorentz covariance and with each other, so they may be considered as a definition of the intrinsic phases involved. However, it is useful to know the connection to field theory. In this appendix we show how 7 H follows from the usual definitions of symmetry operations in terms of creation and annihilation operators.

PROPERTIES

OF THE

CROSSING

527

MATRIX

P, C, and T Let us start by defining the operations P, C, and Tin terms of the annihilation operators A,,(k) and B,,(k) for particles and antiparticles, where wz denotes the spin component with respect to the “fixed z axis,” PA,(k) P-l = qA7,L(K1),

CA,,(k) C-l = rlc4,,w,

(A.11

The annihilation operators a,,(k) and b,,(k), where X is the helicity, are linear combination of A,,,(k) and B,,(k),

where R(k) is the helicity rotation defined in (2.29). From (A.l) and (A.2) we have Pa,(k) P-l = C Df~rl[R-l(k)] P,4,,(k) P-l 111

(A.3a) = qp 2 &,[R-l(k) AT

R(k)] a,,,(l)

where we used (C.2) to evaluate D’[R-l(k) R(L)]. The angle 4(k) is the azimuthal angle in & = (sin 19cos +, sin 0 sin I$, cos 0). Similarly, we get Ca,(b) C-l = 7fb,(k), Ta,(k) T-’ = c Df:[R-l(k)] m

(A.3b) T&(P)

T-l (A.3c)

528

DABOUL

The above relations give CPTa,(k)

pp-lc-1

=

$(-

qS+A

= ~s(-l)~-~

e2i”“(‘:)-~(R))b_,(k)

b-#),

where rf = rfr)pr)T.

(A.4)

Similar equations hold for b,(k), except that we have to replace the intrinsic phase for the particles ?a0 by those for the antiparticles rlzo. The transformation properties of the creation operators a,‘(k) and b,+(k) follow from those for a,(k) and b,(k) by simply taking the Hermitian conjugate of the above equations. To derive the transformation properties of the H amplitudes from those of the creation operators, it is most convenient to use the LSZ formalism with “in” and “out” fields. In this formalism the H amplitudes are proportional to ,,t(34 1 12)in , i.e., HA~A~:A~A~(~ + 20~ (a&

3 + 4, (k,k, ; k&d)

, 4, out) d,(k, . 3, out) &, , d,(k, , 2, in) d,(k, , 1, in) #,),

(A’5)

where &, is the vacuum state and where the numbers i in ali (ki , i, ft) are to distinguish different types of particles from each other. Equations (A.3, a-d) apply equally well to the “in” and “out” fields except that the in fields go into out fields under T and 8. Since the H amplitudes must transform under P, C, T, and 0 as the scalar product ..t(34 / 12)i, does, we get H-JR,

HdCR,

k’) = $$$

K)

=

C-1)

g

HO,

HT,(TR, TK) = J$%$ H&OR, OK) = s = F-1)

2

%--94,)+2&+2&

H,(R,

K),

K),

(F-1) 2s2+2s4H,(R, K),

G4.8)

(- l)z(Si-Ar) H,(R, K) ‘(S,-~,)+2S~+2&

H,(R, K).

(A.9)

Note that the factors for T and 0 are upside down compared to those for P and C, since T and 19are antiunitary operators.

PROPERTIES

OF

THE

CROSSING

529

MATRIX

To get Eq. (A.9) we made use of the following throughout this paper:

definition

that we shall use

7f = (-i)f,

(A.lO)

where f’ is the fermion number of the particle. This definition is adopted from Streater and Wightmann [13], who prove its consistency with normal statistics (see exch,ange symmetries below). Using (A.lO) gives (see Appendix D)

From (A.4) and (A.lO) we get 7f = (-i)f/(?f7f).

(A. 12)

This definition for 7T is different from the Wigner-Eisenbud (qr = -(-i)“S) and the Coester (‘I T = (- l)zS) conventions [9] quoted by TW, which are independent of the product yp+‘. Exckatge Symmetries E,, , Es4 , and E The transformation properties of the H amplitudes under the exchange symmetries E;, , Ex4, and E = Ez4E12follow by commuting the fields. We shall adopt here the so-called “normal statistics,” according to which two fermions anticommute, two bosons commute, and a boson and a fermion also commute [13]. Therefore, the commutation of any two fields ai and aj contribute a factor (- l)“lj defined by (also see Eq. (D.4))

(-l)“zj

_ (7’ ftl

if particles i and .j are fermions, otherwise.

(A.12)

Therefore

Ht&E&

E,,K) == (--1)“” HdR K),

(A.13)

H&&R,

E,,K) == (- 1)“” &(R,

(A.14)

H&ER,

k),

EK) = (- l)01?+u31H,( R, K) = iF(-l)“SIF’S? H,(R, K),

where we used Eq. (D.9b) to get the last equation.

(A.15)

530

DABOUL

The Crossing Synmetry Xl, The crossing relation among the H amplitudes is

(A.16) where the amplitude H,,(XR, XK) is continued from T+ to ST along the following path: First H(XR, K) is continued along the c.m. continuation path PpTMN, which takes K& E Ti- to K&, E S+ and thereby takes all the energy components from real positive values to pure imaginary values (Fig. 6). Next we have to continue K,&, E S+ to XK,“, E S+. For this we recall Eq. (B.23), so that XKkn = (%n-’

Kfm = RI?(~)

Kfm = RL(~;) Kf,,, ,

(A. 18)

where L.(J) is a pure boost (2.28) 6” k (6,) -b), and R = (R(b) R,(a) R3(r))-l. Since the rotation R does not change the energy component of the momenta, we need only consider the analytic continuation of K&, to L(b) K&, . We can do this in a well-defined way by going continuously through the intermediate momenta Q&W Kit, >which depends on the continuous parameter h, where the four-vector b(X) is defined by b,(X) = (2/r-J,

sinh b)

where /3 = --i // b j/ is positive for s = 4p, t = -2p, b(h) is defined such that b2(X) = 1 and

(0 < h < gr), and p large (see Section 7.1).

t, = (1, 0)

for

;\ = 0,

b

for

A==+.

h(X) =

(A.20)

Under this continuation the energy components behave as in Fig. 6 in the complex energy plane. To see this, we recall that the energy component of any boosted vector is given by (L(b)k),, = d . k. Therefore, the energy of particle i (or antiparticle i) is given by E,(x) = (L@(x)) k;J,, = b(h) . kit = dl

- /? sin2 h Wit - sin h b.kii

= 2/l ~ p” sin”% W: + sin h(ciwis - bowit),

(A.21)

PROPERTIES

OF

THE

CROSSING

531

MATRIX

where b . k,I-i is evaluated by using the relation h

The path values of negligible, reactions

. k:,

=

(A&J1

b

.

(/l&J1

kfyi

z

f”

.

&,S

=

EiWiS.

(A.22)

of E(X) in the complex energy plane can be seen most clearly for large s and I, where the difference between the masses becomes comparatively and approaches the case of equal mass reactions. For the equal mass E(h) becomes especially simple, where p = 1 and b,, = 0, so that E,(X) = cos Xi I wit ; + ci sin hwi*

(for equal mass reactions)

which shows clearly that Ei(h) go back to the positive particles 2 and 3 whereas E,(J) for the crossed particles real axis (Fig. 6). This completes the description of H(XR, XK), which we shall call TX. To get (A.16) we start from the crossing relation for

(A.23)

real axis for the uncrossed 1 and 4 go to the negative the continuation path for the A4 functions,

where 7cr is given in (2.44) or (2.45). We assume that M(XR, XK) is continued along P. We can derive (A.24) formally by using the LSZ formalism, without worrying about the continuation path. Bros, Epstein, and Glaser [14] proved the crossing relations of the M functions for massive particles from field theory, but their prolof is very complicated. C-TMN, who quote Bros et al., use a different expression for qcr than ours (Section 2.7). Therefore, we prefer to look upon (A.24) as a symmetry relation for mzssir~eand masslessparticles [ 151 which is compatible with Lorentz covariance and with discrete symmetries; in fact, the verification of conditions (3.1))(3.6) is a consistency check of (A.24). Using I’A.24) and the relationship (2.34) we get H,,,(XR, XK)

For P we have /2(--k) = +h(k) ia, so that

for TX.

(A.25)

532

DABOUL

Finally, we note that if we apply L(6(h)) to (K&JTW the E,(X) will behave as for TX but with a negative imaginary part, i.e., the path of &(A) will now be the reflection of TX about the real axis. The E,(X) and &(X) will follow a path equivalent to the one described by TW and approach the real negative axis from below. However, E,(h) and ES(h) will not go back to their starting point, but rather they would approach the positive real axis from below. However, there might be more complicated parametrization of h(h) which would allow Ez and Es to approach the positive real axis from above and at the same time allow E1 and Ed to follow the path that TW prescribed.

APPENDIX

B.

SYMMETRIES OF THE cm.

HELICITY

AMPLITUDES

In this appendix we derive the symmetry properties of the c.m. amplitudes

.fidOR06,t>)= ch’V%%O, R,3,t)f,,(R,3,t)

(B.1)

from those for the H amplitudes, H,,(OR,

OK) = q/(0,

R, K) HdR, K).

CB.2)

This can be done systematically for all diagonal symmetries by using the following general expression for VCm(O, R, S, t), which we shall derive below:

Vi%@, R s, t> = &md&n

, OR, OKcm(~,t)) &O, R, G&, t>),

(B.3)

where /lFm is the proper Lorentz transformation which takes the transformed momenta OK into the corresponding c.m. momenta K,&O(s, t)) for the reaction OR, i.e., 0 &n(S,

t) O&m(s, t) = K!m(O(s, t)),

(B.4)

when O(s, t) are the invariant energy and momentum transfer variables in OR. Although /I&, depends in general on s and I, we shall simply use /I& without the arguments s and t, if no confusion can arise. A unique A,“, always exists since fl&, relates two sets of four-momenta K = OK,, and K’ = Kcm which have the same scalar products ki . kj = ki’ * kj’ and whose Gram determinants satisfy G(K) = G(K’) f 0 [S]. An exception is the forward and backward scattering where G(K) = G(K’) = 0, but A,“, for these special cases can be defined uniquely as limits of /I& as the scattering angle 0 goes to 0 or r.

PROPERTIES

OF

THE

CROSSING

533

MATRIX

Using (2.36), (B.4), and (B.2) we get fo,t(OR, (?(.K f)) = H&OR,

Km,(O(s, t)))

= g DOh.Oh+crn * OR,

= 1 V;%(O, k’

OKcm(s, t)) HdOR,

OK,,,(s, t))

(B.5)

R, s. t),f;,,(R, s, t).

This proves (B.1) with VCm as given in (B.3). If in (13.1) we use c.m. amplitudes f lw in the JW phase convention, has to be modified accordingly,

V%t@,

R, s, t> = i5~~iW/&?iR))

V;:dO,

R, s, t),

then VCm

(B.6)

where f:,lN(R) : (-1)s2+s4+A2+Aais defined in (2.40). The A,& are rotations for the operations P, T, C, Ejj . Therefore, the corresponding D matrix is diagonal, D .,,o/&c?m

2 OR, O&m) = MO, RI &,t

,

(B.7)

where &(O, R) are phases. Also the D matrix for 0 = Ist is diagonal. For all these operations, (B.l) reduces to (no summation)

iB.8)

with rl?iO,

R) = S,(O, 4 rl,lHK4 R Km).

(B.9)

In Table II we list Ll& , S,(O, R), ~~~~(0, KC&, qirn, and r&W for all important discrete symmetries, where qJhwcorresponds to 7:” in the JW phase convention. In evaluating 8, and pi” one must be very careful to observe the conventions such as the E prescription and the definitions given in Section 2. We shall illustrate some of the techniques and the problems involved by deriving the c.m. symmetry relations explicitly for the operations P, T, Eij , and X. Parity For parity, L& = &(z-) modulo &(277). The ambiguity the symrnetry matrix, since Dh,h,(R2(2r), R, PK) = (-l)“‘i may choose A,‘, = R2(r) to be specific. 595/62/z-20

R,(~z-) does not affect 6,,,, = 6,,,, . So we

534

DABOUL

To evaluate D(R,(r), we have (see (C.5))

R, PK) we note that for any momentum k in the x-z plane,

WH(R2(cx),k) == R,(n2n)

for

k = kser + kzez ,

(B.lO)

where II is the number of times k crosses the negative z axis when it rotates around the ~1axis by angle 01. An even II does not contribute to D(R3(r), R, PK), and an odd ?zcontributes a factor (- l)2si. In rotating PKcm around the 11axis by 7~,- k, and - k, cross the -e, axis once, whereas - k, and - k, do not cross this axis at all. That - k, must cross the -eZ axis is not clear at first. But because of the E prescription, - kz must end up below the z axis (see Fig. 5b). Therefore,

Substituting this and the expression (A.6a) for rlH(P, R, K) in (B.9) gives f-?,(R, s, t) = -%%’ (-l)““i+^i),f;,(R, QPQP

s, t),

which is equivalent to the familiar formula in the JW convention given in Table II. Motion Reversal T For T we have Ll& = R,(B + 2~ - r) R,(n). The necessity of the increment 2~ can be understood by looking at Figs. (5c, d) and noting the E prescription. To evaluate D(ATm , TR, TK), we note that W?R,(n>. k) = R3(n)

for any k

(B.14)

= emi”?3,,,,,,,

(B.15)

and D~~‘S)*(R3(~)) = Di,,,(R,(n)) so

Furthermore, crosses the -e,

in applying Rs(6 + 2~ - rr) on R3(~) TK none of the momenta axis, so that (B.lO) gives DTh&Rl(O

+ 2~ - rr), TR, R3(7r) TK) = a,,,~ .

(B.17)

Therefore, D Th,ThGGn > TR, TK) = (-l)2s3+2s4+‘A” s,,,, .

(B.18)

PROPERTIES

OF

THE

CROSSING

535

MATRIX

Substituting this and the expression (A.8) for qH(7’, R, K) in (B.3) gives (B.19) where X = X, - h, and p = h, - h, . The intrinsic parities qiT are discussed in Appendix A. Note that the product T~‘Q~/(Q~~/~~) need not be equal to +1 for inelastic scattering.

The Exchange symmetries E, El, , and Es1 Using ,/l& = R,(r) and the same arguments as for P and T, we get (in the JW phase convention)

$r(ER,

s, t) = iF(- I)zsL+n~ef,w(R, s, t),

(B.20)

where the normal statistics (Appendix A) were used. This general result reduces to that of JW which was derived for special reactions ER = R, where particles 1 and 2 and also particles 3 and 4 are identical (see Table 11); for obtaining the special result we use iF

=

i(fl+f2+13+f4)

=

i(;?f,+~f,)

=

(_1)2S1+2S3

for

ER = R,

(B.21)

so that f;,:(R,

s, t) = (-l)“-”

fiW(R, s, t)

for

ER = R.

(B.22)

The two symmetries E,, and Es4 differ from P, T, and E in that they map the invariant variables (s, t) not into themselves but rather into (s, u). Jf the reaction R is such that E,?R = R or E,,R = R, then El3 or Ez4 lead to forward-backward symmetry of the differential cross sections du/dQ (0) = dcr/dQ (r - 0). Our method is as easily applicable to E,, and Es4 as it is for P, r, and E. The pertinent information is given in Table II.

The Crossing Symmetry Xl, In the following, we sketch the derivation of the crossing matrix and point out some of the subtleties which lead to the different crossing matrices of TW and C-TMN. We also describe a direct method for finding the overall phase factor of the crossmg matrix. The crossing matrix follows by using the same procedure as for the previous symmetries. The only difference is that (I& is not a simple rotation, but a complicated complex Lorentz transformation. For the C-TMN continuation path, we have A&n = h(b) &(a) &(T)

for

Tc-TMN

(B.23)

536

DABOUL

with 6, := +

s

(k,’ -- k,‘), ,

(B.24)

where kit are the c.m. momenta in the t channel and h(b) is the “helicity boost” (2.27) for 6, . This expression for Af, follows, since (a) LICK takes XK,“, = (ksS, -kl”; -k4$, kz8) into K&t, s) = (kst, kdf; kit, kzf), so it must take the fourvector to = (1,0) = (kls + kes)/d\/s into 6, . Therefore, h-l(b) A& must be a rotation R, since it leaves t, unchanged. (b) Under A& E L+(C) the four-vector w,(K) =

l u,g,,(kls)a (k2”)’

= l ,031 v’skf,(k&

(k3’)?’ = qa&ls

+ kz’)R (kq’)’ (ks’)” (B.25)

sin 8,) = S,,, 4 -\/$(R, s, t)

is mapped into (~:m),v w,(K) = w,(&K)

= c,,0v(-k4t)a (k2”)’ (k3)Y

-E,,&,t

+ k3t)n (k,“)’ (k3)Y = ---E,~~~t/?k~,(k~,

= -6 Lt.312 d+(XR,

sin 0,) (B.26)

t, ~1,

where +(XR, T, S) is the Kibble function for the reaction XR evaluated in S+. Hence, the value of its square root 2/+(XR, t, s) depends on the continuation path from T+ to S+. Noting that 1/$(XR, t, s) > 0 in T+ (see Section 2.2), we find (see Fig. 2) \‘4(XR,

t, s)CpTMN = d+(R, s, t)

d&XR,

t, s)TW = --d$(R,

s, t)

in S+,

(B.27)

in S+.

(B.28)

The first relation (B.27) leads to (B.23), since A,“, reverses the y axis ((e,), = S,,,) for lY In contrast, AtgW) (which takes XK,“, to KtTW evaluated in S+) leaves the y axis unchanged. In this case we have (1X(TW) cm

= h(bTW) R,(olTW)

for rTW,

(B.29)

where bTW = $

(kzt - k,t)TW:

(B.30)

Note that b and bTW in (B.24) and (B.30) are defined in exactly the same way in terms of the t channel c.m. momenta kzt and kdt; the only difference is that these

PROPERTIES

OF

THE

CROSSING

momenta are continued along I’and rTw, respectively. Since K,!!l in S+ (Eq. (7.23)), it follows that hTW = ZstR,(7r)b

537

MATRIX

- Is&(~)

in S+

K&a (B.31)

and /jX(TW) cm

=

I,&(n)

GAl

(B.32)

.

From (B.31) we get h(P)

(B.33)

= Z,,R,(?T) h(b) R,(7r).

By substituting this relation in (B.29) and noting (B.32), we get ,yTW= 77 - cd.

(B.34)

From now on we shall only discuss the derivation of the crossing matrix of C-TMN; the crossing matrix of TW follows from this by using the relations (7.21). In this derivation we use the general helicity amplitudes as an intermediate step (TW’s original method). Although we can avoid using those amplitudes by working directly .with the M functions as C-TMN did, we prefer to use the H amplitudes to emphasize the similarity between finding the crossing matrix and finding the symmetry relations for P, T, E, and Eij . As before, we must now calculate D(L$& , XR, XK&). For this we use WH(&,

, ciki) = J@‘(,z(h) R,(x) R3(n), ciki) = W’(/z(b) R,(a), R,(n) ciki) W”(R,(n), = MQJ

ciki)

(B.35)

&(4,

where Gi can be determined (modulo 2~) using (C.8). The beautiful thing about (C.8) is that it is applicable to all possible analytic continuations: we need only substitute the correct values for 11k 11,11k’ 11,k, , and k,‘. Hence, it can be used for both h(E))R2(a) with rC-=MN and h(bTw) R,(cLTw) with rTw. And since Oi do not depend explicitely on a: or aTW, but depend only on EikfS and kit, we actually get the sanze expressions for Si in terms of s and t for both paths, although naturally the Qi have different numerical vaIues for the two cases. The expressions for cos L’i and sin Qi which follow from (C.8) are essentially the samle as those for wi , except for some signs: cos f2, = - cos 01) cos l2, = cos W?)

sin Q2, = - sin w1 ,

cos 52, = cos w3,

sin 52, = sin w2 , sin L& = - sin wQ ,

cos Q, = -

sin 52, = sin wq ,

cos wq )

(B.36)

538

DABOUL

These equations are only enough to relate Qi and wi modulo 2~. However, we shall prove below that it is consistent to write (4

Q, = WI + 77

(modulo ~QT),

(b)

R, = co2

(modulo 471.),

(cl

L$ = -w3 - 2rr

(modulo 4n),

(4

fin, = 7T~ co4

(modulo 47~).

(B.37)

These relations are enough to determine the crossing matrix uniquely, since the 4~ ambiguities do not affect the overall phase factor since &,,(47~) = S,,,, . By substituting Eqs. (B.37) in (B.35), we get

where l 1 = E~ = -Q = -Ed = -1. Finally, by substituting this and the expression (A.16) for qx(X, R), we immediately get the crossing matrix (2.46). Deternzinatiotl of the overall phase factor In the last section we obtained the crossing matrix in a few steps, but we left out the most difficult step, viz., that of proving Eqs. (B.37). These relations are essential for determining the overall phase factor. TW neglected this factor completely and C-TMN used condition (3.5) and some complicated arguments to determine this factor. We now give a direct method for determining this factor. To prove (B.37) we use the 2 x 2 matrix representation of the Wigner rotations, WH(h(b) R?(a), ERR,

ki”) = I~-‘(k~i) h(b) R?(u) II(EiR,(7T) ki”).

(B.39)

It is well known that this representation is only unique up to a sign, which leads to (-1)“” ambiguities. However, in our case the representations of h(~&~) and h-l(@) are uniquely defined by the analytic continuation of (2.27). Moreover, the sign ambiguity in determining the 2 x 2 matrix for h(b) R,(a) does not affect the overall phase factor, as this matrix is common to all four Wigner rotations.

PROPERTIES

OF

THE

CROSSING

539

MATRIX

In fact, we could choose the sign of h(h) R,( LY ) such that Eq. (B.37a) becomes satisfied. Thus, in principle we can completely calculate the following 2 x 2 matrices:

which in turn determines AQi modulo 2x, so that Qi become determined modulo 4~. The multiplication of the matrices in (B.39) is rather complicated, so instead of doing this, we find relationships among the four matrices for special cases; this is enough to determine the 2n ambiguities. Thus, from (B.39) we have R(!SJ z

BjjR(Qj)

(no summation)

Ajj

(B.41)

where Ajj _

h-l(EjR3(~)

kj”) h(EjR3(7T)kj”),

Bij = I~-‘(ki,) h(kij).

(B.42) (B.43)

We shall evaluate Aji and Btj for equal mass reactions and for two special angles 19~= 0, VT.For these two cases, which we call case 1 and case 2, A,, and Bij are easy to calculate. Case !

0, = 0. Therefore, k,’ -= k:,” so that (C.6a) gives A,, = lzr’(-R3(rr)

k,“) lz(RJrr) kgR) = hr’(-k,“)

h(k,“) = R,(rr).

(B.44)

Since nz3 = llzl it follows that (/Cam),,== (k4t)o , so that k,“?t := kJt. Using (C.3) gives B,, = K1(kg*) h(k,‘) = K1&,‘)

i~(k,~)

= R,(2n) R,(z-)(o”k, -I- g3 I/ k, ii) A = R,(277) R,(r)

(B.45)

ia, = MT),

where we used the facts that $(k4) = r (see Section 2.6), (kdtjo = 0, and )I k \l/rn = i for Case 1. Case 12. 19, = rr. Therefore, 1~~ = k,” and kzS = k,” and consequently kbt = kit and kzt = kst. This immediately tells us that et must be an odd multiple of 57. By following the continuation path rC-TMN, we find that 8, = -n. For

540

DABOUL

Case 2 the boost factors Lpl(k,“) L(Iids) and L~l(k2~“) L(k,“)(x and only the R(lii) factors are left in the following: A,, = P(Rg(7r) k2”) I?(&(X) /y)

= R,'(T)

R&n)

A,, == lzr'(-R3(~) = R-$k,")

= t or s) cancel out,

= Rrn1(R3(7T)k,“) R(R,(n) ksS)

= R,(-274,

(B.46)

kl") I?(-R3(n) kdS) = R-l(-R,(n)

k,") R(-RR,(n) k,")

R(k,") = R;l(O) R,(O) = 1,

B,, = h-l(k,t) i~(k,~) = R-'(k,') = R,'(-n)R,(-TT)

(B.47)

R(kzt) = R,'(&)

R&z-)

= I,

(B.48)

B,, = /~~~(k,~)h(kaf) = R '(kl") R(k,') = R,l(O) R&n

+ 0,) = R,(--2~). (B.49)

Note that although

kzS = k,” (backwards

scattering)

in (B.46), the rotations

R(R,(vr) kz8) and R(R,(n) Iiz3) are not equal because of the E prescription: This gives R(R,(n) h-z”) = R,(+n) and R(R,(n) kz8) = R&n) (see Figs. 5c and Sd). Similar remarks are true for other momenta. In (B.47) we used R(--kJ = R(kJ (see Section 2.4). Substituting the expressions (B.44)-(B.49)

into (B.41) gives the relations

R&&)

= B,,R,(@) A,, = R,(T) R,(Q,) I&(T) = R&T

&(-%I

= B,,R,(Q,) As = R&&J R,(--274

R&Q,) = &R@d

A,, = k-24

- Q,),

Case 1, Case 2,

(B.50)

Case 2,

R&h),

so that (a)

Q, = --x ~ L?,

(modulo 477),

(b) (c)

Sz, = Q, + 2~ sz, = sz, - 2?7

(modulo 4~),

(B.51)

(modulo 4~).

Finally, we have to check whether (B.37) are consistent with (B.51). For this, we recall that for the equal masses we have in general [6] w1 = wq = --x ~ w,

wg = wg = w,

For %,$= 0 and rr one can show that w1 = -n/2

wi=--rr 2 Wl = wq = -Tr,

-rr
and -n,

(B.52)

respectively, so that

for all i,

for 9, = 0,

w2 = wg = 0

for Bs = n.

(B.53)

PROPERTIES

OF

THE

CROSSING

541

MATRIX

Substituting this in (B.37) gives .Q, = in

and

Q,=

~ g-r

Q, = .Qn::= 0

and

Q3 = -Q4

(modulo 4~7) for Case 1, = -23

(modulo 47~) for Case 2.

(9.54)

These values of sZi satisfy relations (B.Sl), which proves that Eqs. (B.37) are correct for special cases. But these equations must also be true in general, since a jump of 27r cannot occur by continuous change of the masses or the variables s and t. Thus our proof of (B.37) is complete.

APPENDIX

C

In the text we need certain Wigner rotations and some products of Lorentz transfor:mations, which we list here without proof for easy reference. The following relations refer to the 2 x 2 matrix representations O’/‘(& of the Lorentz transformations involved. The four-vector k in all the following relations is assumed to be real andi in the physical region (i.e., k, 3 IH and ij k 11> 0). The corresponding relations for other k can be obtained from these by analytic continuation of both sides of the following relations: L-l@) L(k) = L2(k) = u . k,

(C.1) (C.3)

R-l(k) R(k) = R,(2$(k)) R,(T), k’(B) h(k) = &(2+(k)) k> = &kb), WH(R2(a), k) = R,(n277)

~H(~3(zk~),

R,(n)(& + for any k,

~73

/I k ii>/w

for k in the x-z plane,

(C.3) cc.41 CC.5)

where n is the number of times k crosses the negative z axis when it rotates around the y axis by an angle 01, h-l( - k‘) h(k) if k, is continued in the upper half-plane, for TW’s path, (II - k II == 11k II),

(C.6a) (C.6b)

WWst , k) =

h-l(-k) h(k) = R3(r) hTW-I(-k) h(k) = R,(2$(k))

R,(-n)

for [I - k )I = - ‘j k 11, (C.7a) // - k 11= + !I k I/, (C.7b)

where in both cases Ds(Zst) was chosen to be one, WH(L(b) R,(a), k) = R@)

fork and b in the x-z plane,

(C.8)

542

DABOUL

where llz”/jo) -EL

cos 52 = &

(k,k,’

sin J2 = $

(b ;q k’)P = $

where k,’ = (L(b) &(a)),,k,

APPENDIX

D.

-

1)

(bllig’ -- b&r’) a

0,

, k’ = ??z2,k = jl k 11,and k’ s I/ k’ jj.

RELATIONS

INVOLVING

THE COMMUTATION

FACTORS uij

We define the fermion number f by 1 for a fermion, f = 10 for a boson.

(D.1)

We use

F -fi

+fi +f, t-f,

CD.3

to denote the total number of fermions in a reaction. Hence, F must be an even number for all allowed reactions. From (D.l) and (D.2), we note the following simple identities, which we shall need later:

(D.3)

jy==,

uij = jz:fj

((-I>“”

(-1)” 1-2F

=

iF’

(fi

=

defined in (A.12)),

_ (-l)““i,

(-1)2fJ,

4- .h>"

=

fi

=

(-p

.h

+

+

(D.4) (D.5)

=

Xh

1,

?

P.6)

CD.71

so that (-

l)(fi+fj)'

= (- l)ft+fj+2fif,

= (- l)2st+2sj.

CD.81

Note that although (D.5) is always valid, the relation if = i2s is not always true, as, for example, when S = 1. We now prove the following useful relations: (4 (b)

(- 1)%+“13+~14- (- 1)““’ (_ 1p+034 = i”( _ 1)2s1+2s2

(cyclic), (cyclic).

CD.91

PROPERTIES

proo,: (b)

@> (-])%+Ww (_

1 y+‘%

OF

THE

CROSSING

--_ (-1)fh)

543

MATRIX

_ (-1)-f?

_ (-1)““~.

--_ j?flfa+2f3f4 =

i(fl+f2)2+(fg+f4)?~(fl+f?)-(fJ+f4)

_

i(fl+fs)?+(F-(fl+fn))“-F

=

j:!(f,+ft)L+F’--2F(fl+fI)

--_ (-1)

(fl+fJ~

iF i’

_

(-1)“s’i’s2

iF,

where we used Eqs. (D.3)-(D.8).

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15.

G. COHEN-TANNOLJDJI, A. MOREL, AND H. NAVELET, Anlr. P/z~s. (Ncp+r’ Yorkj 46 (1968), 239. T. L. TRUEMAN AND G. C. WICK, Am. Phys. (New Yorh-j 26 (1964), 322. M. JA~COB AND G. C. WICK, Ann. Phys. (New York) 7 (1962), 404. IVAN J. MUZINICH, J. Math. Phys. 5 (1964), 1481. L. L. C. WANG, Phys. Rev. 142 (1965), 1187. J. DABOUL, Phys. Rev. 177 (1969), 2375. M. S. MARINOV AND V. I. ROGINSKII, Preprint from Institute for Experimental and Theoretical Physics, Moscow, 1963; YA. A. SMORODINSKY, Dubna Preprint, 1963. These authors were quoted by TW. J. DABOUL, “Linear Symmetries of Scattering Amplitudes,” to be published in Fortschritte Der Physik. E. WIIZNER AND L. EISENBUD, Phys. Rev. 72 (1947), 29: F. COESTTR, Phys. Rev. 89 (1953), 619. G. C. WICK, Ann. Phys. (New York) 18 (1962), 65. H. P. STAPP, Phys. Rev. 125 (1962), 2139. A. KOTANSICI, Am Phys. PO!. 29 (1966), 699; 30 (1966), 629. R. F. S~REATER AND A. S. WIGHTMAN, “PCT, Spin and Statistics, And All That,” Eq. (4.69), W. A. Benjamin, New York, 1964. J. BROS, H. EPSTEIN, AND V. GLASER, Com,,~nz. hf&h. Phys. 1 (1965), 240. D. Z~ANZIGER, P/7)>s. Rev. B 133 (1964), 1036.