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s- and t-Channel transformation matrices for helicity and invariant amplitudes for γ + N → 1± + N
ANNALS OF PHYSICS 80, 387-396
s- and t-Channel
(1973)
Transformation
Invariant RONALD Center
Amplitudes
Matrices for y + N -
G. PARSONS, BEN L. MANNY,
for Particle
Theory,
for
The University
Helicity
and
I* + N+
AND ROBERT BECK CLARK
of Texas
at Austin,
Austin,
Texas
78712
Received July 14, 1972
The Algebraic transformation matrices, valid for all s and I, connecting the s- and t-channel helicity amplitudes to a corresponding set of gauge invariant covariant amplitudes are presented for the photoproduction processes y + N j l* + N.
I.
INTRODUCTION
This paper is the third and last in a series of papers presenting transformation matrices connecting a set of helicity amplitudes to a set of gauge invariant covariant amplitudes. In this paper the processes N + y + N + (l-, l+) are considered. The transformation matrices for the above processes are given for both the s-channel and t-channel amplitudes. As in the first two papers [ 1, 21, the calculation was performed using the symbolic algebraic computer language REDUCE [3] to perform the necessary matrix multiplications and a variety of substitutions. The program produced output which required only a very small amount of further manipulation by hand to obtain the results shown below. The program is general enough to calculate the transformation matrices for any process of the type A+B+C+D
where A, B, C, and D are any particle or resonance of a given spin and parity and where the set of covariant amplitudes is given. In Section IT, the notation and conventions are established, and, in Section III, the results of the calculation are listed. + Supported in part by the U. S. Atomic Energy Commission under Contract At(40-1)3992.
387 Copyright All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
388
PARSONS,
MANNY,
AND
CLARK
11. NOTATION
We define J@&~,~, to be the helicity amplitude
for the s-channel reaction
1+2-+3+4 with momenta pi and helicities Xi . For the corresponding reaction in the t-channel, 4+2+3+i
with momenta qi and helicites Xi we define the helicity amplitude M!$,,A,A, . The spin-parities Jp of particles 1-4, respectively, are (&+), (I-), (fr+) and (l-, l+). The mass of particle 2 (the photon) is zero. A. s-Channel In the s-channel, the helicity amplitudes Ai by
are related to the covariant amplitudes
There is an implied sum O-3 over repeated greek indices using the metric of Bjorken and Drell [4]. The kinematics are defined in Fig. 1. The polarization four-vector of the photon (particle 2) is c”(p2 , h,) and has components 1 ‘I
+I)
= 2112
II 0
0
1
1 H -1
-i
)
E”(P2 9 -I)=-
-i
0
2
0
where pzw = 1p2 / (1, 0, 0, -1). The spinors u(p, , X,) for nucleon 1 have matrix representations 1 u ( Pl,
+ 2
i
El + m, Ii2 1 2ml
i I 0 IPII 4 + ml 0
,
0 u
(
P1,--
1 0
1 = ( El + ml 1ljz I 21 24 6
I 9
I Pl I + 1711
S- AND t-CHANNEL
TRANSFORMATION
389
MATRICES
FIG. 1. s-Channel center-of-mass kinematics.
where plu = (El , 0, 0, I p1 I). The spinors U(P, , h3) for nucleon 3 have the matrix representations
where p3p = (ES , I p3 1sin 8, , 0, j pS j cos 0,). The polarization particle 4 is EQ~ , h4) where
i I 0
dP4 3 +I)
1
= - 21/2
-cam e,5 i
sin 8,
’
four-vector
for
390
jn14 90) =1 1
PARSONS,
MANNY,
AND
CLARK
-- E4
in4 sin 0
dp4
8
’
,
1i-cos 8, 1 --
,:
c~~
8,
4
0
EYP4 3 -l)‘w
-j
’
sin 8,
and where p4p = (E4, The matrices SiP respect to the photon processes referred to
- ( p4 / sin 8,) 0, - / p4 ( cos 6,). are defined so as to be individually gauge invariant with (particle 2). These matrices are listed below for the various in this paper. We define P = (1/2)(p, + p3).
N+y+N+l(s+) + 1- - (Q+) + Ix;”
= P’(P,
. Pp4u - pz ’
Yy
= pz * p4gvLL
y:
= p2 ’ PgV, - p2”PP
4;”
= gvup’z - p2y
sy
= (g*upLpz . P - Pz”W
p4l-y
- PZ’P,”
$72
Y’,” = P”(PQYs - p2 . Py’) JT
=
P2wf2
sy
=
y*(p2
. PP4@
9;;
=
y’(y”~2
'p4
-
p4?2)
9’;
= y’(yL”p, . P -
P%)
G
= Y”Y”P’,
-
Pz . P4W
S- AND
?-CHANNEL
TRANSFORMATION
MATRICES
N+y+N+l+ (it+) + l- + (4’) + 1+ 9;” = iy,pY(p,
- Pp4@ - pz . p.$Pu)
C
= iyAp
. p4gYu - P2”P49
Sr
= iy,(p,
. PgY” - p2”Pu)
=5T = 9y
~ys(PP2
-
P2W
= iy,(gvGp2 . P - p2yPu) p2
YT = iy5Py(PLIp2 - p2 * Py”) 9;” = iysp2YP2 9:
= iysPvyuP,
-C
= h.yvb2
=%
=
9;:
= iy,y”(y*p,
-0;;
=
iy5yy(yL;I)2
.
PP4” . P4
-
P2
. P4W
P4”p2)
* P - Pup*)
iysy"yup2
FIG.
2.
t-Channel
center-of-mass
kinematics.
391
392
PARSONS, MANNY, AND CLARK
B. t-Channel In the t-channel, the helicity amplitudes are related to the covariant amplitudes Ai by
There is an implied sum O-3 over repeated greek indices using the metric of Bjorken and Drell [4]. The kinematics are defined in Fig. 2. The polarization four-vector for the photon (particle 2) is @(q2, A,) and has components
442 3 $1) = &
i I 0 -cos 6, -i 3
1
l %I22-1) = p
sin Bt
0 cos 8,
-i
i -sin
etI
where q2“ = I q2 I (1, sin 19~~0, cos 0,). The spinors for the antinucleon matrix representation9
2’ (ql ) + ;, = ( E’2fn,m’)1’2
-
El
y2
have the
0 I Ql I 4 f ml 0 1
-
21(ql ) - ;) = ( y+,m1
>
I %
I
+
ml
0 -1 0
where qlu = (4 , (40, I q1 I). 1 This convention for labeling the helicities of the antiparticles, represented by v-spinors, has been adopted because of its widespread usage. The general expression for this convention is
where s = --h and where, for p in the fz direction, Xl/? =
1 0
0
and
s- AND
~-CHANNEL
TRANSFORMATION
MATRICES
393
The spinors for N (particle 3) have the matrix representation 0
1 0 I 43 I E,
+
m3
-1 0 I 43 I E,
+
t%
0
where q3u = (E3, 0, 0, - 1q3 I). The polarization the matrix representations
four-vector
for particle 4 has
0
1
r1 -cos ot
21/2
i
’
sin 8,
--
) fi’4
444 ?0)
-E4 sin fl’4
et
0 --El cos et 1124
dq‘l 3 -1)
-i I 0 -cos et -i 2112 1
'
sin 8,
where q4 = (E4 , - 1q4 1sin et , 0, - I q4 / cos 0,). The matrices Yifl are defined so as to be individually gauge invariant with respect to the photon (particle 2). These matrices are listed below for the various processes referred to in this paper. We defined Q = (1/2)(q3 - ql).
394
PARSONS,
MANNY,
AND
CLARK
l-+y-d+m i=+ 4“
=
-Q”h
sy
=
-q2
#y
=
q2 . Qg”N
Jy
=
gy2
AT’
=
(g%
*
. q4 g”(”
Q”(Q“t2
-fT =
42Y42
$7
=
QyQ2
4,“
=
-y”h -Y”(Y”42
G
y”W%
=
4; =
-
+
-
-
4“ =
s;z=
Qsi’
q2 .
l--($‘)
+ @)
aQ10
42”44”
q2”QG
q2”yU
Q - qx”Q”>42 - q2. QY“)
.
.
QW- q2. qaQ") . q4
.
-
44”42)
Q - Q”42)
Y’Y”d2
1++ 3’1” =
--iysQ”(q2
6?
=
s-92
3:
= kh
FT
= iy5W42
-
Qa”
* 44P .
-
+
92 *
l--+(p)
qdQ”>
q2yq4w)
Qg”” - qt”Q’9 -
q2”y*)
Q - qz”Q’912 9: = b’@@l”dz - % . Qr‘9 3”;” = iy5q2yy~42
4”
= Mgyv,
ST
=
iy5Qyufj2
Ti‘
=
-hiy”(q2
Gl
=
-iy5y”(yV2
J%
=
hW‘q2
Gi
=
iy5yYt2
.
.
Qw -
*
q4
Q-
qaQ’9 - W42) Q“4J -
qz .
+ ($y
S- AND t-CHANNEL TRANSFORMATION MATRICES
III. A. s-Channel The kinematical
RESULTS
invariants in the s-channel are (see Fig. 3): s = (PI + PZ = (P3 + PA25 t = (PI - PA2 = (Pz - P4Y, u = (PI - P*Y = (Pz - P32, s + t + u = ml2 + mS2 + md2, 9& = {(S - (mi -+ mi)“)(s - (mi --
qSij = (s - (mi + mj)2)1/2 & = (s - (mi -- mJ2)l/’ wi = (s + mi2 - mj2)/2s1/2 pii = 9&/29
m2 = 0, WIj)2))‘/2
threshold (i,j) pseudothreshold
(i, j)
i = 1,2-j
= 2,1
1i == 3,4-j
= 4,3
) p1 I = 1p2 1 = p12 = pzl = c.m. initial momentum I p3 / = 1p4 I = pza = pd3 = c.m. final momentum cos e = 2st + S2 s
SZ@?i2
+ (ml2 - m22)(m,2 - rna2) Y;2%4
FIG. 3. Kinematic variables in the s-channel.
B. t-Channel The kinematical
invariants in the t-channel are (see Fig. 4): t =
(q2
+ qd2 = (Sl + qJ2,
s = (41 -
4212
= (93 - qJ2,
ZJ = (41 - %I2 = (42 - qd2, t + s + ii = ml2 + mS2 + md2, m2 = 0,
396
PARSONS,
MANNY,
AND
CLARK
FIG. 4. Kinematic variables in the r-channel.
qj
=
{(t
-
(mi
+
mJ2)(t
~
(rni
-
rnj)“)}‘l”
$ii = (t - (WZ~ + WZ~)‘)I/’
threshold (i, j)
*ii = (t - (W7i-
pseudothreshold (i, j)
WZj)2)1’2
oi = (t + mi2 - mi2)/2t’12 = 9J2P
qij
i=4,2*j==2,4 I i = 3, 1 t) j = 1,‘3
1q2 j = / q4 1 =
qJ2
=
q2a
= c.m. initial momentum
1q1 1 = j q3 ) =
qsl
=
q13
= c.m. final momentum
cos e = 2ts + t2 - t&q2 t
+ (mJ2 - m22)(m32 - ml”) <29&
(s, t) and A!$&s, The helicity amplitudes ~4’l\$,,~~~~ amplitudes Ai(s, t) are listed in Section IV.2
t) in terms of the covariant
REFERENCES
1. ROBERT BECK CLARK, BEN L. MANNY, AND RONALD G. PARSONS, Ann. Phys. 69 (1972), 522. 2. RONALD G. PARSONS,Ann. Phys. 72 (1972), 171. 3. A. C. HEARN, REDUCE, a user-oriented interactive system for algebraic simplification, in “Interactive Systems for Experimental Applied Mathematics” (M. Klerer and J. Reinfelds, Eds.), Academic Press, New York, 1968. 4. J. D. BIORKEN AND S. D. DRELL, “Relativistic Quantum Mechanics,” McGraw-Hill, New York, 1964. 2 See NAPS document # 02167 for 107 pages of supplementary material. Order from ASIS/ NAPS c/o Microfiche Publications, 305 E. 46th St., New York, NY 10017. Remit in advance for each NAPS accession number $1.50 for microfiche or $5.00 for photocopies up to 30 pages, 15e for each additional page. Make checks payable to Microfiche Publications.
s- and t-Channel
(1973)
Transformation
Invariant RONALD Center
Amplitudes
Matrices for y + N -
G. PARSONS, BEN L. MANNY,
for Particle
Theory,
for
The University
Helicity
and
I* + N+
AND ROBERT BECK CLARK
of Texas
at Austin,
Austin,
Texas
78712
Received July 14, 1972
The Algebraic transformation matrices, valid for all s and I, connecting the s- and t-channel helicity amplitudes to a corresponding set of gauge invariant covariant amplitudes are presented for the photoproduction processes y + N j l* + N.
I.
INTRODUCTION
This paper is the third and last in a series of papers presenting transformation matrices connecting a set of helicity amplitudes to a set of gauge invariant covariant amplitudes. In this paper the processes N + y + N + (l-, l+) are considered. The transformation matrices for the above processes are given for both the s-channel and t-channel amplitudes. As in the first two papers [ 1, 21, the calculation was performed using the symbolic algebraic computer language REDUCE [3] to perform the necessary matrix multiplications and a variety of substitutions. The program produced output which required only a very small amount of further manipulation by hand to obtain the results shown below. The program is general enough to calculate the transformation matrices for any process of the type A+B+C+D
where A, B, C, and D are any particle or resonance of a given spin and parity and where the set of covariant amplitudes is given. In Section IT, the notation and conventions are established, and, in Section III, the results of the calculation are listed. + Supported in part by the U. S. Atomic Energy Commission under Contract At(40-1)3992.
387 Copyright All rights
0 1973 by Academic Press, Inc. of reproduction in any form reserved.
388
PARSONS,
MANNY,
AND
CLARK
11. NOTATION
We define J@&~,~, to be the helicity amplitude
for the s-channel reaction
1+2-+3+4 with momenta pi and helicities Xi . For the corresponding reaction in the t-channel, 4+2+3+i
with momenta qi and helicites Xi we define the helicity amplitude M!$,,A,A, . The spin-parities Jp of particles 1-4, respectively, are (&+), (I-), (fr+) and (l-, l+). The mass of particle 2 (the photon) is zero. A. s-Channel In the s-channel, the helicity amplitudes Ai by
are related to the covariant amplitudes
There is an implied sum O-3 over repeated greek indices using the metric of Bjorken and Drell [4]. The kinematics are defined in Fig. 1. The polarization four-vector of the photon (particle 2) is c”(p2 , h,) and has components 1 ‘I
+I)
= 2112
II 0
0
1
1 H -1
-i
)
E”(P2 9 -I)=-
-i
0
2
0
where pzw = 1p2 / (1, 0, 0, -1). The spinors u(p, , X,) for nucleon 1 have matrix representations 1 u ( Pl,
+ 2
i
El + m, Ii2 1 2ml
i I 0 IPII 4 + ml 0
,
0 u
(
P1,--
1 0
1 = ( El + ml 1ljz I 21 24 6
I 9
I Pl I + 1711
S- AND t-CHANNEL
TRANSFORMATION
389
MATRICES
FIG. 1. s-Channel center-of-mass kinematics.
where plu = (El , 0, 0, I p1 I). The spinors U(P, , h3) for nucleon 3 have the matrix representations
where p3p = (ES , I p3 1sin 8, , 0, j pS j cos 0,). The polarization particle 4 is EQ~ , h4) where
i I 0
dP4 3 +I)
1
= - 21/2
-cam e,5 i
sin 8,
’
four-vector
for
390
jn14 90) =1 1
PARSONS,
MANNY,
AND
CLARK
-- E4
in4 sin 0
dp4
8
’
,
1i-cos 8, 1 --
,:
c~~
8,
4
0
EYP4 3 -l)‘w
-j
’
sin 8,
and where p4p = (E4, The matrices SiP respect to the photon processes referred to
- ( p4 / sin 8,) 0, - / p4 ( cos 6,). are defined so as to be individually gauge invariant with (particle 2). These matrices are listed below for the various in this paper. We define P = (1/2)(p, + p3).
N+y+N+l(s+) + 1- - (Q+) + Ix;”
= P’(P,
. Pp4u - pz ’
Yy
= pz * p4gvLL
y:
= p2 ’ PgV, - p2”PP
4;”
= gvup’z - p2y
sy
= (g*upLpz . P - Pz”W
p4l-y
- PZ’P,”
$72
Y’,” = P”(PQYs - p2 . Py’) JT
=
P2wf2
sy
=
y*(p2
. PP4@
9;;
=
y’(y”~2
'p4
-
p4?2)
9’;
= y’(yL”p, . P -
P%)
G
= Y”Y”P’,
-
Pz . P4W
S- AND
?-CHANNEL
TRANSFORMATION
MATRICES
N+y+N+l+ (it+) + l- + (4’) + 1+ 9;” = iy,pY(p,
- Pp4@ - pz . p.$Pu)
C
= iyAp
. p4gYu - P2”P49
Sr
= iy,(p,
. PgY” - p2”Pu)
=5T = 9y
~ys(PP2
-
P2W
= iy,(gvGp2 . P - p2yPu) p2
YT = iy5Py(PLIp2 - p2 * Py”) 9;” = iysp2YP2 9:
= iysPvyuP,
-C
= h.yvb2
=%
=
9;:
= iy,y”(y*p,
-0;;
=
iy5yy(yL;I)2
.
PP4” . P4
-
P2
. P4W
P4”p2)
* P - Pup*)
iysy"yup2
FIG.
2.
t-Channel
center-of-mass
kinematics.
391
392
PARSONS, MANNY, AND CLARK
B. t-Channel In the t-channel, the helicity amplitudes are related to the covariant amplitudes Ai by
There is an implied sum O-3 over repeated greek indices using the metric of Bjorken and Drell [4]. The kinematics are defined in Fig. 2. The polarization four-vector for the photon (particle 2) is @(q2, A,) and has components
442 3 $1) = &
i I 0 -cos 6, -i 3
1
l %I22-1) = p
sin Bt
0 cos 8,
-i
i -sin
etI
where q2“ = I q2 I (1, sin 19~~0, cos 0,). The spinors for the antinucleon matrix representation9
2’ (ql ) + ;, = ( E’2fn,m’)1’2
-
El
y2
have the
0 I Ql I 4 f ml 0 1
-
21(ql ) - ;) = ( y+,m1
>
I %
I
+
ml
0 -1 0
where qlu = (4 , (40, I q1 I). 1 This convention for labeling the helicities of the antiparticles, represented by v-spinors, has been adopted because of its widespread usage. The general expression for this convention is
where s = --h and where, for p in the fz direction, Xl/? =
1 0
0
and
s- AND
~-CHANNEL
TRANSFORMATION
MATRICES
393
The spinors for N (particle 3) have the matrix representation 0
1 0 I 43 I E,
+
m3
-1 0 I 43 I E,
+
t%
0
where q3u = (E3, 0, 0, - 1q3 I). The polarization the matrix representations
four-vector
for particle 4 has
0
1
r1 -cos ot
21/2
i
’
sin 8,
--
) fi’4
444 ?0)
-E4 sin fl’4
et
0 --El cos et 1124
dq‘l 3 -1)
-i I 0 -cos et -i 2112 1
'
sin 8,
where q4 = (E4 , - 1q4 1sin et , 0, - I q4 / cos 0,). The matrices Yifl are defined so as to be individually gauge invariant with respect to the photon (particle 2). These matrices are listed below for the various processes referred to in this paper. We defined Q = (1/2)(q3 - ql).
394
PARSONS,
MANNY,
AND
CLARK
l-+y-d+m i=+ 4“
=
-Q”h
sy
=
-q2
#y
=
q2 . Qg”N
Jy
=
gy2
AT’
=
(g%
*
. q4 g”(”
Q”(Q“t2
-fT =
42Y42
$7
=
QyQ2
4,“
=
-y”h -Y”(Y”42
G
y”W%
=
4; =
-
+
-
-
4“ =
s;z=
Qsi’
q2 .
l--($‘)
+ @)
aQ10
42”44”
q2”QG
q2”yU
Q - qx”Q”>42 - q2. QY“)
.
.
QW- q2. qaQ") . q4
.
-
44”42)
Q - Q”42)
Y’Y”d2
1++ 3’1” =
--iysQ”(q2
6?
=
s-92
3:
= kh
FT
= iy5W42
-
Qa”
* 44P .
-
+
92 *
l--+(p)
qdQ”>
q2yq4w)
Qg”” - qt”Q’9 -
q2”y*)
Q - qz”Q’912 9: = b’@@l”dz - % . Qr‘9 3”;” = iy5q2yy~42
4”
= Mgyv,
ST
=
iy5Qyufj2
Ti‘
=
-hiy”(q2
Gl
=
-iy5y”(yV2
J%
=
hW‘q2
Gi
=
iy5yYt2
.
.
Qw -
*
q4
Q-
qaQ’9 - W42) Q“4J -
qz .
+ ($y
S- AND t-CHANNEL TRANSFORMATION MATRICES
III. A. s-Channel The kinematical
RESULTS
invariants in the s-channel are (see Fig. 3): s = (PI + PZ = (P3 + PA25 t = (PI - PA2 = (Pz - P4Y, u = (PI - P*Y = (Pz - P32, s + t + u = ml2 + mS2 + md2, 9& = {(S - (mi -+ mi)“)(s - (mi --
qSij = (s - (mi + mj)2)1/2 & = (s - (mi -- mJ2)l/’ wi = (s + mi2 - mj2)/2s1/2 pii = 9&/29
m2 = 0, WIj)2))‘/2
threshold (i,j) pseudothreshold
(i, j)
i = 1,2-j
= 2,1
1i == 3,4-j
= 4,3
) p1 I = 1p2 1 = p12 = pzl = c.m. initial momentum I p3 / = 1p4 I = pza = pd3 = c.m. final momentum cos e = 2st + S2 s
SZ@?i2
+ (ml2 - m22)(m,2 - rna2) Y;2%4
FIG. 3. Kinematic variables in the s-channel.
B. t-Channel The kinematical
invariants in the t-channel are (see Fig. 4): t =
(q2
+ qd2 = (Sl + qJ2,
s = (41 -
4212
= (93 - qJ2,
ZJ = (41 - %I2 = (42 - qd2, t + s + ii = ml2 + mS2 + md2, m2 = 0,
396
PARSONS,
MANNY,
AND
CLARK
FIG. 4. Kinematic variables in the r-channel.
qj
=
{(t
-
(mi
+
mJ2)(t
~
(rni
-
rnj)“)}‘l”
$ii = (t - (WZ~ + WZ~)‘)I/’
threshold (i, j)
*ii = (t - (W7i-
pseudothreshold (i, j)
WZj)2)1’2
oi = (t + mi2 - mi2)/2t’12 = 9J2P
qij
i=4,2*j==2,4 I i = 3, 1 t) j = 1,‘3
1q2 j = / q4 1 =
qJ2
=
q2a
= c.m. initial momentum
1q1 1 = j q3 ) =
qsl
=
q13
= c.m. final momentum
cos e = 2ts + t2 - t&q2 t
+ (mJ2 - m22)(m32 - ml”) <29&
(s, t) and A!$&s, The helicity amplitudes ~4’l\$,,~~~~ amplitudes Ai(s, t) are listed in Section IV.2
t) in terms of the covariant
REFERENCES
1. ROBERT BECK CLARK, BEN L. MANNY, AND RONALD G. PARSONS, Ann. Phys. 69 (1972), 522. 2. RONALD G. PARSONS,Ann. Phys. 72 (1972), 171. 3. A. C. HEARN, REDUCE, a user-oriented interactive system for algebraic simplification, in “Interactive Systems for Experimental Applied Mathematics” (M. Klerer and J. Reinfelds, Eds.), Academic Press, New York, 1968. 4. J. D. BIORKEN AND S. D. DRELL, “Relativistic Quantum Mechanics,” McGraw-Hill, New York, 1964. 2 See NAPS document # 02167 for 107 pages of supplementary material. Order from ASIS/ NAPS c/o Microfiche Publications, 305 E. 46th St., New York, NY 10017. Remit in advance for each NAPS accession number $1.50 for microfiche or $5.00 for photocopies up to 30 pages, 15e for each additional page. Make checks payable to Microfiche Publications.