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Concerning the four-quark two-gluon subprocess
Volume 176, number 3,4
PHYSICS LETTERS B
28 August 1986
CONCERNING THE FOUR-QUARK TWO-GLUON SUBPROCESS ¢~ John F. G U N I O N
Department of Physics, Universityof California at Davis, Davis, CA 95616, USA and Z. K U N S Z T 1,2
Institute for TheoreticalPhysics, Universityof Bern, CH-3012 Bern, Switzerland Received 4 June 1986 The extension of the unequal flavor four-quark two-gluon amplitude, given previously, to the case of equal flavor quarks is presented. The result can be obtained using simple permutations of the previous expression, and illustrates nicely the technical advantages of the improved helicity method.
In a previous paper [1] we calculated the four-quark two-gluon amplitude in QCD for unequal quark flavors. We demonstrate here how to use simple permutation identities to obtain the full equal flavor result from the previous expression. In the case of equal quark flavors 36 additional Feynman diagrams contribute. These can be obtained from the diagrams of fig. 1 of ref. [1] by particle label 3 ~ 5 interchange. Under 3 ~ 5 interchange the color structures of table 1 of ref. [1] will transform as
C(i)---'C(Q35(i)),
(1)
where the map Q35(i) is given by
,0 . Q35(i)
=
7
8
10
9
12
11
1
2
4
3
12) 6
"
(2)
Due to the 3 ~ 5 operation, the matrix M appearing in eq. (9) of ref. [1] can be decomposed, in the equal flavor case, as M(X1, X2, ~k3, ~k4, ~k5, ~k6, k l , k2, k3, k4, ks, k6) 12 38
=g4 E
E [D(1, ~kl, )k2, )k3, ~-4, )k5, ~k6, k,, k 2, k 3, k4, k5, k6)(~h3 ' x4~xs,_~6R(l, i)
i=1 /=1
- D ( I , ~kl, X2, X5, ~k4, ~k3, ~k6, k l , k2, ks, k4, k3, k6)Sxs,
x4&3,-x,R( l, Q35(i))]C(i),
(3)
where D denotes the space-time part of a given Feynman diagram (see eqs. (7) and (8) of ref. [1]), R is the color decomposition matrix given in table 2 of ref. [1] and C(i) are the color structures given in table 1 of ~' Work supported, in part, by the Department of Energy. 1 Supported by Schweizer Nationalfonds. 2 On leave of absence from Central Research Institute for Physics, H-1525 Budapest, Hungary.
0370-2693/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
477
Volume 176, number 3,4
PHYSICS LETTERS B
28 August 198B
ref. [1]. The negative sign in front of the second term is due to Fermi statistics. The Kronecker deltas appearing in the above equation are due to helicity conservation along the quark lines. As a result we have 24 non-vanishing helicity amplitudes in the equal flavor case, compared to 16 for the unequal flavor case. In ref. [1], eqs. (11) and (12), we have explained how parity and %parity can be used to reduce the number of independent helicity amplitudes, that need to be analytically computed, by a factor of 4. A further reduction by a factor of 2 can be obtained by using the W-parity properties of the Dirac 7-matrices and massless spinors, see eq. (13) of ref. [1]. This latter operation can be used to flip the helicity of a single quark line, if an appropriate mapping of the numerator functions for the various diagrams is included. This map is given in eq. (14) of ref. [1] ,t Therefore, even in the case of equal quark flavors we can construct all the helicity amplitudes from the two numerator functions Nl(l ), N2(/) given in table 5 of ref. [1] and the set of denominator functions P(l) given in table 7 of ref. [1]. For the sake of clarity we give below explicit expressions for the six helicity amplitudes that can be used to construct all remaining amplitudes via parity and W-parity operations - we suppress the m o m e n t u m arguments, kl, k z, k 3, k4, ks, k6, upon which the functions M, N, P, and E 12 (~) , that appear below, depend:
M(a, +, -, +, _)=g4
y~,
E ~ ) R(l, i ) -
i=1 l=1
M(a, +,
,
12 38
, +) = g4 y~ E t~(l)
tJ [ N a ( m ( l ) ) l s , - , 6 E ~ ) R ( l ,
P(I)
i=1 /=1 12
M(a,
,
E(~ ) L
38
, +, +1 = _g4 ~_, ~., t~(l ) i=1 /=1
R(I, Q35(i)) C(i),
i)C(i),
( [J~a(m(l))]5"*6e~) I R(', ~_.35(i))C(i), P(I) ~
(4)
J3~5
(5)
(6)
"1 3 ~ 5
where the index a takes on the values a = 1, 2 and denotes the two independent gluon helicity combinations - a = 1 corresponds to X1 = X 2 = + and a = 2 corresponds to X1 = - X 2 = + . In the above formulas the 3 *-, 5 and 5 o 6 interchanges apply to m o m e n t u m arguments only. The maps bt(l) and re(l) are given by table 4 of ref. [1]. The map Q35(i) was defined in eq. (2). Finally the E ~ ) of the above equations are defined in terms of those given in eq. (18) of ref. [1] by ~12=(t)= E1 z and a~12]tT('2)__--El2., We would like to make several additional remarks. First, the reference choice for gluon helicity vectors used in ref. [1] in the computation of N2(/), e+(kl, k3), e - ( k 2, k4) :~2, is not symmetric under the 3 ~ 5 interchange. It would appear that this gauge choice does not have the 3 ~ 5 symmetry required to use the permutation techniques described above. However, referring to eq. (4), we note that the sum of contributions from the first term to a given color structure, C(i), is actually gauge invariant on its own. Therefore, the sum of contributions from the second term to any given color structure may actually be computed in a different gauge. We have effectively done this because of the non-3 ~ 5 symmetric choice of gluon reference momenta. It is, of course, important that no extra phase factor is generated by the change of gauge. This is true for our choice of gluon helicity vectors which has the property that
e ~ ) ( k , q) = e(,X)(k, q') + k ~ ( k , q, q'),
(7)
where the function a can easily be computed. In order to check our calculation, in particular the above gauge trick, we recalculated the N 2 set of numerator functions using a different (symmetric) gauge choice, e+(kl, k2), e;(k2, kl), and obtained ,1 We note that the published version of eq. (14) has a misprint. The map given by eq. (14) applies to the numerator functions, NUM (defined in eqs. (15) and (16) of ref. [1]), only, and not to the diagrams as a whole. *~ The first argument of e is the gluon four-momentum; the second argument is the polarization reference momentum.
478
"Volume 176, number 3,4
PHYSICS LETTERS B
28 August 1986
complete agreement with our previous result. Finally, we note that by setting the color sums Eli21 R (l, i ) C ( i ) and Eli2=aR(I, Q3s(i))C(i) to 1 if 1 ~< l ~< 20 and to 0 if / > 20, in eq. (3), we obtain the helicity amplitudes for the Q E D process, eeee~,7, which has been calculated in ref. [2]. We would like to thank the Oregon W o r k s h o p on High Energy Physics for support during the course of this work.
References [1] J.F. Gunion and Z. Kunszt, Phys. Lett. B 159 (1985) 167. [2] CALKUL Collab., F.A. Berends et al., Nucl. Phys. B 264 (1986) 265.
479
PHYSICS LETTERS B
28 August 1986
CONCERNING THE FOUR-QUARK TWO-GLUON SUBPROCESS ¢~ John F. G U N I O N
Department of Physics, Universityof California at Davis, Davis, CA 95616, USA and Z. K U N S Z T 1,2
Institute for TheoreticalPhysics, Universityof Bern, CH-3012 Bern, Switzerland Received 4 June 1986 The extension of the unequal flavor four-quark two-gluon amplitude, given previously, to the case of equal flavor quarks is presented. The result can be obtained using simple permutations of the previous expression, and illustrates nicely the technical advantages of the improved helicity method.
In a previous paper [1] we calculated the four-quark two-gluon amplitude in QCD for unequal quark flavors. We demonstrate here how to use simple permutation identities to obtain the full equal flavor result from the previous expression. In the case of equal quark flavors 36 additional Feynman diagrams contribute. These can be obtained from the diagrams of fig. 1 of ref. [1] by particle label 3 ~ 5 interchange. Under 3 ~ 5 interchange the color structures of table 1 of ref. [1] will transform as
C(i)---'C(Q35(i)),
(1)
where the map Q35(i) is given by
,0 . Q35(i)
=
7
8
10
9
12
11
1
2
4
3
12) 6
"
(2)
Due to the 3 ~ 5 operation, the matrix M appearing in eq. (9) of ref. [1] can be decomposed, in the equal flavor case, as M(X1, X2, ~k3, ~k4, ~k5, ~k6, k l , k2, k3, k4, ks, k6) 12 38
=g4 E
E [D(1, ~kl, )k2, )k3, ~-4, )k5, ~k6, k,, k 2, k 3, k4, k5, k6)(~h3 ' x4~xs,_~6R(l, i)
i=1 /=1
- D ( I , ~kl, X2, X5, ~k4, ~k3, ~k6, k l , k2, ks, k4, k3, k6)Sxs,
x4&3,-x,R( l, Q35(i))]C(i),
(3)
where D denotes the space-time part of a given Feynman diagram (see eqs. (7) and (8) of ref. [1]), R is the color decomposition matrix given in table 2 of ref. [1] and C(i) are the color structures given in table 1 of ~' Work supported, in part, by the Department of Energy. 1 Supported by Schweizer Nationalfonds. 2 On leave of absence from Central Research Institute for Physics, H-1525 Budapest, Hungary.
0370-2693/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
477
Volume 176, number 3,4
PHYSICS LETTERS B
28 August 198B
ref. [1]. The negative sign in front of the second term is due to Fermi statistics. The Kronecker deltas appearing in the above equation are due to helicity conservation along the quark lines. As a result we have 24 non-vanishing helicity amplitudes in the equal flavor case, compared to 16 for the unequal flavor case. In ref. [1], eqs. (11) and (12), we have explained how parity and %parity can be used to reduce the number of independent helicity amplitudes, that need to be analytically computed, by a factor of 4. A further reduction by a factor of 2 can be obtained by using the W-parity properties of the Dirac 7-matrices and massless spinors, see eq. (13) of ref. [1]. This latter operation can be used to flip the helicity of a single quark line, if an appropriate mapping of the numerator functions for the various diagrams is included. This map is given in eq. (14) of ref. [1] ,t Therefore, even in the case of equal quark flavors we can construct all the helicity amplitudes from the two numerator functions Nl(l ), N2(/) given in table 5 of ref. [1] and the set of denominator functions P(l) given in table 7 of ref. [1]. For the sake of clarity we give below explicit expressions for the six helicity amplitudes that can be used to construct all remaining amplitudes via parity and W-parity operations - we suppress the m o m e n t u m arguments, kl, k z, k 3, k4, ks, k6, upon which the functions M, N, P, and E 12 (~) , that appear below, depend:
M(a, +, -, +, _)=g4
y~,
E ~ ) R(l, i ) -
i=1 l=1
M(a, +,
,
12 38
, +) = g4 y~ E t~(l)
tJ [ N a ( m ( l ) ) l s , - , 6 E ~ ) R ( l ,
P(I)
i=1 /=1 12
M(a,
,
E(~ ) L
38
, +, +1 = _g4 ~_, ~., t~(l ) i=1 /=1
R(I, Q35(i)) C(i),
i)C(i),
( [J~a(m(l))]5"*6e~) I R(', ~_.35(i))C(i), P(I) ~
(4)
J3~5
(5)
(6)
"1 3 ~ 5
where the index a takes on the values a = 1, 2 and denotes the two independent gluon helicity combinations - a = 1 corresponds to X1 = X 2 = + and a = 2 corresponds to X1 = - X 2 = + . In the above formulas the 3 *-, 5 and 5 o 6 interchanges apply to m o m e n t u m arguments only. The maps bt(l) and re(l) are given by table 4 of ref. [1]. The map Q35(i) was defined in eq. (2). Finally the E ~ ) of the above equations are defined in terms of those given in eq. (18) of ref. [1] by ~12=(t)= E1 z and a~12]tT('2)__--El2., We would like to make several additional remarks. First, the reference choice for gluon helicity vectors used in ref. [1] in the computation of N2(/), e+(kl, k3), e - ( k 2, k4) :~2, is not symmetric under the 3 ~ 5 interchange. It would appear that this gauge choice does not have the 3 ~ 5 symmetry required to use the permutation techniques described above. However, referring to eq. (4), we note that the sum of contributions from the first term to a given color structure, C(i), is actually gauge invariant on its own. Therefore, the sum of contributions from the second term to any given color structure may actually be computed in a different gauge. We have effectively done this because of the non-3 ~ 5 symmetric choice of gluon reference momenta. It is, of course, important that no extra phase factor is generated by the change of gauge. This is true for our choice of gluon helicity vectors which has the property that
e ~ ) ( k , q) = e(,X)(k, q') + k ~ ( k , q, q'),
(7)
where the function a can easily be computed. In order to check our calculation, in particular the above gauge trick, we recalculated the N 2 set of numerator functions using a different (symmetric) gauge choice, e+(kl, k2), e;(k2, kl), and obtained ,1 We note that the published version of eq. (14) has a misprint. The map given by eq. (14) applies to the numerator functions, NUM (defined in eqs. (15) and (16) of ref. [1]), only, and not to the diagrams as a whole. *~ The first argument of e is the gluon four-momentum; the second argument is the polarization reference momentum.
478
"Volume 176, number 3,4
PHYSICS LETTERS B
28 August 1986
complete agreement with our previous result. Finally, we note that by setting the color sums Eli21 R (l, i ) C ( i ) and Eli2=aR(I, Q3s(i))C(i) to 1 if 1 ~< l ~< 20 and to 0 if / > 20, in eq. (3), we obtain the helicity amplitudes for the Q E D process, eeee~,7, which has been calculated in ref. [2]. We would like to thank the Oregon W o r k s h o p on High Energy Physics for support during the course of this work.
References [1] J.F. Gunion and Z. Kunszt, Phys. Lett. B 159 (1985) 167. [2] CALKUL Collab., F.A. Berends et al., Nucl. Phys. B 264 (1986) 265.
479