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Effects of neutral weak currents in inclusive e+e− annihilation
Volume 66B, number 5
EFFECTS
PHYSICS LETTERS
OF NEUTRAL
WEAK CURRENTS
28 February 1977
IN INCLUSIVE e+e - ANNIHILATION
E. LENDVAI and G. POCSIK Institute for Theoretical Physics, Eftv6s University, Budapest, Hungary
Received 2 January 1977 Inclusive e+e- annihilation into n hadrons is studied in one-photon and one-neutral Z-meson exchange approximation for polarized e+e- beams. The measurability of the hadronic structure functions and weak effects are discussed for arbitrary n.
In the energy range o f PETRA, PEP and at higher energies, the weak interactions are switched on, producing essential interference effects of neutral weak currents with the pure photon exchange. Thus, neutral weak currents can be pursued in high energy hadron production [1, 2] by measuring n-hadron final states in inclusive e+e - annihilation, n = 0, 1, 2, 3 . . . . . Theoretical discussions are available for n = 0, 1 [1] and n = 2 [2]. In the present note we give a simple derivation for the general case o f n spinless hadrons in the presence of arbitrarily polarized incident beams. The independent hadronic structure functions turn out to be measurable and characteristic weak effects appear in the cross section. Let us consider the inclusive production of n spinless hadrons, e - ( n ) + e+(n ') -+ h l ( P l ) + ... + hn(Pn) + X where n, n' mean the polarizations multiplied by the unit vectors representing the spin directions in the rest frame. e - ( e +) moves in the +z ( - z ) direction and q is the incident total energy. Introducing the parameters
e4 a =~
q4'
e2g2 f=
g4
q2(q2 _ m 2)
,
l-
(q2 _ m 2 ) 2 '
and the hadronic tensor Huv(d 1 J2),
Huv(JlJ2)
= ~(2~r)46(4)(q - P t - "'" -Pn
X
- Px)(hl(Pl)"'" hn(Pn)XlJlulO>
"" hn(Pn)XIJ2~lO>*' (2)
in one (~, + Z)-exchange the differential cross section can be written as n
do (n) = 8n2a2 Re T 2 I-I d3pi e4q 2 bi=l 2(2r03p O' Re T 2 =
(3)
ReA~A~*(aH~(77) + 2fgv Ha~(TZ) +tg2 Ha~(ZZ)) + Re 2g A AUA~*(fH~ft(TZ) +tgv Ha~(ZZ))
(4) + Re tg2 A~A~5*Ha~(ZZ). Here, we have assumed the usual weak coupling g z Zu(/'~ + J ~ ) with the neutral weak hadronic (leptonic) current g~(J'~), while the leptonic matrix elements A s, A ~ are A s = ~Tau, A ~ = fr),a(gV + 75gA)u. The distribution [4] presents parity conserving and parity violating leptonic or/and hadronic pieces. Neglecting lepton masses, the leptonic tensors A eA ~* etc. represent divergenceless currents and only their space components are nontrivial, hence 9 elements Huv determine the cross section;/l, u = 1,2, 3. The leptonic and hadronic tensors can be parametrized in matrix form by the system
449
Volume 66B, number 5
=(1 1
PHYSICS LETTERS
=(1-1
P1
01,
p5=I_~
1 ),
=I0
P2
01,
p6=(l
P3
0 11,
1 0 11,
p7:I_l
28 February 1977
P4=(~
0 1),
),
p8=(
(5)
1 10),
For the leptonic matrices we get (AcxA~) _q2 . . . . . (Ac~5A#5)J --7- [(1 + nznz)P 1 + (nyny - nxnx)P 2 ~ (nxny + nynx)P 4
i(n z + nz)P5] ,
(6) (AocA;5) =
q~
t
.
t
t
•
t
t
t
[(n z + nz)P 1 - l(nxny + nynx)P 2 - l(nyny - nxnx)P 4 - i(1 + nznz)P 5 ].
In the hadronic matrices (H,~t~)we separate the parity violating terms A i (Ha0(73")) = Vi(•y)P i,
(Ha0(')'Z)) = (Vi(3,Z) +Ai(vZ))P i,
(Hao(ZZ)) = (Vi(ZZ) +Ai(ZZ))P i,
(7)
(summations are included over i = 1..... 9). The combinations of the hadronic structures emerging in the cross section are V~ = a/I/(73') + 2fg v Vi(TZ) + t(g 2 -+g2) Vi(ZZ),
A i-+ = 2fgvAi(TZ) + t(g2 +g2)Ai(ZZ)'
(8) V 0 = 2fg A Vi(TZ) + 2 t g v g A Vt(ZZ),
A 0 = 2fghAi(TZ ) + 2 t g v g h A l { Z Z ).
Substituting (5)-(8) into (4), we have Re T 2 --~_q2 Re((1 + nznz)[V1 ' + +A1+ + i(V 0 +A0)] +(nz +nz)[V 1 , 0 +A 10+I(V 5 . + +A~)] I
?
t
(9)
t
+ (nyny - nxn x) [V~ ÷A~ - i(V ° +A4°)] - (nxny + nyn x) [V~ +A 4 + i(V ° + A0)] ).
The reality properties of V~ etc. follow from those of H~o's, that is V1.....4,6,8, A I .....4,6,8 are real (complex) for YT, ZZ(3,Z); V5,7,9, A5,7, 9 are imaginary. The 7Z-interference seems to be observable in the region q ~ 30 GeV [2]. At very high energies, q,f, t are proportional to q-4, so that the electromagnetic and weak contributions are commensurable. At high energies with q2 ,~ m 2, the relative strength of the 7Z and electromagnetic contributions grows as q2. The above weak effects differ from those of charmed hadrons and heavy leptons by propagator effects and the form of parity violation which occurs at the hadronic vertex. By measuring the cross section (3), one can obtain at best four combinations of the hadronic structures V i and Ai, respectively, differing in parity properties and determined by various incident polarizations. It is straightforward to show that rotating these combinations in the same way, all the v~'O,A~ '0 can be reached except for Vf,7,9, A5,7, 9. Therefore, for i = 1,2, 3, 4, 6, 8 Vi(77 ), Vi(TZ), V/(ZZ) and for i = 1..... 9Az(TZ),Ai(ZZ) can be experimentally separated at a fixed q2. V5,7,9(,),,), ' "yZ) can also be separated neglecting the pure weak contributions VS,7,9(ZZ). It is expected that the interference terms can be identified beside the ZZ-contributions also by their q2-behaviours as suggested by eq. (8). One can parametrize Huv by Lorentz invariant structure functions, it is easy to see that for vanishing lepton mass the most general decompositions are Huv = IguvW + piuPlvWij
for VV, AA
~euv~P~iWi - (Piuev~ + t"iv " eUl~t:Vlt-'m ~,,~,,'r Wi, lm
(10) for VA,
6u, v = 1, 2, 3; sum over i,j, 1, m = 1..... n). For n >/3 I¢ can be eliminated. Making use ofeq. (7), measurable com450
Volume 66B, number 5
PHYSICS LETTERS
28 February 1977
binations can be extracted from eq. (10). If we decide to introduce the minimum three hadron momenta Piu in (10), then the above discussion shows that for every n all the independent Lorentz scalar structure functions can be experimentally identified. As is seen, the parity violating asymmetries in Re T 2 occur in polarization measurements except gA VS(~'Z) and gv A 1(')'Z) corresponding to the correlations (.Pi X p/) kelectron and (Pl × Pm)(Pix, Ply, 0). In conclusion, the n-hadron inclusive cross section contains a rich structure providing possibilities for studying hadron production and weak effects. The most general kinematics appears already for n = 3 in terms of a few specific kinematical correlations as shown by eqs. (7), (9), (10).
References [1] R. Gatto and G. Preparata, Nuovo Cim. Lett. 7 (1973) 89; A. McDonald, Nucl. Phys. B75 (1974) 343; R. Budny and A. McDonald, Phys. Lett. 48B (1974) 423; G.V. Dass and G.G. Ross, Phys. Lett. 57B (1975) 173; K. Nagy, E. Lendvai and G. P6csik, Phys. Lett. 62B (1976) 426. [2] G.V. Dass and G.G. Ross, CERN-preprint, Th. 2196-CERN (1976).
451
EFFECTS
PHYSICS LETTERS
OF NEUTRAL
WEAK CURRENTS
28 February 1977
IN INCLUSIVE e+e - ANNIHILATION
E. LENDVAI and G. POCSIK Institute for Theoretical Physics, Eftv6s University, Budapest, Hungary
Received 2 January 1977 Inclusive e+e- annihilation into n hadrons is studied in one-photon and one-neutral Z-meson exchange approximation for polarized e+e- beams. The measurability of the hadronic structure functions and weak effects are discussed for arbitrary n.
In the energy range o f PETRA, PEP and at higher energies, the weak interactions are switched on, producing essential interference effects of neutral weak currents with the pure photon exchange. Thus, neutral weak currents can be pursued in high energy hadron production [1, 2] by measuring n-hadron final states in inclusive e+e - annihilation, n = 0, 1, 2, 3 . . . . . Theoretical discussions are available for n = 0, 1 [1] and n = 2 [2]. In the present note we give a simple derivation for the general case o f n spinless hadrons in the presence of arbitrarily polarized incident beams. The independent hadronic structure functions turn out to be measurable and characteristic weak effects appear in the cross section. Let us consider the inclusive production of n spinless hadrons, e - ( n ) + e+(n ') -+ h l ( P l ) + ... + hn(Pn) + X where n, n' mean the polarizations multiplied by the unit vectors representing the spin directions in the rest frame. e - ( e +) moves in the +z ( - z ) direction and q is the incident total energy. Introducing the parameters
e4 a =~
q4'
e2g2 f=
g4
q2(q2 _ m 2)
,
l-
(q2 _ m 2 ) 2 '
and the hadronic tensor Huv(d 1 J2),
Huv(JlJ2)
= ~(2~r)46(4)(q - P t - "'" -Pn
X
- Px)(hl(Pl)"'" hn(Pn)XlJlulO>
"" hn(Pn)XIJ2~lO>*' (2)
in one (~, + Z)-exchange the differential cross section can be written as n
do (n) = 8n2a2 Re T 2 I-I d3pi e4q 2 bi=l 2(2r03p O' Re T 2 =
(3)
ReA~A~*(aH~(77) + 2fgv Ha~(TZ) +tg2 Ha~(ZZ)) + Re 2g A AUA~*(fH~ft(TZ) +tgv Ha~(ZZ))
(4) + Re tg2 A~A~5*Ha~(ZZ). Here, we have assumed the usual weak coupling g z Zu(/'~ + J ~ ) with the neutral weak hadronic (leptonic) current g~(J'~), while the leptonic matrix elements A s, A ~ are A s = ~Tau, A ~ = fr),a(gV + 75gA)u. The distribution [4] presents parity conserving and parity violating leptonic or/and hadronic pieces. Neglecting lepton masses, the leptonic tensors A eA ~* etc. represent divergenceless currents and only their space components are nontrivial, hence 9 elements Huv determine the cross section;/l, u = 1,2, 3. The leptonic and hadronic tensors can be parametrized in matrix form by the system
449
Volume 66B, number 5
=(1 1
PHYSICS LETTERS
=(1-1
P1
01,
p5=I_~
1 ),
=I0
P2
01,
p6=(l
P3
0 11,
1 0 11,
p7:I_l
28 February 1977
P4=(~
0 1),
),
p8=(
(5)
1 10),
For the leptonic matrices we get (AcxA~) _q2 . . . . . (Ac~5A#5)J --7- [(1 + nznz)P 1 + (nyny - nxnx)P 2 ~ (nxny + nynx)P 4
i(n z + nz)P5] ,
(6) (AocA;5) =
q~
t
.
t
t
•
t
t
t
[(n z + nz)P 1 - l(nxny + nynx)P 2 - l(nyny - nxnx)P 4 - i(1 + nznz)P 5 ].
In the hadronic matrices (H,~t~)we separate the parity violating terms A i (Ha0(73")) = Vi(•y)P i,
(Ha0(')'Z)) = (Vi(3,Z) +Ai(vZ))P i,
(Hao(ZZ)) = (Vi(ZZ) +Ai(ZZ))P i,
(7)
(summations are included over i = 1..... 9). The combinations of the hadronic structures emerging in the cross section are V~ = a/I/(73') + 2fg v Vi(TZ) + t(g 2 -+g2) Vi(ZZ),
A i-+ = 2fgvAi(TZ) + t(g2 +g2)Ai(ZZ)'
(8) V 0 = 2fg A Vi(TZ) + 2 t g v g A Vt(ZZ),
A 0 = 2fghAi(TZ ) + 2 t g v g h A l { Z Z ).
Substituting (5)-(8) into (4), we have Re T 2 --~_q2 Re((1 + nznz)[V1 ' + +A1+ + i(V 0 +A0)] +(nz +nz)[V 1 , 0 +A 10+I(V 5 . + +A~)] I
?
t
(9)
t
+ (nyny - nxn x) [V~ ÷A~ - i(V ° +A4°)] - (nxny + nyn x) [V~ +A 4 + i(V ° + A0)] ).
The reality properties of V~ etc. follow from those of H~o's, that is V1.....4,6,8, A I .....4,6,8 are real (complex) for YT, ZZ(3,Z); V5,7,9, A5,7, 9 are imaginary. The 7Z-interference seems to be observable in the region q ~ 30 GeV [2]. At very high energies, q,f, t are proportional to q-4, so that the electromagnetic and weak contributions are commensurable. At high energies with q2 ,~ m 2, the relative strength of the 7Z and electromagnetic contributions grows as q2. The above weak effects differ from those of charmed hadrons and heavy leptons by propagator effects and the form of parity violation which occurs at the hadronic vertex. By measuring the cross section (3), one can obtain at best four combinations of the hadronic structures V i and Ai, respectively, differing in parity properties and determined by various incident polarizations. It is straightforward to show that rotating these combinations in the same way, all the v~'O,A~ '0 can be reached except for Vf,7,9, A5,7, 9. Therefore, for i = 1,2, 3, 4, 6, 8 Vi(77 ), Vi(TZ), V/(ZZ) and for i = 1..... 9Az(TZ),Ai(ZZ) can be experimentally separated at a fixed q2. V5,7,9(,),,), ' "yZ) can also be separated neglecting the pure weak contributions VS,7,9(ZZ). It is expected that the interference terms can be identified beside the ZZ-contributions also by their q2-behaviours as suggested by eq. (8). One can parametrize Huv by Lorentz invariant structure functions, it is easy to see that for vanishing lepton mass the most general decompositions are Huv = IguvW + piuPlvWij
for VV, AA
~euv~P~iWi - (Piuev~ + t"iv " eUl~t:Vlt-'m ~,,~,,'r Wi, lm
(10) for VA,
6u, v = 1, 2, 3; sum over i,j, 1, m = 1..... n). For n >/3 I¢ can be eliminated. Making use ofeq. (7), measurable com450
Volume 66B, number 5
PHYSICS LETTERS
28 February 1977
binations can be extracted from eq. (10). If we decide to introduce the minimum three hadron momenta Piu in (10), then the above discussion shows that for every n all the independent Lorentz scalar structure functions can be experimentally identified. As is seen, the parity violating asymmetries in Re T 2 occur in polarization measurements except gA VS(~'Z) and gv A 1(')'Z) corresponding to the correlations (.Pi X p/) kelectron and (Pl × Pm)(Pix, Ply, 0). In conclusion, the n-hadron inclusive cross section contains a rich structure providing possibilities for studying hadron production and weak effects. The most general kinematics appears already for n = 3 in terms of a few specific kinematical correlations as shown by eqs. (7), (9), (10).
References [1] R. Gatto and G. Preparata, Nuovo Cim. Lett. 7 (1973) 89; A. McDonald, Nucl. Phys. B75 (1974) 343; R. Budny and A. McDonald, Phys. Lett. 48B (1974) 423; G.V. Dass and G.G. Ross, Phys. Lett. 57B (1975) 173; K. Nagy, E. Lendvai and G. P6csik, Phys. Lett. 62B (1976) 426. [2] G.V. Dass and G.G. Ross, CERN-preprint, Th. 2196-CERN (1976).
451