What we can learn about parity violation in neutral current neutrino reactions from observations of the hadronic final state

What we can learn about parity violation in neutral current neutrino reactions from observations of the hadronic final state

Nuclear Physics B 116 (1976) 525-540 © North-Holland Publishing Company WHAT WE CAN LEARN ABOUT PARITY VIOLATION IN N E U T R A L CURRENT NEUTRINO RE...

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Nuclear Physics B 116 (1976) 525-540 © North-Holland Publishing Company

WHAT WE CAN LEARN ABOUT PARITY VIOLATION IN N E U T R A L CURRENT NEUTRINO REACTIONS FROM OBSERVATIONS OF THE HADRONIC F I N A L STATE Otto NACHTMANN Institut fu'r Theoretische Physik der Universitizt Heidelberg

Received 4 June 1976

Even if the weak neutral current is a pure vector or axial vector, i.e. conserves parity, parity violating correlations among ttadrons in the final state can occur. The corresponding asymmetries must, however, be within certain bounds. Explicit expressions for these bounds are given. Experimental violation of these bounds would prove that the neutral current contains both vector and axial vector parts.

1. Introduction One of the big questions in weak interaction physics is whether the hadronic weak neutral current discovered by the Gargamelle group [1] conserves parity or not, i.e. whether it is a pure vector or pure axial vector current or whether it contains both vector and axial vector pieces. Some means to decide this question are well known (see ref. [2] for a review), but it seems that an analysis of what we can learn about this question by observing correlations among the hadrons in the final state is missing. The present article intends to fill this gap. The paper contains only straightforward kinematic considerations, it is addressed to experimentalists, not to theorists. In this spirit we include a complete treatment of the kinematics appropriate for the study of the final hadronic state without supposing knowledge of the analogous problems in electroproduction (see ref. [3] for a review). Sect. 2 contains a discussion of the basic kinematics, sect. 3 gives a recipe for constructing the bounds on parity violating asymmetries under the hypothesis of pure vector or pure axial vector current as well as some examples. The p r o o f o f the recipe is given in appendix B. The very busy reader may find it sufficient to read only the examples.

2. Kinematics We will be concerned with neutrino reactions involving neutral currents, i.e. with 525

O. N a c h t m a n n / Neutral current neutrino reactions

526

v'(k')

Fig. 1. N e u t r a l current n e u t r i n o reaction.

the following two reactions (see fig. 1):

v(k) + N(p) -+ v'(k') + X ( p ' ) ,

(2.1a)

~(k) + N(p) -+ ~'(k') + X ( p ' ) ,

(2.1b)

where v (~) stands for the incoming neutrino (antineutrino) and v' (~') for the outgoing neutrino (antineutrino), N stands for nucleon and X denotes the final hadronic state. The four momenta are indicated in brackets. The outgoing neutrino (antineutrino) may or may not be identical to the incoming one but we assume that it has zero mass. We assume that the reactions eq. (2.1) are described by effective Lagrangians in the following way:

L(V)eff= -~--~Gv'Tx(1 - "Ys)VJx ,

(2.2a)

L(~)eff= -X/~2Ov3'x(1 - 7s)V'Sx? ,

(2.2b)

pr

where G is Fermi's constant. For v = we have of course L(')ere= L(~)errand Jx = JtxThe metric and "},-matrix conventions follow ref. [4]. The relevant kinematic variables are collected in table 1. We will now consider a neutrino reaction where the initial nucleon is unpolarized and where we sum over a given class P of final hadronic states. The cross section for this case is

3o (v'r)(v'r) Ov~Q2

-

G2 1 l ( v ' ~ ) k ° I¢(~,'r)(~'r) , 87r E 2

(2.3)

where l ap is the lepton tensor and Wox the hadron tensor. They are given by the following expressions:

1~o'~) = 2 ( k xf k o + k x k o! - g x o ( k k t ) ¥ i e a x ~ o k ' a k ~} , WO,,r) = ~ ph

(2.4)

P o(2rr)3 VS(p' - p - q ) ( N ( p ) l J t o l X ( p ' ) ) ( X ( p ' ) l J x l g ( p ) )

X~P M

(2.5a)

527

O. N a c h t m a n n / N e u t r a l c u r r e n t n e u t r i n o reactions

Table 1 Kinematic variables for neutrino nucleon scattering (non-covariant variables refer to the laboratory frame) Variable

Meaning

E = pk/M E'= pk'/M 0 q=k-k' Q2 = _ q 2

energy of the initial neutrino energy of the final neutrino scattering angle of the leptons; 0 ~< 0 ~< rr four-momentum transfer four-momentum transfer squared

= 4 E E ' sin2~ 0

energy transfer invariant mass squared of the final hadronic system

u =pq/M=E - E' W 2 = M 2 + 2 M y - Q2

e = [1 + 2 ( 1 + u2/Q 2) t a n 2 ~ 0 ] - 1 polarization parameter

= 4(E - ½u) 2 - (Q2 + u2) 4(E-

½u) 2 + (Q2 + u2) azimuthal angle (see fig. 2); 0 -<<@ ~< 2rr equivalent energy of real photons needed to produce a final hadronic state of mass W

K = v - Q 2 / Z M = (W 2 - M 2 ) / 2 M

rAAg, r),,o~. = ~

X~Y

Po

3

-~-(27r) V 6 ( p ' - p - q ) ( N ( p ) l J o l X ( p ' ) ) ( X ( p ' ) l J t x l N ( p ) ) , ~wt

(2.5b) where V is the normalization volume and our states are normalized to one. An average over the spin directions of the initial nucleon is understood in eq. (2.5). We note that knowledge of the total momentum and total energy of the final state hadrons and of the direction of the incident neutrino beam is in principle sufficient to determine the energies and scattering angle of the leptons. In present experiments, however, these quantities can only be obtained with big uncertainties. We will now concentrate our attention on the hadronic vertex in fig, 1 where the weak neutral current is absorbed by the nucleon. The lepton tensor Ixo of eq. (2.4) describes the state of polarization of this current, the tensor Wp~, describes the absorption of the current by the nucleon. It is convenient to introduce a helicity basis. We work in the laboratory system and choose the basis vectors of our coordinate frame in the following way (see fig. 2):

e3=q/lql,

e2=k'Xk/lk'Xkl,

el=e2Xe

3 .

(2.6)

In this coordinate frame the four-momentum of the nucleon and the four-momentum transfer are as follows:

p~' = (M, O, O, 0),

qX = (u, O, O, Iql).

(2.7)

528

O. Nachtmann / Neutral current neutrino reactions

+

,q

/

4 Fig. 2. Coordinate frame in the laboratory system. The lepton momenta k, k' are in the xz plane The azimuthal angle ~ of an arbitrary vector 19is also indicated.

We will now define polarization vectors for right-handed (+), left-handed ( - ) , longitudinal (0) and scalar (d) polarization of the weak neutral current: e+_x=-7-x/~(0, 1,-+i, 0 ) ,

eox : 1 / Q ( l q l , O, O, v) ,

e~ = 1 / a q x = 1 / a ( v , O, 0, Iql).

(2.8)

The normalization is such that -e*e+=-e*

= e ~ e o = - e ~ e d = 1.

e

(2.9)

The lepton tensor/xo (eq. (2.4)) is now easily expanded in this helicity basis l(~, 7) = ~

~

m=-+,O n=+.,O

(v,7) , o, lm, n en, hem,

(2.10)

where the sums run over -+ and 0 only since massless neutrinos cannot generate scalar (d) polarization. The quantities -ran l ( v ' ~ are best expressed in terms of the polarization parameter e defined in table I. They are listed in appendix A. We will now define cross sections and interference terms for the absorption of the current in definite helicity states following Hand's convention for the flux factor [5] which is commonly used in electroproduction, o(V,u-)w. (v'p)(~'l~)=~ m,n ~ - , Q 2 , v ) = ~K=" *mp w"'ph ~n'

(2.11)

where K is given in table 1. From eqs. (2.5) and (2.11) we find immediately the hermiticity condition O'm, n = On, m .

(2.12)

O. Nachtmann / Neutral current neutrino reactions

529

The cross section eq. (2.3) is easily rewritten as follows: Oo( v ' r ) ( ~ ' r ) _ G ~ K ~ t(v,vO(i; 3v OQ2 8rr2 E 2 m,n "rn,n ~.~' Q2, v) a(mV,'~(I'; Q2, v).

(2.13)

We will now consider rigid rotations of the final hadronic states around the z-axis. Let F 0 be the set of all final hadronic states obtained from the states in IF"by a rigid rotation by an angle q~ around the z-axis in the positive direction (compare fig. 2). By rotational invariance this is equivalent to rotating the density matrix lxo (eq. (2.4)) of the current by an angle -qS. But this rotation is easily performed in the helicity basis (that is why we introduced it). We find for the cross sections (eq. (2.11)) Om,n ( v ' ~ tt--q>,~ ' p ' D2 , V) = e -iO(m -n)Om,n(i~; Q2,/,),

(m, n = +1, 0)

(2.14)

Inserting in eq. (2.13) leads to 0a(v, r q O ( v , r O _ G 2 1 K Q 2 [ 4 ( o + + + o _ ) + e O o 0 g ~ } ( o + + ~v~Q2 47r2 E 2 1 - e + cos ~bX / ~ + e)Re(oo+ - % _ ) ~ cos ~bx/e(1 - e)Re(oo+ + -

- o__)

o o_

)

sin ( ~ b ) x / ~ + e)Im(ao+ + % _ ) -+ sin ( ¢ ) x / e ( l - e)Im(eo+ - e o _ )

- cos(25) e Reo_+ + sin (2~b) e Im e_+} ,

(2.15)

(v,v-)/-. where Om, n in eq. (2.15) is understood as a shorthand for Om, n (1% Q2, v). Note that the ~b-dependence is explicit in eq. (2.15). The fact that the highest Fourier components occurring are cos2~b and sin 2~b reflects the vector nature of the effective Lagrangian (eq. (2.2)) [6]. For a scalar or pseudoscalar current there would be no S-dependence at all. This, as is well known, can be used to check that the effective Lagrangians eq. (2.2) are indeed of the vector type. We have now collected the somewhat lengthy but necessary kinematic formulae and will turn next to our main topic.

3.

Tests for

a pure vector or axial vector weak neutral

current

In this section we will assume that the weak neutral current Jx in eq. (2.2) is a pure vector or a pure axial vector current and consider various ways to test this hypothesis. Let F be a group of final hadronic states as in sect. 2. Consider the group I~ consisting of all final hadronic states obtained from those of I" by a reflection R on the lepton plane, i.e. the xz-plane of the coordinate frame of fig. 2. R corresponds to the

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O. Naehtmann / Neutral current neutrino reactions

following matrix

R =

-1

(3.1) 1

Under our hypothesis of pure vector or pure axial vector current the cross sections (eq. (2.11)) have a simple transformation property underR o(V,g)rr~. m,n kl , Q2 , u) = (--1)m+no(Vm~ - ,"-) n(17R; Q2, v)

m, n = +1, 0

(3.2)

3.1. The O-distribution for a reflection symmetric set o f final states Let P = 17R be a reflection symmetric set of final states and consider the set P~ obtained from 17 by a rigid rotation of the final states by an angle 4~ around the zaxis as explained in sect. 2. What is the 0-dependence of the cross section Oo(v'r~)(v'r~)/Ov~Q 2 (eq. (2.15)) under the hypothesis of pure vector or pure axial vector current? From eqs. (3.2) and (2.12) we find easily for this case o++ (17; Q2, v) = o _ _ (17; Q2, u)

Oo+(17; Q2, u) = - % _ (17; Q2, v),

o+_(P; Q2, v) = o_+(17; Q2, v) = o*_(17; Q2, v).

(3.3)

Inserting in eq. (2.15) we obtain G2

3v~Q 2

1

47r2 E 2

-

o

o

+ c o s ( 0 ) ~ ( ] - + e)2 Re o(0~5)(F; Q2, v) + sin (0) ~/e(1 - e)2 Im O(o~V-)(P; Q2, v) _ cos(20) eo(v_,_~)(r; Q2, v)}.

(3.4)

Comparing with the general expression eq. (2.15) we note the absence of a sin20 term. Experimental observation of such a term would therefore prove the presence of both vector and axial vector contributions to the weak neutral current. Example: Consider the one-hadron inclusive reactions v+N-+v'+h+X,

b- + N - + ~ ' + h + X ,

(3.5)

where h stands for the hadron which we want to observe, e.g. a charged pion. The inclusive cross section for this process is of the form of eq. (3.4) under the hypothesis of pure vector or pure axial vector current. Indeed, let h u denote the four momentum of the hadron h and 0 the azimuthal angle of the three-vector k in the laboratory frame (compare fig. 2 and substitute k foru).

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O. Nachtmann / Neutral current neutrino reactions

The inclusive cross section for the reactions (3.5) can be written as follows:

3-~51VN._,~,hX/=4~2E

2

hO l O÷+

X 2Re a(o~'F) + s i n ( ¢ ) X / ~ e )

+

cos(¢)~/~l+

21m o(0~'F) - cos(2¢)eo(+V-v)},

(3.6)

where urn, _(v,~) n in eq. (3.6) are the helicity cross sections for the inclusive reactions eq. (3.5) with the hadron momentum h at azimuthal angle ¢ = 0, i.e. h lying in the xzplane. Eq. (3.6) follows immediately from eq. (3.4) by observing that the set o f all states with a hadron h of momentum h in the x z plane is reflection invariant. Observation of a sin 2¢ term in the one-hadron inclusive cross section would therefore prove presence of both vector and axial vector contributions ot the weak neutral current * Corollary to eq. (3.4]: If we assume identity of the incoming and outgoing neutrinos in eq. (2.1), i.e. v = we must have Jx = J~ in eq. (2.2) and as a consequence (v)

.

2

a(V-) ( p . 0 2

(3.7)

Om,n(F, Q , v) = vm, n~--, ~ , v) .

F o r a reflection symmetric set of final states F we find then equality of the neutrino and antineutrino cross sections if we integrate eq. (3.4) over all ¢ 2zr

f o

2rr

_

d~b0°(v' pq9 Ov aQ 2

f o

~o(~'r~ )

d~b

~v OQ 2

.

(3.8)

The equality of the cross sections for this case was the standard test for pure vector or axial vector nature of the weak neutral current used up to now. We note that it should in any case be useful to test the vector-axial vector structure of the weak neutral current independent of the assumption of the identity of the incoming and outgoing neutrinos. 3. 2. Parity violating asymmetries a m o n g the final-state hadrons

Consider a group F o f final hadronic states and the group p n of states obtained by a reflection R (eq. (3.1)) on the lepton plane, i.e. the x z plane o f fig. 2. We consider also the groups P~ and Fff obtained from iv' and FR by rigid rotations by an angle ¢ around the z axis as explained in sect. 2. The cross sections for observing the final state hadrons in the groups PO or Fff are easily derived. From eq. (2.15) we

* E. de Rafael informed me that he considered this test also [7]. [:or work on parity violation in electroproduction see ref. [8]. The corresponding test for charged currents has been discussed in ref. [6].

O. Nachtmann / Neutral current neutrino reactions

532

find for the difference of the cross sections the following expressions: ~o(v'l'0 )

3o (v'l-d~O)_ G 2 1 KQ 2

3v3Q 2

3v3Q2

4rr2 E 2 1 - e

X {--X/i-Z-e--'2(o 0') - o~)_) - 2 cos(¢)x/e-O - e ) R e ( o g ) + O(ov)) -

2 sin (¢) V ~

+ e) Im (O(o~) + o ( ~ ) + 2 sin (2¢) e Im o(f)},

3o(v-'r~)

Oo(~'r~ )

G 2 1 KQ 2

3v3Q2

3v3Q2

4rr2 E 2 1 - e

X {X/1- e2(o(~-+) - 0(_7) ) + 2 c o s ( ¢ ) V ~ -

e) Re(o(07+)+ 0(07_))

2 sin ( ¢ ) X / ~ + e ) I m (0(07) + O(o~) + 2 sin (2¢)e Im o(f+)} ,

(3.9)

(v,v-),n. where -m,n•(V'U-) in eq. (3.9) stands for Ore, n [1, Q2, p).We note the absence of a cos 2¢ term in eq. (3.9). Presence of such a term would prove that the weak neutral current is not a pure vector or pure axial vector. If we assume identity of the incoming and outgoing neutrino we must again have

o t"r/~ (v)n = o m(~) ,n

t3.10)

~

in eq. (3.9) and the comparison of neutrino and antineutrino cross sections provides some further tests for our hypothesis. For pure vector or pure axial vector current we must for instance have 27r

(~o(V, F4~)3o(V, rJ~cb)~

2,

(3o(~,pq))

3o(~,ra~4))

Example: Consider some identified hadron momentum h, e.g, the momentum with biggest absolute magnitude among the momenta of the charged particles. Let ¢ be the azimuthal angle of this m o m e n t u m vector (compare fig. 2) and consider the plane spanned by h and the z-axis. Let N+(¢) be the mean number of tracks on one side or the other of this plane, i.e. with positive or negative projection onto the vector h X q. The C-dependence of the difference N+(¢) - N _ ( ¢ ) is as shown in eq. (3.9). Observation of a cos2¢ term would rule out pure vector or axial vector weak neutral current. We will next consider cross sections for the observation of final states in the sets P~ and Pff and integrate over a given interval Axe and over the energy spectrum of incoming neutrinos with some spectral function ~(E), 3o(v, P~) (Y, r~) o (v'F) = f d E ~(E) ~ de

~v~Q 2

,

O. Nachtmann / Neutral current neutrino reactions

o(R"'~-):fdE ~J(~:)f

a~

dg5 3 ° ( v ' l ~ ) ( ~ ' ~ ) 3v 0Q 2

533

(3.12)

Of course the integration over E extends only over those energies which are kinematically allowed for given v and Q:. One might also select events above a certain energy etc. All these cuts are supposed to be built into the function •(E). What can we say on the asymmetry o(v,F) _ o(v,~-) A (v,v) =

(3.12a) o(v, ~)

+

o(V, -a) ,

under the assumption of pure vector or pure axial vector weak neutral current? It turns out that under this assumption the asymmetry A must be within a certain bound IA(V,~) I ~
(3.13)

where u and v are given by the following expressions: U = d + dc2(cos 2~b) (sin~) 2 + db2(cos2q~) (cosq~) 2 - d3(cos2qS) 2 - b2(cos~b)2 - c2(sin~b)2 ,

(3.14)

v = da 2 + 2dc2(sin 2~b)(sinq~) (cos~b) - 2db2(sin 2~) (sin~b) (cos40 + d c 2 ( c o s 2 0 ) ( c o s O ) 2 + db2(cos2dp)(sinc~) 2 + d3(sin2q~) 2 -

2abc[(cosO) 2 + (sinqS) 2] + cZ(cosO) 2 + b2(sin402 ,

(3.15)

/ b= \Ea(1-e)/

d ~

/ \E2( 1-

,

(A

where

IE'2-(~_ eil=fdE~(E)E2(~_e)'

(3.16)

O. Nachtmann / Neutral current neutrino reactions

534

(cosq~) = AI~ f dq~ cosq~, ~q~ A4~

(3.17)

and similarly for the other averages. Note that e depends on E as shown in table 1. If u and v are both identically zero replace u and v by u' and v' defined below u' = 4d + 1 - d2(cos2~b) 2 - 2b2(cos~b}= - 2c2(sin~b} z , V' = a 2 + d2(sin2qS) 2 + 2c2(cos~) 2 + 2b2(sinq~) 2 .

(3.18)

Should u' and o' also turn out to be identically zero then A m a x = 0. The proof of these assertions is given in appendix B. If we want to consider the cross sections eq. (3.12) integrated over some interval in v and Q2 the corresponding bound on the asymmetry is the maximum of the bound eq. (3.13) over this interval in v and Q2. Example 1. Take neutrino events with given Q2 and v and consider the three charged hadrons in the final state with the largest absolute magnitudes of the laboratory momenta. Let l p / 2 , 13 be these momenta, where I!11 > 1121~> 1131 •

(3.19)

How often do ! 1,12, ! a form a right-handed or left-handed system? If the weak neutral current is a pure vector or pure axial vector, we find the following bound for the asymmetry: a((! X 12)13 > 0 ) - o((! 1X 12)13 < 0)

o((11 X 12)!3 ~ ~ + o((11 × 12)! a ~0))

<~a ,

(3.20)

where a is defined in eq. (3,16). Eq. (3.20) follows immediately from eqs. (3.13)-(3.15) and the observation that we do not make any selection on the azimuthal angle ~ and therefore have to integrate over all ~ in eq. (3. l 2), As a numerical illustration we computed the bound eq. (3.20) for a spectral function ~(E) = e x p ( - 0 . 7 E )

(E in G e V ) ,

taking as integration range 5GeV~
Q2 (GeV 2) 0.2

0.19

1.0

0.42

As can be seen from the expression for e in table 1 and from eq. (3.16) the bound

O. N a c h t m a n n / Neutral current neutrino reactions

535

will be the lower the higher E and the lower v and Qz are. Example 2: Consider inclusive production of A hyperons. How often is the longitudinal polarization of the A in the direction of its momentum or opposite to it? For a parity conserving weak neutral current this asymmetry is again bounded by the quantity a of eq. (3.16).

3.3. Parity violating correlations involving hadrons and the direction o f the incoming neutrino We consider two identified hadron momenta 11, 12, e.g. the two momenta with largest absolute magnitude among the momenta of the charged hadrons where I111 > I12 I. Let k be the unit vector in the direction of the neutrino beam. What can we say on the asymmetry

o(,01 ×

> o) - o(i,(tl x 12) < o)

A - a(~¢(!1 × 12) > O) + a(l¢(!1 × 12) < 0) '

(3.21)

for pure vector or axial vector weak current? We assume again that we are considering neutrino events for given v and QZ where we integrate over the energy of the incident neutrino with some spectral function ~(E). The bound on this asymmetry is obtained as follows. Define an angle X depending on E, v, Q2 and a parameter r (0 ~< r < oo) t

X = arcsin -S

for 0 ~< r ~< s

_I

- ~n

for r~> s ,

Q(E z - vE - ~Q2)1/2 s-

+

(0 ~< X < ~ 7r).

2

(3.22)

Define ?~(r, v, Q2) as follows: / 2XX/1 - e2 ~2

X(r, v, QE)= \TrE2(1

e)/

+

~1~-~

/

(3.23)

where the brackets denote integration over E as in eq. (3.17). The bound on the asymmetry eq. (3.21) is ]AI ~< max ()t(r, u, QZ))l/2 .

(3.24)

If we consider neutrino events not for fixed p and Q2 but out of a given region in the u-Q 2 plane we have also to maximize in eq. (3.24) over the values of v and Q2 of that region.

536

O. Nachtmann /Neutral current neutrino reactions

As an example we consider the case of fixed E, v and Qz. The bound eq. (3.24) is then easily evaluated and we find IAI < ~ .

(3.25)

All these assertions are proven in appendix C. This concludes our exposition of the general method and of some simple exampies. We note that time reversal invariance is in ge.,:eral of no use for placing restrictions on correlations among final state hadrons. The two exceptional cases are: (i) a summation over all final state hadrons, (ii) elastic scattering where the final hadronic state consists of a single nucleon. In case (i), however, time reversal invariance gives again no restriction on the cross section for an unpolarized nucleon if the neutrinos are massless. Observing parity-odd correlations in elastic scattering on the other hand seems a remote experimental possibility. In all our discussion we have tacitly assumed Q2 > 0. For Q2 = 0 the transformation to the helicity basis becomes singular as can be seen from eq. (2.8). It is easily seen that for Q2 = 0 there can be no parity violating correlations among final state hadrons if the current is pure vector or pure axial vector. The reasoning is identical to the derivation of Adler's relations for charged current neutrino reactions at Q2 = 0 [9]. Experimentally one deals of course always with finite values of Q2 and the extrapolation to Q2 = 0 is presumably not easily done. Finally we note that the bounds which we have derived are equally valid if the nucleon is replaced by a nucleus as target. For experiments done on nuclei it is, therefore, in principle not necessary to extract cross sections for individual nucleons in order to study the correlations among the final state hadrons. The idea for this investigation arose in conversations with P. Musset at the Rencontre de Moriond 1976. The author is grateful to the organizers of this meeting for their kind invitation, and to D. Haidt and B. Stech for useful discussions. Appendix A The density matrix elements 1- m(v'~) of eq. (2.10) are as follows: n

Q2

0 2

I 5'f) = Y - - 7 I1 +_#-f-z

= 1 - c [1 -T-~/1 - e2] ,

l (v'~) -oo

-

Q2 1 -

=

2e,

l(V+ ~) -

e

=

Wgo

02

[_X/~

+

= l(+ v ' ~ ) -

-

Q2 1 -

l ,

e e

'

,)1, + e) g X/~(~_ e) ]

(A.1)

O. Nachtmann / Neutral current neutrino reactions

537

The quantity e is the fraction of longitudinal polarization compared to transverse,

e-

(v,~-)

1++

(A.2)

+ l (v'~)__ "

Appendix B The p r o o f for the bound on the asymmetry (eq. (3,13)) runs as follows. Let F and FR be as stated in the beginning of sect. 3.2. Denote by Fx, n the amplitude for the absorption o f a current with polarization vector e n which leads to a final state X *. With the help of eq. (2.14) we find easily for the cross sections of eq. (3.12)

o:

~ F~,mMm, n F x , , , , x~_p

oR =

x~"

R F~v,m M m , n F x , n ' *

(B.1)

where we have absorbed some irrelevant factors in the definition ofFx, n and where the matrix Mm, n is defined as follows:

gin, n

= fdL'~'(E) 5~| £

dq5 E 2 ~

e-iOm l(U'~)(Em,n , O2, u) eiepn

m, n = _+1, 0 .

(B.2)

We define also matrices M R, 34+ in the following way:

(MR)m,n = ( - - 1 ) m M m , _ n ( - 1 ) n ,

M+_ = M +-M R .

(B.3)

If the weak neutral current conserves parity, the amplitudes Fx, n must satisfy

ex, n = + - ( - 1 ) n F x R , - n '

(B.4)

where X R is the state obtained from X by a reflection R and the upper (lower) sign is valid for a pure vector (pure axial vector) current. In either case we find for the asymmetry defined in eq. (3.12a)

A -

XCF X~l"

F?(,m(M_)m,nFx, n (B.5)

~,m(M+)m,nFx,

n

* Indices for the polarization state of the target are suppressed since they are irrelevant for the following.

538

O. Nachtmann / Neutral current neutrino reactions

We are interested in the extrema of A where we allow arbitrary amplitudes I x , n, since we make no dynamical assumptions. This problem can be phrased as follows. What are the extrema of FtxM

Fx ,

XEP

under the constraint FtxM+ F x = I , X~I"

where we use matrix notation and F x are arbitrary vectors? A simple application of Lagrange's multiplier technique shows that IAI ~< ~kmax ,

(B.6)

where 2~max is the biggest root of the equation det(M

- kM+) = 0 .

(B.7)

This leads easily to eqs. (3.13)-(3.16). If M+ is a singular matrix we have to project onto the subspace where M+ is nonsingular. This leads easily to eq. (3.18).

Appendix C To prove the assertions of subsect. 3.3 we consider the plane z = 1 in the coordinate frame of fig. 2. The momentum of the incoming neutrino intersects this plane at the point ( - s , 0, 1) where s is defined in eq. (3.22). Consider now the set P(r) of hadronic final states where the vectors I 1 and ! 2 span a plane a normal to the x-z plane such that the intersection of these two planes and of the plane z = 1 is the point (r, 0, 1) with r~> 0. We also require/~(! 1X lz) > O. Now we rotate P(r) rigidly around the z-axis by an angle q~ as explained in sect. 2. A simple geometrical consideration shows that k ( l 1 X 12) stays positive for I~bl ~ ~n + × where X depends on r as shown in eq. (3.22). Denoting by I ~ ( r ) the set of states obtained from P(r) by a reflection on the x-z plane and by Po(r), P~(r) the rotated states we find easily from eq. (2.13)

c2

a°(~'~) ( k ( t t x t2) > o) = - 3vOQ z

x

f

8712

X+n/2

dr(f

0

fdE 44E)

d$/(v,V)m,n°(v'~)(P~(r);m,n w v, Q2)

-x-n/2

--X+3n/2

+f

X+n/2

dO l(V'~) o(V'~)(14~ (r); v, Q2)} gl"l,rl ?n,~l q)

(C.1)

o. Nachtmann / Neutral current neutrino reactions 0o(v,V-)

3v 3Q2

539

(k(q × q) < o) = - - f de ~(~) 8zr2

S

X+rr/2

x

dr(f

dO l(V,V)m,n°(v'-~)(P~(r);m,nw

v, Qz)

0 -X-rr/2 -X+37r/2 +f

dO

t{d,~>o~d:~Y(p~(r);.,0 2 ) } .

(c.2)

x+Tr/2 Use of eqs. (2.14) and (B.4) leads to the following expression for the asymmetry eq. (3.21): oo

f

dr E Ftx(r)~I_FX(r ) X~ P(r)

A - o

f 0

,

(C.3)

dr ~ Ftx(r)f4+Fx(r ) X~ P(r)

where/~/_+ are constructed as shown below: x+rr/2

~lm n =f dEt~(E) &

f_x

doe -i(~m l(V,V)ei(TM -

~' = f d E ~ j Mm n ~(E) '

~

E

Q

~r / 2

X+3rr/2 x+rr/2

dO e-i~,m l @'~) ei~n , m,n

if/_+ = 214 + ~ , R + liar + 3~'.

(C.4)

The matrices 34 R, ~ , R are obtained from M, M' as shown in eq. (B.3). Using the same technique as in appendix B to extremize the asymmetry A of eq. (C.3) we arrive after some algebra at eqs. (3.24) and (3.25). Note added in proof Recently, evidence for reactions of the type vu + N - ~ u -

+e + +X,

(1)

has been reported [10]. If we interpret these reactions as production and subsequent semileptonic decay of charmed particles we-can apply the methods of this article to test for parity violation in the charm-changing weak current. To be specific, let us assume that reaction (1) occurs in the following way

vu + N - + U - + C + X'

(2)

L_, e + + ve + X" , where C stands for some charmed particle. We assume that the relevant effective weak

O. Nachtmann / Neutral current neutrino reactions

540 Lagrangian is

£eff = --~22 {eTX(1 -- 3's) Ue + ~)'a(1 -- Ts) u,} Jtx(AC 4: 0) + h.c.,

where J a ( A C 4: 0) is the charm-changing current and where we assume g-e universality. We ask the question: how can we test whether J x ( A C 4: 0) violates parity? This can again be studied by observing parity violating correlations in the final state of reaction (1). Even if the charm-changing current is pure vector or pure axial vector, there can be parity violating correlations, induced by the lepton current either from the production vertex or from the semi-leptonic decay. We observe, however, that all parity violating effects from the leptons in the decay of the charmed particle C -+ e + + v e + X " ,

(3)

drop out if we sum over all the phase space of these leptons as can be easily verified. If, therefore, we make no cuts on the positron in reaction (1), all the tests described previously for the neutral current can be used to test the hypothesis of pure vector or pure axial vector charm-changing current, neglecting the muon mass. As an additional example to subsect. 3.2 we note the following. Consider the unit vectors k and/¢' in the direction of the incoming neutrino and outgoing muon. Let ! be some identified hadron momentum, e.g. the momentum of the most energetic kaon. If the charm-changing current conserves parity, the following asymmetry of the hadron with respect to the lepton plane A = o((k^ X k')l^ > 0) - o((k^ × k')l^ < 0) ,

(4)

o((k X k')l 3> O) + o((k X k')! < O) must be bounded by IA[ ~< F-da27r2 - 8 a b c + 4e211/2 d--2 _ 4b 2 ,

(5)

where a, b, e, d are defined in eq. (3. ! 6).

References [1] F.J. Hasert et al., Phys. Letters 46B (1973) 138; Nucl. Phys. B73 (1974) 1. [2] J.J. Sakurai, CERN report TH 2099 (1975).

[3l K. Berkelman, Continuum electroproduction by coincidence measurements, Proc. 1971 Int. Syrup. on electron and photon interactions at high energies (Cornell University, 1972) ed. N.B. Mistry. [4] J.D. Bjorken and S.D. Drell, Relativistic quantum fields (McGraw Hill, New York, 1965). [5] L.N. Hand, Phys. Rev. 129 (1963) 1834. [6] A. Pais and S.B. Treiman, Phys. Rev. DI (1970) 907; A. Pais, Ann. of Phys. 63 (1971) 361. [7] E. de Rafael, private communication. [8] C.P. Korthals-Altes, M. Perrottet and E. de Rafael, Nucl. Phys. B87 (1975) 527. [9] S.L. Adler, Phys. Rev. 135 (1964) B963. [101 J. Blietschau et al., Phys. Letters 60B (1976) 207; J. van Krogh et al., Phys. Rev. Letters 36 (1976) 710.