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Space-time structure of neutral currents and y-distributions
Nuclear Physics B107 (1976) 481-492 © North-Holland Publishing Company
SPACE-TIME STRUCTURE OF NEUTRAL CURRENTS ANDy-DISTRIBUTIONS * G. ECKER Institut for Theoretische Physik, Universitdt Wien
Received 17 September 1975
Assuming scaling of the nucleon structure functions, we derive relations for the ydistributions in deep inelastic neutrino scattering for the most general local interaction of neutrinos with hadrons via neutral currents with V, A, S, P and T pieces. In addition, optimal bounds for the y-distributions are presented which follow from positivity requirements for the structure functions and which may serve to discriminate between pure V, A currents and the presence of S, P, T couplings. The influence of the predictions of the spin-~ parton model on the bounds is investigated. All relations and bounds depend only on the ratio of total neutrino and antineutrino cross sections. 1. Introduction Although neutral currents were suggested long before the advent of gauge theories, the experimental confirmation of neutral current phenomena was considered by many people to give support to unified theories o f weak and electromagnetic interactions. It is therefore only natural that the prevailing opinion has been so far that neutral currents are vectors and/or axial vectors. More recently, several authors have argued [ 1 - 7 ] that the space-time structure o f neutral currents should be investigated experimentally and various tests have been proposed for the presence of scalar, pseudoscalar and tensor couplings. Another closely related question concerns the validity of scaling in deep inelastic scattering. The quark parton model has been impressively successful in describing deep inelastic scattering of electrons and neutrinos via charged currents. However, in these reactions presumably only the charged constituents of the nucleon take part in the interaction. On the other hand, the quark parton model also predicts that roughly half the momentum of the nucleon is carried by neutral gluons which are usually assumed to have integer spin. It is an interesting question whether these neutral particles participate in the neutral current interaction. It has been shown by Pakvasa and Rajasekaran [7] that if neutral currents exist which are built up o f neutral particles of integer spin and which * Supported in part by Fonds zur F6rderung der wissenschaftlichen Forschung, Projekt Nummer 1905.
481
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have S, P or T pieces, then scaling is necessarily violated since the dimension of these currents is in general different from 3. The purpose of this paper is to investigate the y-distributions for inclusive scattering of (anti)neutrinos on nucleons to get information on the two questions just raised: is scaling valid and, if so, is there any evidence for S, P, T neutral current couplings? Relations and bounds for the y-distributions will be derived which follow from scaling and positivity constraints on the structure functions and which will depend only on the ratio of the total cross sections. Positivity bounds for the y-distributions in inclusive neutrino scattering have been derived before for V, A charged weak currents [13]. The bounds allow to check for the presence of S, P, T neutral currents without making detailed fits to the y-distributions. The results of this paper supplement other tests for the Lorentz structure of the neutral current interaction that have already been proposed [ 1 - 7 ] . Although the main application in the immediate future will be for inclusive scattering, our results for the spin-~- parton model are directly applicable to the y-distributions for the purely leptonic processes vu + e - ~ vu + e - and flu + e - ~ flu + e-. This will be an interesting and independent test for the space-time structure of neutral currents when more data on these reactions will become available. In sect. 2 we define the y-distributions for inclusive scattering in the scaling limit for the most general case where V, A, S, P and T couplings are present. We derive a relation between the neutrino and antineutrino y-distributions and make definite predictions at a certain fixed point in y. These relations test scaling only. In sect. 3 the positivity requirements for the structure functions are used to derive bounds for the y-distributions. We distinguish between pure V, A currents and S, P, T currents and investigate the modifications of the bounds if spin-½ partons are assumed. For two typical values of the cross section ratio the results are also presented graphically. In sect. 4 the results of the paper are discussed and summarized. Finally, an appendix contains the general neutral current Lagrangian, the nucleon structure functions including positivity constraints, the differential cross sections in the scaling limit and the predictions of the spin-~- patton model.
2. Relations for y-distributions The most general local Lagrangian for neutrinos interacting with hadrons has vector, axial-vector, scalar, pseudoscalar and tensor pieces. The Lagrangian and the double differential cross sections in the scaling limit are given in the appendix together with definitions of the various structure functions of the nucleon. In deriving the cross sections explicit use has been made of the scaling assumption for structure functions as given in eqs. (A.8). From (A.9) and (A. 10) we obtain the differential cross sections with respect toy,
G. Ecker/ Neutralcurrents
483
do
(~) + 16f13(1 - - y ) - 8f14Y2 + 8f15Y 2 -Y-4(f9 +flO)(2y_y2),
(2.1)
where
G2EM e
~- ~ JF.(x)a~, G2EM
6- ~
fx~.(~)ax,
i = L , R,S, 13 , / ' = 7 , 8 , 9 , 10, 11, 12, 15,
f14 - G2EM 1"x2F14(x) dx
(2.2)
8rr .:
The total cross sections are
Ou = 2 f L + } f R +2fS+~(f7+f8)+ ~'f12+8f13--sf14 (~) (R) (L) + 82315 g ~(f9 + f l 0 ) "
(2.3)
We now eliminate fs, fg, fl0 and I"13 from the differential cross sections using the total cross sections. One arrives at the following expressions for the normalized y-distributions: do _--
_
1
o(~)
-
-
(~)
dy
'
[~0') = 1 + ~-y- ~_y2 + r(1 - [y + ~y2) 1
+--(3y2+2y--2)(fL +fR +f7+f8+4f12--Sf14+Sf15), 3ov f ~ ( y ) = 1 + l y _ ~v 2 + r - l ( 1
(2.4a)
__Sy2 + 3y2)
+ 31- (3y 2 +2y_2)OCL+fR+f7+f8+4f12_8f14+SflS),
(2.4b)
v
where r is the cross section ratio o~/o v. In this section we want to draw two conclusions from eqs. (2.4). By eliminating the unknown quantity in brackets one obtains a relation between the y-distributions,
rf~(v)-f~(v) = (r- 1) - ~ y ( 2 - y ) .
(2.5)
It has already been observed [ 1,2] that the difference between the neutrino and
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G. Ecker /Neutral currents
antineutrino cross sections is proportional to y (2 - y ) ; the relation (2.5) shows that the proportionality constant is determined by the total cross sections. Relation (2.5) is a useful test of the scaling hypothesis. Of course, it may also be used to deduce the cross section ratio r from the y-distributions, once scaling is established. Besides checking the energy dependence of total cross sections, relation (2.5) is another practicable test for scaling in the near future. The y-distributions (2.4) show another interesting feature. The quadratic function 3y 2 + 2y - 2 = 0 has a root in the physical domain of y, namely for (2.6)
Y =Yl = ~(X/~-1) ~ 0.55.
At this fixed point the y-distributions are completely determined by the ratio r, fu(Yl ) =Yl + -~ + r(3 --3Yl) ~ 1.05 -- O.15r ,
f (vi) =Yi +
+ r-l(
(2.7)
-3Yl) •
As for the relation (2.5) the equalities (2.7) are valid for a general V, A, S, P, T interaction and depend only on scaling. Moreover, (2.5) and (2.7) together should also be very useful in the case of cuts in y, when only the shape, but not the absolute scale of the y-distributions is known. Since cuts usually have to be applied for small y, the fixed point Yl is in the experimentally accessible region. Note that the values of f(u/o)(Yl) are relatively insensitive to variations of r. It should also be emphasized that both (2.5) and (2.7) are genuine predictions of the scaling model. For instance, for spin-0 or spin-1 constituents taking part in the neutral current interaction, both results are changed.
3. Bounds fory-distributions Besides the relations obtained in the last section, the y-distributions are subject to certain restrictions following from the positivity requirements on the structure functions. Those constraints are given in the appendix and are easily translated into inequalities for the integrated quantities
I7 >o, --fll +f12-- 2f14 + 2f15 ~> 0 ' f l l + }f13 + 2f14 -- 2f15 >~ 0 , --fll +~f13 1 ~>0,
&>o,
Ii1 >o,
G. Ecker / Neutral currents
485
(fll + ~-fl3 + 2f14--2f15)(--fll + lf13)>/(lf13--f15 )2 , f7(--fll +f12--2f14 + 2f15) ~>f~ ,
fsf11>~ f2 0 •
(3.1)
The problem we propose to solve is to extremize the quantity
e = f L +fR +f7 +f8 + 4f12--8f14 + 8f15 '
(3.2)
subject to the inequalities (3.1) and the equalities (2.3) for the total cross sections. Depending on whether y is below or above the fixed point y 1 a minimal value o f f will give an upper or lower bound for the y-distributions and vice versa for a maximum. A convenient approach to solve the problem is provided by the method of Lagrangian multipliers [8]. The calculations are straightforward and only the results will be presented for the various cases of interest. All bounds to be discussed in the following are optimal for the assumptions used.
3.1. V, A neutral currents In this case only fL, fR andfs may be different from zero and F = fL + fR •
(3.3)
The equalities for the cross sections and the positivity constraints fL ~> 0,
fR ~> 0,
fs ~> 0 ,
(3.4)
lead to
I%- a~-I ~< F.<
](%+ a~).
(3.5)
Since ~ ~
lev - % 1 <~ ~(e v + ov)
(3.6)
and the bounds coalesce for r = ~ or 3.
3.2. V, A neutral currents and spin-~ constituents For spin-~ constituents the Callan-Gross relation [9] implies the additional constraint fs = 0 .
(3.7)
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486
F is now completely determined by the total cross sections and turns out to be equal to the maximal value in subsect. 3.1, F-- ~(o v + o~) .
(3.8)
A detailed discussion of the y-distributions in this case has already been presented [101.
3.3. S, P, T neutral currents It is known [ 1,2] that any y-distribution for V, A currents can also be obtained from certain combinations of S, P, T currents, while the converse is, of course, not true. It is therefore clear that the bounds derived for S, P, T neutral currents are the same as for the most general case of V, A, S, P, T. We have to distinguish between different regions for r in this case. For } ~r~< 3, 43-t% - o~t ~< F ~< (24(% + or) + 9x/22ovcr ~- - 302 - 3 0 2 ) / 2 8 .
(3.9)
The condition for the maximum to be real is 0.14 ~-- ~(11 - 4X/7) ~
(3.10)
which is just the allowed domain in r for the general case of V, A, S, P, T couplings [1, 2, 4]. For ~(11-4,,/7) ~< r ~< ~ and 3 ~< r ~< ~(ll + 4~/7) the lower bound in (3.9) is not optimal, but instead one finds (24(o u + crg)- 9X/22ovo-ff - 3o 2 - 3o2)/28 ~
3.4. S, P, T neutral currents and spin-~ constituents As before, the bounds are the same as for V, A, S, P, T with spin -1 partons. The parton model relations given in the appendix lead to f12 =f14 = 0,
f13 = 2fll = 2f15 '
F = f L + r e +f7 + f 8 + 8f15 '
(3.12) (3.•3)
and the inequalities (3.1) simplify considerably. In this case (24(0 u + o ~ ) - 9 2x/~ouo ~ - 3 0 2 - 3o 2 )/28 ~< F ~< (24(op + cry) + 9x//22ovo~- 302- 302)/28 ,
(3.14)
G. Ecker / Neutral currents
487
i~(y) /
f---~ . j
L-
/ /
/
/
/ / / / / t
/ / /
Fig. 1. Bounds for the neutrino y-distribution for r = o-ff/a v = 0.5. V, A currents: bounded by (a) and (c). V, A + spin-~ partons: (c). V, A, S, P, T: bounded by (a) and (d). V, A, S, P, T + spinpartons: bounded by (b) and (d).
t~(Y)
! / / !
/
,'d /
/// i II iii
.......S
~Y
Fig. 2. As fig. 1, for bounds for the antineutrino y-distribution for r = 0.5.
488
G. Ecker / Neutral currents
f~ (y) / /
r'=]
i
11
/ / /
~
z
\
1!
/ I
.~
.;
.~
~ ~ ,v
Fig. 3. As fig. 1, for bounds for the y-distribution for r = 1 (neutrino and antineutrino distributions are identical).
i.e. the more restricted bounds of the last paragraph are now valid for all allowed values o f r . As emphasized by the authors of ref. [7], the spin-~- patton model does not further restrict the allowed range for r as given by eq. (3.10). In order to discuss these bounds further let us turn to experiment to get some information about the ratio r. Unfortunately, the existing experimental results [11] for the total cross sections are not very precise, yet. While b o t h the Gargamelle experiment and the preliminary results from the CALTECH-FNAL experiment favour a value o f r around 0.5, the Harvard-Pennsylvania-Wisconsin group find r more in the vicinity o f 1. We use r = 0.5 and 1 as two typical values and obtain the bounds shown in figs. 1, 2, 3. For r = 1, the y-distributions for neutrino and antineutrino scattering are identical.
4. Discussion and summary The bounds and relations derived in this paper should be useful for two reasons. As mentioned in the introduction, it is not at all clear whether only the spin-~- constituents o f the nucleon take part in the neutral current interaction. Pakvasa and
G. Ecker / Neutral currents
489
Rajasekaran [7] have stressed the fact that neutral currents built up of integer spin constituents (gluons) will lead to a violation of scaling. One obvious test for scaling is to check the energy dependence of the total cross sections. The relations and bounds presented in the last two chapters provide additional and independent tests of the scaling assumption. These tests will be especially stringent for y near the fixed point y 1 --~ 0.55. The forthcoming results of the CALTECH-FNAL experiment will give a first answer to these questions. As already mentioned, our results should also be useful to analyze data with cuts in y. A second application of the results of this paper will be to check whether or not S, P, T neutral current couplings are present and, if not, whether the Callan-Gross relation is also valid for neutral currents. These checks are possible without detailed fits to the y-distributions which may not be very conclusive in view of the limited statistics of the present experiments. As can be seen from the figures, the bounds " for V, A currents are quite restrictive even for r = 1, the least favourable case. Our results supplement tests proposed earlier [1, 2, 4, 7] to detect S, P, T couplings in inclusive scattering such as risingy-distributions and r < ~- or r > 3. To summarize, we have obtained the following restrictions on the y-distributions for inclusive neutrino scattering on nucleons via neutral currents from the assumption of scaling of structure functions: (a) a relation between neutrino and antineutrino y-distributions; (b) definite values for the two y-distributions at a fixed point; (c) positivity bounds, with and without S, P, T couplings, with and without assuming spin- ~ constituents. All relations and bounds depend only on the ratio of antineutrino to neutrino total cross sections. I would like to thank Dr. H. Kiihnelt for reading the manuscript and the "Theodor. K6rner-Stiftungsfonds zur F6rderung yon Wissenschaft und Kunst" for financial support.
Appendix The most general local Lagrangian for the interaction of incoming left-handed neutrinos and right-handed antineutrinos with hadrons which is time-reversal invariant can be written [7] "~int = ~ G
{v3/a p(Va +Aa) + ~pS + i-PTsvP + -po opT a~ } .
(A.1)
In writing down this Lagrangian it has been assumed that incoming and outgoing neutrinos are identical which implies that the neutrino currents are diagonal or, in other words, that the neutral intermediate boson is hermitian. Therefore, the hadronic vector, axial-vector, scalar, pseudoscalar and tensor currents V~, A a, S, P, T ~# are also hermitian.
G. Ecker / Neutralcurrents
490
To write down the cross sections for inclusive scattering of (anti) neutrinos on nucleons the following structure functions are needed (kinematical and other conventions are those of ref. [ 12] ) (2n) 6 ~M ~ (NI Vv+ Avli)(iI
Mv~[
/quqv \ +Au IN) 64(pi-P-q)=~--~-guu}W1 (q2' u)
Mv'~W2(q2, v)
W3(q2, v)
+ terms which do not contribute to cross sections for massless neutrinos,
(A.2) (2Ir) 6 ½M ~ (NISli) (/]SIN) 64(pi- p - q ) = W7(q2, v) , i
(A.3)
(2n) 6 XM~. (NIPIi)qlPlN)64(pi-p-q) = W8(q2, v),
(A.4)
l
W9(q2, v) (2n) 6 -~M ~(NISIi)~ (ilTvlN)64(pi-p-q) = i(q~pv-qvpu)
(2n) 6 ~M ~lPli)i
(i[T~vlN)64(pi-P-q) = ie~3P°'q~
M2
Wl O(q2, V) M2 '
(A.5)
(A.6)
(270 6 {M .~(NI T v 1i) (il Ta3 IN) 64(p i- p - q) = (g#ag~¢-g~t~3) W11(q2, v) l
+ (puqv-qupv)(paq~-q~p~)
w~2(q2, ~,) M4
(g~Pd°~-g~PuP~-g~oPvP~+g~cPuPa)
Wl3(q 2, v)
X
M2
W14(q 2, v)
(g~aqvq3- g~quq ~ - gu3qvq~ +g~quq~)
M2
W15(q2, v)
-(guapvq3- gvapuq3-gu~pvqa +gv~puqa + (19~ q)) -
M2
(A.7)
Hermiticity and T-invariance imply that all structure functions are real. In the Bjorken limit _q2, v ~ oo (x = -q2/2Mv, y = vie fixed) the following limiting functions are supposed to exist (scaling assumption) W 1 ~ F 1 (x),
vW2/M -~ F2(x), vW3/M -+ F3 (x), W7 -~ F7 (x),
~W~JM -~ F~o(X), Wll -+ Fll (x), v 2 w t 2/M 2 ~ F 12(x), v W t f f M - * Fi3(x),
G. Ecker/ Neutral currents
491
uWI4/M ~ F14(x), uW15/M-+F15(x ).
(A.8)
W8 ~ FS(X),
vW9/M-+ F9(x ) ,
Note that scaling will not hold for S, P, T structure functions, if constituents of integer spin take part in the neutral current weak interactions [7]. In the scaling limit the differential cross sections for neutrino scattering are given by d2o
@) G2EM {2XFly2 +
dx dy -
87r
2F2(1 -y) ~ xF3(2y-y2) + x(F7 +F8)y2
+ 4xFlz(2-y)2 + 16F13(1 -y)-8x2F14 y2 + 8xF15y2 ~ 4x(F9+Flo)(2y-y2)}. (A.9) The structure functions obey certain positivity conditions. If we define for convenience FR(X ) -- FL(X ) = xF3(x),
FR(x) + FL(X)
=
2xFl(x),
2Fs(x) = F2(x) - 2XFl(X),
(A.10)
the positivity constraints may be written [7] FL~>0,
FR~>0,
FT~>0,
F8~>0,
FS~>0, Fll ~>0,
- F I 1 + F12-2xF14 + 2F15 >~0, FI3 Fll + ~ - - + 2x F14-2F15 >/0, F13
-Fll +~-- ~>0, 2.x
2x >~4xz(F13-2xF15 ) 2 '
F 7 ( - F l l + F12 -2xF14 + 2F15) >/F 2 , F8Fll ~f2 0 •
(A.1 I)
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G. Ecker / Neutral currents
In the p a t t o n model with spin-~- constituents the following relations hold F s = 0,
F12 = F14 = 0,
FI3 = 2FllX = 2F15x.
(A.12)
References [1] B. Kayser, G.T. Garvey, E. Fischbach and S.P. Rosen, Phys. Letters 52B (1974) 385. [2] R.L. Kingsley, F. Wilczek and A. Zee, Phys. Rev. D10 (1974) 2216; R.L. Kingsley, R. Shrock, S.B. Treiman and F. Wilczek, Phys. Rev. DII (1975) 1043. [3] T.C. Yang, Phys. Rev. D10 (1974) 3744. [4] M. Gronau, Technion preprint (1975). [5] B. Kayser, S.P. Rosen and E. Fischbach, Phys. Rev. DI1 (1975) 2547. [6] G.V. Dass and G.G. Ross, CERN preprint TH-1985 (1975). [7] S. Pakvasa and G. Rajasekaran, Phys. Rev. DI2 (1975) 113. [8] M.B. Einhorn and R. Blankenbecler, Ann. of Phys. 67 (1971) 480. [9] C. Callan and D. Gross, Phys. Rev. Letters 22 (1969) 156. [10] G. Ecker and H. Pietschmarm, Acta Phys. Austr., in press. [ 11] Proc. Colloque sur la physique du neutrino ~ haute dnergie, Paris, 1975. [ 12] H. Pietschmann, Formulae and results in weak interactions (Springer, Vienna-New York, 1974). [13] A. de Rtljula and S.L. Glashow, Phys. Rev. D9 (1974) 180; C.P. Korthals bates, M. Perrottet and E. de Rafael, Phys. Letters 57B (1975) 260.
SPACE-TIME STRUCTURE OF NEUTRAL CURRENTS ANDy-DISTRIBUTIONS * G. ECKER Institut for Theoretische Physik, Universitdt Wien
Received 17 September 1975
Assuming scaling of the nucleon structure functions, we derive relations for the ydistributions in deep inelastic neutrino scattering for the most general local interaction of neutrinos with hadrons via neutral currents with V, A, S, P and T pieces. In addition, optimal bounds for the y-distributions are presented which follow from positivity requirements for the structure functions and which may serve to discriminate between pure V, A currents and the presence of S, P, T couplings. The influence of the predictions of the spin-~ parton model on the bounds is investigated. All relations and bounds depend only on the ratio of total neutrino and antineutrino cross sections. 1. Introduction Although neutral currents were suggested long before the advent of gauge theories, the experimental confirmation of neutral current phenomena was considered by many people to give support to unified theories o f weak and electromagnetic interactions. It is therefore only natural that the prevailing opinion has been so far that neutral currents are vectors and/or axial vectors. More recently, several authors have argued [ 1 - 7 ] that the space-time structure o f neutral currents should be investigated experimentally and various tests have been proposed for the presence of scalar, pseudoscalar and tensor couplings. Another closely related question concerns the validity of scaling in deep inelastic scattering. The quark parton model has been impressively successful in describing deep inelastic scattering of electrons and neutrinos via charged currents. However, in these reactions presumably only the charged constituents of the nucleon take part in the interaction. On the other hand, the quark parton model also predicts that roughly half the momentum of the nucleon is carried by neutral gluons which are usually assumed to have integer spin. It is an interesting question whether these neutral particles participate in the neutral current interaction. It has been shown by Pakvasa and Rajasekaran [7] that if neutral currents exist which are built up o f neutral particles of integer spin and which * Supported in part by Fonds zur F6rderung der wissenschaftlichen Forschung, Projekt Nummer 1905.
481
482
G. Ecker / Neutral currents
have S, P or T pieces, then scaling is necessarily violated since the dimension of these currents is in general different from 3. The purpose of this paper is to investigate the y-distributions for inclusive scattering of (anti)neutrinos on nucleons to get information on the two questions just raised: is scaling valid and, if so, is there any evidence for S, P, T neutral current couplings? Relations and bounds for the y-distributions will be derived which follow from scaling and positivity constraints on the structure functions and which will depend only on the ratio of the total cross sections. Positivity bounds for the y-distributions in inclusive neutrino scattering have been derived before for V, A charged weak currents [13]. The bounds allow to check for the presence of S, P, T neutral currents without making detailed fits to the y-distributions. The results of this paper supplement other tests for the Lorentz structure of the neutral current interaction that have already been proposed [ 1 - 7 ] . Although the main application in the immediate future will be for inclusive scattering, our results for the spin-~- parton model are directly applicable to the y-distributions for the purely leptonic processes vu + e - ~ vu + e - and flu + e - ~ flu + e-. This will be an interesting and independent test for the space-time structure of neutral currents when more data on these reactions will become available. In sect. 2 we define the y-distributions for inclusive scattering in the scaling limit for the most general case where V, A, S, P and T couplings are present. We derive a relation between the neutrino and antineutrino y-distributions and make definite predictions at a certain fixed point in y. These relations test scaling only. In sect. 3 the positivity requirements for the structure functions are used to derive bounds for the y-distributions. We distinguish between pure V, A currents and S, P, T currents and investigate the modifications of the bounds if spin-½ partons are assumed. For two typical values of the cross section ratio the results are also presented graphically. In sect. 4 the results of the paper are discussed and summarized. Finally, an appendix contains the general neutral current Lagrangian, the nucleon structure functions including positivity constraints, the differential cross sections in the scaling limit and the predictions of the spin-~- patton model.
2. Relations for y-distributions The most general local Lagrangian for neutrinos interacting with hadrons has vector, axial-vector, scalar, pseudoscalar and tensor pieces. The Lagrangian and the double differential cross sections in the scaling limit are given in the appendix together with definitions of the various structure functions of the nucleon. In deriving the cross sections explicit use has been made of the scaling assumption for structure functions as given in eqs. (A.8). From (A.9) and (A. 10) we obtain the differential cross sections with respect toy,
G. Ecker/ Neutralcurrents
483
do
(~) + 16f13(1 - - y ) - 8f14Y2 + 8f15Y 2 -Y-4(f9 +flO)(2y_y2),
(2.1)
where
G2EM e
~- ~ JF.(x)a~, G2EM
6- ~
fx~.(~)ax,
i = L , R,S, 13 , / ' = 7 , 8 , 9 , 10, 11, 12, 15,
f14 - G2EM 1"x2F14(x) dx
(2.2)
8rr .:
The total cross sections are
Ou = 2 f L + } f R +2fS+~(f7+f8)+ ~'f12+8f13--sf14 (~) (R) (L) + 82315 g ~(f9 + f l 0 ) "
(2.3)
We now eliminate fs, fg, fl0 and I"13 from the differential cross sections using the total cross sections. One arrives at the following expressions for the normalized y-distributions: do _--
_
1
o(~)
-
-
(~)
dy
'
[~0') = 1 + ~-y- ~_y2 + r(1 - [y + ~y2) 1
+--(3y2+2y--2)(fL +fR +f7+f8+4f12--Sf14+Sf15), 3ov f ~ ( y ) = 1 + l y _ ~v 2 + r - l ( 1
(2.4a)
__Sy2 + 3y2)
+ 31- (3y 2 +2y_2)OCL+fR+f7+f8+4f12_8f14+SflS),
(2.4b)
v
where r is the cross section ratio o~/o v. In this section we want to draw two conclusions from eqs. (2.4). By eliminating the unknown quantity in brackets one obtains a relation between the y-distributions,
rf~(v)-f~(v) = (r- 1) - ~ y ( 2 - y ) .
(2.5)
It has already been observed [ 1,2] that the difference between the neutrino and
484
G. Ecker /Neutral currents
antineutrino cross sections is proportional to y (2 - y ) ; the relation (2.5) shows that the proportionality constant is determined by the total cross sections. Relation (2.5) is a useful test of the scaling hypothesis. Of course, it may also be used to deduce the cross section ratio r from the y-distributions, once scaling is established. Besides checking the energy dependence of total cross sections, relation (2.5) is another practicable test for scaling in the near future. The y-distributions (2.4) show another interesting feature. The quadratic function 3y 2 + 2y - 2 = 0 has a root in the physical domain of y, namely for (2.6)
Y =Yl = ~(X/~-1) ~ 0.55.
At this fixed point the y-distributions are completely determined by the ratio r, fu(Yl ) =Yl + -~ + r(3 --3Yl) ~ 1.05 -- O.15r ,
f (vi) =Yi +
+ r-l(
(2.7)
-3Yl) •
As for the relation (2.5) the equalities (2.7) are valid for a general V, A, S, P, T interaction and depend only on scaling. Moreover, (2.5) and (2.7) together should also be very useful in the case of cuts in y, when only the shape, but not the absolute scale of the y-distributions is known. Since cuts usually have to be applied for small y, the fixed point Yl is in the experimentally accessible region. Note that the values of f(u/o)(Yl) are relatively insensitive to variations of r. It should also be emphasized that both (2.5) and (2.7) are genuine predictions of the scaling model. For instance, for spin-0 or spin-1 constituents taking part in the neutral current interaction, both results are changed.
3. Bounds fory-distributions Besides the relations obtained in the last section, the y-distributions are subject to certain restrictions following from the positivity requirements on the structure functions. Those constraints are given in the appendix and are easily translated into inequalities for the integrated quantities
I7 >o, --fll +f12-- 2f14 + 2f15 ~> 0 ' f l l + }f13 + 2f14 -- 2f15 >~ 0 , --fll +~f13 1 ~>0,
&>o,
Ii1 >o,
G. Ecker / Neutral currents
485
(fll + ~-fl3 + 2f14--2f15)(--fll + lf13)>/(lf13--f15 )2 , f7(--fll +f12--2f14 + 2f15) ~>f~ ,
fsf11>~ f2 0 •
(3.1)
The problem we propose to solve is to extremize the quantity
e = f L +fR +f7 +f8 + 4f12--8f14 + 8f15 '
(3.2)
subject to the inequalities (3.1) and the equalities (2.3) for the total cross sections. Depending on whether y is below or above the fixed point y 1 a minimal value o f f will give an upper or lower bound for the y-distributions and vice versa for a maximum. A convenient approach to solve the problem is provided by the method of Lagrangian multipliers [8]. The calculations are straightforward and only the results will be presented for the various cases of interest. All bounds to be discussed in the following are optimal for the assumptions used.
3.1. V, A neutral currents In this case only fL, fR andfs may be different from zero and F = fL + fR •
(3.3)
The equalities for the cross sections and the positivity constraints fL ~> 0,
fR ~> 0,
fs ~> 0 ,
(3.4)
lead to
I%- a~-I ~< F.<
](%+ a~).
(3.5)
Since ~ ~
lev - % 1 <~ ~(e v + ov)
(3.6)
and the bounds coalesce for r = ~ or 3.
3.2. V, A neutral currents and spin-~ constituents For spin-~ constituents the Callan-Gross relation [9] implies the additional constraint fs = 0 .
(3.7)
G. Ecker / Neutral currents
486
F is now completely determined by the total cross sections and turns out to be equal to the maximal value in subsect. 3.1, F-- ~(o v + o~) .
(3.8)
A detailed discussion of the y-distributions in this case has already been presented [101.
3.3. S, P, T neutral currents It is known [ 1,2] that any y-distribution for V, A currents can also be obtained from certain combinations of S, P, T currents, while the converse is, of course, not true. It is therefore clear that the bounds derived for S, P, T neutral currents are the same as for the most general case of V, A, S, P, T. We have to distinguish between different regions for r in this case. For } ~r~< 3, 43-t% - o~t ~< F ~< (24(% + or) + 9x/22ovcr ~- - 302 - 3 0 2 ) / 2 8 .
(3.9)
The condition for the maximum to be real is 0.14 ~-- ~(11 - 4X/7) ~
(3.10)
which is just the allowed domain in r for the general case of V, A, S, P, T couplings [1, 2, 4]. For ~(11-4,,/7) ~< r ~< ~ and 3 ~< r ~< ~(ll + 4~/7) the lower bound in (3.9) is not optimal, but instead one finds (24(o u + crg)- 9X/22ovo-ff - 3o 2 - 3o2)/28 ~
3.4. S, P, T neutral currents and spin-~ constituents As before, the bounds are the same as for V, A, S, P, T with spin -1 partons. The parton model relations given in the appendix lead to f12 =f14 = 0,
f13 = 2fll = 2f15 '
F = f L + r e +f7 + f 8 + 8f15 '
(3.12) (3.•3)
and the inequalities (3.1) simplify considerably. In this case (24(0 u + o ~ ) - 9 2x/~ouo ~ - 3 0 2 - 3o 2 )/28 ~< F ~< (24(op + cry) + 9x//22ovo~- 302- 302)/28 ,
(3.14)
G. Ecker / Neutral currents
487
i~(y) /
f---~ . j
L-
/ /
/
/
/ / / / / t
/ / /
Fig. 1. Bounds for the neutrino y-distribution for r = o-ff/a v = 0.5. V, A currents: bounded by (a) and (c). V, A + spin-~ partons: (c). V, A, S, P, T: bounded by (a) and (d). V, A, S, P, T + spinpartons: bounded by (b) and (d).
t~(Y)
! / / !
/
,'d /
/// i II iii
.......S
~Y
Fig. 2. As fig. 1, for bounds for the antineutrino y-distribution for r = 0.5.
488
G. Ecker / Neutral currents
f~ (y) / /
r'=]
i
11
/ / /
~
z
\
1!
/ I
.~
.;
.~
~ ~ ,v
Fig. 3. As fig. 1, for bounds for the y-distribution for r = 1 (neutrino and antineutrino distributions are identical).
i.e. the more restricted bounds of the last paragraph are now valid for all allowed values o f r . As emphasized by the authors of ref. [7], the spin-~- patton model does not further restrict the allowed range for r as given by eq. (3.10). In order to discuss these bounds further let us turn to experiment to get some information about the ratio r. Unfortunately, the existing experimental results [11] for the total cross sections are not very precise, yet. While b o t h the Gargamelle experiment and the preliminary results from the CALTECH-FNAL experiment favour a value o f r around 0.5, the Harvard-Pennsylvania-Wisconsin group find r more in the vicinity o f 1. We use r = 0.5 and 1 as two typical values and obtain the bounds shown in figs. 1, 2, 3. For r = 1, the y-distributions for neutrino and antineutrino scattering are identical.
4. Discussion and summary The bounds and relations derived in this paper should be useful for two reasons. As mentioned in the introduction, it is not at all clear whether only the spin-~- constituents o f the nucleon take part in the neutral current interaction. Pakvasa and
G. Ecker / Neutral currents
489
Rajasekaran [7] have stressed the fact that neutral currents built up of integer spin constituents (gluons) will lead to a violation of scaling. One obvious test for scaling is to check the energy dependence of the total cross sections. The relations and bounds presented in the last two chapters provide additional and independent tests of the scaling assumption. These tests will be especially stringent for y near the fixed point y 1 --~ 0.55. The forthcoming results of the CALTECH-FNAL experiment will give a first answer to these questions. As already mentioned, our results should also be useful to analyze data with cuts in y. A second application of the results of this paper will be to check whether or not S, P, T neutral current couplings are present and, if not, whether the Callan-Gross relation is also valid for neutral currents. These checks are possible without detailed fits to the y-distributions which may not be very conclusive in view of the limited statistics of the present experiments. As can be seen from the figures, the bounds " for V, A currents are quite restrictive even for r = 1, the least favourable case. Our results supplement tests proposed earlier [1, 2, 4, 7] to detect S, P, T couplings in inclusive scattering such as risingy-distributions and r < ~- or r > 3. To summarize, we have obtained the following restrictions on the y-distributions for inclusive neutrino scattering on nucleons via neutral currents from the assumption of scaling of structure functions: (a) a relation between neutrino and antineutrino y-distributions; (b) definite values for the two y-distributions at a fixed point; (c) positivity bounds, with and without S, P, T couplings, with and without assuming spin- ~ constituents. All relations and bounds depend only on the ratio of antineutrino to neutrino total cross sections. I would like to thank Dr. H. Kiihnelt for reading the manuscript and the "Theodor. K6rner-Stiftungsfonds zur F6rderung yon Wissenschaft und Kunst" for financial support.
Appendix The most general local Lagrangian for the interaction of incoming left-handed neutrinos and right-handed antineutrinos with hadrons which is time-reversal invariant can be written [7] "~int = ~ G
{v3/a p(Va +Aa) + ~pS + i-PTsvP + -po opT a~ } .
(A.1)
In writing down this Lagrangian it has been assumed that incoming and outgoing neutrinos are identical which implies that the neutrino currents are diagonal or, in other words, that the neutral intermediate boson is hermitian. Therefore, the hadronic vector, axial-vector, scalar, pseudoscalar and tensor currents V~, A a, S, P, T ~# are also hermitian.
G. Ecker / Neutralcurrents
490
To write down the cross sections for inclusive scattering of (anti) neutrinos on nucleons the following structure functions are needed (kinematical and other conventions are those of ref. [ 12] ) (2n) 6 ~M ~ (NI Vv+ Avli)(iI
Mv~[
/quqv \ +Au IN) 64(pi-P-q)=~--~-guu}W1 (q2' u)
Mv'~W2(q2, v)
W3(q2, v)
+ terms which do not contribute to cross sections for massless neutrinos,
(A.2) (2Ir) 6 ½M ~ (NISli) (/]SIN) 64(pi- p - q ) = W7(q2, v) , i
(A.3)
(2n) 6 XM~. (NIPIi)qlPlN)64(pi-p-q) = W8(q2, v),
(A.4)
l
W9(q2, v) (2n) 6 -~M ~(NISIi)~ (ilTvlN)64(pi-p-q) = i(q~pv-qvpu)
(2n) 6 ~M ~lPli)i
(i[T~vlN)64(pi-P-q) = ie~3P°'q~
M2
Wl O(q2, V) M2 '
(A.5)
(A.6)
(270 6 {M .~(NI T v 1i) (il Ta3 IN) 64(p i- p - q) = (g#ag~¢-g~t~3) W11(q2, v) l
+ (puqv-qupv)(paq~-q~p~)
w~2(q2, ~,) M4
(g~Pd°~-g~PuP~-g~oPvP~+g~cPuPa)
Wl3(q 2, v)
X
M2
W14(q 2, v)
(g~aqvq3- g~quq ~ - gu3qvq~ +g~quq~)
M2
W15(q2, v)
-(guapvq3- gvapuq3-gu~pvqa +gv~puqa + (19~ q)) -
M2
(A.7)
Hermiticity and T-invariance imply that all structure functions are real. In the Bjorken limit _q2, v ~ oo (x = -q2/2Mv, y = vie fixed) the following limiting functions are supposed to exist (scaling assumption) W 1 ~ F 1 (x),
vW2/M -~ F2(x), vW3/M -+ F3 (x), W7 -~ F7 (x),
~W~JM -~ F~o(X), Wll -+ Fll (x), v 2 w t 2/M 2 ~ F 12(x), v W t f f M - * Fi3(x),
G. Ecker/ Neutral currents
491
uWI4/M ~ F14(x), uW15/M-+F15(x ).
(A.8)
W8 ~ FS(X),
vW9/M-+ F9(x ) ,
Note that scaling will not hold for S, P, T structure functions, if constituents of integer spin take part in the neutral current weak interactions [7]. In the scaling limit the differential cross sections for neutrino scattering are given by d2o
@) G2EM {2XFly2 +
dx dy -
87r
2F2(1 -y) ~ xF3(2y-y2) + x(F7 +F8)y2
+ 4xFlz(2-y)2 + 16F13(1 -y)-8x2F14 y2 + 8xF15y2 ~ 4x(F9+Flo)(2y-y2)}. (A.9) The structure functions obey certain positivity conditions. If we define for convenience FR(X ) -- FL(X ) = xF3(x),
FR(x) + FL(X)
=
2xFl(x),
2Fs(x) = F2(x) - 2XFl(X),
(A.10)
the positivity constraints may be written [7] FL~>0,
FR~>0,
FT~>0,
F8~>0,
FS~>0, Fll ~>0,
- F I 1 + F12-2xF14 + 2F15 >~0, FI3 Fll + ~ - - + 2x F14-2F15 >/0, F13
-Fll +~-- ~>0, 2.x
2x >~4xz(F13-2xF15 ) 2 '
F 7 ( - F l l + F12 -2xF14 + 2F15) >/F 2 , F8Fll ~f2 0 •
(A.1 I)
492
G. Ecker / Neutral currents
In the p a t t o n model with spin-~- constituents the following relations hold F s = 0,
F12 = F14 = 0,
FI3 = 2FllX = 2F15x.
(A.12)
References [1] B. Kayser, G.T. Garvey, E. Fischbach and S.P. Rosen, Phys. Letters 52B (1974) 385. [2] R.L. Kingsley, F. Wilczek and A. Zee, Phys. Rev. D10 (1974) 2216; R.L. Kingsley, R. Shrock, S.B. Treiman and F. Wilczek, Phys. Rev. DII (1975) 1043. [3] T.C. Yang, Phys. Rev. D10 (1974) 3744. [4] M. Gronau, Technion preprint (1975). [5] B. Kayser, S.P. Rosen and E. Fischbach, Phys. Rev. DI1 (1975) 2547. [6] G.V. Dass and G.G. Ross, CERN preprint TH-1985 (1975). [7] S. Pakvasa and G. Rajasekaran, Phys. Rev. DI2 (1975) 113. [8] M.B. Einhorn and R. Blankenbecler, Ann. of Phys. 67 (1971) 480. [9] C. Callan and D. Gross, Phys. Rev. Letters 22 (1969) 156. [10] G. Ecker and H. Pietschmarm, Acta Phys. Austr., in press. [ 11] Proc. Colloque sur la physique du neutrino ~ haute dnergie, Paris, 1975. [ 12] H. Pietschmann, Formulae and results in weak interactions (Springer, Vienna-New York, 1974). [13] A. de Rtljula and S.L. Glashow, Phys. Rev. D9 (1974) 180; C.P. Korthals bates, M. Perrottet and E. de Rafael, Phys. Letters 57B (1975) 260.