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Muon-proton scattering and the effect of neutral currents in the scaling region
8.C.3 [
Nuclear Physics B49 (1972) 513-524. North-Holland Publishing Company
MUON-PROTON EFFECT OF NEUTRAL
SCATTERING
CURRENTS
AND THE
IN T H E S C A L I N G R E G I O N
A. LOVE and G.G. ROSS Rutherford High Energy Laboratory D.V. NANOPOULOS University of Sussex Received 14 August 1972 Abstract: We discuss the problem of testing weak-interaction models with ~ttreqaext generation of accelerators, and conclude that muon-proton scattering must be considered. The crosssections for both polarised and umpolarised targets are obtained in the scaling region. The feasibility of detecting neutral weak currents is examined and optimum kinematical conditions to do this are found. Finally we suggest ways to isolate the neutral weak current.
1. INTRODUCTION There has recently been much renewed interest in unified models of the weak and electromagnetic interactions of leptons [1 ]. Such models spontaneously break an underlying gauge symmetry of the Lagrangian thus giving a mass to some of the gauge fields. A feature of these theories is the postulated existence of a massive neutral vector boson, the Z, which gives rise to a neutral current coupled to the leptons. In Weinberg's model [ 1], the gauge symmetry is SU 2 ® U 1 and the leptons transform as left-handed doublets. ½(1-~5)(Ve) e
,
l(1-75)(~uu ) , /a
and right-handed singlets ½(1 +3,5) e ,
21-(1 +75)/~ .
It is of great interest to find observable effects due to this neutral current and several authors have analysed neutrino-electron [2] and electron-positron [3] scattering to this end. However, purely leptonic interactions are Jimited in the c.m. energy which can be achieved (without using colliding beams) because of the small mass
514
A. Love et al., Muon-proton scattering
of the lepton target. Moreover, in order to test the group structure of the model fully it is necessary to have information on scattering involving both the doublet members. For these reasons it would be of considerable interest to look for the effects of the neutral vector meson in semi-leptonic processes. Unfortunately, the extension of the model to hadrons is not straight-forward because of the necessity of incorporating strange particles [4]. The simplest approach is to group the p, n and X quarks into a left-handed doublet
~(1-3,5
,(
p
ncosO c + x s i n O c
)
three right-handed singlets ½(1 + 3,5) p ,
1(1 +3,5) (n cos 0C + X sin Oc),
½(1 +3,5) ( - n sin 0 C + X cos Oc) , and a left-handed singlet 1(1 -3,5) ( - n sin 0 c +X cos Oc) . Here 0 C is the Cabbibo angle. This gives rise to a neutral strangeness changing current which must somehow be suppressed. Glashow, Ilipoulos and Maiani [5] have suggested a mechanism to do this by introducing a fourth quark, the p'. In this approach, there are two left-handed doublets
(
p
½(1-3,5) ncosOc+XsinO c
)'
(
P'
½(1-75 ) _nsinOc+XcosO c
)
and four right-handed singlets ½(1 +3,5) P ,
1 (1 +3,5) P' '
½(1 +3,5) (n cos Oc+ X sin Oc) ,
½(1 + 3,5) ( - n sin Oc+ X cos Of) . This ensures that the neutral current contains no strangeness changing component. The picture is further clouded by the fact that various charge assignments for these quarks are possible. Despite the confusion over the details of the semi-leptonic Lagragian, there is always a neutral component predicted for the hadronic current, unless new leptons are postulated [6]. Obviously the first place to look for this current is in neutrino-nucleon scattering. However, owing to the uncertainties in the model this is not necessarily
A. Love et al., Muon-proton scattering
515
sufficient. Indeed, in a variation due to Prentki and Zumino [7] the neutrino decouples entirely from the neutral current. In any case it is useful to look at scattering involving the other member of the lepton doublet. For these reasons we suggest that one would look at muon-nucleon scattering. Although the effect of the Z will be masked by the purely electromagnetic term, if we go to sufficient momentum transfer the latter may be sufficiently suppressed to allow Z effects to be become significant. Since elastic form factors are very small at large momentum transfers, we must consider inelastic scattering in the scaling region where the nucleon behaves as though it were composed of point-like consituents. In sect. 2 we compute the differential cross-section for inelastic muon-proton scattering in the scaling region assuming the partons are of spin ½. In sect. 3 we consider the feasibility of observing the effects due to the Z. For a Z mass of 86 GeV and incident muons of 200 GeV/c we find a maximum effect of magnitude 8% of the purely electromagnetic term. This should be observable since it has a characteristic nonscaling behaviour. In sect. 4 we extend the analysis to scattering off polarised targets. Finally in sect. 5 we consider ways to isolate the neutral current and minimise theoretical uncertainties.
2. INELASTIC/~-p SCATTERING. We consider inelastic muon-proton scattering ~ - p -~/~- P
,
where F is an unspecified multiparticle final state. We take the muon to be longitudinally polarised with negative helicity as will be the muons produced at N.A.L. by K decay. The kinematics [8] is indicated in fig. l, where we are working in the lab frame. As is conventional, we introduce the invariants Q2 and v where Q2 - _ q 2 = _ (k_k,)2 ,
(1)
v = (P. q)/M
(2)
.
The scaling region is given by v and Q2 both large. The differential cross section may be written in the form d2o
ot2
dE-E-rd~'
4E2sin 4 ~ 0
E+E'
{W2 c°s2 10 + (2W1 +---M--- W3) sin2 ½0} ,
where the structure functions WI, W2 and W3 are functions of v and Q2.
516
A. Love et al., Muon-proton scattering
P
p'
Fig. 1. Kinematics of deep inelastic muon-proton scattering
We shall calculate the structure functions within the framework of the parton model [9]. In this model the photon or Z interacts with one of the partons while the rest remain undisturbed during the interaction. The interaction with the patton is as if the parton were a free structureless particle. This is summarised in fig. 2. It is straightforward to calculate the contribution to the structure functions coming from a parton, i, with fractiony of the proton momentum. The lowest order purely electromagnetic contribution is [9]
w~(i)
2Y
W~r(t3 = 0 ,
(4)
where Qi is the charge of the parton, i, and Q2 = q2. The contribution coming from the cross terms involving both a photon and a Z propagator is
2wZ(t)= 2 ( g v + g A ) Q i G V M- 6
ty
-
[ 0 2 / ( 0 2 + m 2 Z )] ,
(5a)
A. Love et al., Muon-proton scattering
517
k k'
Fig. 2. Parton model of the scattering.
wz,i)=
Y 6 Qi Giv -ff
( Q2)
[Q2/(Q2+m2)] ,
(Sb)
2(gv+gA)QiGiA_~ 1 6(yQ~__~)[Q2/(Q2 + m 2 ) ] .
(5c)
wzo9 = 2(gv + gA)
---~
Here we have written the Z coupling at the lepton vertex as e7" (gv - g A T S ) , and at the hadron vertex the Z coupling to the ith parton is eT u (G{¢ - G ~ , 7 5 ) In Weinberg's model [1,4] the couplings are given by g v = (1 - 2 cos 20W)/(2 sin20w) , (6) gA = - 1/(2 sin 2 0W) , 7 ~ (a~, - a iA 75)
= (i
IJ~ - 2 sin 2 0 w
JUli>/sin 20 w ,
(7)
J~ and ju are the hadronic weak and electromagnetic currents defined by Weinberg [4] and ju depends on the particular charge assignment made for the partons. 0 w is the mixing angle introduced by Weinberg. In order to compute the structure functions we must sum over configurations of N partons for all values of N with probabilitY P(N) and integrate over the momentum fractiony with probability density f~¢ (y) for parton i. This gives
A. Loveet al., Muon-protonscattering
518
N 2W~=~
N
i N
W~=1 ~ P(N) ~ Q2x f~C(x)=-l F'r (x) , N i w~ = o ,
(8)
N 2WlZ -_~ -2( g v +gA) (02/(Q2 + m2)) ~ P(N) ~ N i
Qi GV i f'N i (x)
N i w z = 2 (gv +gA)(Q2/(Q2 +m2)) ~ P(N) ~ QiGvxflN(x ) N i
1 FZ (x) (Q2/ (Q2 + m2)) , =--1)
N wZ =7(g 2 v +gA) (Q2/(Q2 + rn2z)) ~ P ( N ) ~
N
i
i i (x ) QiGAf'N
=- -1 GZ (x) (Q2/ (Q2 + m2))
(9)
/)
where x = Q2/(2My), and we have introduced scaling functions Fy (x), F z (x) and C z (x). The relationship between the pure photon term and the cross term evidently depends on the m o m e n t u m distribution functions f ~ (x). If we assume that they are independent of parton type then we have
N i P(N) fN (x) ~ Qi GV N i (Q2/(Q2 + m2)) Wit , WZ = 2(g V + gA ) N P(N)fN(x) ~ Q2i N i N i P(N) f N (X) ~ Qi GV N i (Q2/(Q2 + m2)) w~ , WZ = 2(g V + gA ) N
e U)iu(x)
N
O,2
i
(lOa)
(10b)
A. Love et al., Muon-proton scattering
519
N 4M wZ = ~ ( g v + g A
)
N
i
(Q2/(Q2 +m2)) W~ .
N
B
(10c)
02
N
i
This assumption, although probably not exact, enables us to estimate the relative size of the cross term to the photon term. The structure functions W~r and W~ may be determined from low energy data where the propagator suppresses the Z term.
3. DETECTION OF THE NEUTRAL CURRENT The differential cross section of the previous section may be conveniently written in the form d2o dE'd~2'
~2p2 F (l~pp)2 ] x2M2(1 _p)3 E l1_+ D(x,p,E) ,
(11)
where
D(x, p,E) - F ~r(x) + (Q2/ (Q2 + m2)) [F Z (x) + 1 _ p 2 x a z (x)] ,
(12)
1 +0 2
and we have used the variables
x =-Q2/(2Mp),
P - E'/E .
(13)
In the absence of Z exchange D would be a function o f x alone. The second of the two terms on the r.h.s, of eq. (12), which corresponds to the cross term, will violate this scaling prediction. The relative magnitude of the two terms may be determined if we assume that the patton momentum distribution function is independent of patton type. Then, using eqs. (8), (9) and (10), eq.:(12)becomes 2(gv + gA)
D = F "r (x) I1 + (Q2/(Q2 + m2))
N N
e(X)& (x) Z: QiGiv i
i
+ x ( 1 - p 2) 1 +p2
~
N
e(N)fN(X ) ~
QiGiA
. (14)
A. Love et al., Muon-proton scattering
520
Barring an accidental cancellation, the maximum relative effect of the Z will occur when P is small and x is near 1. Then,
4 M_E _ (gV + g A ) D'~FT(x) I 1 + m2z X
P(g)fu (~) ~ i
-F'Y(x) [1+
4ME
R]
[~N p(N)fN(x ) ~N Q 2il -1 i
i + ~ P(N) fN (X) Qi G Q~GA N "
.
(15)
The ratio R may be estimated using a specific parton model. If we assume the partons are conventional quarks, or Glashow et al. [5] quarks with charges ~,3,2 2 _ and - ~ for p, p', n and X, and ignore possible quark-antiquark pairs in the proton, then in both cases R = - cos 20 w [~ - 2 sin 2 0W] / [sin 20 w ] 2 ~
y7 ,
(16)
where we have taken 0 w = 30 ° (see refs. [2, 10]). With that value of Weinberg's angle, and the assumed charge assignment for the quarks, the couplings to the Z are 1
gv:0'
1
g A - N/r3'
_
G P - 3N/~'
1
n_
GPA N/~'
GV
2
3N/'3'
n_
GA
1
N/~ '
For an incident muon energy of 200 GeV and a Z mass of 86 GeV, the cross term an its maximum will be 8% of the purely electromagnetic term in D. If we include the effect of a possible quark-antiquark sea then R may be written in the form
I N3 i < ~
P(N) N
+
/----4
Qi(GiA+G
_
fN(X)
< ~ Qi (G~A+ Gtv ) > ?=-1
R =R 0 - + N
L
3
<~ Q2 )sea /=-4
3
/=I
(17)
l
fN (x)
A. Love et aL, Muon-proton scattering
521
where R 0 is the ratio obtained above assuming no sea. It is apparent from eq. (17) that the effect of the sea depends on the details of the distribution functions. If the sea dominates then the ratio R will be reduced relative to R 0 by a factor 4, for conventional quarks, or s for Glashow et al. quarks. Recent analyses [9, 14] suggest the proportion of sea near x = 1 is small therefore the reduction of R is not great. A further complication is the possibility of the existence of vector or scalar neutral gluons [14]. Although the gluons may affect the distribution functions they will not contribute directly to the cross section and consequently the ratio R will be unaffected by their presence. There remains the difficulty o f separating the effect due to Z-photon interference from background or effects due to breakdown of scaling. The background from two photon exchange may be expected by naively counting powers of a to be less than 1% of the one photon exchange term. More detailed model dependent analysis [11 ] supports this view and, moreover, the dependence on u and q2 is expected to be similar to the lowest order term. It is clear from eq. (t4) that a characteristic of the Z-7 interference term is that it depends on the variable p whereas the purely electromagnetic term does not. More specifically, as p ~ 1 with x fixed the interference term vanishes as the sum of a term linear in (1 - p ) a n d a term quadratic in (1 - O ) . This characteristic dependence on p may be a useful test of whether the non-scaling term is indeed due to photon-Z interference.
4. SCATTERING FROM POLARISED PROTONS We now consider deep inelastic scattering o f # - mesons from protons with polarisation vector N in the lab frame. In addition to the momentum distribution functions it is necessary to introduce a spin distribution [ 12] siN(a)which is the probability for the parton in the N parton configuration to have spin parallel a = +1 or anti-parallel a = - 1 to the proton polarisation direction. It is then straightforward to compute the differential cross section from the ith parton. Summing incoheretatly as before we obtain in addition to the polarisation terms the following contribution to the cross section: d2o dE'd~'
-
oL2
(
4EZMp sin 2 ~-0
-N'(k+k')
~ P(N) Q2fiN(x)SNi (o) o N,i,o
Q2 2 IN . (k+k')N,i,a~P(N)Qi GiV f~i (x) Sf(a) +2(gv+gA) Q2+m
+g . ( k - k ' )
N,i,o ~ e(N)QiGiAfiN(x)SN (°)° ] } .
(18)
A. Love et aL, Muon-proton scattering
522
It may be observed that the Z-photon interference term gives an asymmetry with a characteristic non-scaling behaviour. Unfortunately it is not possible to choose the polarisation direction to greatly enhance the cross term over the electromagnetic term, and consequently for the purposes of detecting the Z term there is no gain in polarising the target. Moreover, because of the diminution due to incomplete polarisation the asymmetric interference term is likely to be less than 1% of the total cross section. The situation is further complicated by the unknown spin-distribution functions. It therefore seems unprofitable to try to observe Z effects with a polarised target.
5. ISOLATION OF THE NEUTRAL CURRENT In the discussion of sect. 3 we considered the feasibility of detecting the Z-photon interference term in the presence of the purely electromagnetic term for the process # - + p-+/z- + F. In any detailed application of the parton model the uncertainties of the "quark-antiquark sea" and the unknown parton distribution cloud the issue. Since the "quark-antiquark sea" provides an isoscaYar contribution it may be eliminated by subtracting the inclusive cross section ~- +n-~-
÷F .
Although this has the disadvantage of introducing relative normalisation errors, the characteristic non-scaling behaviour of the interference term should help to distinguish a true Z-contribution from background. However, upon making the subtraction the ratio R decreases to ~- correspondingly reducing the ratio of the cross term to the background. Another possible source of information is the inclusive cross section/a + + p~/~+ + F. Since the/2 + mesons will be produced by K + decay, they will be longitudinally polarised with positive helicity, whereas the/2- mesons had negative helicity. This differential cross section is still given by eqs. (11), (14) and (15) provided we replace the plus sign in front of the term involving ~ Qi GiA by a minus sign. Taking the difference of the differential cross section wltti/2- and/~+ beams, gwes for the scaling function D, .
.
.
.
/
Du - - Du÷ = 4(gA + gv) Fw (x) (Q2/(Q2 + rn2))
(1 +p2)
(19) ~N
i
The absence of a one-photon exchange term makes this combination a very useful one for looking for the Z-photon interference term. There is the further advantage
A. Love et al., Muon-proton scattering
523
that there is only one cross term which has a definite non-scaling behaviour, decreasing as ( 1 - p2) as p -+ 1 at fixed x. Finally, in order to eliminate the contribution o f the "quark-antiquark sea" one can take the combination o f inclusive cross section do(/.t+p) - d a ( / a - p ) - do(/~+n) + da(/~-n) However, as this involves the combination of four measurements it is unlikely to be determined to very great precision.
6. CONCLUSIONS F r o m a discussion of the possible ways in which the neutral weak current predicted by unified models can be detected experimentally, we concluded that it would be desirable to have information on deep-inelastic muon-proton scattering at high energies. Parton model calculations with 200 GeV/c muons showed that the Z-photon interference term i n / l - p scattering could be expected to be 8% o f the purely electromagnetic term in a suitable kinematical region. If such an effect were found it would not only be important evidence in support of unified models, but its absolute magnitude and variation with target type could help to choose between models. Information on b o t h / a - and/~+ scattering should provide the most definite p r o o f o f the existence of the neutral current. One o f us (D.V.N.) would like to thank Dr R.J.N. Phillips for the hospitality of the Rutherford Laboratory while this calculation was being performed.
REFERENCES [1] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264; S. Weinberg, Phys. Rev. Letters 27 (1971) 1688; A. Salam and J. Strathdee, Trieste preprint IC/71/145; J. Schecter and Y. Ueda, Phys. Rev. D2 (1970) 736. [2] G. 't Hooft, Phys. Rev. 37B (1971) 195; B.W. Lee, Phys. Rev. D5 (1972)823. [3] J. Godine and A. Hankey, M.I.T. preprint 279 (1972); A. Love, Rutherford preprint RPP/T/11 (1972). [4] S. Weinberg, Phys. Rev. D5 (1972) 1412. [5] S.L. Glashow, J. llipoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [6] H. Georgi and S.L. Glashow, Phys. Rev. Letters 28 (1972) 1494. [7] J. Prentki and B. Zumino, CERN preprint TH. 1504 (1972). [8] S.D. Drell and J.D. Walecka, Ann. of Phys. 28 (1964) 18. [9] R.P. Feynman, Phys. Rev. Letters 23 (1969) 1415; J.D. Bjorken and E.A. Paschos, Phys. Rev. 185 (1969) 1975.
524
A. Love et al., Muon.proton scattering
[10] S. Weinberg, On the mixing angle in renormalizable theories of weak and electromagnetic interactions, M.I.T. preprint (December 1971). [ 11 ] J. Kingsley, Cambridge preprint D.A.M.T.P. 72/1. [12] M. Gourdin, Nucl. Phys. B38 (1972)418. [13] N. Booth, T° Quirk, A. Skemja and W. Williams, NAL proposal 98, RHEL proposal 96. [14] J. Kuti and V.F. Weisskopf, M.I.T. preprint 1971; Phys. Rev. D4 (1971) 3418; P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B28 (1971) 240; Phys. Letters 34B (1971)621.
Nuclear Physics B49 (1972) 513-524. North-Holland Publishing Company
MUON-PROTON EFFECT OF NEUTRAL
SCATTERING
CURRENTS
AND THE
IN T H E S C A L I N G R E G I O N
A. LOVE and G.G. ROSS Rutherford High Energy Laboratory D.V. NANOPOULOS University of Sussex Received 14 August 1972 Abstract: We discuss the problem of testing weak-interaction models with ~ttreqaext generation of accelerators, and conclude that muon-proton scattering must be considered. The crosssections for both polarised and umpolarised targets are obtained in the scaling region. The feasibility of detecting neutral weak currents is examined and optimum kinematical conditions to do this are found. Finally we suggest ways to isolate the neutral weak current.
1. INTRODUCTION There has recently been much renewed interest in unified models of the weak and electromagnetic interactions of leptons [1 ]. Such models spontaneously break an underlying gauge symmetry of the Lagrangian thus giving a mass to some of the gauge fields. A feature of these theories is the postulated existence of a massive neutral vector boson, the Z, which gives rise to a neutral current coupled to the leptons. In Weinberg's model [ 1], the gauge symmetry is SU 2 ® U 1 and the leptons transform as left-handed doublets. ½(1-~5)(Ve) e
,
l(1-75)(~uu ) , /a
and right-handed singlets ½(1 +3,5) e ,
21-(1 +75)/~ .
It is of great interest to find observable effects due to this neutral current and several authors have analysed neutrino-electron [2] and electron-positron [3] scattering to this end. However, purely leptonic interactions are Jimited in the c.m. energy which can be achieved (without using colliding beams) because of the small mass
514
A. Love et al., Muon-proton scattering
of the lepton target. Moreover, in order to test the group structure of the model fully it is necessary to have information on scattering involving both the doublet members. For these reasons it would be of considerable interest to look for the effects of the neutral vector meson in semi-leptonic processes. Unfortunately, the extension of the model to hadrons is not straight-forward because of the necessity of incorporating strange particles [4]. The simplest approach is to group the p, n and X quarks into a left-handed doublet
~(1-3,5
,(
p
ncosO c + x s i n O c
)
three right-handed singlets ½(1 + 3,5) p ,
1(1 +3,5) (n cos 0C + X sin Oc),
½(1 +3,5) ( - n sin 0 C + X cos Oc) , and a left-handed singlet 1(1 -3,5) ( - n sin 0 c +X cos Oc) . Here 0 C is the Cabbibo angle. This gives rise to a neutral strangeness changing current which must somehow be suppressed. Glashow, Ilipoulos and Maiani [5] have suggested a mechanism to do this by introducing a fourth quark, the p'. In this approach, there are two left-handed doublets
(
p
½(1-3,5) ncosOc+XsinO c
)'
(
P'
½(1-75 ) _nsinOc+XcosO c
)
and four right-handed singlets ½(1 +3,5) P ,
1 (1 +3,5) P' '
½(1 +3,5) (n cos Oc+ X sin Oc) ,
½(1 + 3,5) ( - n sin Oc+ X cos Of) . This ensures that the neutral current contains no strangeness changing component. The picture is further clouded by the fact that various charge assignments for these quarks are possible. Despite the confusion over the details of the semi-leptonic Lagragian, there is always a neutral component predicted for the hadronic current, unless new leptons are postulated [6]. Obviously the first place to look for this current is in neutrino-nucleon scattering. However, owing to the uncertainties in the model this is not necessarily
A. Love et al., Muon-proton scattering
515
sufficient. Indeed, in a variation due to Prentki and Zumino [7] the neutrino decouples entirely from the neutral current. In any case it is useful to look at scattering involving the other member of the lepton doublet. For these reasons we suggest that one would look at muon-nucleon scattering. Although the effect of the Z will be masked by the purely electromagnetic term, if we go to sufficient momentum transfer the latter may be sufficiently suppressed to allow Z effects to be become significant. Since elastic form factors are very small at large momentum transfers, we must consider inelastic scattering in the scaling region where the nucleon behaves as though it were composed of point-like consituents. In sect. 2 we compute the differential cross-section for inelastic muon-proton scattering in the scaling region assuming the partons are of spin ½. In sect. 3 we consider the feasibility of observing the effects due to the Z. For a Z mass of 86 GeV and incident muons of 200 GeV/c we find a maximum effect of magnitude 8% of the purely electromagnetic term. This should be observable since it has a characteristic nonscaling behaviour. In sect. 4 we extend the analysis to scattering off polarised targets. Finally in sect. 5 we consider ways to isolate the neutral current and minimise theoretical uncertainties.
2. INELASTIC/~-p SCATTERING. We consider inelastic muon-proton scattering ~ - p -~/~- P
,
where F is an unspecified multiparticle final state. We take the muon to be longitudinally polarised with negative helicity as will be the muons produced at N.A.L. by K decay. The kinematics [8] is indicated in fig. l, where we are working in the lab frame. As is conventional, we introduce the invariants Q2 and v where Q2 - _ q 2 = _ (k_k,)2 ,
(1)
v = (P. q)/M
(2)
.
The scaling region is given by v and Q2 both large. The differential cross section may be written in the form d2o
ot2
dE-E-rd~'
4E2sin 4 ~ 0
E+E'
{W2 c°s2 10 + (2W1 +---M--- W3) sin2 ½0} ,
where the structure functions WI, W2 and W3 are functions of v and Q2.
516
A. Love et al., Muon-proton scattering
P
p'
Fig. 1. Kinematics of deep inelastic muon-proton scattering
We shall calculate the structure functions within the framework of the parton model [9]. In this model the photon or Z interacts with one of the partons while the rest remain undisturbed during the interaction. The interaction with the patton is as if the parton were a free structureless particle. This is summarised in fig. 2. It is straightforward to calculate the contribution to the structure functions coming from a parton, i, with fractiony of the proton momentum. The lowest order purely electromagnetic contribution is [9]
w~(i)
2Y
W~r(t3 = 0 ,
(4)
where Qi is the charge of the parton, i, and Q2 = q2. The contribution coming from the cross terms involving both a photon and a Z propagator is
2wZ(t)= 2 ( g v + g A ) Q i G V M- 6
ty
-
[ 0 2 / ( 0 2 + m 2 Z )] ,
(5a)
A. Love et al., Muon-proton scattering
517
k k'
Fig. 2. Parton model of the scattering.
wz,i)=
Y 6 Qi Giv -ff
( Q2)
[Q2/(Q2+m2)] ,
(Sb)
2(gv+gA)QiGiA_~ 1 6(yQ~__~)[Q2/(Q2 + m 2 ) ] .
(5c)
wzo9 = 2(gv + gA)
---~
Here we have written the Z coupling at the lepton vertex as e7" (gv - g A T S ) , and at the hadron vertex the Z coupling to the ith parton is eT u (G{¢ - G ~ , 7 5 ) In Weinberg's model [1,4] the couplings are given by g v = (1 - 2 cos 20W)/(2 sin20w) , (6) gA = - 1/(2 sin 2 0W) , 7 ~ (a~, - a iA 75)
= (i
IJ~ - 2 sin 2 0 w
JUli>/sin 20 w ,
(7)
J~ and ju are the hadronic weak and electromagnetic currents defined by Weinberg [4] and ju depends on the particular charge assignment made for the partons. 0 w is the mixing angle introduced by Weinberg. In order to compute the structure functions we must sum over configurations of N partons for all values of N with probabilitY P(N) and integrate over the momentum fractiony with probability density f~¢ (y) for parton i. This gives
A. Loveet al., Muon-protonscattering
518
N 2W~=~
N
i N
W~=1 ~ P(N) ~ Q2x f~C(x)=-l F'r (x) , N i w~ = o ,
(8)
N 2WlZ -_~ -2( g v +gA) (02/(Q2 + m2)) ~ P(N) ~ N i
Qi GV i f'N i (x)
N i w z = 2 (gv +gA)(Q2/(Q2 +m2)) ~ P(N) ~ QiGvxflN(x ) N i
1 FZ (x) (Q2/ (Q2 + m2)) , =--1)
N wZ =7(g 2 v +gA) (Q2/(Q2 + rn2z)) ~ P ( N ) ~
N
i
i i (x ) QiGAf'N
=- -1 GZ (x) (Q2/ (Q2 + m2))
(9)
/)
where x = Q2/(2My), and we have introduced scaling functions Fy (x), F z (x) and C z (x). The relationship between the pure photon term and the cross term evidently depends on the m o m e n t u m distribution functions f ~ (x). If we assume that they are independent of parton type then we have
N i P(N) fN (x) ~ Qi GV N i (Q2/(Q2 + m2)) Wit , WZ = 2(g V + gA ) N P(N)fN(x) ~ Q2i N i N i P(N) f N (X) ~ Qi GV N i (Q2/(Q2 + m2)) w~ , WZ = 2(g V + gA ) N
e U)iu(x)
N
O,2
i
(lOa)
(10b)
A. Love et al., Muon-proton scattering
519
N 4M wZ = ~ ( g v + g A
)
N
i
(Q2/(Q2 +m2)) W~ .
N
B
(10c)
02
N
i
This assumption, although probably not exact, enables us to estimate the relative size of the cross term to the photon term. The structure functions W~r and W~ may be determined from low energy data where the propagator suppresses the Z term.
3. DETECTION OF THE NEUTRAL CURRENT The differential cross section of the previous section may be conveniently written in the form d2o dE'd~2'
~2p2 F (l~pp)2 ] x2M2(1 _p)3 E l1_+ D(x,p,E) ,
(11)
where
D(x, p,E) - F ~r(x) + (Q2/ (Q2 + m2)) [F Z (x) + 1 _ p 2 x a z (x)] ,
(12)
1 +0 2
and we have used the variables
x =-Q2/(2Mp),
P - E'/E .
(13)
In the absence of Z exchange D would be a function o f x alone. The second of the two terms on the r.h.s, of eq. (12), which corresponds to the cross term, will violate this scaling prediction. The relative magnitude of the two terms may be determined if we assume that the patton momentum distribution function is independent of patton type. Then, using eqs. (8), (9) and (10), eq.:(12)becomes 2(gv + gA)
D = F "r (x) I1 + (Q2/(Q2 + m2))
N N
e(X)& (x) Z: QiGiv i
i
+ x ( 1 - p 2) 1 +p2
~
N
e(N)fN(X ) ~
QiGiA
. (14)
A. Love et al., Muon-proton scattering
520
Barring an accidental cancellation, the maximum relative effect of the Z will occur when P is small and x is near 1. Then,
4 M_E _ (gV + g A ) D'~FT(x) I 1 + m2z X
P(g)fu (~) ~ i
-F'Y(x) [1+
4ME
R]
[~N p(N)fN(x ) ~N Q 2il -1 i
i + ~ P(N) fN (X) Qi G Q~GA N "
.
(15)
The ratio R may be estimated using a specific parton model. If we assume the partons are conventional quarks, or Glashow et al. [5] quarks with charges ~,3,2 2 _ and - ~ for p, p', n and X, and ignore possible quark-antiquark pairs in the proton, then in both cases R = - cos 20 w [~ - 2 sin 2 0W] / [sin 20 w ] 2 ~
y7 ,
(16)
where we have taken 0 w = 30 ° (see refs. [2, 10]). With that value of Weinberg's angle, and the assumed charge assignment for the quarks, the couplings to the Z are 1
gv:0'
1
g A - N/r3'
_
G P - 3N/~'
1
n_
GPA N/~'
GV
2
3N/'3'
n_
GA
1
N/~ '
For an incident muon energy of 200 GeV and a Z mass of 86 GeV, the cross term an its maximum will be 8% of the purely electromagnetic term in D. If we include the effect of a possible quark-antiquark sea then R may be written in the form
I N3 i < ~
P(N) N
+
/----4
Qi(GiA+G
_
fN(X)
< ~ Qi (G~A+ Gtv ) > ?=-1
R =R 0 - + N
L
3
<~ Q2 )sea /=-4
3
/=I
(17)
l
fN (x)
A. Love et aL, Muon-proton scattering
521
where R 0 is the ratio obtained above assuming no sea. It is apparent from eq. (17) that the effect of the sea depends on the details of the distribution functions. If the sea dominates then the ratio R will be reduced relative to R 0 by a factor 4, for conventional quarks, or s for Glashow et al. quarks. Recent analyses [9, 14] suggest the proportion of sea near x = 1 is small therefore the reduction of R is not great. A further complication is the possibility of the existence of vector or scalar neutral gluons [14]. Although the gluons may affect the distribution functions they will not contribute directly to the cross section and consequently the ratio R will be unaffected by their presence. There remains the difficulty o f separating the effect due to Z-photon interference from background or effects due to breakdown of scaling. The background from two photon exchange may be expected by naively counting powers of a to be less than 1% of the one photon exchange term. More detailed model dependent analysis [11 ] supports this view and, moreover, the dependence on u and q2 is expected to be similar to the lowest order term. It is clear from eq. (t4) that a characteristic of the Z-7 interference term is that it depends on the variable p whereas the purely electromagnetic term does not. More specifically, as p ~ 1 with x fixed the interference term vanishes as the sum of a term linear in (1 - p ) a n d a term quadratic in (1 - O ) . This characteristic dependence on p may be a useful test of whether the non-scaling term is indeed due to photon-Z interference.
4. SCATTERING FROM POLARISED PROTONS We now consider deep inelastic scattering o f # - mesons from protons with polarisation vector N in the lab frame. In addition to the momentum distribution functions it is necessary to introduce a spin distribution [ 12] siN(a)which is the probability for the parton in the N parton configuration to have spin parallel a = +1 or anti-parallel a = - 1 to the proton polarisation direction. It is then straightforward to compute the differential cross section from the ith parton. Summing incoheretatly as before we obtain in addition to the polarisation terms the following contribution to the cross section: d2o dE'd~'
-
oL2
(
4EZMp sin 2 ~-0
-N'(k+k')
~ P(N) Q2fiN(x)SNi (o) o N,i,o
Q2 2 IN . (k+k')N,i,a~P(N)Qi GiV f~i (x) Sf(a) +2(gv+gA) Q2+m
+g . ( k - k ' )
N,i,o ~ e(N)QiGiAfiN(x)SN (°)° ] } .
(18)
A. Love et aL, Muon-proton scattering
522
It may be observed that the Z-photon interference term gives an asymmetry with a characteristic non-scaling behaviour. Unfortunately it is not possible to choose the polarisation direction to greatly enhance the cross term over the electromagnetic term, and consequently for the purposes of detecting the Z term there is no gain in polarising the target. Moreover, because of the diminution due to incomplete polarisation the asymmetric interference term is likely to be less than 1% of the total cross section. The situation is further complicated by the unknown spin-distribution functions. It therefore seems unprofitable to try to observe Z effects with a polarised target.
5. ISOLATION OF THE NEUTRAL CURRENT In the discussion of sect. 3 we considered the feasibility of detecting the Z-photon interference term in the presence of the purely electromagnetic term for the process # - + p-+/z- + F. In any detailed application of the parton model the uncertainties of the "quark-antiquark sea" and the unknown parton distribution cloud the issue. Since the "quark-antiquark sea" provides an isoscaYar contribution it may be eliminated by subtracting the inclusive cross section ~- +n-~-
÷F .
Although this has the disadvantage of introducing relative normalisation errors, the characteristic non-scaling behaviour of the interference term should help to distinguish a true Z-contribution from background. However, upon making the subtraction the ratio R decreases to ~- correspondingly reducing the ratio of the cross term to the background. Another possible source of information is the inclusive cross section/a + + p~/~+ + F. Since the/2 + mesons will be produced by K + decay, they will be longitudinally polarised with positive helicity, whereas the/2- mesons had negative helicity. This differential cross section is still given by eqs. (11), (14) and (15) provided we replace the plus sign in front of the term involving ~ Qi GiA by a minus sign. Taking the difference of the differential cross section wltti/2- and/~+ beams, gwes for the scaling function D, .
.
.
.
/
Du - - Du÷ = 4(gA + gv) Fw (x) (Q2/(Q2 + rn2))
(1 +p2)
(19) ~N
i
The absence of a one-photon exchange term makes this combination a very useful one for looking for the Z-photon interference term. There is the further advantage
A. Love et al., Muon-proton scattering
523
that there is only one cross term which has a definite non-scaling behaviour, decreasing as ( 1 - p2) as p -+ 1 at fixed x. Finally, in order to eliminate the contribution o f the "quark-antiquark sea" one can take the combination o f inclusive cross section do(/.t+p) - d a ( / a - p ) - do(/~+n) + da(/~-n) However, as this involves the combination of four measurements it is unlikely to be determined to very great precision.
6. CONCLUSIONS F r o m a discussion of the possible ways in which the neutral weak current predicted by unified models can be detected experimentally, we concluded that it would be desirable to have information on deep-inelastic muon-proton scattering at high energies. Parton model calculations with 200 GeV/c muons showed that the Z-photon interference term i n / l - p scattering could be expected to be 8% o f the purely electromagnetic term in a suitable kinematical region. If such an effect were found it would not only be important evidence in support of unified models, but its absolute magnitude and variation with target type could help to choose between models. Information on b o t h / a - and/~+ scattering should provide the most definite p r o o f o f the existence of the neutral current. One o f us (D.V.N.) would like to thank Dr R.J.N. Phillips for the hospitality of the Rutherford Laboratory while this calculation was being performed.
REFERENCES [1] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264; S. Weinberg, Phys. Rev. Letters 27 (1971) 1688; A. Salam and J. Strathdee, Trieste preprint IC/71/145; J. Schecter and Y. Ueda, Phys. Rev. D2 (1970) 736. [2] G. 't Hooft, Phys. Rev. 37B (1971) 195; B.W. Lee, Phys. Rev. D5 (1972)823. [3] J. Godine and A. Hankey, M.I.T. preprint 279 (1972); A. Love, Rutherford preprint RPP/T/11 (1972). [4] S. Weinberg, Phys. Rev. D5 (1972) 1412. [5] S.L. Glashow, J. llipoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [6] H. Georgi and S.L. Glashow, Phys. Rev. Letters 28 (1972) 1494. [7] J. Prentki and B. Zumino, CERN preprint TH. 1504 (1972). [8] S.D. Drell and J.D. Walecka, Ann. of Phys. 28 (1964) 18. [9] R.P. Feynman, Phys. Rev. Letters 23 (1969) 1415; J.D. Bjorken and E.A. Paschos, Phys. Rev. 185 (1969) 1975.
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[10] S. Weinberg, On the mixing angle in renormalizable theories of weak and electromagnetic interactions, M.I.T. preprint (December 1971). [ 11 ] J. Kingsley, Cambridge preprint D.A.M.T.P. 72/1. [12] M. Gourdin, Nucl. Phys. B38 (1972)418. [13] N. Booth, T° Quirk, A. Skemja and W. Williams, NAL proposal 98, RHEL proposal 96. [14] J. Kuti and V.F. Weisskopf, M.I.T. preprint 1971; Phys. Rev. D4 (1971) 3418; P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B28 (1971) 240; Phys. Letters 34B (1971)621.