A model for dileptons in neutrino reactions

Nuclear Physics B108 (1976) 483-513 © North-Holland Publishing Company

A MODEL FOR DILEPTONS IN NEUTRINO REACTIONS * L.M. SEHGAL and P.M. ZERWAS

111. Physikalisches lnstitut, Technische Hochschule, Aachen, 14/.Germany Received 12 March 1976 We have constructed a detailed parton model for dileptons originating in the production and decay of "charmed" particles in high-energy neutrino interactions. Distributions are calculated for the primary and secondary lepton, and for the hadronic shower accompanying them. The model is able to account successfully for all prominent features of the dimuon data obtained so far, and yields significant clues to the structure of the charm-changing current underlying this phenomenon.

1. Introduction The reported characteristics of the dimuon events observed in the HarvardPennsylvania-Wisconsin-Fermilab(HPWF) experiment [1 ] appear to establish a prima facie case for the production of a new hadronic quantum number in neutrino interactions [2]**. A plausible interpretation of these events is the sequence of reactions (fig. 1) v/.t + N ~ / ~ - + C +... / L U + + v/z + ....

(1)

where C is a charmed particle of mass 2 - 3 GeV whose production and decay are governed by a AC = AQ selection rule. We present in this paper a model of dilepton production, based on the charm interpretation; that aims at a consistent account of the observed characteristics of these events. The model is sufficiently detailed to predict the distribution of both the primary and secondary lepton and the hadronic shower that accompanies them. An attempt is made to deduce, from the existing data, the structure of the charm-changing interaction underlying the dilepton phenomenon. (A brief description of our model has been published earlier [6].) The model, discussed below, is designed as an asymptotic description of the reac-

* Work supported by the W. German Bundesministerium fiir Forschung und Technologie. ** Evidence for dimuons has also been obtained in the Caltech-Fermilab experiment [3]. Dilepton events have recently been observed in bubble chamber experiments at CERN [4] and Fermilab [5]. 483

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L.M. Sehgal, P.M. Zerwas / A model for dileptons

Hodrons

Fig. 1. Sketch of reaction (1). tion (1) valid at energies where scale invariance can be applied to both inclusive and semi-inclusive cross sections. The language used is that of the quark parton and quark fragmentation models. Certain non-scaling effects associated with the mass of the charmed hadron have been incorporated. Our attempt has been at a simple and transparent description whose consequences can be derived analytically. Because of the approximate scale-invariant character of the model, several of the distributions obtained depend only weakly on the incident neutrino spectrum. These can be immediately compared with existing data that have been obtained from experiments with wide-band neutrino beams. The characteristics of the dimuons are determined, not only by the dynamics of the production and decay process, but also by the structure of the charm-changing interaction. This raises the expectation that a detailed analysis of the dimuon characteristics can delineate the nature of the charmed current and possibly discriminate between various theoretical proposals [7]. Rather than investigate the consequences of each of these schemes, we propose a classification based on the chirality of the charmed current and the charge of the charmed quark. This classification embraces most existing models and provides a natural basis for a phenomenological determination of the charmed current structure. Sects. 2 and 3 describe in detail our model for charm production and charm decay. The procedure for obtaining dimuon distributions is given explicitly in sect. 4. In sect. 5, we discuss the influence of experimental constraints, such as angle and energy acceptance, on the dimuon distributions, and show how these can be incorporated into the model. Sect. 6 contains the results of the model for various dimuon characteristics, and a comparison with the data of the HPWF experiment. Explicit formulas are given, so that future measurements involving new experimental conditions can easily be compared with the predictions of the model. In sect. 7, we state our conclusions concerning the probable nature of the charmed current, and indicate how further experiments could help to resolve the uncertainties and ambiguities that still exist. We list in ref. [8] some theoretical papers on the subject of dimuons that con-

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tain alternative approaches to the subject or general discussions. Ref. [9] lists some of the experimental reports from which we have drawn information about the nature of the experimental set-up and various aspects o f the data.

2. Charm p r o d u c t i o n The basic point o f view adopted in this paper is that the production o f the charmed hadron C in the first step o f reaction (1) takes place in a deep inelastic collision (large v, large Q2) to which the ideas o f scaling and the parton model may reasonably be applied. In support of this view, we quote the following experimental facts. (i) The dimuon events possess a high inelasticity: the average visible energy o f the events is (Evis) -~ 115 GeV, while that of the primary muon ~ - ) is ( E _ ) -~ 60 GeV. On average, therefore, at least 55 GeV energy is transferred from the neutrino beam to the hadron target. (ii) The v distribution o f the primary muon is fairly broad, the average value being (v) ~ 0.1. Since v = Q2/2ME v and since E v is o f the order o f 100 GeV, the average Q2 is of order 20 GeV 2, which is again quite large. From the fact that only a few events of the type/a-/a- have been observed, we conclude that most of the dimuon (/~-/a +) events represent the production o f a single charmed particle involving the action of a charm-changing (AC = 1) current. The charm-changing cross section may be written as do(AC = I) _ G2MEv

dxdy

~

F(x,y),

(1)

where the form of F(x,y) is determined by the structure of the charmed current. In table 1, we list six possible forms for the charmed current and the corresponding expressions for F(x, y) in the parton model. These currents are constructed from a charmed quark of charge Qc = ~ or - ~-, and are characterized by a definite chirality Table 1 Parton model expressions for F(x, y) in various models Xc

Class

F(x, y)/x

+]

-

L

u +d

+~

-

L

2s

c-3,#(1 - 3,s)n

2

+ -~

+

R

(u + dr) (1

c-7/~(1 - 3"S)h

+ ~-

+

R

2s (1 - y)2

ff~#(1 + 7s)c

- ]1

-

R

(~-+ d) (1

pq't~(I - "Ys)c

-½

+

L

~-+d-

Charmed current

Qc

c-7#(1 + "rs)n ~-~#(1 + -rs)X

- y)2

- y)2

Qc = charge of charmed quark, xc = chirality of charmed current; u, d, s ... = patton densities in proton. Analogous functions F(x, y) in ~ reactions are obtained by interchanging (u, d, s) ~ Ca, d, s3.

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486

×c = --1 (left-handed) or +1 (right-handed). Any current involving only quarks of the above two charges may be written as a linear combination of these currents. The classification according to Qc and Xc has the advantage that the resulting y distributions have a direct relation to the product Qcxc. For the cases where Qcxc is negative (class L) the y distribution is constant, while for QcXc positive (class R) the distribution is (1 - y)2 . Our interest is in the inclusive distribution of the charmed hadron C produced in reaction (1). An important clue is afforded by certain characteristics of the secondary muon ~÷) that arises in the decay of C. The energy of this muon is found to be large ((E+) -~ 10 GeV), yet its momentum transverse to the v - #- plane is low (~<1 GeV). This suggests strongly that the parent hadron C is produced with high energy (E c >>MC) in the approximate direction of the momentum transfer vector q. This is precisely the hallmark of a current fragment in a deep inelastic collision *. Such fragments are expected to manifest themselves (in the lab) as particles of high energy emanating in the forward direction (relative to q) with hmited transverse momenta. In the parton model charmed current fragments have a simple interpretation [10]: they are the bremsstralalung products of the charmed quark c produced in the collision of the incident neutrino with a quark constituent of the nucleon. Assuming this emission process to be scale-invariant one is led to the conclusion that the distribution of the charmed hadron C (integrated over its transverse momentum) should depend only on the Lorentz invariant ratio z = h'p/q'p where h and p are the fourmomenta of C and of the nulceon, and q is the momentum transfer. In the lab frame, this ratio reduces to

z = EC/V,

(2)

E c being the energy of the charmed particle and v the energy transfer. Thus the inclusive cross section for C-production, in the current fragmentation region, should have the form da(C) _ G2MEv - F(x, y ) D(z)

dxdydz

n

(3)

D(z) being the so-called fragmentation function. To complete our model for charm production, we must specify the form of the function D(z). We assume, for definiteness, that C is a charmed meson (belonging, for instance, to the family F, D, F*, D* etc.). The theoretical prejudice is that, asymptotically, D(z) behaves as z -1 for z -+ 0 and vanishes as a power of (1 - z) for z ~ 1. These expectations are borne out by measurements of rr-(zr+) multiplicity in g(v) interactions, which determine the analogous function D~(z) for pion production [11]. As can be seen in fig. 2, the data can be fairly represented by the simple

A more quantitative criterion for a current fragment is discussed below.

L.M. Sehgal, P.M. Zerwas / A model for dileptons

487

I !

5.(

I I

t,.O I

Drtlz}

3 .G

2.C

\

\

\

1.0

0.1 0.2 0.3 O.t, 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 2. Fragmentation function Drr(z) obtained from ~r- multiplicity in ~ + N ~ or- + .... (Barish et al., ref. [11]).

form z 1 (1 -- z) for values o f z not too close to the origin *. We are thus encouraged to adopt for D(z) the same simple form

D(z) = K z -1 (1 - z)

(4)

recognizing that some deviations from this form are to be expected in the small z region. Note that the integral f D ( z ) d z is a measure of the charmed particle multiplicity. Therefore, the modification of the form (4) at small z will be such as to make this integral finite. The possibility that charmed baryons are also produced appreciably among the current fragments cannot be excluded, but experience with baryon production vis-a-vis meson production in ordinary reactionssuggests otherwise. Quark counting rules would predict that the function D(z) in this case would behave as z - 1 (1 - z ) 3 instead of the form (4). We have based our considerations on the assumption of charmed meson production only, but our results are not qualitatively changed if charmed baryon production is included. The cross-section formula (3) is applicable to an energy domain in which all masses may be neglected. At energies of present interest, constraints due to nonzero masses and thresholds are of some significance. We have incorporated into the model one such constraint which represents a "minimal" scale-breaking correction. * Further comparisons of the formula z -1 (1 - z) with data on pion production are made in the first paper of ref. [12].

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This constraint is the requirement E C/> M C, which in terms of the scaling variables y and z, implies a multiplicative factor OO, z - A ) ,

a = MclE ~ .

(5)

Because our model for D(z) emphasizes low z values, this factor effectively dampens the cross section at smally. An independent constraint comes from the threshold requirement y(1 - x ) > 6,6 = (M C + M)2/2MEz,. For practical purposes, the effect of this constraint also is to' cut off small y values, an effect that is to a large extent already contained in the factor (5). Taking account of (5), the inclusive cross-section formula for the production of the charmed hadron C reads da(C) _ G2ME

dx dy dz

zr

v F(x,y) D(z) O(yz - A).

(6)

We refer to the consequences of this formula as "scaling limit" results, despite the fact that a minimal scale-breaking correction is included. Some important implications of eq. (6) may be noted immediately. (i) The y-distribution of the dimuon events will have the form do

( log (y/A) -- 1 + A / y ,

- - ~

(I _ y ) 2 (log ( y / A ) -

dy

t

(L) ; 1 + A/y),

(7)

(R).

As can be seen from fig. 3a, these distributions show a depletion at small ),-values and are significantly different from the naive expectations of constant for L and

B 0

SCALING LIMIT

SCALING LIMIT

.° :o.o:

I

Eo

b:o.o

1.0

do" dy

tJik\

0.5-

" 0.2

o'.~

t;.o

y

Jl o'.8

1:o

0.2

o.,

0.6

0.8

1~

z

Fig. 3. (a) y dependence of semiinclusive C production in L and R type theories. (b) z dependence of C production.

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L.M. Sehgal, P.M. Zerwas / A model for dileptons

(1 _ y ) 2 for R. The average values o f y are

rlog(1/A)

(Y)L = ½ [_log (l/A)

+ O(A log A),

(8)

R = r/ og

q_l+ o(A log ,,),

t.~(l/a)

which for a typical choice/x = s~ are (Y)L = 0.63, (Y)R = 0.43. (ii) Shown in fig. (3b) is the z dependence of the cross section for C production, given by

do

(D(z)(1 - A/z) O(z- A),

(L) ;

--co ~

dz

(9)

~ D(z) (1

A/z) 3 0(z - A ) ,

(R).

This figure illustrates the effect of the constraint O(yz - A) in eq. (6), in the absence of which the z dependence is just D(z). (iii) The energy dependence of the 2# cross section relative to the 1/a cross seciion can be extracted from eq. (6). The result is gCt/-/d+)

o(lu)

flog (i/A) - 2 + O(A log A),

(L) ;

[log (l/A)

(R).

(10)

(2(

~17 d- O(A log A ) .

As seen from the plot in fig. 4a the dimuon to single muon ratio is predicted to be

I

SCALING LIMIT Mr=2GeV

25

t

2.5-

/

HPWF CUI"S

/ 2.0

,,R

/

/L

20-

1.5-

s

~R

t.s 1,,..

Q:

1.0-

0.5"

I.O

', so

bib

loo

,so Ev ( G e V )

2bo

2~o ~-

b}

z/

a)

0

~o

t 'oo

,~o

E v (GeV)

ioo

iso

---

Fig. 4. (a) Predicted energy dependence of a(2~)/o(l#) in the scaling limit, normalized to unity at E v = 100 GeV. (b) Same ratio for the conditions of the HPWF experiment.

490

L.M. Sehgal, P.M.. Zerwas / A model for dileptons

an increasing function of energy in the energy domain of current interest, and rises by a factor ~ 2 from E u = 50 GeV to E v = 150 GeV. We have also shown in fig. (4b) the energy dependence of o(2/a)/o(l~) appropriate to the HPWF experiment, with the acceptance o f this experiment taken into account (see sect. 5). The dependence here is even steeper. In view of this strong energy dependence, the HPWF measurement of o(2t0/o(1/a) -~ 1% at 100 GeV, can be reconciled with the upper limit of 10 -3 obtained at the Serpukhov [13] energies o f E v -~ 20 GeV. (iv) The energy spectrum of the charmed particle implied by eq. (6) is (e -= E c / E v ) do

( (e-1 - 1 - log e) 0(e - A ) ,

(L) ;

de

l (e -1 -- ~ + 3 log e -

(R) .

3

3e + ~ e 2) 0(e - A ) ,

(11)

The e -1 behaviour at small e reflects the z -1 behaviour of D(z). (A more realistic choice of D(z) may be expected to temper this behaviour somewhat.) Finally, in order to define the domain of applicability of the model, we give a criterion by which "current fragments" may be separated from particles of other types. A reasonable (but not sufficient or unique) condition is to require that in a frame in which the momentum transfer vector q has components (qo = 0, q), the charmed particle be produced in the forward hemisphere relative to q. This translates into the Lorentz invariant requirement (12)

h'q
In the lab frame, this reduces to the following constraint on the velocity of the charmed particle: /)

/3c >

+Q2

(13)

As an illustration, parameters typical of the HPWF experiment are ~, = 50 GeV, Q2 = 20 GeV 2. Assuming M c -~ 2 GeV, the condition (13) implies E c ~> 5 GeV. (As will be seen in sect. 3, the corresponding constraint on the energy of the secondary muon is (E+) ~> 2 GeV.) Most of the dimuon events in the HPWF data do conform to these criteria, thus inducing confidence in the applicability of the model to this experiment. In other experimental situations, eq. (13) can be used as a guideline for defining events that fall within the purview of the model. The model is not well adapted to describe the low-energy fragments (E C -~Mc, E+ <~ 1 GeV), which represent small values of z, at which our choice of D(z) is unrealistic. This completes our elaboration of the charm production process, and we proceed to a discussion o f the decay.

3. Charm decay There are two general possibilities for the decay of C leading to a final/a*: (i) two-

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491

body decay C -~ la÷ + Pu' (ii) n-body decay C ~ #+ + uu + hadrons. (A third possibility is the sequential decay C ~ L+UL, L ÷ -+/.t+VjL , where L ÷ is a heavy lepton with a mass ~
(14)

where k is the/a ÷ m o m e n t u m and F the width of the decay C ~/~÷ + .... The complete dimuon distribution, therefore, is do(/.t-U +) _ do(C) B(C -+/a +) 1 dr cb:dydzd3k/ko dxdydz F d3k/ko '

(15)

B(C ~ p+) denoting the branching ratio of C into a given muonic decay mode. (A sum over each type of particle C, as well as a sum over all muonic channels will be understood in eq. (15).) In the case of the two-body decay, the spectrum of the p÷ has an unambiguous form, given by 1

dF

P d3k/ko

-

1

~ [(h - k) 2]

(two-body),

(16)

n

h being the four-momentum of the decaying particle. No unique answer exists, however, for the n-body decay. We have adopted the view that the essential features in this case can be abstracted from the four-fermi interaction c~q+p+

+v u

(17)

describing the transformation of the charmed constituent of C into an uncharmed quark q. Taking the mass of c to be close to that of the meson and neglecting other masses, we obtain 24 k . h 1

2k'h]

(n-body; L) ;

dI"

P d3k/ko

(18)

I

nbody .)

According to our prescription, the/a + spectrum in the n-body case coincides with the electron spectrum in/~- decay, provided we use a Michel parameter ~- for class R and zero for class L models. Since the decaying particle, in our model, is assumed to travel approximately in the plane defined by the incident v u and the o u t g o i n g / i , it is convenient to resolve the ~u+ m o m e n t u m into components perpendicular (k±) and parallel (kll) to

L.M. Sehgal, P.M. Zerwas / A model for dileptons

492

.L~n-body

decay

I.--

z I.U

I.H

~ N\ N

0.4

0.8

1.2

1.6

k . (GeVlc)

Fig. 5. (a) Kinematics of C decay. (b) k£ distribution of #+. this plane (fig. 5). We may therefore write

d3k/ko = dk 0 dk± d e ,

(19)

where ¢ is the angle between kit and h and k 0 = x/~12 + k 2. To obtain the distribution in k±, it is simplest to go to the rest frame of the decaying particle. Integrating the expression for ( l / F ) dF/d3k/k 0 over ¢ and k 0, we obtain 0(1 - ~±),

(two-body) ;

1 dr' = H(~±) = / 6(1 - ~2) _ 4(1 - ~-3), P d~'±

/

[3(1 - ~2) _ 4(1

~3),

(n-body; L) ; (n-body; R ) ,

(20)

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493

where ~'± -- 2 k z / M C. A comparison with the data is shown in fig. 5. The data are clearly compatible with an n-body decay, or possibly a combination of n-body and two-body decays. The observed cut-off in k± cannot be directly converted into a mass for the decaying particle, for two reasons. First, the charmed hadron C will inevitably possess some transverse momentum associated with the production process. In the case of ordinary current fragments such as pions and kaons, this transverse momentum is known to be of order 3 0 0 - 4 0 0 MeV. It is conceivable that the corresponding momentum for charmed particles is higher (as appears to be the case, for instance, in q, production in strong interactions [14]). The observed ks distribution of the/1 ÷ is thus a convolution of the kz distribution generated by the decay and that associated with the decaying particle. An additional correction to the k± distribution in fig. 5 can arise from the non-zero mass of the hadronic decay products [12]. (A typical hadronic system might consist of two or three n's and/or K's.) This correction will tend to lower the end-point of the kz spectrum, and thus will work in the opposite direction to the effect coming from transverse momentum of the decaying particle. While a precise estimate of the two corrections is not possible, it seems reasonable to interpret the data in fig. 5 as indicating a charmed particle mass between 2 and 3 GeV. The curves L and R in fig. 5 are consequences of our specific four-fermi model for the semileptonic decay of C. Their general shape, however, is similar to what one would obtain from a different prescription such as a three-body decay analogous to K -+ rr I p [ 12]. The difference between the L and R curves shown in this figure has probably no detailed significance, and plays a negligible role as far as the dimuon characteristics are concerned. Finally, we ask for the distribution of the/l + longitudinal momentum kit. In the relativistic limit E c >>Mc, we have ktl >> ks, and this distribution becomes related to the transverse momentum distribution given in eq. (20). Defining -- k,,/L c

u+/E c ,

(21)

the result is * 1 dF

1 dI'[

=H(~').

(22)

Combining eq. (22) with eq. (6), we obtain the complete dimuon differential cross section do

dx dy dz d~

G2ME _

lr

u F(x, y ) O(z) B H ( f ) 0 (vz - A ) .

* To derive this, it is useful to note that ~"~ (ko + kll)/Mlrest frame of C •

(23)

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4. Calculation o f d i m u o n distributions

Eq. (23) is the central formula of this paper. It permits the calculation of any distribution which depends on the variables x, y, z, ~'. As an example, a variable a = f(x, y, z, ~') will have a distribution given by 1

1

1

l

do =fdxfdyfd fdf d x d y d z d~" a ( a - / ( x , y , z, ¢)) 0 0 0 0

do

= , d x j d y j dz 0 0 0 dxdydzd~" D f ( x , y , z , ¢ ) / a f l

.

(24)

a=f(x,y,z,f)

In sect. 6, we will derive a number of such distributions and compare them with the data. Here, we summarize how experimentally measured quantities are related to the scaling variables that appear in the dimuon formula. The quantities that are measured directly are the lab energies of the two muons (E+, E ) and their angles (0+, 0 ) relative to the incident beam. In addition, the hadronic energy E H is measured in a certain fraction of the events. The energies E+ and E _ are related to x , y , z and ~"by E _ / E v = (1 - y ) ,

(25)

E+/E v = y z f .

The expressions for E H depend on whether the decay is a two-body one or an n-body one. In the former case, the hadrons in the final state are all produced at the vertex at which the particle C is created. In this case, E H / E v = y(1

z),

(two-body).

(26a)

In the n-body case, a part of the hadronic energy is generated in the decay of C. We have estimated this by appealing to the four-fermi process c -+ q +/l ÷ + v u, which suggests that in the decay C -+/1 + + uu + hadrons, the energy difference (E C - E+) will on average be shared equally between the hadrons and the neutrino. Accordingly, we have for the full hadronic energy the approximate expression E H / E v =y(1 -- z) + } y z (1 -- ~'),

(n-body).

(26b)

Note that because our model favours low values o f z, the bulk o f the hadronic energy comes from the first shower. Thus our approximation for the energy produced in the decay is fairly innocuous. The angle 0 has a simple relation to the variables x a n d y : 1

M

xy

2 sin 2 ]0_ -" Ev 1 - y

(27)

On the other hand, 0+ can be expressed in terms of the scaling variables only if we permit ourselves a special approximation, namely, that the direction of/l ÷ coincides with that of the momentum transfer vector q. Even within our interpretation of C

L.M. Sehgal, P.M. Zerwas / A model for dileptons

495

as a current fragment, this is an extreme assumption, since, as explained in sect. 3, a certain transverse momentum is to be expected. Granting that this assumption may have some validity on the average, however, we find 2 sin 2 ½ 0+

M x(1 - y )

Ev

(28)

Y

It may be useful to note that in the approximation in which eq. (28) is true, the variables x and y can be deduced from measurements of 0+, 0_ and E _ , without any knowledge o f the incident neutrino energy. The relation is E x = 2 ~ - sin -~ 0_ (sin ~- 0+ + sin ~ 0 _ ) , sin ~1 0_ l 1 s i n ~ 0 _ + s i n ~ 0+

Y=

(29)

(It would be interesting to know from experiment what the distributions of these approximate x a n d y variables look like.) Besides the directly measured quantities E+, E , E H, 0+ and 0_, there are a number o f derived variables whose distributions are of interest. Among these are the variables v+ and v_, defined as E+ 2 v+ = 2 ~ - s i n ~ 0_+.

(30)

These are related to the scaling variables by

V_ = x y ,

V+ = XZ~'(1 - - y ) .

(31)

An easily accessible kinematical variable is the invariant mass of the two muons

Muu, given by 2ME v

(32)

- xz~.

Finally, another variable of interest is Wmin, the minimum invariant mass recoiling against the primary muon. It is given by WZmm= 2114(1 - xy) (E H + E+) + M 2 - 2ME_xy ,

(33)

and in terms of the scaling variables, it has the expression Wm

In

j'y [1 - x - z (1 - ~') (1 - xy)] ,

2MEv I,y[1 - x - ~z (1 - ~') (1 - xy)] ,

(two-body) ; (n-body) •

(34)

(Note that the scaling variable expressions for v+, Wmin and Muu are approximate in the same sense as eq. (28).)

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Many o f the results to be obtained in sect. 6 require no explicit knowledge o f the parton densities u, d, s etc. Where such knowledge is needed, we have used the parametrisation o f Wilson [15], derived from fits to Gargamelle and SLAC data. This parametrisation reads

u(x) + d(x) = 2 (qV(x) + q C ( x ) ) , ~-(x) + d(x) = 2s(x) = 2~-(x) = 2 q C ( x ) , 2qV(x ) = 4.7 (1 - x ) 3 + 3.1 (1

1.35 -X) 4 + ~ -

(1 - x ) 5 ,

0.32 (1 - x) 7 2qc(X) = - 7 -

(35)

The procedure outlined in this section yields distributions which we refer to as "scaling limit" results. These results have a weak dependence on the parameters M C and E v, and in most cases involve only the ratio A = Mc/E v. The values we use are M C = 2 GeV, E v = 100 GeV. F o r comparison of our model with the data o f the HPWF experiment, it is necessary to take account of biases introduced b y the experimental set-up. We proceed to discuss these effects in sect. 5.

5. Effects of experimental acceptance The dimuon distributions measured in the HPWF experiment (and other similar set-ups) are subject to distortions, arising from the limited acceptance o f the detector. There are two types of acceptance problems involved. (a) Angle acceptance. The detector can receive only muons whose angle relative to the forward direction is, on average, not greater than 00 = 225 mr. This constraint on 0+ and 0_ translates into a constraint on the variables x a n d y , as can be seen from eqs. (27) and (28), and may be expressed as

0(8

x ( 1f- Y ) ) o ( ~ - ~ xy ) ,

8=xEv ~0~.

(36)

While this is, in general, a complicated constraint, its principal effect is to cut away events correspgnding to small as well as large y values. We have simulated this effect by incorporating into our calculation an effective angle acceptance function e(y) * (b) Energy acceptance. Another important bias arises from the requirement that both muons penetrate a certain minimum thickness o f iron in order to be recorded in the magnetic spectrometer. In the case of the HPWF experiment, the thickness involved is approximately 1.25 m, and its effect is to remove muons with energy • The function chosen had a smooth fall-off from one at y = 0.7 to zero at y -- 0.9. Its effect is to cut-off large y values. The corresponding cut at small y is largely superseded by the factor O(yz- A) already present in the "scaling limit" formula eq. (6).

L.M. Sehgal, P.M. Zerwas / A model for dileptons

497

lower than about 4 GeV *. This cut is particularly severe for the secondary muon p*, since, in our model, the energy spectrum of the p* tends to be soft. We have taken account of this effect by including in the model a multiplicative factor

0 (yz~ - Emin

E-~-v ) '

Emin = 4 GeV.

(37)

The results of sect. 6 labelled "HPWF cuts" are based on the inclusion of angle and energy acceptance constraints in our dimuon formula eq. (23). It is clear that constraints applicable to any other experimental set-up can be incorporated into the model by multiplying this equation by the appropriate acceptance function.

6. Results and comparison with data

6.1. TheE+ - E asymmetry One of the most interesting features of the dimuon data is a pronounced asymmetry between the energies of the two muons, E being larger than E÷ in the majority of events. One of the first challenges to our model is to reproduce this asymmetry quantitatively. We have calculated the distribution of p = E + / E within our model. Noting that p = yzf/(1 - y ) , this distribution may be obtained from the general formula (24). The result, in the limit A = 0, is

1 f dy[1-a+alogp]g(y), p/(1 + p) __~

dp

f

dy

(two-body),

+ 2 a log (PO0 + a ( - ~- - Coo02 + ~1 (po03

,

(n-body, L) ;

p/(1 +p)

1 1 +p)

(n-body, R ) ,

(38)

where a = (1 - y ) / y , and gfy') = 1 for L, (1 _ y ) 2 for R. These distributions are plotted in fig. 6a. (The two-body and n-body results are very similar and have not been shown separately.) Inclusion of the factor O(vz - A), A 4: O, makes only a

* This conclusion was reached after consulting our experimental colleagues E. Radermacher and H. Reithler, whom we wish to thank.

L.M. Sehgal, P.M. Zerwas / A model for dileptons

498

SCALING L/HIT

I0

1

8

a)

6"

Lu ~k L

2 0

0

~2

o:.

oJs

de

,.'o

,.'2

E+ /E_ I I

i li~ 15 ! i

Ev. IO0 GeV HPWF CUTS .....

R

b)

\ - ~ _ ~ b

o'.2

oi.

es

E,/E"

..... ae

I.o

.

....... -,

L2

I.,

~--

Fig. 6. (a) Predicted distribution of E+/E_in the scaling limit. (b) Same distribution, including HPWF acceptance, compared with data.

slight correction, the two-body results being modified to (~)corr = (~)zx_0 - f

dYg(Y) I 1 - a (Y) + °~l°g P(1£ Y)1 "

(39)

l - I,/o

The correction is of relative order A/p, and affects only the very small p region. In order to confront the HPWF data, we have calculated the p distribution, including the acceptance cuts described in sect. 5, The results are shown in fig. 6b. The agreement is remarkably good. It is clear that both the L and R cases are viable. The L curve appears to reproduce better the tail of events with p > 1, but these may well contain a substantial number of events due to b- contamination of the p beam.

499

L.M. Sehgal, P.M. Zerwas / A model for dileptons

300

1

250-

SCALING LIMIT

12-

Mc = 2 GeV

SCALING LIMIT Nc = 2 OeV

10-

200-

a)

b)

150-

i p

W V

/

I 00-

//I/

/// //

i

£.0

/L

/N÷ 6vu.I ~-

s

//¢ /

50"

216o

260

"

-

E. (OeV)

lbO

2oo

E v (OeV)

Fig. 7. Average values of (a) E_, (b) E+, as functions of E v, in the scaling limit. A related question concerns the average muon energies ( E ) and (E+). In the scaling limit, we obtain the results shown in fig. 7a, b, and fig. 9a. A close analytic approximation to these results is = t 1 (log 1//` - ~)/(log 1//` - 2) + 0 ( / , log/,) < E )IE~

t 3(logl//,

- i37 ~ ) / ( l o g 1/A -- ~2) + 0 ( / ` log A)

3 (log 1//,

2) 1 + O ( / , l o g / , )

(L), (R)

(L), (40)

(E+)IE v =

(log 1//, - ~ )

1 + O(A log A)

[ ~ (log 1/A -- s) + O(A log A)

~: >/<&>=IN ~ (log 1//, - ~) + o(/, log/,)

(R), (L),

(R).

T o obtain results that can be compared to the HPWF measurements, we have calculated the same quantities, including the experimental acceptance. (Recall that the energy acceptance is particularly important for E+.) These results are shown in fig. 8(a, b) and fig. 9(b). The general agreement with the data is quite satisfactory. The cases L and R are, however, not clearly resolved. It may be noted that for E v ~ 30 GeV, (an energy appropriate to the Fermilab

500

L.M. Sehgal, P.M. Zerwas / A mode/for dileptons

HPWF

CU TS

t.

HPWF

•

CUTS

eL

250.

12 ,I

•

l ,°

i!R

* ..." " I

200.

I

*

8

/ ~ Zz~/ " "

a)

t~

/,¢. /

,,."

A V

,R

÷

-~ 150

L

tO0

,L 50-

0

o

io

,~o

Ibo

2'oo

2~o

~o

Ev (GeV)

Ibo

I~o

E ~ (GeV)

250

2bo =

Fig. 8. Average values of (a) E+, (b) E_, appropriate to the HPWF experiment. Crosses show data points. (Ev values were obtained assuming E u ~ Evis + (E+)).

28-

/

/R

28.

I //

21,.

z /,

l

SCALING LIMIT Me = 2 GeV

/I

20

21.-

HPWF

l

CUTS

÷

2o.

/

a)

b)

/

/

/~ 16.

sR s

12 i

v

" 8

~

f

I* /

,¢ 0

L

8.

0

=

I00

2'00 E,(Oev)

0

*

/,/

4

/

÷

*

ibo

2bo E~ ( GeV )

Fig. 9. Variation of IE)/ with Eu, (a) in scaling limit, (b) with HPWF cuts included.

L.M. Sehgal. P.M. Zerwas / A model for dileptons

501

bubble chamber experiment [5]) the scaling limit curves in figs. 7a, 7b and 9a predict, for the L case, the values (E+) -~ 2 GeV, (E_) -~ 10 GeV, (E_)/(E+) -~ 5. While the relevance of the model at such energies is debatable, it is interesting that these numbers are in accord with the parameters of the dilepton events in ref. [5].

6.2. The distribution Qf EH/E+ and E /E H The ratio ~ = EH/E+ has within our model the expression (1 - z)/zf,

(two-body),

[(1 - z ) + ~ z (1 - f)]/zf,

(n-body),

,~ =

(41)

and its distribution should be a test of our mechanism for charm decay. The scaling limit result for the n-body case is shown in fig. 10a, where we have suppressed the small difference between L and R. (Formulas for the ~ distribution are given in the appendix.) The results with cuts included are shown in fig. 10b, together with the data (based on 17 events). There is a qualitative agreement. Some discrepancy can be noticed at the larger values of ~. This is just the region most sensitive to the small z behaviour of D(z). With a less singular choice of D(z), the discrepancy at large ~ is found to decrease. We have examined, similarly, the distribution of the ratio ~ = E / E H , related to the scaling variables by 1- y

(two-body) "

y ( 1 - z) '

77=

(42)

1-y y[1 -z

(n-body).

+ ~-z (1 - ~')]'

SCALING

LIMIT

25:

a)

s

o o

k

i

~ ~N/E,

HPWF

i

k --

I'o

I

CUTS

E~.IOOGeV

b)

z s i ~ s

0,~-" , 0

2

EH/E+

i

I

~

6 L

Fig. 10. Distribution o f the ratio ~ = EH/E +, (a) in scaling limit, (b) including HPWF cuts.

L.M. Sehgal, P.M. Zerwas / A model for dileptons

502

SCALING

LIMIT

HPWF

I

T

Q)

IO

CUTS

E~ = tO0 GeV

tO t f l

b)

8

6Or. m[

\\\

Q:

,~

\

i,...

Q: 2. %..

o

0

.~

i

i

E_/E"

lb

~

0

2

~

L

s

e

E_/E H

._

lo

Fig. 11. Distribution o f the ratio n = E_/E H, (a) in scaling limit, (b) including HPWF cuts.

The scaling limit results (see appendix) are shown in fig. 11 a, and those with HPWF cuts in fig. 1 lb *. (We have shown again the n-body case.) Once again, the data are in agreement with the model, but no clear discrimination between the L and R cases emerges. * The curve " R " shown in fig. l l a was misdrawn in ref. [6].

HPWF

SPECTRUM +CUTS

- - L

t

I0

. . ~ -

- - - R

Lu Lu

E_ (GeV)

=

Fig. 12. Distribution o f p - energy E _ .

L.M.Sehgal,P.M.Zerwas/ A modelfordileptons I' Cut-off

HPWF SPECTRUM

20-

+ CUTS

15z

503

---- R

10"~

uJ

5

0 5 10 15 20 25 30 35 /.0 45 50 E. (GeV) Fig. 13. D i s t r i b u t i o n o f tz ÷ e n e r g y E+.

HPWF

SPECTRUM +CUTS

10 .....

L R

1 u~

3=

5

00

~

,

~0

~80

.

120 EH

160

(GeV)

?00

-~.-

Fig. 14. D i s t r i b u t i o n o f h a d r o n e n e r g y E H.

6.3. The spectrum orE_, E+ and E H The model makes predictions for the distribution of the ratios E_/Ev, E+/E, and EH/E v in the scaling limit (see appendix). By folding in the neutrino spectrum and cuts appropriate to the HPWF experiment, we obtain the energy distributions shown in figs. 1 2 - 1 4 . The data are well-reproduced in all cases. In the case o f the E_ distribution, the L case appears to give a better fit to the data, and is the first evidence for a charmed current belonging to the L type models. However, the uncertainties in-

L.M. Sehgal, P.M. Zerwas / A model for dileptons

504

herent in the neutrino spectrum do not permit a firm conclusion on this point.

6.4. Distribution of dimuon mass Muu The model enables us to calculate the distribution of the "reduced" dimuon mass

Muu _ _

mug -=-2 ~

v

= x/xz~",

(43)

in the scaling limit. The result, (given in the appendix) is peaked at small values of muu and decreases almost exponentially with increasing muw For comparison with the HPWF data, we have assumed a mean neutrino energy of 100 GeV, and folded in the acceptance cuts. These yields the Muu distribution shown in fig. (15), which is in excellent accord with the data. (We have shown results for the case where the x dependence is (u + d); the alternative case of a sea distribution (~- + d) gives a similar curve, with the peak slightly shifted to lower values). It is important to stress that the Mu# distribution is strongly affected by the energy cut E+ > 4 GeV, the position of the peak being closely correlated with the value of the cut-off. The fact that the model reproduces the peak in the observed distribution, as well as the general shape, must be considered a success. It is clear, that the Muu distribution does not help much to discriminate between the various models for the charmed current structure. Ev = 100 GeV

15

~

HPWF CUTS L; (u+d)

----- R;(u+dl

10"

I P-

uJ

5-

M~ (GeV) Fig. 15. Distribution o f dimuon mass MUU.

L.M. Sehgal, P.M. Zerwas / A model for dileptons

505

6.5. D i s t r i b u t i o n in Wmin

In the scaling limit, we can calculate the distribution of the dimensionless variable =- W m i n / N / s ,

(44)

S = 2ME v .

Following our general procedure, the result is shown in fig. 16a. (The relevant formula is given in the appendix.) Observe that the distribution peaks at a value -~ 0.35 in the R case, and at co ~ 0.75 in the L case. Computing the Wmin distribution appropriate to the HPWF experiment (with E v -~ 100 GeV), we obtain the results shown in fig. 16b. Note that the distinction between the L and R cases is no longer so sharp (principally because of the cut E+ > 4 GeV). Further, the theoretical curves are found to differ little between models of the type (u + d) and (h- + d). The data are in good agreement with the theoretical expectations *. 6.6. T h e d i s t r i b u t i o n o f v+

Fig. 17 shows the predictions of our model for the v+ distribution expected in the HPWF experiment. (The scaling limit formula is given in the appendix.) The L and R cases are practically indistinguishable, and are not shown separately. The data are in fair agreement with the model, with the "u + d" case being slightly preferred to the "~ + d". * Note from eq. (34), that since x, z, and ~"tend to be small (on average), w 2 is approximately equal to y. Thus the distribution of w is closely related to the y distribution.

SCALING

L/NIT

N¢=2GeV,

- - L

- - L

....

I ::)

R

o)

I /sl

3.

II/1

R

....

g,

o:

Ey=IOOGeV

~X

~~" /0 tu

b)

:% /

Q:

5

I

/

O 0

.

0.2

.

0.~

.

.

0.6

Wm~n/k~ ~

O.O =

"

~ 1.0

0

,

2

b

12

W,,~in (GeV)

[""]

lk ------m-

Fig. 16. Distribution of Wmin, (a) in scaling limit, (b) including HPWF acceptance.

2b

L.M. Sehgal, P.M. Zerwas / A model for dileptons

506

~o-

- -

(u+ d)

....

(~÷~)

20.

tu tu

II

1o.

o

at

o

/12

Fig. 17. Distribution of o+.

6. 7. The distribution o f Yvis and Yvis We turn n o w to variables that can, potentially, be of i m p o r t a n c e in distinguishing b e t w e e n the L and R classes of models. These variables are E H + E+ Yvis -

Evi s

EH ,

Yvis = Evis '

(45)

and t h e y are related to the scaling variables b y 1 - z (1 - ~')

(two-body)

1 - ~1 z (1 - ~')

(n-body),

Y 1 - y z (1 - ~') Yvis

(46)

Yl-~yz(1-~') 1- z

Y 1 -yz(1-

~)

(two-body) (47)

Yvis =

1 - ~-z(1 + ~')

(n-body).

Y 1--~-z(1-~') Because of the p r o p o r t i o n a l i t y to y , these variables reflect strongly the y distributions, and hence are sensitive to the distinction b e t w e e n L and R. However, the acceptance cuts affect these distributions strongly, and t e n d to blur the L / R distinction.

507

L.M. Sehgal, P.M. Zerwas / A model for dileptons HPWF

CUTS

HPWF

E~ - tO0 GeV

t 1o 'o"

dyvi"

CUTS

E,~= I00 GeV

a)

8

b)

R,, "

~

t

61

i I1

't//

g-

•

,

O2

0,~

1

i

i

0.6

00

tO

Yvi=- E~d'E+

Evi$

ogl 0

,

a2

O.g

i

0.6

0.8

= -Evis

Fig. 18. Distribution of (a) Yvis and (b) Yvis" Predictions compared with HPWF data. (Data include correction for geometrical losses.) Shaded events are those with E+ > E _ .

The scaling limit predictions are given in the appendix. The predictions, including the HPWF acceptance, are shown in figs. 18a and 18b, along with the data. The data are almost equivocal with respect to the L and R cases. In fig. 18b, however, the data in the region Yvis > 0.6 seem to be difficult to explain on the basis of an R type model. (It may be pointed out here that if the charmed current is a mixture of L and R types, the predicted distributions will be weighted averages of the L and R distributions.)

6.8. Distribution of o_ We consider, finally, the distribution of the variable u (= xy) whose determination requires measurement of the primary lepton only (see eq. (30)). The scaling limit results (see appendix) are shown in fig. 19a for four types of models, characterized by L or R and by x dependences of the type (u + d) or (~ + d). With the inclusion of the HPWF cuts, these distributions are modified to those shown in fig. 19b. The influence of the cuts is particularly noticeable in the "u + d" type models. The data in fig. 19b are ambiguous with respect to the (u + d) and (~ + d) cases. In order to test two currently popular models of the charmed current, we have also calculated the v distribution in the GIM [16] and DGG [17] models. As is seen from fig. 19c, both models are, at present, viable as far as the v_ distribution is concerned.

~l

SCALING

LIMIT

I

2~"

20,

',,\d/

16,

12,

8'

u+d

0 0.1

0

0.2

0.3

E v = I 0 0 GeV -- -u+d 20-

1

NPWF

CUTS

~.....

R

b)

16.

Lu Lu

8" •

i

0 0

nl

O.2 v_

E~. = 100 GeV HPWF l

CUTS

c)

GIM

20

DGG 16

~J

t2

#

0.2

O.I

0.3

v_

Fig. 19. (a) ~ distributions in scaling limit, for various types of models. (b) Modified v_ distributions, taking account o f HPWF acceptance, compared with data. (c) Distributions expected in the GIM and DGG models, under HPWF conditions, compared with data.

L.M. Sehgal, P.M. Zerwas / A model for dileptons

509

7. Conclusions

The systematic agreement between the data and the calculated distributions of our model provides compelling support to the hypothesis that the dimuon phenomenon is a manifestation of new hadron production in neutrino reactions. The quantitive success of the model in reproducing various dimuon characteristics indicates that our parton description of charm production and decay has at least a modicum o f truth. On the basis of our calculations and the comparison with data, we are able to make the following statements: (i) The transverse momentum characteristics of the secondary muon indicate that the parent charmed particle has a mass 2 - 3 GeV, and that its probable decay modes are multibody decays. (ii) Present data do not permit an unequivocal statement about the L or R nature of the charmed current, although the E distribution and the Yvis distribution show a slight preference for the L type models. (Note, in this connection, that the GIM model is L-type, while the DGG model is, to a good approximation, R type.) (iii) The v distribution of the primary muon is better described by a superposition of valence and sea distributions (u + d and ~- + d, respectively) than by either of these alone. Both the GIM and DGG models are compatible with the data. (iv) The possibility that the charmed current is not chiral, but a superposition of L and R currents, cannot be ruled out. Similarly, the possibility that more than one kind o f charm is being excited cannot be excluded, as long as the masses of the charmed particles involved are not much higher than 3 GeV. (v) The HPWF measurement of o(2/2)/o(1/2) -~ 1% is, within our model, an underestimate of the true dimuon rate, because o f the limited energy acceptance of the detector. The true rate could be higher by a factor of two, although we do not consider the model to be quantitatively reliable in the kinematical domain which is excluded by the cuts. (vi) The o(2/2)/o(1/2) ratio should be an increasing function of energy in the energy domain of current interest. We believe it is possible to reconcile the upper limit on dimuons obtained at Serpukhov [13] with the measured ratios at the higher energies. A rise by a factor of 2 is expected in the HPWF experiment between E v = 50 and E v = 150 GeV. (vii) All our results can be transcribed to dimuons produced by antineutrinos, by interchanging the/2÷ and/2- labels. The L and R labels in the various figures remain u n c h a n g e d . In cases where a distribution depends on the parton densities (such as the o and Mint distributions), predictions for ~ are obtained from the p predictions by interchanging (u + d) ~ (~ + d). While some dimuon events have been observed with v-, the present statistics are too low to draw serious conclusions. Measurements with ~ beams would serve as an important check of the consistency of the model, and would help to elucidate further the nature of the charm-changing interaction.

510

L.M. Sehgal,P.M.Zerwas/ A modelfor dileptons

While preparing this account we were informed of a parallel effort by Derman and Llewellyn Smith, applying a patton model to dimuons with special attention to threshold effects. A part of this work has appeared in two recent preprints by Derman [18]. We thank C.H. Llewellyn Smith for a correspondence in this connection. We are indebted to Professors H. Faissner and R. Rodenberg and all our colleagues for encouragement and support. Finally our thanks to Irmgard L. Zerwas for assistance in the computation.

Appendix We collect in this appendix a number of formulas for dimuon distributions, alluded to in sect. 6. Several of the results are given in the form of integrals which may be computed numerically. In some cases, analytic simplification is possibe, but has not been done to avoid lengthy expressions. We use the notation F(x,y) = xQ(x)gfy), with 1

g(Y) = { ( 1 - y)2} f°r ( L ) t y p e m°dels •

A.1. Distribution of ~ = EH/E+ (two-body) ,

docc

+~

(A.1)

1

d~

_1

f dzg(~)D(z).(fO) ( 1 _ ~ ) ,

+

zo

(n-body),

with z 0 = max (A, 1/(1 + ~)), ~0 = z-1(2 - z)/(2~ + 1).

A.2. Distribution of ~ = E

do =

d,

/E H

1 1 (1 - y f dyg(v) \ ~ - ) (1+ an)/(1 +~)

y l - y ( 1 +~7)'

(two-body), (i.2)

+ ~ d y dz g(y) D(z) H(~o) l~zY O(yz_ A) O(fo) O(1- fO) , (n-body).

with ~'0

2

=- 1

(xy)

+2_ 1 z

yr/

.

L.M. Sehgal, P.M. Zerwas / A model for dileptons

511

A.3. Distribution o r e _ = E / E v, e+ = E+/E v, e h = E H / E v

A 1 + -1--e -

do cc[In l ~ e

,

(L),

(A.3) - /[ e _2~Fn 1 -Xe de-e

de+

!-: eh

l,

(., (A.4)

yz

1

+ / d y g ( y ) y ~ ~ eh) ,

(two-body),

d~c~. deh

(A.S) [fdydzgQvVz)D(z)H(fo)O(vz_A)O(fo)O(l_fo)

X.4. Distribution o f m2 1

dm 2 ~g

'

(n-body).

= M2uu/(2MEv) 1

m ## 2

\ xz I "

mZ,*u,/x

(A.6)

A.5. Distribution o f co2 = W 2 i n / ( ~ E v )

~g(Y) rdxx0(x)f dyy(l_ x y ) fJd z

D(z) z O(yz -

A) 0(~-0) 0(1 -- ~'0)'

with ~2 _ y(1 - x)

~o = 1 + yz(1 - x y ) do

(two-body) ;

'

(x

(A.7)

dco2

r~xQ(x~f d~y ~ f

~ ~z) .(~o~O~z- ~O(~o~O0- ~o~,

vith 2(co 2 - y ( 1 ~0 = 1 + y z ( 1 - x y ) A. 6. Distribution o f o+

x)) '

(n-body)

512

L.M. SehgaL P.M. Zerwas / A model for dileptons

with V+ ~0 - x z ( 1 - y )

(A.8)

I

A. 7. Distribution o f Yvi s do

1

(1-ly~s)2fdYdz~)

dYvis

y

Y D(z) H(fO) O(yz - A) 0(~'0).0(1 - ~'0)' z

with Yvis yz(1 - Yvis) ' Y -

1 --

~0 =

(two-body),

2(y - Yvis)

(n-body).

1 -

yz(1 - Y v i J '

(A.9)

A.8. Distribution o f Yvis

~s2sidy dz g(y) D(z) -1T- - z 0(~'0) 0(1 - ~'0) O(yz - A),

with

( ~I l(~ ~s)

~'0 = 1 1 do

-z

,

(two-body), (A.10)

dYes 1

+y(1 -z))H(~o)O(~o)O(1 - ~o)'O(yz- A)

fdydzg(y)D(z)l(1

(1 + Yvis)2 with 2 ~'o - z ( 1 + Y ~ s )

[1

Yvis -

7 -

~z(1 -

~

Y~s)] -

(n-body) '

A. 9. Distribution o f v_ x0

do ~

do_ with

J

dx Q(x) g

0

x0:'nl' ~1

(~)[,og k - ( ")1 1 - ~

,

(A.11)

L.M. Sehgal, P.M. Zerwas / A model for dileptons

513

References [1] [2] [3] [4] [5] [6] [7]

[8]

[9]

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