PHYSICAL REVIEW

LETTERS

VOLUME 44 21 APRIL 1980 NUMBER 16

Dimuon Measurements and the Strange-Quark Sea

Raymond Brock Carnegie -Mellon University, Pittsburgh, Pennsylvania 15213

(Received 17 December 1979)

This paper discusses a method for the extraction of the ratio fs - jxs(x)dx/$xd(x)dx from dimuon rate measurements in neutrino and antineutrino scattering, which takes into account the threshold effects for heavy-quark production. From the y dependence of D scattering to dimuon final states the parameter mc

2 , which appears in the defini­tion of the slow rescaling variable £ = x + mc

2/2EMpy, is estimated. This paper further discusses the crucial uncertainties inherent in the measurement of fs .

PACS numbers: 12.40.Cc, 13.15.+ g

One r e c u r r i n g p rob lem in the par ton in t e rp re t a ­tion of neutr ino sca t t e r ing is our meager knowl­edge of the amount of s t r a n g e - q u a r k sea in the proton. As the sea content a t low Q2 is an input to most quantum-chromodynamics (QCD) ca lcu la ­t ions , l a rge uncer ta in t ies in this quantity a r e bound to have se r ious consequences . I d i scuss in this Le t t e r a method of easi ly accounting for r e sca l i ng effects in the ext rac t ion of the quantity S= jxs(x) dx f rom cha rm product ion by neutr inos and ant ineutr inos a t modera te ene rg ie s where the threshold effects might dominate any QCD co r r ec t ions of the pa r ton d is t r ibut ions .

We a r e motivated by the r ecen t observat ion by the Columbia-Brookhaven National Labora to ry Bubble-Chamber Group1 working a t F e r m i l a b that the excitat ion function for [iTe + events in neutr ino c h a r g e d - c u r r e n t in te rac t ions has a g rad ­

ual energy dependence (see Fig. 1). This i s the f i rs t observat ion of this phenomenon in the bubble chamber where momentum cuts on the leptons a r e not s e v e r e and s e r v e s , then, a s the f i r s t observat ion of the threshold suppress ion in c h a r m e d - p a r t i c l e production by neut r inos . As has long been acknowledged,3 the neutr ino p r o d u c ­tion of c h a r m is a potentially good tool for d e t e r ­mining the s t r a n g e - q u a r k sea because of the Cabibbo-allowed s-c (s-~c in u) t r ans i t ion . I take a s Ansatz for this product ion the modif ica­tion of the naive qua rk -pa r ton model f i r s t de ­sc r ibed by Barnet t 4 within the context of the p o s ­s ible r ight-handed v production of heavy quarks of a few y e a r s ago. Those ideas , a s re levant to c h a r m e d - p a r t i c l e product ion by neutr inos and ant ineut r inos , suggest that the c r o s s sec t ion i s bes t r ep r e sen t ed by4

^ = ^T(x,y,E)[aMsin26d+ts(0cos*ej6

where a=l for v, a = 0 f o r ^ ; dR = T(x,y,E)£d(g)9(l- £), sR = T{x, y, E)£s{g)0(l - 0 and I a s s u m e

xs(x) =xs(x).

( i)

0d r e p r e s e n t s the mixing angle for the d-~c, AS=0 t rans i t ion and 0s is the angle for the AS = 1 t r a n s i ­tion. The var iab le £ is the so -ca l l ed "slow re sca l ing v a r i a b l e " and is the fract ion of momentum of the t a rge t which is c a r r i e d by the s t ruck quark (ei ther down or s t range) . It is defined a s 5

i;=x+mc2/2EMpy. (2)

1027

VOLUME 44, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 21 APRIL 1980

0 50 100 150 200 250

E^(GeV)

FIG. 1. The ratio adu"g+)/cr(ju") as a function of energy in neutrino scattering from an isoscalar t a r ­get. The data come from the Columbia-Brookhaven National Laboratory collaboration (Ref. 1). The curves are the shapes predicted for this quantity using mc

= 1.5 GeV/c2. Curve a was calculated using the Field and Feynman (Ref. 2) normalizations for the parton distributions and corresponds to a branching ratio of c—- M+ of approximately 14%. Curve b was calculated using the same parton distributions with twice the amount of strange-sea quarks as in Ref. 5. Curve b corresponds to a branching ratio of approximately 11%.

This c r o s s sect ion is supp re s sed re la t ive to the r e s c a l e d form (at E — <*>) by the functional depen­dence of the par ton densi t ies on | r a t h e r than A;; by the factor T(p,y,E), which is

Tb9y,E) = l-y+xy/i; (3)

and by the 0(1 - £) function which r educes the x-y phase space to

x<l-mc2/2E][^y (4a)

and

y>mc2/2EMp. (4b)

A prob lem in a l l approaches to the s t r a n g e - s e a m e a s u r e m e n t through a c h a r m signal is that the ext rac t ion of S from the m e a s u r e d quanti t ies i s complicated by the p r e s e n c e of T(x,y,E) and £s(£) r a t h e r thSLnxs(x) a lone.

Apar t f rom whatever fundamental uncer ta in t ies t h e r e may be in the bas ic r e sca l ing approach, the only uncer ta in p a r a m e t e r in the model is m c . I

R\

0 50 100 150 200 250

Ep(GeV)

FIG. 2. Mean value of y as a function of antineutrino energy as defined in Eq. (5) for ^-induced MV" events. The datum point is <y) = 0.57± 0.03 at <£) = 85± 6 GeV for M+M" events from antineutrino scattering in the experiment of Molder et al. (Ref. 6). The curves are (y) calculated using various values of mc : solid curve, mc = 1.5 GeV/c2; dashed curve, mc = 2.0 GeV/ c 2 ; and dash-dotted curve, mc =2.5 GeV/c 2 .

take the approach that mc is s imply a p a r a m e t e r defined so that the maximum value of | i s unity [see Eq. (3.12b) in Albright and Shrock in Ref. 4 ] . This p r e s e r v e s the useful identification of i; a s a momentum fraction and al lows us to t r e a t mc a s a measurab le phenomenological p a r a m e t e r . In the following I a s s u m e that the shape of xs{x) [=xs{x)] can be reasonably taken a s that of Field and Feynman2 (FF) , which i s (1 -x)8. Assuming th is , I will descr ibe a s imple method of obtaining the coefficient of this function (which i s 0.1 in FF) . In o rder to do th is , I use (y) vs E in v s ca t t e r ing to JU + JLT final s t a t e s to give a reasonab le value for mc. v data a r e used because

(y)= fysRdxdy/fsRdxdy (5)

is independent of the normal iza t ion for the s t range quark pa rame t r i za t ion . Figure 2 shows the r e ­sult of the p r e sen t calculat ion of this quantity for var ious values of mc. Also shown a r e the r e s u l t s of Holder et al.Q for JU + JLL" events at the ave rage energy of 85 ±6 GeV. A value for the p a r a m e t e r mc of 1.5 GeV is reasonable and is used in the following.

One can now invest igate the amount of s u p p r e s ­sion that would be exper ienced in the neutr ino and ant ineutr ino production of cha rm. This sup­p re s s ion is r a t h e r s e v e r e ( l a rger values of mc

leading to s t i l l l a r g e r suppress ion) . I demon-

1028

VOLUME 44, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 21 APRIL 1980

rp 0.5 h-

250

E(GeV)

1.0

0.5

250

E(GeV)

Fig. 3. (a) rd and (b) rs , as a function of energy. rd and rs are defined in Eqs. (6a) and (6b) which are normalized to unity at asymptotic energies. The shapes of the individual parton distributions are from Ref. 2, and the normalizations of the distributions are arbitrary when displayed as rd and rs .

s t r a t e this suppress ion by plotting the r a t io s

rs=S~1JsRdxdy (6a)

and

rd=D~1jdRdxdy (6b)

[with D = Jxd(x)dx] in F igs . 3(a) and 3(b). Note that at 50 GeV the down quark is only 75% a c t i ­vated while the s t r ange sea is l e s s than half a c t i ­vated.

In Fig. 1, curve a is the shape of the total c r o s s sect ion for c h a r m e d - q u a r k production u s ­ing F F normal iza t ions . Curve b i s the s a m e quantity with twice the F F s t r a n g e - q u a r k sea . The data a r e uncer ta in enough a t high ene rg ies to accommodate e i ther of these normal i za t ions . Clear ly , th is i s not a useful method of d e t e r m i n ­ing S.

A bet te r method for accompl ishing this is to de te rmine the quantity

FS(E) = fsRdxdy/JdRdxdy (7)

in dimuon product ion by both neutr inos and a n t i -neu t r inos . This can be obtained by r a t e s a t a pa r t i cu l a r energy—the r e sca l ing effects making FS(E) an energy-dependent quantity. One a s s u m e s that the c r o s s sect ions for dimuon product ion via cha rm for neut r inos and ant ineut r inos can be schemat ica l ly r ep re sen t ed by

o( ji jit, v) oc v s in2 0d JdR dx dy

+ s cos26C J sRdx dy , (8a)

o(MM, v) ccscos29cfsdxdy , (8b)

where v, s, and s a r e the semileptonic branching r a t i o s for cha rmed p a r t i c l e s or iginat ing from valence d quarks , s sea quarks , or s sea quarks where dR and sR a r e defined in Eq. (6) (I a s s u m e that sR= sR). By a m e a s u r e m e n t of

R21(E) = o"(uifi)/^(^)9 (9a)

521(E) = a i 7( /x/x)/^(M+) , (9W

and

R{E) = &]{ii+)/ov{ii-) (9c)

and a s imple manipulation of Eqs . (8a) and (8b), the following is obtained:

(v/s)(sm9d/cos6s)2

FS(E)- (10) ' (l/R)(s/s)[R21(E)/R21(E)]- 1 *

In pr inc ip le , this can vary with energy and hence i s not the des i r ed quantity which is

fs = s/D = Jxs(x) dx/fxd(x) dx . (11)

We can, however, sca le FS(E) to asympto t ic en­e r g i e s by cons t ruct ing the function shown in Fig. 4 which is the ra t io of the cu rves in Fig. 3. We then obtain

fs = [rd(E)/rs(E)]Fs(E). (12)

The normal iza t ion of S can then be fixed by some other de terminat ion of D.1

There a r e t h r ee uncer ta in r a t io s in Eq. (10). They a r e (a) v/s, (b) s/s, and (c) tan2# = s i n 2 # d / cos 2 # s . The r a t io s of branching r a t i o s [(a) and (b)] a r e in pr inc ip le a cce s s ib l e by studying c h a r m e d - p a r t i c l e decays i n £ + e ~ exper imen t s and cha rmed-ba ryon production in neutr ino i n t e r ­ac t ions . Evidence f rom D° decays sugges ts that the value for s i n # d / c o s # s may not be the Cabibbo tangent.8 An acceptable value for this quantity is

1029

VOLUME 44, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 21 APRIL 1980

LP

250

E (GeV)

FIG. 4. rd /rs as a function of energy. The curve is the ratio of the curves of Figs. 3(a) and 3(b) used to scale Fs (E) to its asymptotic value as in Eq. (11).

not yet de termined, while many efforts a r e under ­way.9 The range for th is value has been ca lcu­lated within the context of the Kobayashi-Maskawa model with s ix quarks 1 0 to va ry widely from t a n # c , where 6C is the s tandard Cabibbo angle . One approximat ion to Eq. (10) would be to a s s u m e v = s a n d / o r s = s a n d tan2fl = t a n 2 0 c . One could then make only the co r r ec t i on outlined he re for the r e sca l ing effects. However, because of the potentially huge uncer ta in t ies in these r a t i o s , I have ref ra ined from doing so .

I thank P r o f e s s o r Lincoln Wolfenstein for encouraging th is idea and for many valuable d i s ­cuss ions , I a l so thank P r o f e s s o r C. Baltay, Dr . M. Murtagh, and Mr. M. Hibbs for p e r m i s ­sion to quote from unpublished r e s u l t s and d i s ­

cuss ions of the Columbia-Brookhaven National Labora tory exper iment . This work was sup­por ted by the U. S. Depar tment of Energy.

H am grateful to Professor C. Baltay for permission to first show these data in publication. They were previously reported in Proceedings of the Lepton-Photon Symposium, Fermilab, August 1979 (unpub­lished) .

2R. D. Field and R. P. Feynman, Phys. Rev. D L5, 2590 (1977).

3M. Holder^ at., Phys. Lett. 69B, 377 (1977); A. Benvenuti etal., Phys. Rev. Lett. ^ 1 , 1204 (1978); S. J. Barish etal., ''Study of Charmed Quark Produc­tion by Antineutrinos" (to be published).

4R. M. Barnett, Phys. Rev. Lett. 36, 1163 (1976); R. M. Barnett, Phys. Rev. D 14, 70 (1976); E. Derman, Nucl. Phys. B110, 40 (1976); C. Albright and R. Shrock, Phys. Rev. D^U), 575 (1977); V. Barger, T. Gottschalk, and R. J. N. Phillips, Phys. Rev. D 16, 746 (1977); J . Kaplan and F. Martin, Nucl. Phys. 115, 333 (1976).

5Here x ^Q2/2MP v and y = v/Ev, where Q2 is the four-momentum transfer, v is the energy transfer, and Ey is the neutrino energy.

6See Holder etal., Ref. 3. 7I have investigated the effects of asymptotic free­

dom (AF) by repeating the calculation with the Q2 -dependent parton distributions of A. J. Buras and K. J. F. Gaemers [Nucl. Phys. B132, 249 (1978)]. 1 find that AF effects change the curves of Fig. 4 by less than 5%. I have also checked that the results are relatively insensitive to assumptions about the shape of xs(x). By changing the exponent in the sea-quark parametrization from 8 to 6 or from 8 to 10 results in changes to rs of less than 10%.

8G. S. Abrams, Phys. Rev. Lett. 43, 477 (1979). 9L. Wolfenstein, Carnegie-Mellon University Report

No. C0O3066-134, 1979 (unpublished); R. E. Shrock etal., Phys. Rev. Lett. 42, 1589 (1979); V. Barger etal., Phys. Rev. Lett. 42, 1585 (1979); R. E. Shrock and L.-L. Wang, Phys. Rev. Lett. 41, 1692 (1978).

10M. Kobayashi and K. Maskawa, Prog. Theor. Phys. _49, 652 (1973). With the now-standard notation, tan0

: S H C :/(' .CiCoCo "r SyS> 9ib ).

1030