P I I Y S I C A L R E V I E W D V O L U M E 2 1 , N U M B E R 7 1 A P R I L 1 9 8 0

Heavy-quark production by neutrinos and antineutrinos

David M. Scott and K. Tanaka Department of Physics, The Ohio State University, Columbus, Ohio 43210

(Received 5 September 1979; revised manuscript received 3 December 1979)

The rates for producing t and b quarks in, respectively, neutrino and antineutrino interactions with aucleons are estimated. Experimental quark-parton distribution functions, SU(2) X SU(2) X U(l) gauge group mixing angles, and threshold suppression thrcugh rescaling are used in the calculation. The ratios to total cross sections of b-quark production by ij, R 1, and t-quark production by v, R '; , are, respectively, R = and R 7 = 10- for an incident energy of 200 GeV.

With s e v e r a l high-energy neutrino-beam experiments i n p r o g r e s s a t CERN and Fermi lab , t h e r e is con- s iderable in te res t i n the question of production r a t e s of heavy quarks i n such beams. In th i s study, we e s - t imate the r a t e of t- and b-quark production in, respectively, neutrino and antineutrino interact ions with nucleons. We use the s tandard six-quark model with theoretically predicted mixing angles, a s imple quark-parton model, and a rescal ing approach to include threshold suppression effects. Both r a t e s a r e s m a l l a t p resen t experimental energ ies and become accessible experimentally only f o r incident energ ies above about 500 GeV. Because we include a heavy-quark sea , our resu l t s will exceed the l imi t s suggested by Ell is et al.' a t incident energies above 1000 GeV. On the other hand, our resu l t s a r e s m a l l e r than those obtained by ~hi1lips'-due pr imar i ly t o the difference in mixing angles.

We s t a r t the descript ion of our calculation by noting that f o r an i sosca la r t a rge t , the differential c r o s s sect ions f o r neutrino and antineutrino interact ion a r e given i n t e r m s of the quark-parton model by

where K= G ~ W E / ~ = (1.5 x cmZ) X [ E ( G ~ v ) J . Here G is the weak-interaction constant, M is the nucleon m a s s , E is the energy of the incident neutrino (antineutrino), and u, d , s , etc., a r e quark-distribution func- tions. ri2 is the sum of the s q u a r e s of a row o r column of the Kobayashi-Maskawa weak-current matr ix,3 f o r example rU2 = f ,,' + f,,' + r,:, rd2 = I,,,' + r c d 2 + rtd2, and r,, is the m a t r i x element connecting the up and down quarks. Since th i s mat r ix i s unitary, a l l I?,'= 1, However, we include these t e r m s t o show explicitly where heavy-quark production occurs .

The differential c r o s s sect ions f o r t-quark production f r o m v interact ion and b-quark production f r o m i7 interact ion a r e easi ly excerpted f r o m Eqs. ( la) and ( lb) :

Note that the l a s t t e r m i n each expression gives the r a t e f o r "spectator" production of heavy quarks, This occurs when a n incident neutrino o r antineutrino in te rac t s with the antiquark of a vir tual quark-antiquark pa i r producing a Light antiquark and the spectator heavy quark.

The quark distribution functions (u, d, . . . ) depend, i n general , on both the scaling var iab les x = ~'/2;21u and y = V/E. A s usual, Q' is the modulus of the four-momentum transfer. squared and v is the energy t r a n s f e r i n the laboratory f rame. We have chosen t o use the experimentally determined distribution func- t i o n ~ ~ - ~ f o r the valence and light-sea (up, down, and s t range f lavors) quark distributions which a r e ex- p r e s s e d as functions of x. We wri te the total c r o s s sect ions f o r producing heavy quarks a s

21 - 1771 O 1980 The American Physical Society

1772 D A V I D M . S C O T T A N D K . T A N A K A 21 -

The lower limit on each integration with respect t o x accounts for threshold suppression and will be discussed below.

Fi rs t , we shall write down the expressions for the quark distributions; for the up and down quarks, they can be rewritten i n t e r m s of valence and sea components: u(x) = u,(x) + u,(,~) and d(x) = d,(x) +d,(x). The valence and sea quark distribu- tions a r e , respectively,

The slopes of the total c ros s sections4 a s a func- tion of energy a r e normalized to aU"/E, (GeV) = 0.61 x cmZ and oa/E5- (GeV) = 0.29 x cm2. Then the Fermilab-Harvard-Ohio State- Pennsylvania-Rutgers-Wisconsin (FHOPRW) data5 give G=3.82, 6=3.7, H = 0.39, and y=4.6; the CERN-Dortmund -Heidelberg-Saclay (CDHS) data6 give G = 3.73, 6 = 3.5, H = 0.496, and ~ ~ 6 . 7 ; and the Big European Bubble Chamber (BEBc) data6 give

for x > 0.3 and H = 0.38, y = 4.9. According to the quark-parton model, the total number of valence quarks in a nucleon i s

The value obtained from Eq. (4) with the FHOPRW and CDHS data i s 3.2. We now neglect the contri- bution of the charmed quark sea to this expres- sion to obtain the distribution for each of the light s ea quarks where the assumption of an SU(3)-sym- metric s ea i s used,

The heavy-quark components of the sea (which may come from gluon fusion) a r e included by a s - suming they satisfy the same distribution (6) but occur at some fraction of the light sea quarks. We find that when we se t this fraction to 20% and use the same mixing angles and quark masses a s Phillips, we essentially reproduce the production ra tes he calculatedZ with the gluon-fusion model.

We impose threshold suppression by using the rescaling variable7

where m, i s the mass of the quark being produced. The effect of this rescaling is to reduce the do- main of integration8 for the original scaling vari- able x, a,& x-' 1. This restr ict ion occurs when heavy quarks a r e produced from Light quarks (either valence or sea quarks). However, when

FIG. 1. Ratio of b-quark production c ros s section in 7 N irteractions to total D1V c ros s section. The contri- butions to this ratio from u , c , t , and b quarks in the nucleon a r e also displayed.

heavy quarks a r e produced from other heavy sea quarks, kinematics produces a further inhibiting effect, the production of a heavy spectator such a s i n the gluon-fusion model. This reduces the x domain even f ~ r t h e r . ~ The lower limit now be- comes

FIG. 2. Ratio of t-quark production c ros s section in v N interactions to total v N cross section. The contri- butions to this ratio from d , s , b , and r quarks in the nucleon a r e also displayed.

21 - H E A V Y - Q U A R K P R O D U C T I O N B Y N E U T R I N O S A N D . . . 1773

FIG. 3 . Ratio of b-antiquark production c ros s section in vN interactions to total vN c ros s section. The con- tributions to this ratio from F, F, r, and b quarks in the nucleon a r e also displayed.

To perform the calculations, we used quark masses of m, = 1.87 GeV, m, = 4.75 GeV, and m, = 15.0 GeV and the mixing angles of Fritzschlo: rUb=0.008, rCb=0.16 , r, ,=0.03, r , ,=-0.15, and rtb =0.98. The valence and sea contributions of the CDHS and FHOPRW quark distributions a r e s imi lar , therefore the heavy-quark production ra tes a r e similar. For illustrative purposes, we use the lat ter data. The results shown in Figs. 1 and 2 indicate that the c ros s section for t-quark production in neutrino beams and b-quark produc- tion i n antineutrino beams will be on the order of and of the total c ros s section, respec- tively, a t 200-GeV incident energy. The b-quark production includes contributions from the u, c , t , and b quark seas and the t-quark production in- cludes contributions from the d, s, b, and tqua rk seas , a s well a s those from valence quarks. Fur- thermore, a t energies above 500 GeV for the neu- tr ino beams and 800 GeV for antineutrino beams, the dominant form of heavy-quark production is from the heavy-quark sea even when i t s s ize i s 20% of that of the light-quark sea. The reason for this dominance i s the large magnitude of r,, relative t o the other mixing angles. At lower en-

FIG. 4. Ratio of F-antiquark production c ros s section in N interactions to total T N c ros s section. The con- tributions to this ratio from a, s, b, and 1- quarks in the nucleon a r e also displayed.

ergies, however, threshold suppression causes heavy-quark production to occur primarily from light ones.

Using the same method, we have also calculated the heavy-antiquark production in neutrino- and antineutrino-nucleon interactions. These results a r e presented in Figs. 3 and 4.

In this investigation, we have obtained numeri- ca l est imates for the ra tes of heavy-quark produc- tion in neutrino and antineutrino beams. We have used experimental quark distributions and theo- ret ical est imates of the weak-interaction quark mixing angles found through use of an SU(2) xSU(2) x ~ ( 1 ) gauge group. The results should serve a s a reasonable estimate for experimental planning until more definitive information on quark masses , mixing angles, and the quark-parton model i s available.

The authors a r e grateful to N. W. Reay for vari- ous discussions including the observation of the importance of this problem. One of us (KT) thanks C, Quigg for his hospitality a t Fermilab. This r e - search was supported by the U.S. Department of Energy under Contract No. EY-76-C-02-1545.

'J. Ell is , M . K. Gaillard, D. V. Nanopoulos, and S. Rudaz, Nucl. Phys. w, 285 (1977).

'R. J. N. Phillips, Nucl. Phys. M, 475 (1979); G, 418 (1979). See also J. Smith and C. H. Albright, Phys. Lett. =, .I19 (1979).

3 ~ . Kobayashi and K. Maskawa, Prog. Theor. Phys. $, 652 (1973).

4 ~ . C . Bosetti et a l . , Phys. Lett. E, 273 (1977); B. C . Barish et al. , Phys. Rev. Lett. 2, 1595 (1978); M. Holder, in Proceedings of the Stanford Linear

1774 D A V I D M . S C O T T A N D K . T A N A K A 2 1 -

Accelerator Center Summer Study, 1978 (unpublished). 5 ~ . Benvenuti et n l . , Phys. Rev. Lett. g, 1317 (1979). 6 ~ . Tittel, in Proceedings of the 19th Inten~ational Con- ferelzce on High Energy Phys ics , Tokyo, 1975, edited by S, flomma, M. Kawaguchi, and H. Miyazawa (Phys. Soc. of Japan, Tokyo, 1979).

'R. M. Barnett, Phys. Rev. Lett. 23, 1163 (1976); H. Georgi and H. 11. Politzer, e'bid. ;,'6, 1281 (1976).

his interpretation of rescaling is equivalent to the

second case investigated by C . H. Albright and H. E. Shrock, Phys. Rev. D 16, 575 (1977).

his effect is discussed by J. P. Leveille and T. Weiler, Nucl. Phys. E, 147 (1979).

'OH. Fritzsch, Phys. Lett. s, 317 (1978) and Nucl. Phys. E, 189 (1979). The maximum allowed values of the mixing angles a r e taken. Smaller values Tub ~ i l rtd N 5.6 x rtS a r e obtained in T. Kitazoe and K. Tanaka, Phys. Rev. D 18, 3476 (1978).