LETTERE AL I~UOVO CIM]~I~TO VOL. 32, .'~. 5 3 0 t tobre 1981

Gluon Radiative Effect on the Two-Body Mesic Decays of Heavy Quarks.

S. IwAo

Department o] I~hysics, College o] Liberal Arts , Kanazawa University . Kanazawa 920

(ricevuto il 13 Agosto 1981)

I t is well accepted that hadrons arc composite states of quarks and gluons. From the quark-lepton correspondence the fundamental electroweak interactions proceed through the quark current and the lepton current coupled to the relevant vector fields. The strong interaction is mediated by the coupling between the gluon and the quark in the framework of QCD.

Analysing the gluon contribution in the two-body mesie decay of heavy quarks in terms of QCD, we expect to get some new information on the structure of these interactions.

The QCD effect in the semi-leptonic decay of heavy quarks (1-4) has been investi- gated in analogy to the QED radiative correction to the leptonic decay of the muon (6-s). The two-body mesic decay rates have been studied experimentally for D-meson decays (9,1o).

The purpose of this paper is to develop the proper account of the two-body mesic decay of heavy quarks based on QCD. In what follows the gluou effect will modify the result predicted by the Cabibbo angle and the kinematical factor so as to make them coincide with the experimental data. The effective interaction range of quark and antiquark recombination in momentum space is predicted to be (0.34--0.38)M~, Mo being the mass of the charmed quark. The branching probabilitics for the two-body mesie decay rates on B-meson are estimated in this theory.

(l) ~. SUZUKI: Nucl. Phys. B, 145, 420 (1978). ( ' ) N. CABIBBO and L. 1V[AIANI: Phys. Left..B, 79, 109 (1979); N. CABIBBO, G. CORBO and L. MAIANI: Nucl. Phys. B, 155, 93 (1979). (') A. ALI and E. PIETARINEN: NuCL Phys. B, 154, 519 (1979). (') S. IwAo: Quark fragmentation in the semi-leptonlc decay o] heavy quarks, HPICK-089, preprint (Ju]y 1981). (5) R. :E. BEHRENDS, 1-t. J. FINKELSTEIN and A. SIRLI.~': Phys. Rev., 101, 866 (1956). (s) S. M. BERMAN: Phys. Rev., 112, 267 (1958). (') T. KXNOSHITA and A. SIRLL~: Phys. Rev., 113, 1652 (1959). (D IL J. PLAt'O: Phys. Rev., 119, 1400 (1960). (~) I~. L. :KELLY, C. P . ]~ORNE, M. J . LOSTY, A. I~.ITTENBERO, T. SH1MADA, T. C,. TRIPPE, C. G. WOHL, G. P . YOST, N. BARASH-SCIIMIDT, C. BRICMA.N', C. DIONISI, M. MAZZUCATO, L. MONTANET, S . L. CRAW- FORD, M. ROOli~ and B. ARMSTRONG: Rev. Mod. Phys . , 52, S1 (1980). (10) H. TRILLING: The ~ro~p~rties ol charmed particles, LBL-12283, preprint (February 1981).

145

146 s. iwxo

We shall call the ant iquark bound to the ini t ial heavy quark the spectator quark. The gluon contr ibution to the spectator quark is only impor tant in the final-state interaction.

Two-body mesic decay of heavy quarks to two quarks and one ant iquark me- d ia ted by the weak intermediate boson requires the recombination of the quark and the ant iquark in the final state.

The decay under consideration may be dcscribed by

(1) Q -* q - q' + q" --~ q § ps-meson.

Let us denote the momenta by P , q, 1vl, P2 and p for thc Q, q, q', ~" and the final ps-meson, respectively. The recombination of q' and ~ may bc described by a delta-function, viz.

(2) rar M~/iS(pl--p2),

where rcM c denotes the effective range of interact ion in the momentum space. The differential decay spectra for the two-body mesic decay of a heavy quark with

mass M may be defined by

(3) 1 1

d F d3qd3pld3p2 s 3 3 -- rcMr (Pl--P2)" (2~) 5 2 ' MEaE1E 2

[ 1 l ]2 �9 d3Po ~ ' ( P - - q--Pl- -P2--PQ) IM21 �9 6 ' ( P - - q - - P ' - - P z ) ( I M ~ + IMl[2) + i2:~i ~ 2Eo

Here M o, M 1 and M 2 are the matr ix elements for the bare-quark decay, the interfer- ence term between the bare-quark decay and the gluon vertex correction and the gluon bremsstrahlung, respectively (a), and Po is the gluon momenta. We shall add only the effect arising from the radia t ive correction in the final-state interactions to M o in the first order of the QCD fine-structure constant (see, later discussion). In order to proceed further, we shall introduce an addit ional simplifying assumption for the recombincd state, viz.

(4) Ep = E 1 § E z and P ---- P l + P2,

which will be simplified by assuming E1 = E2 = E~,/2. After a lengthy but s traightforward calculation we find the integrated par t ia l -decay

width in the form

1 (5) ~ F = re § 71 + 72,

where A is given by

(6) A = ]cacB[ ~ G M roMe ~,,

Here c a and c~ are the Cabibbo angle factors (or their extension by KOBAYASHI and

GLUON RADIATIVE EFFECT ON TIIE TWO-BODY MESIC DECAY ETC. 147

MASKAWA (Xl)) associated with the QqW and the q'q"W vertex, respectively, W being the fielcl of the in termediate boson, 2(a, b, c) = a s + b ~ + c ~ - 2ab -- 2be-- 2ca. e~ = m J M , e = re~M, m being the ps-meson mass and x = 2E~/M. The absolute square of the reduced widths ~o, 71 and Y2 may be given by

(7) 7~ = x(x~ + x x a - 4 e ~ ) ( l + ~ ~'(Q~) + ~ ~ Q ~ ) ~

x. = 2E~/M, Q~ and Q,~ arc the square of the sum of four-momenta between the quark q and tha t in the ps-meson and the similar quant i ty relevant to the spectator quark, respectively, a~(Q ~) being the QCD fine-structure constant,

(8) 7 1 =

and

�9 ~a~ 4x(x~+xxq--4e~) G I + G 5 - M2(2G~--G 3 + G 4) +

+ ' ~ ~ ~ 4 e ~ ) + 2 x ~ ] + 4M G~[e xq--x(x~ + xx~--

+ 4 ~ I ~ G ~ [ ~ ~ - 2~x~ + x (~ + xx.-- ~ ) -- 3x ~] +

+ M~Gs(x~ + xx . - - 4e~)[x~ +

2 " , [ + 4 2 ] ~ G 4 eq[ - 4e" + e2x~ + x 2] + M~G~[4C'x~ - (x~ -~ xx~- -4e~) 2]

i

(9)

l ~ e |

Z J dy

2 e ~ l

�9 i l - - y / 2 ) ( 1 - - y + ~ - - e 2) - - ~ 1 - - ( 1 - - y + e ~ - - e 2 ) - - 2 e ~ ( 1 - - y + e ~ ) - -

] , - - ' - - ( 1 - - Y + e ~ + e 2 ) + 1 - - y + s q 2 - - E 2

. [ e 2 y ( l ~ y + e ~ ) + y ( l y ) ( l _ y + e 2 q + e ~ ) 2~2y(1 - y / 2 ) ( 1 - y + e ~ ) 1

8(1 - - Y/2)~ e 2) [ _ e2(y_ 2e~)(1 _ y + e ~ _ e 2) _L (v - 2~i(i-:- V + ~ - -

+ 4 e 2 e ~ ( l _ y + 2 , 2 + e ~ + e 2 ) ] / . eq) -- eq(y -- 2eQ) (1 - - y J

_ _ _ _

Here F R in the summation indices means the Fierz rearrangement (12), which may be performed by the simultaneous interchange m~ ~-~ m r, and of the ps-meson by the corresponding one. However, we assume tha t the mult ipl icat ive factor in front of the

(ix) ~'~I. KOBXYAmlI a n d K..hIAS~:XWX: Prog. Theor. Phys., 49, 652 (1973); L.-L. C. WANO: Quark /favour mixing and its ~)hysical implications, in Proceedings ot the X X International Con/erence on High Energy Physics, Madison, IVis., 1980, ed i t ed b y L. DURAND a n d L. G. 1)ONDROM (Madison, Wis . , 1981), p. 510, a n d re ferences t h e r e i n conta ined . (It) :R. E. MARSHAK, I~,IAZUDDIN and C. P. RYAN : Theory of IVeak Interactions (New York , N. Y. , 1969).

148 s. lWA0

summat ion indices is not affected by this t r ea tmen t . T h e funct ions G I . . . . . G 5 are defined in ref. (a). We have chosen the upper l imi t of the in tegra l in eq. (10) in order to nega te the infra-red divergence. Th is choice is done somewhat arbi t rar i ly . We then assume t h a t t he to ta l infra-red d ivergence cancel out comple te ly by the appropr ia te choice of t he gluon mass in the d ivergen t t e rms in t he funct ion G 1.

I n the numer ica l e s t ima te we have chosen Q~ = Q2 = M,. Unde r th is simplifica- t ion the to ta l r ad ia t ive effect becomes 4a,(M ~) in t he last bracket in eq. (8). The QCD scale pa rame te r is chosen to be A ~ 0.178 GeV as before (4).

The processes s tudied are

(10)

and

(11)

DO-~ K-~x+, r,-r, § K - K +

BO"'~ D'g-, D+K - , 7~+~ -, ~ K -

decays. I n the fo rmer decays t he combined effect ar is ing f rom the sum of 7i and 73 shows good fea tures so as to accomodato wi th the exper imenta l findings. A s imilar t r end has been obta ined for t he l a t t e r decays. I n these s tudies we have inves t iga ted two typ ica l cases i) by choosing the r ad ia t ive correct ion to 7o to be zero and if) by the inclusion of i t as s t a ted above. T h e pred ic ted branching probabi l i t ies for the ~o decay are no t affected by these choices apprec iab ly excep t the th i rd one in eq. (12).

We are now in a posi t ion to discuss t he numer ica l analysis more in detai l . F i r s t of all, wc have chosen the quark masses fol lowing a recent ar t ic le by

MARTIN" (la), v i z . m b = 5.174, m~ = M~ ~ 1.8, m, = 0.518 and m d ~ ~n u ~ r a p ~ 3 - ~

= 0.313 GeV. Choosing the branching probabi l i t ies for the semi- leptonic decay for D O and ~o to be 16.4 and 18.8% (i4), respect ively , we find Ftot(D ~ = 1.17-10 -I~ and / 'tot(B ~ = 1.08.10 -I1 GcV, so t h a t the i r l i fe t imes become T(D ~ = 5.61.10 -13 and T(B ~

6.07" 10 - I ' s, respect ively . Fo r definiteness we g ive the to ta l w id th and l i fe t ime of t he t - qua rk by assuming i ts mass to be 25 GeV and its semi- leptonic branching probabi l i ty to be 18o/o. T h e y become ] " to t (T e) = 8.17" 10 -7 GeV and T(T ~ = 8.06.10 -19 s. The expe r imen ta l va lues of D O and D + l i fe t imes are repor ted to be (3.2-o.7)'+1"~ 10-I3 and ~10.o_,.2j.~+I'~ 10_i3 s, respec t ive ly (15). Recent ly , the ]~EBC-EHS col laborat ion has repor ted the mesurcd proper l i f e t ime of D O and ~o to be (2 .1+0 .1 ) .10 -is and (5 .9 • -is s, respect ively(Ie) .

(1,) A . MARTIN: Phys. Lett. B, 93, 338 (1980); 100, 511 (1981); 103, 51 (1981). (1,) K . CHADWIK, P. GANCI, H . KAOAN, R . KASS, F. LABKOWICZ, A. C. MELISSIONO, S. L. OLSEN, R . POLING, C. ROSEN'FELD, G. RL'CINSKI, E. H . THORNDIKE, J . J . MUELLER, D. POTTER, F. SANNE8, P . SKUBIC, R . STONE, A. BRODY, A. CHEN, 1~[. GOLDBERG, N. HORWITZ, 1~. S. ALAM, S. E. CSORNA, ]~. S. PANVINI, J . S. POUCHER, D. ANDREW9, K . BERKELMAN, R . CABENDA, D. G. CASSEL, J . W . DEWIRE, R . EHRLICH, T. FERGUSON, T. GENTILE, ~L G. D. GILCHRIESE, ]3. GITTELMAN, D. L. HARTILL, D. HERRUF, M. YIERZLINGER, D. L. KREINICK, ~ . S . MISTRY, E. NORDBERG, R . PERCHONOK, l~. PLUI~'KETT, ]~. A. SH]NSKY, ]~. H . SIEMANN, A. SILVERMAN, P. C. STEIN, S. STONE, R . TALMAN, H . G. TRONEMANN, I). WEBER, C. BEDEK, J . HAGGERTY, J . 1~. ]ZEN, C. LONGUEMARE, ~V. A. LOOMIS, W . W . MACKAy, F. 1~[. PIPKIN', J . ]:tOELLF, A. J . SADOFF a n d D. L. BRIDGES: Phys. Rev. Leit., 46,

88 (1981). (11) R . SIDWELL: Study Of neutrino produced short-llved particles in a tagged emut.~ion spectrometer, i n 1981 I N S Symposium on Quark and Lepton Physics, he ld June 25-27, 1981, Tokyo, Japan. (x*) B. ADEVA, 1~. AG~ILAR-BENITEZ, W . ALLISON, P . BAGNAIA, B. BALD0, L. !~ARONE, W . BARTL, A. BERGIER, A. BETTINI, R . BIZZARRI, M. BORATAV, O. BORREANI, ~'. BRUYANT, E. CASTELLI, p . CHECCHIA, P . CHLIAPNIKOV, G. CIAPETTI, G. COOREMANS-BERTRAND, I). CRENNEL, M. CEESTI, F . CRIJNS, H . DIBON, E. DI CAPUA, C. DIONISI, J . DOLBEAU, J . DUMARCHEZ, F. •TINNE, A. FERRAND0, C. FISHER, R . FR(YWIRTH, L. OATIGNON, F . GRARD, F. HARTJES, P . HERQUET, A. IIERV1~, S. HOLM- GREN, J . HRUBEC, P . HUGHES, E . JOHANSSON', J . KESTEMAN, E. KISTENEV, W . KITTEL, D. KUHN, N. KURTZ, P. LADRON DE OUEVARA, P . LECOQ, J . LEMONNE, J . LESCEUX, H . LEUTZ, P . LIPARI, T. MOA, L. MONTANET, (~. XEUHOFER, H . NGUYEN, S. ~'ILSSON, K . ~'~ALER, D. PASCOLI, L. PERUZZO,

GLUON RADIATIVE, EFFECT ON TI lE TW0-BODY MESIC DECAY ETC. 149

According to the t ime-reversa l invar iance of weak in te rac t ion these l i fe t imes should coincide wi th each other. T h e y concluded tha t the ques t ion of the D o l i fe t ime is stil l an open one. I n the opinion of the present au thor the game of the quark masses and the exper imenta l l i fe t imes are t end ing to converge to a un ique answer.

The exper imenta l b ranching probabi l i t ies for the two-body mesic decays of D O are B(K-r . +) = (1 .8 • 10 -2, B(u-,~+) = (5.9--3.2) . 10 -4 and B ( K - K +) ~ (2.0~-0.8). 10 -a. As we a l ready s ta ted, t he to t a l cont r ibut ions of ;,1 ~ ;,2 in all these decays show good signs so as to expla in the above values. However , we h a v e to enhance the QCD fine- s t ruc ture cons tant appear ing in ;'1 and ;,2 by 21.,_25.~+1a'7 ~/1~--o-~3.al "+ 9.,~ for the exp lana t ion of the rat io of the first pai r of da t a by keeping the QCD cons tan t in the QCI)-corrected (bare) ;,o. More or less the s imilar enhancement is r equ i red in order to exp la in t he last pair . F o r the m o m e n t t he expe r imen ta l error is so large t h a t we cannot conclude the necessi ty of the enhancemen t fac tor defini t ively. We find re = 0.34 (0.38) by f i t t ing to the absolute branching probabi l i ty .

In case of Be decay the cont r ibu t ion f rom ;,i ~- ;,2 is large enough re la t ive to Yo and no enhancement fac tor seems to be neCessary. The sign of t he fo rmer shows s imilar features as in the case of D o decay. The branching probabi l i t ies are p red ic ted to be B(D*r, -) ~ 3.55 (3.14). 10 -4, B(D+K -) -- 2.38 (2.32). 10 -5, B(,-:+~ -) ~ 6.75 (2.65).

�9 10 -5 and B(r.*K -) = 8.06 (7.41). 10 -6 for the QCD-correctcd (bare) ;,o. E x c e p t for the th i rd exanlplc two choices of ;,o do no t affect to the final answer. Not ice tha t the cor- responding two values of re compensa te thc change due to the re la t ive magni tude of 70 and ;,1 - - ;'2 in B o decays.

The observa t ion of t he branching probabi l i t ies of two-body mesic decays for ]~o seems to g ive a useful in format ion on the s t ruc ture of the re levant in teract ions .

The nunmrical work has been pe r fo rmed wi th NEC PC-8001.

:P. I)ILETTE, G. PIREDDA, B. I)OLJ'AKOV, •. 1)OPI'LETO.~, P. I)OROPAY, P. PORTII, ~[. I~.EGLER, S. t~EU- CROFT, L. I{()liB, P. ROSSI, J . ]~YBIO, G. SARTORI, l~{. SESSA, A. STER(~IOI~, A. SUBRAMANIAN, S. TA- VERNIER, JD. TOET, M. TOUBOUL, A. TOUCIIARD, C. TRONCON, M. VAN IMMER~EEL, ]J. VENTI~'RA, I ). VILAIN, C. VOLTOLI.~I, J . ~VICKENS, D. Z.a, NELL(), L. ZANELLO, (J. ZHOLOI|OV, P. ZOTTO an d G. ZU- MERLE** Phys . Left. B , 102, 285 (1981),