Upper bounds on the top quark mass from Bs-B s mixing

Volume 162B, number 4,5,6

PHYSICS LETTERS

14 November 1985

UPPER BOUNDS ON THE TOP QUARK MASS FROM Bs-Bs MIXING I.I. B I G I

Institut ff~r Theoretische Physik E, RWTH Aachen, D-5100 Aachen, Fed. Rep. Germany Received 10 July 1985

When one succeeds in obtaining experimental limits on Bs-Bs mixing, then one can place rather conservative bounds on the top quark mass. For example, non-maximal mixing of Bs mesons would imply re(top)~< 90 GeV. Such connections between Am(Bs) and re(top) exist strictly only within the standard model with three families, yet will very likely hold also in a four-family ansatz.

1. Introduction. The UA1 Collaboration has recently reported evidence for top quarks with a mass m t = 40 + 10 GeV [1 ]. Nevertheless it appears somewhat premature to conclude that top has been discovered and its mass established. Independent o f these Findings there is considerable circumstantial evidence for the existence o f top quarks: specific topless models are ruled out by the observed pattern in B decays [2] ; secondly jets produced in e+e - annihilation that are initiated by b quarks seem to exhibit a f o r w a r d - b a c k w a r d asymmetry as predicted for a fermion with weak isospin [3] I w = 1/2, Iw,3 = - 1 / 2 . However even if there is only limited doubt on the existence o f top quarks, the actual value o f their mass is a crucial parameter. A lower bound is obtained from PETRA experiments m t > 22.5 GeV.

(1)

Theoretical considerations have been invoked to derive also upper limits: (i) Heavy t o p quarks would generate significant changes in the p parameter that measures the strength o f neutral to charged current interactions [4] :

p ~-- 1 + 3m2GF/87r2x/~.

(2)

From the world average Pexp = 1.01 -+ 0.02,

(ii) Beg et al. have made the interesting observation that an upper bound on m t can be obtained from the requirement o f a non-trivial Higgs sector. They

rind [51 m t ~< 168 GeV

(5)

Virtual top quark exchanges contribute to AmK, eK, e', K L ~ / i f / a - and K + -~ rr+vV. Yet the theoretical uncertainties inherent in these calculations do not at present allow us to get meaningful bounds on m t.

2. B O-frO mixing and m(top). Top exchanges have a much stronger and more direct impact on B " - B 0 mixing: since m b >> AQCD one can calculate Am(B) via the quark box diagram in a reliable way; the top contribution will then dominate the result. Besides m t (and m c for AI'B) two other parameters enter the computation in a prominent way: (a) the KM mixing angles U(tq); (b) hadronic matrix elements of four fermion operators whose size is commonly parametrized in terms o f B B X IfB 12. [a) The K M mixing angles. No direct information exists on the KM angles U(tq), q = b, s, d. Yet employing the standard model with just three families allows to place significant constraints on them * 1.

(3)

I U(tb)l "" 0.997 - 0.999 "~ 1,

(4)

*1 Similar numbers can be found in ref. [6].

(6a)

one deduces m t ~ 430 GeV.

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

383

Volume 162B, number 4,5,6 IU(ts)l = I U(bc)l = 0 . 0 5 -+ 0.01,

(6b)

IU(td)l = I U(bu)l < 0.008.

(6c)

Comparing (6b) with (6c) shows that

BR(B + --* r+vr) "" 1.3 X 10-'4(fB/200 MeV) 2.

(8)

Theoretical estimates o f f a using different approaches do not show much variation any more: results based on QCD sum rules read [8] fBs = 210 -+ 30 MeV,

(9a)

(9b)

(10a)

in bag models [10] while (10b)

in static potential models [11 ]. Some numerical Monte Carlo studies suggest B B "-0.7 [12], yet at present this should be taken at most as an indication. We will use

384

2

(12)

Putting everything together (including radiative QCD corrections) we find with m t = 35 GeV as reference point x s = Am(Bs)/F(Bs) "" (0.8-3.5) × G ( m t ) / G ( 3 5 ). (13) The observable r s = F(B s -~ ~+X)/r(B s -~ ~-X), can be expressed in terms of Xs: rs = x2/(2 + x2),

(14)

where we have already used that Ys = AP(Bs)/2r(Bs) will be very small 0's < 0.1). Strictly speaking, "maximal mixing", i.e. r = 1 will thus be attained only for x 2 ~ 0-. Such a notion is not very useful and instead we will use "near-maximal" mixing r0. 9 = 0.9.

(15)

Experimentally it is almost as accessible as maximal mixing, yet it provides more meaningful restrictions. For (16a)

implies

We will use (9a) for our subsequent analysis. Estimates for B a show much more variation; e.g.

BBs ~ 0.4--1.

2

x t = m t/M~v.

r ~ 0.9,

whereas potential models yield somewhat smaller numbers [9]

BBs = 1,

G(mt) = x t [~ + 9/4(1 - xt) - 3[2(1 - xt) 2 ]

(7)

with Bq - (bq-). Furthermore (6c) leads only to an upper limit on Am(Bd); the CP violation observed in K L decays implies I U(bu)l > 0 in the standard model, yet the resulting lower limit on I U(bu)[ and thus on I U(td)l and Am(Bd) is so loose [7] that we will not discuss Bd - B d mixing any further. One should also note that to a good accuracy I U(ts)l "" I U(bc)l, as already stated in (6b). Thus the ratio x s = Am(Bs) [ F(Bs) which appears in the observables (see below) is very stable against variations in B meson lifetimes. (b) Hadronic matrix elements. The gravest uncertainties enter into the computation of the hadronic matrix elements of the AB = 2 transition operator. The decay constant fB could in principle be measured in the decay B + -~ r+v~. Yet the expected branching ratio is presumably too small to be observable:

BBs "~ 0.5,

(c) Results on B s - B s mixing. The dependence on

m t enters via the GIM factor

- ~ [xt3/(1 - x t ) 3 ] l o g x t ,

Am(Bd) ,~ Am(Bs),

fBs "" 175 MeV.

14 November 1985

PHYSICS LETTERS

(11)

x 2 ~< 18.

(16b)

Condition (16b) is met for Bs mesons if m t ~ 9 0 GeV.

(17)

Turning the argument around one can say: if one can rule out near-maximal mixing experimentally, then the interesting bound (17) on m t follows. If on the other hand r s = 0.9 were observed then one concludes from (13), (16b) m t ~ 40 GeV.

(18)

Similarly one finds: r s ~<0.7 [0.5],

(19a)

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PHYSICS LETTERS

Volume 162B, number 4,5,6

2 2 the b o x diagram that contains both x H = mH/M~v; internal t and H quarks shows a more complicated mass dependence [14]

0.9"

G(xt, XH) = xtx H {(x t - XH) -1 [~ + 3/2(I -- XH) 0.5'

-- 3/4(1 -- x t ) 2 ] log x H + (Xt ,a, XH )

0.3

- 3/4(1 - xt)(1 - X H ) ).

2'o

s'o

~o

"t tG.vJ

Fig. 1. curve I: upper bound on m t as a function of the upper bound on rs; curve II: lower bound on m t as a function ofr s. implies m t <~60 [48] GeV.

(20)

holds then to a good accuracy. These constraints lose their rigour when another quark-(lepton) family is introduced. Thus there exists the algebraic possibility that 0 ) I U(ts)l "~ I U(cb)l and/or (ii) the top contribution to Am(Bs) is largely cancelled by the contribution of a fourth quark H with electric charge 2/3. Both o f these possibilities appear rather unlikely: ad (i): as long as the various mixing angles are small it follows even in a four-family ansatz [13] that U(ts) ~ - U * ( c b ) which implies (20). ad (ii): since nothing is a priori known about the mass o f the quark H, mH, or its KM angles, one has to rely on educated guesses. In this spirit we will assume m H ~ 2 5 0 - 7 0 0 GeV.

Varying m t between 90 and 150 GeV and m H between 250 and 700 GeV one finds G(xt) ~ 0 . 8 8 - 2 ,

G(XH) ~ 4 . 3 - 2 2 . 6 ,

G(xt, XH) "~ --(6.2--11.6). (19b)

These relations are summarized in fig. 1: Curve I gives the upper bound on m t as a function o f the upper bound on rs; curve II gives the lower limit on m t as a function o f r s. (d) Impact of a fourth family. The connection between r s and m t which was derived above rested on the three-family ansatz which put severe constraints on the KM angles U(tq). In particular I U(ts)l ~- I U(cb)l,

(22)

(21)

To obtain larger quark masses in the standard model one had to invoke Yukawa couplings o f order one. The b o x diagram with two internal H quarks contains the GIM factor o f eq. (12) with x t replaced by

(23)

The extra contributions due to the presence of H quarks, G(XH) and G(xt, XH) , can reduce Am(Bs) by (partially) cancelling the t o p contribution - thus vitiating (17) - if some relations hold between U(ts), U(Hs) and U(Hb) [15] . 2 :

1 ,),2 cos 251 ~ ~'202 cos(252 - 53), 1"-6 and/or 3' cos 251 ~ ro cos (51 + ~2 - 53),

(24)

with U(ts) ~- 7e i8 l ,

U(ns) "~ (rei82 ,

U(Hb) m re i83.

Although there is no hard evidence against such relations, it seems unlikely that U(Hq) would be that large. As an example to illustrate that point: generalizing the Wolfenstein ansatz o f the KM matrix to four families as done in ref. [17] one gets ~,~--02 "~0.05,

o -.~ - ~ivt~4 e " 10 - 3 ,

~ '~- - 0 3c " 0 . 0 1 . (25)

To conclude: although the connection between r s and m t as expressed in (16), (17) could in principle be broken if there are more than three families, it will very probably continue to hold.

3. On measuring B s - B s mixing. The sensitivity required to rule out or find near-maximal B s - B s mixing experimentally should be attainable in the near future, t2 For a similar analysis see ref. [16]. 385

Volume 162B, number 4,5,6

PHYSICS LETTERS

both in e + e - annihilation and at p ~ colliders. Unless one studies B s produced just above their threshold as could be done at CESR - one has to deal with the complication that considerably more B u and B d mesons are produced along with the B s mesons. Yet, as already stated, the standard model allows for very small B d - B d mixing; upper limits have thus to rest solely on B s mixing and the sensitivity obtained from a given data sample is reduced. To sketch just one example: B 0 - B ° mixing would show up in e + e - annihilation at PETRA/PEP energies via a smaller than naively expected f o r w a r d - b a c k w a r d asymmetry of b o t t o m jets [18] AbB(observed) = (1 + ?-)-lAbB(stand.-mod.),

(26)

where ?-denotes a weighted average o f B d and Bs mixing. For r d = 0 one f'mds

?-=2Vpsrs/(1 +rsX1

+ I0,

(27)

with Ps describing the Bs abundance relative to the B u abundance and V = r(Bs)/r(Bu) "" r(Bd)/r(Bu). So far, b y averaging over all measurements, one has found agreement with the naive expectation to within 20%:

A~B(Observed)/AbB(stand.-mod.) =

1.00 + 0.20. (28)

If, as seems feasible, this accuracy could be improved to the 10% level then a bound ?-<0.13,

(29)

could be attained at the 90% CL and one could conclude r s <~ 0.9 if Ps > 0 . 1 4 [ ( 1 + V)/V].

(30)

One typically estimates Ps ~ ~ - ½ , which is obtained from a comparison o f lr and K multiplicites, and V 0 . 8 - 1 ; these values satisfy the constraint (30). If what is unlikely, yet not inconceivable - V ~ 0.5 then near-maximal B s mixing could be ruled out b y a 10% accuracy i n A ~ d if Ps ~ 0.42. -

4. Summary. Upper

386

bounds on B s mixing will al-

14 November 1985

low us to place interesting upper bounds on rot, n a m e ly m t ~< 90 GeV. This result which was obtained in the standard model with three families will very likely hold also in a four-family ansatz. The limit on m t is rather conservative, yet not completely failureproof: one can in principle entertain the idea that B B < 0.2 for some obscure reason; however no calculation is known that could yield a rationale for such low values. In this respect the situation for B mesons is quite different from that for kaons. I am grateful for the prodding b y an experimental colleague from Berkeley at the PEP Hilum Workshop that initiated this work. I want to thank H. Paar for organizing such a fine meeting and A. Sanda, L. Reinders and H. Rubinstein for useful discussions.

References [1 ] G. Arnison et al., Phys. Lett. 147B (1984) 493. [2] S. Stone, Weak decays of heavy quarks, in: Proe. 1983 Lepton-photon Symp. (Cornell University, Ithaca), eds. D.G. Cassel and D.L. Kreinick. [3] W. Bartel et al., Phys. Lett. 146B (1984) 437. [4] W.M. Veltman, Nucl. Phys. B123 (1977) 89. [5] M.A.B. Beg et al., Phys. Rev. Lett. 52 (1984) 883. [6] E.A. Paschos, invited talk XXth Rencontre de Mofiond, Heavy quarks, flavottr mixing and CPviolation (1985), preprint DO-TH 85/9. [7] See for example: A. Buras et al., Nucl. Phys. B245 (1984) 369. [8] LJ. Reinders et al., CERN preprint TH 4079/84, Phys. Rep., to be published. [9] H. Krasemann, Phys. Lett. 96B (1980) 397. [10] I.I. Bigi and A.I. Sanda, Phys. Rev. D29 (1984) 1393. [11] F. Wagner, preprint MPI-PAE/PTh 89/83. [12] A. Soni, private communication. [13] See for example: I.I. Bigi, Z. Phys. C27 (1985) 303; E.A. Paschos, preprint DO-TH 85/9. [14] T. Inami, C.S. Lim, Prog. Theor. Phys. 65 (1981) 297. [15] I.I. Bigi, Z. Phys. C27 (1985) 303. [16] U. T~ke et al., preprint DO-TH 84/26 (1984); M. Gronau and J. Sehechter, SLACopUB-345 (1984); X.-G. He and S. Pakvasa, preprint UG-511-553-85 (1985). [17 ] U. Baryer et al., preprint MAD/PH/150(I 984). [18] I.I. Bigi, Phys. l.ett. 155B (1985)125.