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Constraints on technicolor with scalars from b→Xcτν̄ and b→Xsγ
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Nuclear Physics B 561 Ž1999. 3–16 www.elsevier.nlrlocaternpe
™
Constraints on technicolor with scalars from b b X sg
™ X tn and
Zhaohua Xiong a , Hesheng Chen b, Liang Lu b
c
b
a CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, People’s Republic of China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, People’s Republic of China
Received 16 March 1999; accepted 11 August 1999
Abstract
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The O Ž a s . corrections to the inclusive decay of b X ctn and b X sg are calculated in technicolor with scalars model, and the latest experimental results from ALEPH and CLEO are used in determining the Ž s , h. parameter space. We find that in contrast with the enhancement of the branching ratio of b X ctn relative to that of the standard model is very small, the b X sg decay width normalized by the standard model result can be reduced by as much as y50% when the ALEPH experimental value is used. In particular, the CLEO measurement which corresponds to the y33% reduction gives a more stringent constraint than B y B mixing. The branching fraction B Ž b X sg . computed in the model in part of its parameter space with sizable deviation from the standard model prediction can be observable in the forthcoming few years with the upgraded CLEO detector and B-factories at SLAC and KEK. q 1999 Elsevier Science B.V. All rights reserved.
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PACS: 12.60.Nz; 12.38.Bx; 13.25.Hw Keywords: Technicolor with scalars; Decay b
™ X tn ; Decay b ™ X g ; QCD corrections c
s
1. Introduction Technicolor—a strong interaction of fermions and gauge bosons at the scale L TC f 1 TeV—is a scenario for the dynamical breakdown of electroweak symmetry to electromagnetism w1,2x. Based on the similar phenomenon of chiral symmetry breakdown in QCD, technicolor is explicitly defined and completely natural. To account for the mass of quarks, leptons, and Goldstone ‘‘technipions’’ in such a scheme, technicolor, ordinary color, and flavor symmetry are embedded in a large gauge group, called extended technicolor ŽETC. w3,4x. Because of the conflict between constraints on flavor-changing neutral currents ŽFCNC. and the magnitude of ETC-generated quark, lepton and technipion masses, classical technicolor was superseded by a ‘‘walking’’ technicolor and ‘‘multiscale technicolor’’ w5–12x. The incapability to explain the top quark’s large 0550-3213r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 5 1 5 - 5
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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mass without afoul of either cherished notions of naturalness or experiments from the r parameter and the Z bb decay rate by ETC w13,14x led to the topcolor-assisted technicolor w15–19x and the technicolor with scalars model w20–24x. In the standard model ŽSM. of electroweak interactions decays of B mesons with a t lepton in the final state constitute a significant fraction of all semileptonic B decays. These decays are interesting as they can probe certain form factors that are inaccessible in decay modes with light leptons, because they give effects of order m2l w25x. A collective theoretical effect has led to the determination of b semileptonic decays up to order a s for arbitrary b and c masses with massless leptons w26–30x and the perturbative QCD correction to b ctnt w31–33x. Great progress has been made in determining the branching ratio B Ž b X sg . in both theories and experiments in past two years. Instead of leading-order ŽLO. calculation w34–39x, the branching ratio has been estimated at next-leading-order ŽNLO. level w40–46x. The inclusive branching ratio Ž2.32 " 0.57 " 0.35. = 10y4 extracted four years ago from CLEO measurement of weak radiative B-meson decay w47x is replaced by the latest result from CLEO w48x
™™
BŽ b
™ X g . s Ž 3.15 " 0.35 " 0.26. = 10
y4
s
.
Ž 1.1 .
There also exists a comparable result by the ALEPH Collaboration with larger error w49x BŽ b
™ X g . s Ž 3.11 " 0.80 " 0.72. = 10
y4
s
.
Ž 1.2 .
The recent world average experimental measurements of the branching ratio of b gives w50x BŽ b
™ X tn . s 2.6 " 0.4%,
™ X tn c
Ž 1.3 .
c
while the standard model prediction is 2.3 " 0.25% w31,32x. Large errors in experimental values and discrepancies between the measured values and the results calculated in the SM such as mentioned above allow to reveal physics beyond the SM. Therefore, the inclusive processes b X ctn and b X sg may provide powerful tools to limit the range of unknown parameters in technicolor model. In this paper, we will calculate a s correction to the decay b X ctn . Considering the progress mentioned above, although the leading-order result of b sg has been used in limiting the parameter space in the technicolor with scalars model w23x, it is valuable that one considers the NLO calculation in the model. We reconstrain the parameter space by comparing the theoretical results with the latest experimental measurements. The remainder of this paper is organized as follows. A brief review of the technicolor with scalars model is contained in Section 2. We calculate the decay rates of b X ctnt in Section 3 and b X sg in Section 4, and present some constraints on the technicolor with scalars in Section 5. Finally, some remarks are made in Section 6.
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2. Technicolor with scalars model In the technicolor with scalars model, the processes we will study receive contributions from not only particles existing in the SM, but at least one physical charged scalar. In this section we will focus on the quantities which are needed in our calculation. More details of the model have been described in Refs. w20–22x.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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The gauge structure of the technicolor with scalars model is simply the direct product of the technicolor and standard model gauge groups: SUŽ N . TC = SUŽ3. C = SUŽ2.W = UŽ1. Y w21,22x. The technicolor singlet, i.e. ordinary fermions, are exactly those of the standard model. The technicolor sector consists of two techniflavors p and m that also transform under SUŽ2.W . In addition to the above particle spectrum, there exists a scalar doublet f to which the ordinary fermions and technifermions is coupled. When the technifermions condensate, the scalar develops a vacuum expectation value ŽVEV. that generates mass terms for the ordinary fermions. Mixture will happen between the technipions Žthe isotriplet scalar bound states of p and m. and the isotriplet components of f . One linear combination becomes the longitudinal component of the W and Z, and the orthogonal linear combination remains in the low-energy theory as an isotriplet of physical scalars pp . The Lagrangian of the charged scalars pp" coupled to the fermions in the technicolor with scalars model is given by ig f q Lp f s y '2 mW f X pp UL i Vi j m D j DR j q UR j mU j Vji DL i
½
5
qm l i n L i l R i q h.c. ,
Ž 2.1 .
where U, D,l and mU ,m D ,m l represent the column vector and the diagonal mass matrix for the up, down quarks and leptons, respectively, f denotes the technipion decay constant, f X is the scalar VEV, pp stands for the scalar field, Vi j are the elements of the CKM matrix. The chiral Lagrangian analysis in Refs. w21,22x allows us to estimate the mass of the charged scalar. At lowest order, mp2 p s 2'2 Ž 4p frf X . Õ 2 h,
Ž 2.2 . X
where Õ s 246 GeV is the electroweak scale, and f is constrained together with f by f 2 qf X2 sÕ2 . Ž 2.3 . Here h s Ž hqq hy .r2 is the average of the scalar’s Yukawa couplings to p Ž hq . and mŽ hy ., and c1 is a coefficient of order unity. In general, f and f X can depend on hq, hy, Mf and l, where Mf is the mass of the scalar doublet f , and l is the f 4 coupling. Two limits of the model have been studied previously in the literature: Ži. the limit in which l is small and can be neglected w20x, and Žii. the limit in which Mf is small and can be neglected w21,22x. The advantage of working in these two limits is that at the lowest order the phenomenology depends on h, not on the difference of hq and hy. When the largest Coleman–ewinberg corrections for the s field Žisoscalar component of f . are included in the effective chiral Lagrangian w21,22x, one obtains the constraint M˜f f X q
l˜
f X 3 s 8'2 p c1 hf 3 2 and the isoscalar mass ms2 s M˜f2 q
1 3p 2
3
mt
ž / fX
Ž 2.4 .
4
q 2 Nh4 f X 2
Ž 2.5 .
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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in limit Ži., and ms2 s
3 2
l˜ f X 2 y
1
3
8p 2
4
mt
ž / fX
q 2 Nh4 f X 2
Ž 2.6 .
in limit Žii.. Here m t is the pole mass of top quark, M˜f , l˜ are the shifted scalar mass and coupling. The phenomenology can be described in terms of Ž M˜f ,h. in limit Ži. and Ž l˜ ,h. in limit Žii.. In this paper, we choose two physical parameters Ž ms ,h. in both limits of the model. There are some advantages of this choice. First, it enables us to visualize what the parameter space looks like for a fixed not-so-small Mf Ž l.; second, when the limit on the isoscalar mass which comes from the limit on the mass of the Higgs in the SM changes, we simply need to move the vertical line to get an updated parameter space without carrying out a lengthy computation.
3. QCD corrections to b
™ X tn c
Since the physical scalar pp couple to fermions in technicolor with scalars is similar to the interaction of a Higgs doublet with fermions in the type-I two Higgs doublet model Ž2HD. Žsee Eq. Ž2.1.., we can adopt the formalism from Ref. w33x and obtain the explicit expressions particular to this model. As pointed out in Ref. w51x, the n Ž n s 3,4.-body phase space can be decomposed into a product of an n y 1-body phase space and a two-body phase space. The QCD correction to b X ctnt in the technicolor with scalars model can be obtained via the known QCD corrections of t bWq and t bHq decays. Generalizing the method proposed in the SM calculation w32x, one can easily extend to this case in this model by working in the Landau gauge. To the process b X ctn , there are three diagrams contributing to b X ctn corresponding to W-exchange, Goldstone boson ŽGB. exchange which is identified with the exchange of an un-physical scalar, charged scalar exchange w33x. Note that the coupling of the Goldstone boson to matter fields is always the same as that of the SM. The lepton tensor L is obtained by integrating over the leptonic two-body phase,
™
™
™
™
™
ž
L Ž rq . ; 1 y
rt rq
2
rt
/ ž rq
1qi
f rq f X rp p
2
/
,
Ž 3.1 .
™
where r i s m2i rm2b , r q s q 2rm2b and m b Ž m c . is the pole mass of the bottom Žcharm. quark. Defining Ga b c ' G Ž a bc . and c1 s 2 Ž 1 q r c . 1 y
c 2 s y4 1 y
f
ž / f
X
y 4 i Ž 1 y rc .
f fX
,
2
ž /
rc ,
fX
c 3 s Ž 1 y rc . 1 y
2
f
f
ž / f
X
2
y 2 i Ž 1 q rc .
f fX
,
Ž 3.2 .
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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we express G b c S as Gt b H in Ref. w52,53x. Here, we use the result of Ref. w52x. It reads
G b c S s G b c S Ž r q , rc ;c1 ,c 2 ,c 3 . s G b0c S q a s Ž m b . G b1c S , 0
Ž 3.3 .
1
where G stands for the tree-level decay rate and G represents the corresponding O Ž a s . correction, S denotes an effective scalar which includes the contributions of both the GB and the charged scalar. Taking the effective scalar S to be off-shell and integrating over its momentum one can obtain the contribution to the b X ctn decay rate from S as
™
GS Ž b
™ X tn . s 4m'2Gp 2 b
Hr 1y'r Ž
F
c
2
c
.2
G b cS Ž r q , rc ;c1 ,c 2 ,c3 . L Ž r q . d r q .
Ž 3.4 .
t
From Eq. Ž3.4., and considering G b c S to be linear in the relevant c1 ,c 2 ,c 3 term in Eq. Ž3.2., we get the scalar and the interference terms between GB and scalar in the total rate as
Gp pŽ b
™ X tn . s 4m'2Gp 2 b
c
™ X tn . s y m2 Gp 2 b
c
Hr 1y'r Ž
ž / ž / ž /H ' ž / 2
f
2
f
2
X
f
2
rq
rt r q rp2 p
rq
rt
c
.2
G b cS r q , rc ;2 Ž 1 q rc . ,4 rc ,y 1 q rc
t
F
= 1y
4. Rare radiative decay b
X
rt
= 1y
GI Ž b
4
f
F
2
d rq ,
Ž1y r c . 2
'
rt
rt
rp p
Ž 3.5 . G b cS r q , rc ;2 Ž y1 q rc . ,0,1 q rc
d rq .
Ž 3.6 .
™X g s
™
We begin with the Wilson coefficients Ci Ž m . in the effective Hamiltonian of b X sg . They can be expanded in powers of a s and split into Ci Ž m . SM and Ci Ž m . TC term as follows: Ci Ž m . s Ci0 Ž m . q
asŽ m. 4p
Ci1 Ž m . q . . . s Ci Ž m . SM q Ci Ž m . TC ,
Ž 4.1 .
where Ci0 Ž m . are the LO Wilson coefficients, Ci1 Ž m . appear in the NLO calculation, TC stands for the extra contributions from the technicolor with scalars model, and SM stands for the standard model. The running Wilson coefficients can be obtained by evolving the subtraction point m down from the matching scale mW to the low-energy scale m b according to the Renormalization Group Equations w54x. In general, in theories beyond the SM there will be additional contributions, which are characterized by the Õalues of the coefficients Ci at the perturbatiÕe scale mW ; the ‘‘effective coefficients’’ defined in Refs. w40–42,54–56x at m s m b in the technicolor with scalars model are exactly expressed as those in the SM. Here we only give the
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additional contributions to the coefficients from the charged scalar pp in the technicolor with scalars model. At m s mW the LO coefficients are expressed as w34,35x Ci0 Ž mW . TC s 0 for i s 1, . . . ,6, C 70 Ž mW . TC s C80
Ž mW . TC s
Ž 4.2 .
2
f
ž / ž /
B m 2t rmp2 p y 16 A m 2t rmp2 p
/
E m2t rmp2 p y 16 D m2t rmp2 p
/
ž
fX
/
ž
,
Ž 4.3 .
2
f
ž
fX
/
ž
Ž 4.4 .
with functions A, B, D, E given in Refs. w34,35x. The charged scalar contributions to the NLO Wilson coefficients at the mW scale are w43x Cii Ž mW . TC s 0 for i s 1,2,3,5,6, C41 Ž mW . TC s C 71 C81
Ž mW . TC s Ž mW . TC s
2
f
ž / ž / ž /
G m2t rmp2 p ,
ž
fX f
2 2 I1 m 2t rmp2 p q I2 m2t rmp2 p ln Ž m 2t rmW . ,
ž
fX f
/
/
ž
/
2 2 K 1 m2t rmp2 p q K 2 m2t rmp2 p ln Ž m2t rmW . ,
ž
fX
/
ž
Ž 4.5 .
/
where GŽ x . s
I1 Ž x . s
x
6 Ž 3 x y 2.
36 Ž x y 1 . x 9 Ž x y 1.
4
3
xy1
ln x q 7x 2 y 29 x q 16 ,
2 Ž y8 x 3 q 35 x 2 y 62 x q 24 . Li 2 1 y
ž
y279 x 3 q 975 x 2 y 995 x q 299 q 9 Ž x y 1.
94 x 4 y 13 x 3 y 45 x 2 y 247x q 211 54 Ž x y 1 . x 1296 Ž x y 1 .
q
4
x
/
ln x
q
K 1Ž x . s
1
,
216 Ž y4 x 3 q 25 x 2 y 31 x q 36 . Li 2 1 y
ž
36 Ž y144 x 3 q 1569 x 2 y 3245 x q 1820 . xy1
q247x 3 y 6480 x 2 q 9837x y 3604 .
1 x
/
ln x
Ž 4.6 .
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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The I2 and K 2 terms in Eq. Ž4.5. result when expressing the running m t Ž mW . in the terms of the pole mass m t in the lowest order coefficients by w43x
asŽ m.
mq Ž m. s mq 1 q
p
ž
ln
m 2q
m
2
4 y 3
/
,
Ž 4.7 .
and 2x
I2 Ž x . s
4
6 Ž 3 x 2 y 7x q 2 . ln x y 17x 3 q 51 x 2 y 39 x q 5 ,
27 Ž x y 1 . x K2Ž x . s 42 Ž x y 2 . ln x y 35 x 3 q 168 x 3 y 273 x q 140 . 4 36 Ž x y 1 .
Ž 4.8 .
we use the calculated coefficients at m s m to obtain the decay rate of ™AtXthisg . Itpoint, can be written as w55,56x G a
b
s
2 F
em 2
s
2 tb
) ts
5 b
2
Ž 4.9 . with D s C 7eff Ž m b . q
a s Ž mb . 4p
8
=
½Ý
Ci0,eff Ž m b . ri q g i70,eff ln
is1
As
a s Ž mb . p
mb
mb
16 y 3
8
Ý i , js1,i(j
5
C 70,eff Ž m b . ,
Re Ci0,eff Ž m b . C j0,eff Ž m b .
½
)
5
fi j .
Ž 4.10 .
™
The contribution denoted by < D < 2 comes from the decay b sg , and the A-dependence term represents the bremsstrahlung contribution which corresponds to the process of b sg g w57–61x. g i0,eff are the anomalous dimension matrix which governs the j evolution of the effective Wilson coefficient from mW to m b w43x. The virtual correction functions ri and the bremsstrahlung functions f i j are exactly the same as given in Refs. w43,55,56x. Following Ref. w43x, we do not introduce a cutoff when the photon gets soft w54x, and absorb the virtual photonic correction to the decay b sg g into the quantity f 88 .
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5. Results
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™
In the past several years the non-perturbative bound state corrections to b X ctn and b X sg up to order O Ž1rm2b . were completed in Refs. w31,62–66x. In contrast to the QCD correction to b X ctn , the O Ž1rm2b . correction to the free quark decay model is small; it only reduces the prediction for the rate by approximately 4–8% for the process b X ctn w31,62x. The analysis of the inclusive rates is, however, affected by the uncertainty in the b mass and by bound state corrections. This problem can be partly
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Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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circumvented by fixing m b y m c through the difference of bottom and charmed meson masses and by relating the bound state effects to phenomenological constants that can be determined from other observables in the context of the heavy quark effective theory ŽHQET. w67x. In this paper we consistently restrict our attention to O Ž a s . corrections, setting the O Ž a s2 . terms to zero, and take m b y m c s 3.4 GeV in calculations w32,54,63–66x. 5.1. b
™ X tn
™
c
We consider the difference between the decay branching rate of b X stn calculated in the technicolor with scalars model and the standard model. It may be normalized to the SM one, to cancel the large uncertainty due to m 5b and the elements of the CKM matrix. We obtain the normalized result as
dG Ž b
™ X tn . s G
GI 0
c
0 SM
1 q Ž 0.1720 " 0.0043 . a s Ž m b .
Gp0p
1 q Ž 0.2310 . 0.0033 . a s Ž m b . . Ž 5.1 . 0 G SM It is evident that the O Ž a s . corrections practically cancel in the normalized ratio dG Ž b X ctn .. From Eq. Ž5.1. we arrive at the conclusion that the main uncertainties in the calculation come from the unknown O Ž a s2 . corrections because the ratio in Eq. Ž5.1. is rather insensitive to the exact choice of a s Ž m b .. Here we use the QCD coupling constant a s Ž m b . s 0.218 from a s Ž m Z . s 0.119 " 0.04 w43x and m Z s 91.187 GeV, m b s 4.8 GeV w54x via a s Ž mZ . b 1 Õ Ž m . lnÕ Ž m . asŽ m. s 1y . Ž 5.2 . ÕŽ m. b 0 4p Õ Ž m . q
™
with Õ Ž m . s 1 y b0
a s Ž mZ .
ln
mZ
ž /
,
b0 s
23
,
b1 s
116
. 2p m 3 3 The shifted branching ratio normalized by the SM result dG Ž b X ctn . and allowed parameter space is plotted in Fig. 1 in limit Ži. and Fig. 2 in limit Žii.. The allowed region in Figs. 1 and 2 are already determined by the B 0 –B 0 mixing w20x Žthe region above and to the left of the ‘‘B line’’. and the upper limit on the mass of mp p. There are general reasons to believe that m H in the SM is less than or on the order of 1 TeV. In this work, we simply take mp p,ms ( 1 TeV, the region to the right of the curve mp p s 1 TeV in Fig. 1 and the region outside the curve in Fig. 2 are excluded. We also show the curve mp p s m t y m b in the figures with m t s 175 GeV. If the top quark does not decay to pq p b, in contrast to the area outside this curve in Fig. 1 is excluded, the excluded parameter space in Fig. 2 is the area inside the curve. Note that the chiral Lagrangian analysis break-down w21,22x only constrain the parameter space in limit Ži. w21,22x, the area above and to the left of the hf X s 4p f line is excluded because the technifermion current masses are no longer small compared to the chiral symmetry breaking scale. A similar situation occurs in the lower limit on mp p s 54.5 GeV under ms 0 77.5 GeV w50x. From Figs. 1 and 2 we find the 0.1% contour of dGrG to lie almost below the region allowed by B 0 –B 0 mixing, and the branching ratio increases when we move
™
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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™
Fig. 1. The shifted decay width b X ctn normalized by the SM result in limit Ži. Ždashed lines.. The allowed X parameter space is bounded by the B line, the hf s 4p f line, the mp ps 54.5 GeV line, and the mp ps1 TeV line. The world average experimental upper limit lies outside the allowed region.
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™
from upper left to lower right in the allowed region of the parameter space. This is similar to the bounds from Z bb w23x and B X s mq my w24x. 5.2. b
™X g s
™
In this subsection we present some numerical results of b X sg in the technicolor with scalars model. To determine the Wilson coefficients at m s mW in Section 4, we take mW s 80.33 GeV w50x, and m b g w2.4, 9.6x w43x. For a s Ž m b ., we use the expression
™
Fig. 2. The shifted decay width b X ctn normalized by the SM result in limit Žii. Ždashed lines.. The allowed parameter space is bounded by the B line, the ms s 77.5 GeV line, and the mp ps1 TeV line. The world average experimental upper limit lies outside the allowed region.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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in Eq. Ž5.2. at NLO level, but put b 1 to zero for LO results. Following Ref. w43x, we take a em s 130.3 " 2.3, < Vt s) Vt brVb c < 2 s 0.95 " 0.03. Now we express the branching ratio of b X sg as functions of the Wilson coefficients at scale mW . It reads
™ B Ž b ™ X g . s 0.97 < C Ž m 0 7
s
W
. < 2 y 0.02 < C80 Ž mW . < 2 q 0.44C 70 Ž mW . C80 Ž mW .
q Ž 0.09C 70 Ž mW . q 0.01C 70 Ž mW . y 0.02 .Ž y0.58C41 Ž mW . q1.42C 71 Ž mW . q 0.20C81 Ž mW . . y 0.38C 70 Ž mW . y 0.09C80 Ž mW . q0.03 = 10 4
Ž 5.3 .
™
where the central value is obtained by setting m b s m b s 4.8 GeV. The SM prediction for the inclusive decay rate is B Ž b X sg . s Ž3.37 " 0.27. = 10 4 , which implies that the two-loop correction gives a 22.5% increase to the one-loop result. The branching ratios B Ž b X sg . for various frf X and mp p at LO and NLO levels are plotted in Figs. 3a and 3b. In each figure, two horizontal dashed lines indicate the upper and lower experimental bounds from ALEPH Žsee Eq. Ž1.2.. at 95% C.L. It is noted that each curve in Fig. 3 crosses the experimental allowed region twice. Fig. 4 shows that the coefficients of Ž frf X . 2 as functions of mp p in C 70,eff Ž m b ., C 7eff Ž m b ., and Re D Žin fact, the imaginary part of D does not contribute to the branching ratio B Ž b X sg . in the approximation a s2 s 0. are always positive for any mp p. Hence a cancellation mechanism can occur, and this cancellation forces the curves in Fig. 3 to
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™
™
X Fig. 3. The branching ratios B Ž b X sg . for various f r f and mp p at Ža. LO level and Žb. NLO level. The solid lines correspond to mp ps100,250,500,1000 GeV. Two horizontal dashed lines indicate the upper and lower bounds of the experimental measurements from ALEPH at 95% C.L.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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X Fig. 4. The coefficients of Ž f r f . 2 as functions of mp p in C 70,eff Žsolid line., C 7eff Ždashed line., and Re D Ždotted line..
bend down through the allowed region. It is shown that the experimental limits on frf X in the branching ratio calculated at LO level may give a large increase to the value calculated at NLO level with the same mp p. The shifted branching ratio normalized by the SM result dG Ž b X sg . and the allowed parameter space are drawn in Fig. 5 in limit Ži. and Fig. 6 in limit Žii.. It is clear from Figs. 5 and 6 that dGrG and the allowed parameter space are different from the case of b X ctn in two ways. First, the branching ratio decreases when we move from upper left to lower right in the area bounded by the ‘‘B line’’, the mp p s 1 TeV curve, together with the mp p s 54.5 GeV
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™
Fig. 5. Contours of B Ž b X sg . in the Ž ms ,h. plain in limit Ži. Ždashed lines.. A parameter space bounded X by the B line, the hf s 4p f line, the mp ps 54.5 GeV line, and the mp ps1 TeV line is plotted. The experimental lower limit from CLEO corresponds to the y33% contour, and therefore lies inside the parameter space, while the lower limit from ALEPH lies outside the allowed region.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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™
Fig. 6. Contours of B Ž b X sg . in the Ž ms ,h. plain in limit Žii. Ždashed lines.. A parameter space bounded by the B line, the ms s 77.5 GeV line, and the mp ps1 TeV line is plotted. The experimental lower limit from CLEO corresponds to the y33% contour, and therefore lies inside the parameter space, while the lower limit from ALEPH lies outside the allowed region.
line, the hf X s 4p f line in limit Ži. and the ms s 77.5 GeV line in limit Žii.; second, the 95% confidence level lower bound by CLEO w48x which corresponds to the y33% contour of the plot lies inside the area. However, the 95% confidence level lower bound by ALEPH w49x, which corresponds to the y78% contour of the plot, lies outside the bound. We can see that the y50% contour of dGrG Žcorresponding to the y39% LO result. is never in conflict with the experimental limit on B 0 –B 0 mixing, and therefore, the lower bound on G Ž b X sg . by ALEPH. Since the region of the parameter space in which the correction to b X sg less than y1% includes a light, extremely narrow technirho and has been discussed in Ref. w68x. The part of the parameter space in which the correction to b X sg between y1% and y50%, or conservatively, y1% and y33% bounded by CLEOw48x is significant.
™
™™
6. Discussion and conclusion
™
™
In the context of the technicolor with scalars model we have calculated the QCD corrections up to a s to the inclusive process b X ctn and b X sg . In studying the process b X ctn , the calculation of the lowest order rate as well as the corrections were related to the corresponding calculation for the decay into a virtual W boson and a charged scalar with the subsequent integration over the mass of the tn system by using the Landau gauge. The contributions arising from the exchange of the scalars and interference term have been included. Differing from the virtual effects of charged scalars through the loop diagrams in most cases, which can be partially canceled by the effects of the other particles in the theory, in the process of b X ctn , the charged scalars contributions to b X ctn studied in this paper occur at tree level; therefore, their effects can not be canceled by other new particles in the theory and do not
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Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
15
disappear. Following a general procedure, at LO level we presented the additional contributions to the Wilson coefficients Ci Ž mW . TC from the charged scalars, and used them in expressing the branching ratio B Ž b X sg .. We found that the correction to B Ž b X ctn . relative to the SM result is positive throughout the allowed region of the parameter space in technicolor with scalars in both limits, and almost smaller than 0.1%. However, the correction to B Ž b X sg . is negative, and can be up to y33% of the SM prediction bounded by CLEO, and therefore provides a more stringent constraint than B 0 – B 0 mixing. The maximum decrease of the branching ratio relative to its SM counterpart is about y50% without conflicting with B 0 – B 0 mixing and the ALEPH measurement. Up to now, the SM prediction for the branching ratio of b X sg has an accuracy of below 10%, and in the forthcoming few years, the expectation that the measurements from the upgraded CLEO detector, as well as from the B-factories at SLAC and KEK with the same accuracy, is conceivable. Therefore, the sizable decrease of the branching ratio of b X sg predicted in the technicolor with scalars model in part of its parameter space will distinguish the SM from this model, or make a final decision of the window for the model.
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Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No.19755001, 19555003 and 19675047. We would like thank Dr. Zhentao Wei for useful discussions.
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Nuclear Physics B 561 Ž1999. 3–16 www.elsevier.nlrlocaternpe
™
Constraints on technicolor with scalars from b b X sg
™ X tn and
Zhaohua Xiong a , Hesheng Chen b, Liang Lu b
c
b
a CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, People’s Republic of China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, People’s Republic of China
Received 16 March 1999; accepted 11 August 1999
Abstract
™
™
The O Ž a s . corrections to the inclusive decay of b X ctn and b X sg are calculated in technicolor with scalars model, and the latest experimental results from ALEPH and CLEO are used in determining the Ž s , h. parameter space. We find that in contrast with the enhancement of the branching ratio of b X ctn relative to that of the standard model is very small, the b X sg decay width normalized by the standard model result can be reduced by as much as y50% when the ALEPH experimental value is used. In particular, the CLEO measurement which corresponds to the y33% reduction gives a more stringent constraint than B y B mixing. The branching fraction B Ž b X sg . computed in the model in part of its parameter space with sizable deviation from the standard model prediction can be observable in the forthcoming few years with the upgraded CLEO detector and B-factories at SLAC and KEK. q 1999 Elsevier Science B.V. All rights reserved.
™
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PACS: 12.60.Nz; 12.38.Bx; 13.25.Hw Keywords: Technicolor with scalars; Decay b
™ X tn ; Decay b ™ X g ; QCD corrections c
s
1. Introduction Technicolor—a strong interaction of fermions and gauge bosons at the scale L TC f 1 TeV—is a scenario for the dynamical breakdown of electroweak symmetry to electromagnetism w1,2x. Based on the similar phenomenon of chiral symmetry breakdown in QCD, technicolor is explicitly defined and completely natural. To account for the mass of quarks, leptons, and Goldstone ‘‘technipions’’ in such a scheme, technicolor, ordinary color, and flavor symmetry are embedded in a large gauge group, called extended technicolor ŽETC. w3,4x. Because of the conflict between constraints on flavor-changing neutral currents ŽFCNC. and the magnitude of ETC-generated quark, lepton and technipion masses, classical technicolor was superseded by a ‘‘walking’’ technicolor and ‘‘multiscale technicolor’’ w5–12x. The incapability to explain the top quark’s large 0550-3213r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 5 1 5 - 5
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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™
mass without afoul of either cherished notions of naturalness or experiments from the r parameter and the Z bb decay rate by ETC w13,14x led to the topcolor-assisted technicolor w15–19x and the technicolor with scalars model w20–24x. In the standard model ŽSM. of electroweak interactions decays of B mesons with a t lepton in the final state constitute a significant fraction of all semileptonic B decays. These decays are interesting as they can probe certain form factors that are inaccessible in decay modes with light leptons, because they give effects of order m2l w25x. A collective theoretical effect has led to the determination of b semileptonic decays up to order a s for arbitrary b and c masses with massless leptons w26–30x and the perturbative QCD correction to b ctnt w31–33x. Great progress has been made in determining the branching ratio B Ž b X sg . in both theories and experiments in past two years. Instead of leading-order ŽLO. calculation w34–39x, the branching ratio has been estimated at next-leading-order ŽNLO. level w40–46x. The inclusive branching ratio Ž2.32 " 0.57 " 0.35. = 10y4 extracted four years ago from CLEO measurement of weak radiative B-meson decay w47x is replaced by the latest result from CLEO w48x
™™
BŽ b
™ X g . s Ž 3.15 " 0.35 " 0.26. = 10
y4
s
.
Ž 1.1 .
There also exists a comparable result by the ALEPH Collaboration with larger error w49x BŽ b
™ X g . s Ž 3.11 " 0.80 " 0.72. = 10
y4
s
.
Ž 1.2 .
The recent world average experimental measurements of the branching ratio of b gives w50x BŽ b
™ X tn . s 2.6 " 0.4%,
™ X tn c
Ž 1.3 .
c
while the standard model prediction is 2.3 " 0.25% w31,32x. Large errors in experimental values and discrepancies between the measured values and the results calculated in the SM such as mentioned above allow to reveal physics beyond the SM. Therefore, the inclusive processes b X ctn and b X sg may provide powerful tools to limit the range of unknown parameters in technicolor model. In this paper, we will calculate a s correction to the decay b X ctn . Considering the progress mentioned above, although the leading-order result of b sg has been used in limiting the parameter space in the technicolor with scalars model w23x, it is valuable that one considers the NLO calculation in the model. We reconstrain the parameter space by comparing the theoretical results with the latest experimental measurements. The remainder of this paper is organized as follows. A brief review of the technicolor with scalars model is contained in Section 2. We calculate the decay rates of b X ctnt in Section 3 and b X sg in Section 4, and present some constraints on the technicolor with scalars in Section 5. Finally, some remarks are made in Section 6.
™
™
™
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™
2. Technicolor with scalars model In the technicolor with scalars model, the processes we will study receive contributions from not only particles existing in the SM, but at least one physical charged scalar. In this section we will focus on the quantities which are needed in our calculation. More details of the model have been described in Refs. w20–22x.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
5
The gauge structure of the technicolor with scalars model is simply the direct product of the technicolor and standard model gauge groups: SUŽ N . TC = SUŽ3. C = SUŽ2.W = UŽ1. Y w21,22x. The technicolor singlet, i.e. ordinary fermions, are exactly those of the standard model. The technicolor sector consists of two techniflavors p and m that also transform under SUŽ2.W . In addition to the above particle spectrum, there exists a scalar doublet f to which the ordinary fermions and technifermions is coupled. When the technifermions condensate, the scalar develops a vacuum expectation value ŽVEV. that generates mass terms for the ordinary fermions. Mixture will happen between the technipions Žthe isotriplet scalar bound states of p and m. and the isotriplet components of f . One linear combination becomes the longitudinal component of the W and Z, and the orthogonal linear combination remains in the low-energy theory as an isotriplet of physical scalars pp . The Lagrangian of the charged scalars pp" coupled to the fermions in the technicolor with scalars model is given by ig f q Lp f s y '2 mW f X pp UL i Vi j m D j DR j q UR j mU j Vji DL i
½
5
qm l i n L i l R i q h.c. ,
Ž 2.1 .
where U, D,l and mU ,m D ,m l represent the column vector and the diagonal mass matrix for the up, down quarks and leptons, respectively, f denotes the technipion decay constant, f X is the scalar VEV, pp stands for the scalar field, Vi j are the elements of the CKM matrix. The chiral Lagrangian analysis in Refs. w21,22x allows us to estimate the mass of the charged scalar. At lowest order, mp2 p s 2'2 Ž 4p frf X . Õ 2 h,
Ž 2.2 . X
where Õ s 246 GeV is the electroweak scale, and f is constrained together with f by f 2 qf X2 sÕ2 . Ž 2.3 . Here h s Ž hqq hy .r2 is the average of the scalar’s Yukawa couplings to p Ž hq . and mŽ hy ., and c1 is a coefficient of order unity. In general, f and f X can depend on hq, hy, Mf and l, where Mf is the mass of the scalar doublet f , and l is the f 4 coupling. Two limits of the model have been studied previously in the literature: Ži. the limit in which l is small and can be neglected w20x, and Žii. the limit in which Mf is small and can be neglected w21,22x. The advantage of working in these two limits is that at the lowest order the phenomenology depends on h, not on the difference of hq and hy. When the largest Coleman–ewinberg corrections for the s field Žisoscalar component of f . are included in the effective chiral Lagrangian w21,22x, one obtains the constraint M˜f f X q
l˜
f X 3 s 8'2 p c1 hf 3 2 and the isoscalar mass ms2 s M˜f2 q
1 3p 2
3
mt
ž / fX
Ž 2.4 .
4
q 2 Nh4 f X 2
Ž 2.5 .
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
6
in limit Ži., and ms2 s
3 2
l˜ f X 2 y
1
3
8p 2
4
mt
ž / fX
q 2 Nh4 f X 2
Ž 2.6 .
in limit Žii.. Here m t is the pole mass of top quark, M˜f , l˜ are the shifted scalar mass and coupling. The phenomenology can be described in terms of Ž M˜f ,h. in limit Ži. and Ž l˜ ,h. in limit Žii.. In this paper, we choose two physical parameters Ž ms ,h. in both limits of the model. There are some advantages of this choice. First, it enables us to visualize what the parameter space looks like for a fixed not-so-small Mf Ž l.; second, when the limit on the isoscalar mass which comes from the limit on the mass of the Higgs in the SM changes, we simply need to move the vertical line to get an updated parameter space without carrying out a lengthy computation.
3. QCD corrections to b
™ X tn c
Since the physical scalar pp couple to fermions in technicolor with scalars is similar to the interaction of a Higgs doublet with fermions in the type-I two Higgs doublet model Ž2HD. Žsee Eq. Ž2.1.., we can adopt the formalism from Ref. w33x and obtain the explicit expressions particular to this model. As pointed out in Ref. w51x, the n Ž n s 3,4.-body phase space can be decomposed into a product of an n y 1-body phase space and a two-body phase space. The QCD correction to b X ctnt in the technicolor with scalars model can be obtained via the known QCD corrections of t bWq and t bHq decays. Generalizing the method proposed in the SM calculation w32x, one can easily extend to this case in this model by working in the Landau gauge. To the process b X ctn , there are three diagrams contributing to b X ctn corresponding to W-exchange, Goldstone boson ŽGB. exchange which is identified with the exchange of an un-physical scalar, charged scalar exchange w33x. Note that the coupling of the Goldstone boson to matter fields is always the same as that of the SM. The lepton tensor L is obtained by integrating over the leptonic two-body phase,
™
™
™
™
™
ž
L Ž rq . ; 1 y
rt rq
2
rt
/ ž rq
1qi
f rq f X rp p
2
/
,
Ž 3.1 .
™
where r i s m2i rm2b , r q s q 2rm2b and m b Ž m c . is the pole mass of the bottom Žcharm. quark. Defining Ga b c ' G Ž a bc . and c1 s 2 Ž 1 q r c . 1 y
c 2 s y4 1 y
f
ž / f
X
y 4 i Ž 1 y rc .
f fX
,
2
ž /
rc ,
fX
c 3 s Ž 1 y rc . 1 y
2
f
f
ž / f
X
2
y 2 i Ž 1 q rc .
f fX
,
Ž 3.2 .
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
7
we express G b c S as Gt b H in Ref. w52,53x. Here, we use the result of Ref. w52x. It reads
G b c S s G b c S Ž r q , rc ;c1 ,c 2 ,c 3 . s G b0c S q a s Ž m b . G b1c S , 0
Ž 3.3 .
1
where G stands for the tree-level decay rate and G represents the corresponding O Ž a s . correction, S denotes an effective scalar which includes the contributions of both the GB and the charged scalar. Taking the effective scalar S to be off-shell and integrating over its momentum one can obtain the contribution to the b X ctn decay rate from S as
™
GS Ž b
™ X tn . s 4m'2Gp 2 b
Hr 1y'r Ž
F
c
2
c
.2
G b cS Ž r q , rc ;c1 ,c 2 ,c3 . L Ž r q . d r q .
Ž 3.4 .
t
From Eq. Ž3.4., and considering G b c S to be linear in the relevant c1 ,c 2 ,c 3 term in Eq. Ž3.2., we get the scalar and the interference terms between GB and scalar in the total rate as
Gp pŽ b
™ X tn . s 4m'2Gp 2 b
c
™ X tn . s y m2 Gp 2 b
c
Hr 1y'r Ž
ž / ž / ž /H ' ž / 2
f
2
f
2
X
f
2
rq
rt r q rp2 p
rq
rt
c
.2
G b cS r q , rc ;2 Ž 1 q rc . ,4 rc ,y 1 q rc
t
F
= 1y
4. Rare radiative decay b
X
rt
= 1y
GI Ž b
4
f
F
2
d rq ,
Ž1y r c . 2
'
rt
rt
rp p
Ž 3.5 . G b cS r q , rc ;2 Ž y1 q rc . ,0,1 q rc
d rq .
Ž 3.6 .
™X g s
™
We begin with the Wilson coefficients Ci Ž m . in the effective Hamiltonian of b X sg . They can be expanded in powers of a s and split into Ci Ž m . SM and Ci Ž m . TC term as follows: Ci Ž m . s Ci0 Ž m . q
asŽ m. 4p
Ci1 Ž m . q . . . s Ci Ž m . SM q Ci Ž m . TC ,
Ž 4.1 .
where Ci0 Ž m . are the LO Wilson coefficients, Ci1 Ž m . appear in the NLO calculation, TC stands for the extra contributions from the technicolor with scalars model, and SM stands for the standard model. The running Wilson coefficients can be obtained by evolving the subtraction point m down from the matching scale mW to the low-energy scale m b according to the Renormalization Group Equations w54x. In general, in theories beyond the SM there will be additional contributions, which are characterized by the Õalues of the coefficients Ci at the perturbatiÕe scale mW ; the ‘‘effective coefficients’’ defined in Refs. w40–42,54–56x at m s m b in the technicolor with scalars model are exactly expressed as those in the SM. Here we only give the
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
8
additional contributions to the coefficients from the charged scalar pp in the technicolor with scalars model. At m s mW the LO coefficients are expressed as w34,35x Ci0 Ž mW . TC s 0 for i s 1, . . . ,6, C 70 Ž mW . TC s C80
Ž mW . TC s
Ž 4.2 .
2
f
ž / ž /
B m 2t rmp2 p y 16 A m 2t rmp2 p
/
E m2t rmp2 p y 16 D m2t rmp2 p
/
ž
fX
/
ž
,
Ž 4.3 .
2
f
ž
fX
/
ž
Ž 4.4 .
with functions A, B, D, E given in Refs. w34,35x. The charged scalar contributions to the NLO Wilson coefficients at the mW scale are w43x Cii Ž mW . TC s 0 for i s 1,2,3,5,6, C41 Ž mW . TC s C 71 C81
Ž mW . TC s Ž mW . TC s
2
f
ž / ž / ž /
G m2t rmp2 p ,
ž
fX f
2 2 I1 m 2t rmp2 p q I2 m2t rmp2 p ln Ž m 2t rmW . ,
ž
fX f
/
/
ž
/
2 2 K 1 m2t rmp2 p q K 2 m2t rmp2 p ln Ž m2t rmW . ,
ž
fX
/
ž
Ž 4.5 .
/
where GŽ x . s
I1 Ž x . s
x
6 Ž 3 x y 2.
36 Ž x y 1 . x 9 Ž x y 1.
4
3
xy1
ln x q 7x 2 y 29 x q 16 ,
2 Ž y8 x 3 q 35 x 2 y 62 x q 24 . Li 2 1 y
ž
y279 x 3 q 975 x 2 y 995 x q 299 q 9 Ž x y 1.
94 x 4 y 13 x 3 y 45 x 2 y 247x q 211 54 Ž x y 1 . x 1296 Ž x y 1 .
q
4
x
/
ln x
q
K 1Ž x . s
1
,
216 Ž y4 x 3 q 25 x 2 y 31 x q 36 . Li 2 1 y
ž
36 Ž y144 x 3 q 1569 x 2 y 3245 x q 1820 . xy1
q247x 3 y 6480 x 2 q 9837x y 3604 .
1 x
/
ln x
Ž 4.6 .
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
9
The I2 and K 2 terms in Eq. Ž4.5. result when expressing the running m t Ž mW . in the terms of the pole mass m t in the lowest order coefficients by w43x
asŽ m.
mq Ž m. s mq 1 q
p
ž
ln
m 2q
m
2
4 y 3
/
,
Ž 4.7 .
and 2x
I2 Ž x . s
4
6 Ž 3 x 2 y 7x q 2 . ln x y 17x 3 q 51 x 2 y 39 x q 5 ,
27 Ž x y 1 . x K2Ž x . s 42 Ž x y 2 . ln x y 35 x 3 q 168 x 3 y 273 x q 140 . 4 36 Ž x y 1 .
Ž 4.8 .
we use the calculated coefficients at m s m to obtain the decay rate of ™AtXthisg . Itpoint, can be written as w55,56x G a
b
s
2 F
em 2
s
2 tb
) ts
5 b
2
Ž 4.9 . with D s C 7eff Ž m b . q
a s Ž mb . 4p
8
=
½Ý
Ci0,eff Ž m b . ri q g i70,eff ln
is1
As
a s Ž mb . p
mb
mb
16 y 3
8
Ý i , js1,i(j
5
C 70,eff Ž m b . ,
Re Ci0,eff Ž m b . C j0,eff Ž m b .
½
)
5
fi j .
Ž 4.10 .
™
The contribution denoted by < D < 2 comes from the decay b sg , and the A-dependence term represents the bremsstrahlung contribution which corresponds to the process of b sg g w57–61x. g i0,eff are the anomalous dimension matrix which governs the j evolution of the effective Wilson coefficient from mW to m b w43x. The virtual correction functions ri and the bremsstrahlung functions f i j are exactly the same as given in Refs. w43,55,56x. Following Ref. w43x, we do not introduce a cutoff when the photon gets soft w54x, and absorb the virtual photonic correction to the decay b sg g into the quantity f 88 .
™
™
5. Results
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™
In the past several years the non-perturbative bound state corrections to b X ctn and b X sg up to order O Ž1rm2b . were completed in Refs. w31,62–66x. In contrast to the QCD correction to b X ctn , the O Ž1rm2b . correction to the free quark decay model is small; it only reduces the prediction for the rate by approximately 4–8% for the process b X ctn w31,62x. The analysis of the inclusive rates is, however, affected by the uncertainty in the b mass and by bound state corrections. This problem can be partly
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circumvented by fixing m b y m c through the difference of bottom and charmed meson masses and by relating the bound state effects to phenomenological constants that can be determined from other observables in the context of the heavy quark effective theory ŽHQET. w67x. In this paper we consistently restrict our attention to O Ž a s . corrections, setting the O Ž a s2 . terms to zero, and take m b y m c s 3.4 GeV in calculations w32,54,63–66x. 5.1. b
™ X tn
™
c
We consider the difference between the decay branching rate of b X stn calculated in the technicolor with scalars model and the standard model. It may be normalized to the SM one, to cancel the large uncertainty due to m 5b and the elements of the CKM matrix. We obtain the normalized result as
dG Ž b
™ X tn . s G
GI 0
c
0 SM
1 q Ž 0.1720 " 0.0043 . a s Ž m b .
Gp0p
1 q Ž 0.2310 . 0.0033 . a s Ž m b . . Ž 5.1 . 0 G SM It is evident that the O Ž a s . corrections practically cancel in the normalized ratio dG Ž b X ctn .. From Eq. Ž5.1. we arrive at the conclusion that the main uncertainties in the calculation come from the unknown O Ž a s2 . corrections because the ratio in Eq. Ž5.1. is rather insensitive to the exact choice of a s Ž m b .. Here we use the QCD coupling constant a s Ž m b . s 0.218 from a s Ž m Z . s 0.119 " 0.04 w43x and m Z s 91.187 GeV, m b s 4.8 GeV w54x via a s Ž mZ . b 1 Õ Ž m . lnÕ Ž m . asŽ m. s 1y . Ž 5.2 . ÕŽ m. b 0 4p Õ Ž m . q
™
with Õ Ž m . s 1 y b0
a s Ž mZ .
ln
mZ
ž /
,
b0 s
23
,
b1 s
116
. 2p m 3 3 The shifted branching ratio normalized by the SM result dG Ž b X ctn . and allowed parameter space is plotted in Fig. 1 in limit Ži. and Fig. 2 in limit Žii.. The allowed region in Figs. 1 and 2 are already determined by the B 0 –B 0 mixing w20x Žthe region above and to the left of the ‘‘B line’’. and the upper limit on the mass of mp p. There are general reasons to believe that m H in the SM is less than or on the order of 1 TeV. In this work, we simply take mp p,ms ( 1 TeV, the region to the right of the curve mp p s 1 TeV in Fig. 1 and the region outside the curve in Fig. 2 are excluded. We also show the curve mp p s m t y m b in the figures with m t s 175 GeV. If the top quark does not decay to pq p b, in contrast to the area outside this curve in Fig. 1 is excluded, the excluded parameter space in Fig. 2 is the area inside the curve. Note that the chiral Lagrangian analysis break-down w21,22x only constrain the parameter space in limit Ži. w21,22x, the area above and to the left of the hf X s 4p f line is excluded because the technifermion current masses are no longer small compared to the chiral symmetry breaking scale. A similar situation occurs in the lower limit on mp p s 54.5 GeV under ms 0 77.5 GeV w50x. From Figs. 1 and 2 we find the 0.1% contour of dGrG to lie almost below the region allowed by B 0 –B 0 mixing, and the branching ratio increases when we move
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Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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™
Fig. 1. The shifted decay width b X ctn normalized by the SM result in limit Ži. Ždashed lines.. The allowed X parameter space is bounded by the B line, the hf s 4p f line, the mp ps 54.5 GeV line, and the mp ps1 TeV line. The world average experimental upper limit lies outside the allowed region.
™
™
from upper left to lower right in the allowed region of the parameter space. This is similar to the bounds from Z bb w23x and B X s mq my w24x. 5.2. b
™X g s
™
In this subsection we present some numerical results of b X sg in the technicolor with scalars model. To determine the Wilson coefficients at m s mW in Section 4, we take mW s 80.33 GeV w50x, and m b g w2.4, 9.6x w43x. For a s Ž m b ., we use the expression
™
Fig. 2. The shifted decay width b X ctn normalized by the SM result in limit Žii. Ždashed lines.. The allowed parameter space is bounded by the B line, the ms s 77.5 GeV line, and the mp ps1 TeV line. The world average experimental upper limit lies outside the allowed region.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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in Eq. Ž5.2. at NLO level, but put b 1 to zero for LO results. Following Ref. w43x, we take a em s 130.3 " 2.3, < Vt s) Vt brVb c < 2 s 0.95 " 0.03. Now we express the branching ratio of b X sg as functions of the Wilson coefficients at scale mW . It reads
™ B Ž b ™ X g . s 0.97 < C Ž m 0 7
s
W
. < 2 y 0.02 < C80 Ž mW . < 2 q 0.44C 70 Ž mW . C80 Ž mW .
q Ž 0.09C 70 Ž mW . q 0.01C 70 Ž mW . y 0.02 .Ž y0.58C41 Ž mW . q1.42C 71 Ž mW . q 0.20C81 Ž mW . . y 0.38C 70 Ž mW . y 0.09C80 Ž mW . q0.03 = 10 4
Ž 5.3 .
™
where the central value is obtained by setting m b s m b s 4.8 GeV. The SM prediction for the inclusive decay rate is B Ž b X sg . s Ž3.37 " 0.27. = 10 4 , which implies that the two-loop correction gives a 22.5% increase to the one-loop result. The branching ratios B Ž b X sg . for various frf X and mp p at LO and NLO levels are plotted in Figs. 3a and 3b. In each figure, two horizontal dashed lines indicate the upper and lower experimental bounds from ALEPH Žsee Eq. Ž1.2.. at 95% C.L. It is noted that each curve in Fig. 3 crosses the experimental allowed region twice. Fig. 4 shows that the coefficients of Ž frf X . 2 as functions of mp p in C 70,eff Ž m b ., C 7eff Ž m b ., and Re D Žin fact, the imaginary part of D does not contribute to the branching ratio B Ž b X sg . in the approximation a s2 s 0. are always positive for any mp p. Hence a cancellation mechanism can occur, and this cancellation forces the curves in Fig. 3 to
™
™
™
X Fig. 3. The branching ratios B Ž b X sg . for various f r f and mp p at Ža. LO level and Žb. NLO level. The solid lines correspond to mp ps100,250,500,1000 GeV. Two horizontal dashed lines indicate the upper and lower bounds of the experimental measurements from ALEPH at 95% C.L.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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X Fig. 4. The coefficients of Ž f r f . 2 as functions of mp p in C 70,eff Žsolid line., C 7eff Ždashed line., and Re D Ždotted line..
bend down through the allowed region. It is shown that the experimental limits on frf X in the branching ratio calculated at LO level may give a large increase to the value calculated at NLO level with the same mp p. The shifted branching ratio normalized by the SM result dG Ž b X sg . and the allowed parameter space are drawn in Fig. 5 in limit Ži. and Fig. 6 in limit Žii.. It is clear from Figs. 5 and 6 that dGrG and the allowed parameter space are different from the case of b X ctn in two ways. First, the branching ratio decreases when we move from upper left to lower right in the area bounded by the ‘‘B line’’, the mp p s 1 TeV curve, together with the mp p s 54.5 GeV
™
™
™
Fig. 5. Contours of B Ž b X sg . in the Ž ms ,h. plain in limit Ži. Ždashed lines.. A parameter space bounded X by the B line, the hf s 4p f line, the mp ps 54.5 GeV line, and the mp ps1 TeV line is plotted. The experimental lower limit from CLEO corresponds to the y33% contour, and therefore lies inside the parameter space, while the lower limit from ALEPH lies outside the allowed region.
Z. Xiong et al.r Nuclear Physics B 561 (1999) 3–16
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™
Fig. 6. Contours of B Ž b X sg . in the Ž ms ,h. plain in limit Žii. Ždashed lines.. A parameter space bounded by the B line, the ms s 77.5 GeV line, and the mp ps1 TeV line is plotted. The experimental lower limit from CLEO corresponds to the y33% contour, and therefore lies inside the parameter space, while the lower limit from ALEPH lies outside the allowed region.
line, the hf X s 4p f line in limit Ži. and the ms s 77.5 GeV line in limit Žii.; second, the 95% confidence level lower bound by CLEO w48x which corresponds to the y33% contour of the plot lies inside the area. However, the 95% confidence level lower bound by ALEPH w49x, which corresponds to the y78% contour of the plot, lies outside the bound. We can see that the y50% contour of dGrG Žcorresponding to the y39% LO result. is never in conflict with the experimental limit on B 0 –B 0 mixing, and therefore, the lower bound on G Ž b X sg . by ALEPH. Since the region of the parameter space in which the correction to b X sg less than y1% includes a light, extremely narrow technirho and has been discussed in Ref. w68x. The part of the parameter space in which the correction to b X sg between y1% and y50%, or conservatively, y1% and y33% bounded by CLEOw48x is significant.
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6. Discussion and conclusion
™
™
In the context of the technicolor with scalars model we have calculated the QCD corrections up to a s to the inclusive process b X ctn and b X sg . In studying the process b X ctn , the calculation of the lowest order rate as well as the corrections were related to the corresponding calculation for the decay into a virtual W boson and a charged scalar with the subsequent integration over the mass of the tn system by using the Landau gauge. The contributions arising from the exchange of the scalars and interference term have been included. Differing from the virtual effects of charged scalars through the loop diagrams in most cases, which can be partially canceled by the effects of the other particles in the theory, in the process of b X ctn , the charged scalars contributions to b X ctn studied in this paper occur at tree level; therefore, their effects can not be canceled by other new particles in the theory and do not
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disappear. Following a general procedure, at LO level we presented the additional contributions to the Wilson coefficients Ci Ž mW . TC from the charged scalars, and used them in expressing the branching ratio B Ž b X sg .. We found that the correction to B Ž b X ctn . relative to the SM result is positive throughout the allowed region of the parameter space in technicolor with scalars in both limits, and almost smaller than 0.1%. However, the correction to B Ž b X sg . is negative, and can be up to y33% of the SM prediction bounded by CLEO, and therefore provides a more stringent constraint than B 0 – B 0 mixing. The maximum decrease of the branching ratio relative to its SM counterpart is about y50% without conflicting with B 0 – B 0 mixing and the ALEPH measurement. Up to now, the SM prediction for the branching ratio of b X sg has an accuracy of below 10%, and in the forthcoming few years, the expectation that the measurements from the upgraded CLEO detector, as well as from the B-factories at SLAC and KEK with the same accuracy, is conceivable. Therefore, the sizable decrease of the branching ratio of b X sg predicted in the technicolor with scalars model in part of its parameter space will distinguish the SM from this model, or make a final decision of the window for the model.
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Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No.19755001, 19555003 and 19675047. We would like thank Dr. Zhentao Wei for useful discussions.
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