Net-to-leading order QCD corrections for the B0B0 mixing with an extended Higgs sector

13 ELSEVIER

Nuclear Physics B 523 (1998) 40-58

Next-to-leading order QCD corrections for the B°B ° mixing with an extended Higgs sector J. Urban, F. Krauss, U. Jentschura, G. Soft lnstitut fiir Theoretische Physik, Technische Universit~t Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany

Received 20 October 1997; revised 27 November 1997; accepted 9 January 1998

Abstract We present a calculation of the B°B6 mixing including next-to-leading order (NLO) QCD corrections within the Two-Higgs Doublet Model (2HDM). The QCD corrections at NLO are contained in the factor denoted by r/2 which modifies the result obtained at the lowest order of perturbation theory. In the Standard Model case, we confirm the results for r/2 obtained by Buras, Jamin and Weisz [Nucl. Phys. B 347 (1990) 491 ]. The factor r/2 is gauge and renormalization prescription invariant and it does not depend on the infrared behaviour of the theory, which constitutes an important test of the calculations. The NLO calculations within the 2HDM enhance the LO result up to 18%, which affects the correlation between MH and Vt~. @ 1998 Elsevier Science B.V. PACS: 12.15.Ff; 12.60.Fr; 14.80.Cp; 12.38.Bx Keywords: Mixing; Oscillations; 2HDM; CP violation; CKM matrix

1. Introduction The B-meson plays an important role in present day physics. B ° B ° mixing as well as B-meson decays can be used to determine the CKM elements and to investigate C P violation within the Standard Model ( S M ) . Two large experiments, namely the B A B A R Collaboration at SLAC [2] and the Belle Collaboration at K E K [3], will start taking data in the near future. The major subject of research in those experiments is the determination of the C K M matrix elements. This may also provide first hints on physics beyond the Standard Model. In general terms B ° B ° mixing is a flavour changing neutral current ( F C N C ) process generated through weak interactions. In the SM, the process is generated at the lowest order of perturbation theory via the box diagrams displayed in Fig. 1. Due to the 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S0550-3213(98) 00043- 1

J. U r b a n et a l . / N u c l e a r b

d

..

41

Physics B 523 (1998) 40-58 b

, ,

d

t

f:

- -[ff+- d

b

b

d

d

b

d

b

Fig. 1. Box diagrams for the BOB ° mixing in the framework of 2HDM. We have also taken into account crossed diagrams, which are related to the original ones by a Fierz transformation.

large difference in the masses of heavy particles in the box (W +, H ± and t-quarks) and much lighter external particles (b- and d-quarks), it is possible to disentangle long- and short-distance effects. This can be accomplished within an effective theory. The proper separation of short- and long-distance QCD effects at next-to-leading order was first presented for the SM in [ 1]. We are utilizing here the same framework of renormalization group improved perturbation theory. Our purpose is to include two Higgs-boson doublets as an extended Higgs-boson sector. Such models are usually called Two-Higgs Doublet Models (2HDM) [4,5]. Two particular models which are usually considered (so-called Model I and Model II) differ in the couplings of the charged Higgs bosons to fermions (see for example Ref. [4]):

£Hua - gwV~d [m. cot(/3)PL -- md cot(/3)PR] , x/2rnw

(1)

1I gwVua [m, cot(/3)PL + md tan(/3)PR] . £H,,d- v~mw

(2)

Here gw is the weak coupling constant, mw is the W-mass, and the projectors PL,R are defined by

PL,R =

1 TY5 2

(3)

The subscripts u and d denote up- and down-type quarks, respectively. As we neglect all quark masses except mr, both models are identical in our case. Thus our calculations are valid for all values of tan/3 for the Model I, but only for tan/3 << mr/rob in the Model II. Looking at the Feynman rules for the Higgs-quarkquark coupling, it is easy to verify that in Model II, the term proportional to mb becomes important for large values of tan/3. Due to the projectors in the vertex, we would end up with different operators for the tan/3 and cot/3 parts. However, it should be stressed here that Model II is more interesting, because this is the choice favoured by the Minimal

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

42

Supersymmetric Standard Model [4,6]. Large values of tan/3 will be considered in a future publication. We continue our discussion with a short review of the leading order (LO) calculations. The basic idea is rather simple. First, one has to calculate the box diagrams in Fig. 1. They can be evaluated in the Feynman-'t Hooft gauge for the W-boson. This leaves us with a physical W-boson and a would-be Goldstone boson (unphysical scalar Higgs boson) of the same mass and with couplings proportional to the quark masses. As all quark masses except the top-quark mass are equal to zero to a good approximation, t is the only active flavour in boxes containing Higgs particles or unphysical scalars. In the remaining box diagrams, the effects of u- and c-quarks are taken into account by the GIM mechanism. Evaluating the box diagrams in this framework leads to the well-known Inami-Lim functions [7-10]. The scaling behaviour from the matching scale /,t0 = m w of the full and effective theory down to a lower scale is then determined by the one-loop QCD correction to the effective vertex generated by the previous procedure of integrating out the internal heavy degrees of freedom. The renormalization group equation gives us the factor ~LO describing the effect of scaling [ 11 ]. The effective Hamiltonian for B°B ° mixing at LO reads IG~, ~, , 2_ Heff = ~ 77.~m~t VtdVtt,) T]LoS(Xw,XH)(~LL,

(4)

where

and

S(xw, XH) = Sww(Xw) + 2SwH(XW, XH) + SHH(XH).

(6)

The LO Inami-Lim functions Sww(Xw), Swtf(xw,xH) and SHH(XH) can be found in Appendix A. The arguments of the Inami-Lim functions will be denoted XW,H = 2

2

lll t / m W . t l .

The factor ~LO is determined by the /3-function [12] and the anomalous dimension 3' to be obtained by evaluating the diagrams of Fig. 2. It reads

~LO

-_

L ~ J

-

Ions(m,)

~ 0.85.

(7)

For later convenience we define r/LO = aS(row)(6/23). To evaluate the matrix element of Heft we have to employ

(/~° f('-SLLIB°)

2 ~ = 5 BB (/~ ) fbm-8,

(8)

where fB is the B-meson decay constant and BB(tz) parameterizes deviations from the vacuum insertion approximation.

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

43

d

b

d

b

b

d

b

d

d

b

d

b

Fig. 2. QCD corrections to the effective four-quark interaction. They are needed in the calculation of the anomalous dimension at LO and for the correct separation of long- and short-distance contributions.

2. Explicit QCD corrections The perturbative result for O(o~s) corrections in 2HDM will be presented in this section. The results can be obtained by evaluation of the diagrams of Figs. 3 and 4. We performed our calculation in an arbitrary covariant ~-gauge for the gluon and we employed the Feynman-'t Hooft gauge for the W-boson. As one easily notices, the diagrams a, b and f-i have the "octet" structure 7~, ® 7%', whereas the diagrams c-e and j have "singlet" structure ] ® ] in colour space. The double penguin diagram k contributes to the considered process at the order of O( l / m 4 ) which is negligible for our purpose. Diagrams containing vertex and self-energy corrections (diagrams c, d and el) lead to UV-divergent integrals and hence they have to be regularized and renormalized. We are using dimensional regularization with anticommuting ys. This corresponds to the NDR scheme ("naive" dimensional regularization scheme) [ 13,14]. We performed the necessary renormalization in the MS scheme [ 13,15]. All other diagrams are UV-finite. Furthermore, one notices that the diagrams g-j contain infrared divergences. To deal with them we keep the external quark masses whenever necessary. As we will see, the external quark masses do not affect the final result for the Wilson coefficient. This justifies the method we have chosen. We could also treat the infrared divergences with the method of dimensional regularization [16], but our intention was to verify the results provided by Buras and collaborators in [ 1] and to extend these calculations to the 2HDM. The (0(ors) corrections to the Hamiltonian (4) have the following structure: (dHeff) = 1 G ~ m2 4 7r2 w

where

k

~(VtdWt*b)2U(xw,XH),

(9)

44

,I. Urban et al./Nuclear Physics B 523 (1998) 40-58 b

d

W*

b

d

W+

a)

b

b)

d

b

d

C,t

d

W+

~

U,C~t

b

W+

d

~)

b

W-

d

b

d)

d

b

b

dW

~)

W-

-

-

.....__. d

b

f)

Fig. 3. Diagrams for the NLO calculation in the SM, which are convergent in the limit of vanishing external masses. Additional diagrams with one Higgs boson and two Higgs bosons have to be calculated in the 2HDM. Furthermore, we have to consider crossed diagrams.

with k E { L L , 1,2,3}. C F is the colour factor defined by C F Nc is the number of colours. The operators (~i read

=

(N~ -

1)/(2No) and

11) 'PR),

o : : [deLhi lap.b] + laP.b] [aPLb],

12)

13) 14)

The operator (.01 stems from diagrams g and h in Fig. 4, whereas the operators 02 and 03 follow from diagrams i and j, respectively. Since the relevant operator (-OLLis self-conjugate under Fierz transformations, we will use

J. Urban et al./Nuclear PhysicsB 523 (1998) 40-58 b

d

W+

d

b

W-

b

d

W

g)

d

45 ...,.... d

~

b

h)

W+

b

i)

d

W+

b

j)

b

d

d

b

k) Fig. 4. Diagrams for the NLO calculation in the SM, in which infrared divergences appear when one sets external masses to zero. The last diagram is the so-called double penguin, which does not contribute to the mixing at the leading order in (mB/mw)2. Additional diagrams with one Higgs boson and two Higgs bosons have to be calculated in the 2HDM. Furthermore, we have to consider crossed diagrams.

T,,®Ta=CAi®i=~

1-

i®i

(15)

to retain only one operator. Nevertheless, to keep the calculations transparent, we will abandon the distinction between octet and singlet at the very end only. The coefficient functions ¢ in Eq. (10) can be decomposed as • rb!i) ~-.1 = XI ''sM)(xw) + Xj( i , H ) ~,Xw,XH).

(16)

The functions X.I/'sM).(xw) were already given in Ref. [1 ]. We have recalculated them and we confirm these Standard Model results, X(1,SM) LL ( x w ) = L(I,SM) ( x w ) + 2 ( + 2~:gm + 2 ~ : l n ( x ~ o ) +61n(x.o)Xw--z--- Su~(xw),

(17)

46

J. Urban et al./Nuclear Physics B 523 (1998) 4 0 - 5 8 X (8'SM) (

LL

XW) = L (8'SM) ( XW) + [ 2 ( + ( 3 + ( ) l n ( x t , x,1) + 2(giRl Sww(xw),

XIIs'sM) (xw) = - ( 3 + ()Sww(Xw), (8,SM) ~

X2

(18) (19)

~

~xwj:-2X~'sM~(xw) : - ( 3 + ~ - )

~,

Dlbllld

- -Tl -n-(- X2 ~ - ~ h ) S w w ( X w ) , m~

-

(20)

mb

with all other X equal to zero. We have introduced the following abbreviations:

XW.H= glR =

m? ~ , rn~.H

-Vd.t, =

x,t ln(xa)

m J,/, ,~ , rn[v

Xm =

#6 , m~v

(21)

x/, ln(xt,)

(22)

Xd -- X h

Here. mr stands for the top quark mass renormalized at the scale ;zo in the MS scheme. The function X LL ~j'n) reads X(1,II) CL (.vW, xtt) : L (I,H) (Xw,.~:H)

f

oJ

+ 2 ( + 2 ( g i n + 2 ( I n ( x ~ , , ) + 6 ln(xm) Z x i - i=H.w Oxi x(2SwH(Xw, XH) + SHH(xw,X.)).

(23)

The analytical expressions for the functions Sww, SWH, SHH and L (i'SM'H) a r e listed in Appendix A. Note that powers of tan/3 enter in the definition of the various S and L indicating the number of Higgs bosons involved. The remaining newly calculated functions XLL (8.14), XI 8''~), X~8'H)_ and X~ J'H) can be obtained respectively from XLLC8,SM) (18), XI s'sM) (19), X (8'sM)2 (20) and X~ l'sM) (20) by changing L (i'sM) (xw) to L (i'H) ( x w , XH) and Sww (Xw) to (2Swm(xw, xm) + SHH(X,q) ). We have obtained the function L (i'SM'H) by using computer algebra systems, especially the program FORM to evaluate the Dirac structure and Mathematica 3.0 [17] for summarizing all terms. Maple [18] was used for some integrations. Note that the results of the particular diagrams exhibit non-trivial dependence on the gauge parameter ( and the IR masses, but the functions L (i'SM'H) a r e gauge as well as renormalization prescription invariant. They do not depend on the external quark masses either. Terms depending on sc,xh,x,l, or xt,o are proportional to the LO Inami-Lim functions Sww, SwH or SHH. This reflects the exact factorization of long- and shortdistance effects.

3. Matching and running We start this section with the observation that Eq. (10) is obviously unphysical because the coefficient functions X~i) are gauge dependent. Actually this is nothing but an artifact of the specific way we have regularized the IR divergences. Other regularization

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

47

schemes being not based on the use of small masses for the external quarks would change the obtained result. Dimensional regularization or a gluon mass, for example, would leave us with only one operator OLL at this stage. To take proper account of this fact, we have to evaluate the matrix element of the physical operator OLL up to order O ( ~ s ) using the same IR regularization prescriptions as before. Yet the one-loop amplitude of OLL as given by Fig. 2 results in the same unphysical operators Oi,2.3 with the same coefficients,

(OLL ( U'O ) ) Hoop = O~S(/ZO)

47r

r

(1)( o

-

"(s)(/zo)]i@i(Ok)tree,

k (24)

where the sum over k = LL, 1,2, 3 is understood. The functions Xa,k(/~o ) (i). are 1

X(X) -[xd ln(xa) -- Xb ln(xb) ] A,LL _-- - 3 + 2~:ln(xu,,) + 2~ - 2 ~ Xd -- Xb

(25)

X (8) A,LL - - - 6 1 n ( x ~ o ) - - 5 + 2 ~ + ( 3 + ( ) l n ( x d x h ) -

-

--2~

1

[Xd l n ( x d ) -- x b l n ( x b ) ] ,

Xd -- Xh

(1)--lx(8)(3+()mdmb_ m X4,3----'~ d,2-2 _ _d2m~ -

X ,a,1 (8) = - ( 3

-

In

( xd )

(26)

,

+().

(27)

(28)

Eq. (24) allows us to write for the matrix element of the effective Hamiltonian up to order O ( a s ) in the following form: (Heee) = /\"eft" ~ ( o ) 4- An~ff (1~) - 1 G2m2(V, dVt*b)2CLL(tzo)(OI.L( #0) ) vJoop

4 zr~

(29)

with (OLL(/,ZO))I.loop as given in Eq. (24) and CLL(/zo) reads

CLL(Izo) = S(xw,x14) + ~as(tzo) D~~x w , x H , X~,o).

(30)

The change due to the inclusion of the charged physical Higgs results in replacing S w w ( x w ) and D s M ( x w ) by S2HDM ( Xw, XH ) = SWW( XW ) + 2Swrl( xw, XH ) 4- Sml( X~t ) , D2HDM(Xw, X•, Xm) = DSM (Xw, Xz,o) + DH(Xw, XH, X~o),

where DSM (XW, X~,o)

= CF

~ L (l'sM) (xw) k

(31) (32)

J. Urbanet al./NuclearPhysicsB 523 (1998)40-58

48

+CA{L(8'SM)(xw)+ [6ln(xlzo)+51Sww(xw) } , DH ( xH, xw, x~o) = CF { L (1'/4) (Xw, xH ) +

o?

xi-- + 3 i=H,w Oxi

61n(x~,) ~

2

(33)

SwH(xw, xH) + Sml(xw, xH))

+ CA {L(S'm(xw,x.) + [6 ln(x,~,,) + 5] (2 Swn(Xw, xH)

+ SHH(Xw, xH))} .

(34)

To obtain this, we have used Eq. (15) and the behaviour of OLL under Fierz transformation. The finite quantities in front of the LO Inami-Lim functions in Eqs. (33) and (34) (i).) are the remnants of the functions Xj,kttzo of Eq. (24) The coefficient CLL(#0) in Eq. (29) exhibits no dependence on the choice of the gauge and the external quark states. What remains to be done is the running of our effective theory down to a lower scale. In Ref. [ 1] it was shown that a good choice for this matching scale is/-to = mw resulting in x~,, = 1 and that the physical observables do not depend on the choice of this scale. The scaling down from this matching scale to the mesonic scale is pursued by means of the renormalization group equation. The renormalization group equation for the Wilson coefficient CLL(#) in the SM reads

[#dlz

Y(g)l CLL(#) =0,

(35)

with the initial condition CLL(/-t0 = mw) given in Eq. 30). Expansion of the anomalous dimensions and the/3-function yields g3 /3 (g) = -/3o (,:i.-~Tr)2

y(g) =y(0)

/3, (4n.)4 gS

....

(36)

f g4 +y(l/+... (477")2 (477")4

(37)

The coefficient y (I) was calculated by Buras and Weisz in Ref. [13]. Note that y(J) is renormalization scheme dependent [ 13,19,20]. The coefficients relevant in our further calculation are 1

/3o : ~ ( 11N,. - 2hi), 34

~

(38)

10

/31 = --~N~ - ~-Ncnf

-

(39)

2Cynf,

y(o) = 6No - 1

Nc

[

(40)

y ( l ) _ N o - l -21 + - - - - - N o +

2~

Nc

3

41

5 nf '

(41)

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

49

where Nc is the number of colours and nf is the number of active flavours. The solution to the renormalization group equation for the Wilson coefficient reads

ge(mw)

CI.L(t-t) = exp

g ~---~] LL(mW).

--

(42)

[¢(iz) The evolution from the initial or matching scale mw to a lower one may be performed as in a massless theory because the top quark has been integrated out previously and appears in the matching condition only. At the NLO, Eq. (42) can be approximated by

CLL(/-t) ~ exp

'2(mw) y(o)f/(47r) z + y(l)g4/(477"14] / df /30gS/(41r)2-~_ ~ jCu,(mw) ~(#) as(row) +/31/(4,rr)2a} 2 + /30(4~-12 +/3lot}

as(Ix)

CLz,(mw), (43)

where we have used Eqs. (36) and (37). After elementary integration, we obtain CLL(/X) =~LO 1 +

2/302/31

CLL(mw),

(44)

where ~Lo is the well known LO scaling factor given in Eq. (7). In conclusion we find the following expression for Heff at NLO: 1

2

~i,~ Heft = ~ ~GFm2wt~V,rdVtb)2~?2(Xw, xH)S2HDM(XW, XH)fgLL

(45)

7]2(XW'XH)=Ols(rtlW)Y'°l/(2fl°)[ Iq-OIS(mW)(D2HDM(XW'XH)4~---~--~S2HDM(xw,xH) q-Z)] ,

(46)

with

where T( l ) Z - 2/30

T(o) 2/3~/31.

(48)

Neither the factor r/z, nor the matrix element of (DLL are dependent on the low-energy scale /.t ~ me, up to O ( a ~ ) .

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

50

0.012 0.0115 0.011 0.0105 0.01 "1:3

0.0095 0.009 0.0085 0.008 0.0075 0.007 0.16 0.17 0.18 0.19 0.2

fB BB 1/2

0.21 0.22 0.23 0.24

[GeV]

Fig. 5. The CKM element Vt,t is plotted versus the B-meson decay constant times the square root of the B-parameter. The shaded areas represent the allowed values when errors of Arab and mr are included. There is a small overlap between the LO and the NLO region shown in the picture.

4. R e s u l t s

We want to consider now the mass splitting Arab between the electroweak mass eigenstates BH and BL. The mass splitting Ant8 is directly given by the B°B ° mixing amplitude 1

~,,,8 = - - I < B ° l H e f f l ~ > l I1l B

G2 = 6rrF mw ( VtdVa,). 2S2HDM(Xw, xH)rl2(Xw, xu)BBf'Bme.~

(49)

We obtained the result in Eq. (49) by using Eq. (8). Be is the renormalization prescription independent B-parameter. It is defined by [ 1 ]

Be(I-t)Ots(l~)

6/23 [1

~s(/,z) (,y(I) 4zr \ ~flo

,,if(o)

~]

7~o13'j J "

(50)

The term proportional to a s results from NLO corrections. We use Be = 1.31 i 0.03 [21 ] in our numerical calculation. The current experimental mean value of the mass splitting is given by Ame = (3.05 ± 0.12) x 1()-13 GeV [22]. We use the top-quark mass m pole r = 175 ± 6 GeV [23] which corresponds to mr(row) = 176 GeV. Furthermore, we employ mw = 80.33 GeV, V,, -- 0.9991 [24] in all plots. The meson decay constant is set to 175-t-25 MeV [21]. From Fig. 5 we can deduce that the difference between the LO calculation and the pole corresponding NLO calculation for Vm is approximately 7%, when mt is used in the LO expression. Hence, the NLO calculation enhances the result for the mass splitting up to 15%. This can also be concluded by a direct comparison of r/LO = a s ( M w ) 6/23 and r/2 within the SM. We have found r/2 = 0.4942 and r/LO = 0.575 l.

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

51

8 .10 "13 ',, ,

-Vtd = 0.004 ......... Vtn = 0.005 - - Vt~ = 0 . 0 0 6

7 .10 "13

> t,~

6 .10 "13 ',

. . . ',,,

..... Vtd=0.007 ......

5 .10 -T3

....

Vtd = 0.008 Vt,j = 0.009

'-'~ 4 .10 "13

E

'<1 3 .10 -13 2 .10 -13

~

;...~.i.21 ..~;.??.i..21.~i..?~.L-i ~L~.LL~: -i

100 200 300 400 500 600 700 800 900 1000 mR [GeV]

Fig. 6. The mass splitting within the 2HDM including NLO corrections for different values of Vtd and for tan/3 = 1. The top-quark mass is set t o m p ° l e = 175 GeV. The shaded strip is the experimentally allowed region. The factor fBB~/2 is fixed to 0.2 GeV in this figure. The limit mH ----'cx~ yields the Standard Model results.

> c~

3 . 1 0 "la

i:~i~iiiiiili!:i:~!ii~i~i~i:i:i::iiii~ii?!~i ~il;~ ~i!i i~iiii~i!~i :~I~iil:i:!iiii~i!~iiiii :i ! ii!ii!i~iiiii :i:!:~i:i~L!i:!~i?i~i' i:i:~

•' . .

E 2.10 -la <1 ........... m t = 175 GeV ] --- mt=181GeV.L 1 -10 -la

100 2 0 0 3 0 0 4 0 0 5 0 0 600 7 0 0 8 0 0 9 0 0 1 0 0 0

mR [GeV]

Fig. 7. The mass splitting within the 2HDM for different values of the top-quark pole mass. In this figure, r r~l/2 Vtd = 0.009, tan/3 = 1, jBo n = 0.2 GeV. If we c o n s i d e r an e x t e n d e d H i g g s sector within the f r a m e w o r k of the 2 H D M we have to d e c r e a s e the related C K M elements, to get an overlap b e t w e e n the allowed range for

dmB and the H i g g s mass. W e r e c o g n i z e for e x a m p l e in Fig. 6 that we can not find a physical H i g g s b o s o n with a mass smaller than 1 TeV for Vta = 0.0086, a s s u m i n g that the ratio o f v a c u u m e x p e c t a t i o n values tan/3 = 1 and fBv/B-BB = 0.2 GeV. If we decrease tan/3 we have to d e c r e a s e Vtd even more. Fig. 5 is in full a g r e e m e n t with Fig. 6 in the S M limit mH ~

exp. I f we assume Vtd = 0.009 and if we furthermore take into account

the errors for f R y / f i R and mt the 2 H D M can not be distinguished f r o m the S M for H i g g s - b o s o n masses larger than 1 TeV. In Fig. 7 it is indicated that we have a relatively sensitive relation b e t w e e n the t o p - q u a r k mass and the possible H i g g s - b o s o n mass. The c o m p a r i s o n o f N L O and L O calculation is shown in Fig. 8. We have plotted the mass splitting o v e r the H i g g s mass for two typical values o f Vta and a ratio of

52

J. Urban et al./Nuclear Physics B 523 (1998) 4 0 - 5 8

5 .10 "~3

>

J

........... Vta = Vtd = Vtd = Vt~ =

4 .10 13 t\.k

t',,

0.005 0.005 0.007 0.007

LO NLO LO NLO

3 .10 "13

E <1

2 .10 -13

1 -10"13

lo0 200 300400

6o0 r0o 800 9001000

mR [GeV] Fig. 8. A comparison between L O calculations and NLO calculations within the framework of the 2HDM. _,, h i / 2

Here, tan/3 = 1.25, j B O B

pole

= 0.2 GeV and m t

= 175 GeV.

the vacuum expectation values tan/3 = 1.25. The difference between NLO and LO calculation approximately amounts to 18%.

5. Conclusions B

We have calculated the B ° B ° mixing within next-to-leading order with the inclusion of two charged Higgs bosons. The NLO calculation leads to an effect of approximately 7% for Vtd in comparison to the LO calculation within the SM. This is indicated in Fig. 5. We have verified that the scheme developed by Buras, Jamin and Weisz [ 1,13] for a proper separation of long- and short-distance effects in QCD remains valid in our 2HDM calculation. This can be considered as a good cross-check of the calculations presented here. The inclusion of NLO corrections lowers the mass splitting in the SM as well as in the 2HDM. We find, for example, a difference between the pure LO and the NLO calculations of approximately 18% for mr-/= 200 GeV and 17.5% for mH = 400 GeV. Hence the NLO contributions play an important role in the 2HDM as they correct the LO result by about 18%. The inclusion of the two Higgs bosons leads to an increase of the calculated amplitude, which depends on the mass and the vacuum expectation values of two Higgs doublets. In all practical calculations, the lower bound of the Higgs-boson mass was assumed to be 100 GeV and the upper bound was 1 TeV. To obtain a mass splitting within the experimental allowed region, it is necessary to decrease the CKM matrix elements for small values of tan/3 and mr1. Our calculations are not valid for higher values of tan/3 (region near tan/3 = 40), because it is then necessary to introduce new operators. In the limit of very heavy Higgs bosons we verify the well-known results for the Standard Model.

J. Urban et al./Nuclear Physics 13 523 (1998) 40-58

53

Acknowledgements We are grateful for fruitful discussions with K. Schubert, B. Spaan, R. Waldi, Th. Mannel and U. Nierste. J. U. thanks M. Misiak for valuable discussions and many useful advices. We would like to thank S. Zschocke and Ch. Bobeth for critically reading of the manuscript. We acknowledge support by DFG, GSI (Darmstadt) and BMBE

Appendix A A. 1. Special functions After carrying out the second integration, the dilogarithm or Spence function appears in our results. It is defined by

L i 2 ( x ) = _ f dt l n ( 1 - t ) - ~-~ xn n--5-, Ix I < 1. 0

(A.1)

n=l

The following useful relations hold: Li2( 1 - x) + Li2( 1 - 1/x) = __1 ln2(x) ' 2

(A.2)

77-2

Liz(x) + Li2(1 - x) = - ~ - in(x) ln(1 - x).

(A.3)

A.2. lnami-Lim functions and NLO in the SM We list now the Inami-Lim functions and the well-known functions

L (i'SM) ( x w ) .

3

Sww(xw)=xw,

+4(1-xw)

2(1-Xw) 2

2(1-xw)3J '

1 xnxw ( 1 + XH 2xHIn(xH) SHt4(xn)-- tan4(/3) 4 , ( 1 -- xn) 2 + ( i - - x - - - ~ J '

(A.4) (A.5)

1 xHxw f (2Xw - 8x,4) In(XH) 2SwH(Xw,XH) -- tanZ(fl ) 4 [,, (1 ~ x~)2-T~n---~ x--~w) 6xw ln(xw) + (1 - X w ) 2 ( x n - x w )

8 -- 2xw - (1-xH)(1-xw)

)

(A.6)

"

The function SwH(Xw, XH) is obtained by including the charged would-be Goldstone bosons or in a more direct way, by choosing the unitary gauge for the W-boson right from the beginning. The functions L (i'SM) c a n be decomposed as L (i'SM) ( x w ) = W W (i) ( x w ) q- 2 W q b(i) ( x w ) q- ¢I9~ (i) ( x w ) ,

where

(A.7)

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

54

WW(1) ( xw ) = WW~,' ) ( xw ) - 2WW:,] ) ( Xw ) + WW,(,~ ) ( Xw ),

(A.8)

WW[,~)(xw) = (4Xw + 38x W + 6X~v)ln(xw) (Xw - 1 )4 (12xw + 48x W + 1 2 x 3 ) L i 2 ( l - 1 / x w )

+

(XW 9

--

I) 4

•

")

(24xw + 48x~v)L12( 1 - x w )

+

(Xw-

3 + 28Xw + 17X~v ( X w - 1) 3

1) 4

2WW~,])(xw ) _ 2 ( 3 + 13Xw) ( x w - 1) 2

(A.9)

2Xw(5 + l l x w ) l n ( x w ) ( x w - 1) 3

12xw(l + 3xw)Li2(l - l/xw) ( X w - 1) 3

24xw(l + xw)Li2(l -xw) ( x w - 1) 3

'

(A.IO)

WW,~,], ) (Xw) = 3,

(A.11)

X~v(7 ' + 52xw - 1 lx~v ' )

ch@( 1~( xw ) =

4(Xw-

1) 3

+

3x~v(4 + 5xw - x~¢) ~ ln(xw)

3X~v(3 + 4Xw - x~v)Li2( 1 ( X w - - 1) 4

4

2(Xwl/xw)

+

1) 4

18x~vLi2(1 - Xw) ( X w - - 1) 4 (A.12)

2 w q ~(1) (Xw) =

4X2w(l I + 13Xw) 2Xw(5 + X w ) ( l - 9 x w ) + (XW - - 1) 3 ( X W - - 1) 4

ln(xw)

24X~v(I + 4Xw + x w ) L i 2 ( l - 1 / x w )

( x w - 1 )4 48Xw(1 + 2 x w ) L i 2 ( l - Xw)

(xw

-

(A.13)

1 )4

i4q+'(S)(Xw) = WW~tS) ( xw ) - 2WW}.S) ( Xw ) + WW~.(8) ( Xw ),

ww~,S' ( x w ) = 2 x w ( 4 - 3Xw) l n ( x w ) (xw

q-

-

(12xw - 12x w - 8X3w)Li2(l - 1/Xw)

1 )3

(xw

(8 - 12xw + 1 2 x 2 ) L i 2 ( l - Xw) (xw-

1) 4

(A.14)

-

1 )4

(23 - x w )

(Xw-

1) 2

(A.15)

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

2WW[8) (xw) -

2(2 - Xw)Tr2

(8 - 5xw) ln(xw)

3xw

(xw - 1 )2

(6xw + 4x2)Li2(1 - 1/xw)

+

(8 + 12xw - 6x2)Li2( 1 - Xw)

xw(xw-

1) 2

XW(XW-

15

(8)

4

2

,

(A.17)

1 l x 2 ( 1 + Xw)

'I)q)(8) (Xw) =

4(Xw-

1) 2 (A.16)

(xw - 1) '

WW,,, (xw) = - 2 3 + ~ r

55

+

3Xw) ln(xw) 2(Xw- l) 3

x3(4-

1) 2

x 3 ( 3 - 3xw + 2x2)Li2(1 - 1/xw) ( X w - 1) 4 q- X~V(2 + 3XW -- 3 x 2 ) L i 2 ( l - xw) (XW-

2W4) (s) (xw) -

30x2 ( X w -- 1) 2

(A.18)

1) 4

12x~vLi2( 1 - 1/xw)

+ 12x 3 ln(xw) (XW -- 1) 3

( X w -- 1) 4

12Xw(2 - x 2 ) L i 2 ( 1 - xw)

(A.19)

(X W -- 1 )4

A.3. NLO in the extended Higgs sector The L (i'H) c a n be decomposed in the following way 2

L li'/4) (xw, Xn) = tan2 (fl) W H (i) ( x w . XH) +

4)H (i) (Xw, xH) + tan4(fl--------~HH(i)(xtt).

(A.20)

We list now the newly obtained results for the explicit QCD corrections in the extended Higgs sector:

HH(I)(xi) =

Xw

" 4)4)(I)(XH) + 6 (ln(x~/) - - l n ( x w ) )

xH

(2x~4 ( 13 + 3xt4) In(x/4)

2 W H ( l ) ( x i ) = x w \ (x/4-- 1)3(XH--~X~w)

--

3 x i - - Smt(xi) , (A.21) 3xi i=/4,W Z

2xi4(9 + 7xtt + 7xw - 23xwxH) (Xw-- 1)2(X/4 -- 1) 2

2X~4( 18 -- 6XH -- 44Xw + 13XHXw + 9X/4XZw) ln(xw) (XH-- 1)2(Xw-- I ) 3 ( X H - - X w )

56

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

2XHXw( 5 -- 27Xw + 6X~v + 6XHX~v ) ln(xw) ( X H - - 1)2(Xw-- 1)3(XH--Xw)

24x}tln(xH) ln(xw)

24x2Li2(1 - 1/XH)

+

( X H - - 1)2(XH--Xw)

(XH-- 1)3(XH--Xw)

24XnXw(1 + xw)Li2(1 - 1/Xw) ( X w - - 1)3(XH--Xw)

4 8 x w x H L i 2 ( 1 -- x w ) ¢-;7=

(A.22)

-

'

( X H ( 3 1 -- 1 5 X H - 1 5 X w - XHXW) 5~-~H~-i)~(-7-~W-- ~jS

2(PH{1)(xi) =X2w \

XH(7 + 21XH -- 12X 2) ln(xH)

2(x.-+ q-

I ) 3 ( X H - - XW)

XH(7 -- 9Xw + 36X~v -- 18X3w) ln(xw) 2 ( X H - - 1)2(XH--Xw)(Xw-- 1) 3

XH(8 -- 36Xw + 9X 2 + 3X 3 ) ln(xw) (X.--

1)2(XH--Xw)(Xw-- 1) 3

x~4(ll - 4 5 X w + 18x w) ln(xw) 2 ( X H - - I)2(XH--Xw)(Xw-- 1) 3

+

6XH In(XH) ln(XW) (XH -- 1)3(XH -- XW)

6XH(1 + XH -- x 2 ) L i 2 ( 1 - I / x H ) (XH-- 1)2(xH--xw)

q-

6xH(1 + 2x 2 -

x3)Li2(1 - l/xw)

(XH -- X W ) ( X W -- 1) 3

12XHLi2 ( 1 -- Xw)

"X

(A.23)

HH(S) ( xi) = xW cD(1)(8) ( xtf ) + 6 (ln(xn) -- ln(xw)) SHH( Xi ) , XH

( 2 4 x H x w L i 2 ( 1 -- x w )

2 4

6x2H(5Xw - XH + 3X~vXH)Li2( I -- l / x w ) (X u -- 1)2(XH- X w ) ( X W -- 1)2Xw

6XH(2X~v -- IOXHXw 4- XHx2)Li2( 1 -- 1/Xw) (XH-

I)2(XH- X w ) ( X w -

1) 2

-+ 6 x Z ( 5 x w - XH -- 8X 2 + 2XHXw)Li2( 1 -- XH) (XH -- I)2(XH -- X w ) ( X w -- I)2Xw

q- 6(X 2 -- XttXw + 2x2x~v)Li2( 1 - XH) (XH -- 1)2(Xt/ -- X w ) ( X w -- 1) 2

(A.24)

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

6x2(--x~/+

57

5xw)Li2( 1 - 1/xN)

(XH -- I)2(XH -- X w ) X w 6x2(5xw

- x1-1 - 8x 2 + 2xHx~v)Li2( 1 - x t t / x w )

1)2(XH- Xw)(Xw-

(XH-

1)2Xw

(XH -- I)2(XH -- XW) (Xw -- 1)2

6XH( l -- XH -- l n ( X H ) ) (X H -- 1 ) 2 ( X w -

6x2 ( 5Xw -

xH

+

6 X H ( 2 X w -- l ) l n ( x w )

1)

(XH -- 1 ) ( X w -

1) 2

8X~v)ln(xn) ln(xw)

-

(X14 -- 1)2(Xt4 -- X W ) ( X w -- 1)2XW 12x2(__xt/_x__ww+ x~)_ l n ( x H ) ln(_xw)

(A.25)

-} (XH -- 1)2(XH -- X w ) ( X w -- 1) 2 J

(

+

2 ~ H ( 8 ) ( x i ) =X2w \ 2 ( x r - / -

(2x~

-

1)(Xw-

1)Xw

7XHXw + 2 x Z x w + 2X 2w + XHXw) 2 ln(xn)

-

2(xH-

1)2(XH- Xw)(Xw-

I)Xw

XH(7 -- 7Xu + 4Xw -- 6X~v) l n ( x w ) 2 ( X H -- I ) ( X H -- X w ) ( X w -- 1) 2 +

-

(XH -- 1)(XH -- X w ) ( X w -- 1)2Xw

X 2 ( 4 -- 6Xw + 3XHXw) i n ( x H ) l n ( x w ) (XH -- I)2(XH -- X w ) ( X w -- 1)2Xw -t x H ( x ~ - 3X~v + 6 x 3 - 3x 4 ) l n ( x H ) l n ( x w ) (XH-- I)2(XH--Xw)(Xw--

1)2X 2

X n ( 3 X 2 + 2Xt-lXW(2 + XW) -- X 2 ( 1 + 2 X w ) )Li2( 1 - l / x , q ) (XH -- 1)2(XH -- Xw)X2w

2 x 2w _ x 2 ) L i 2 ( 1 - Xt-l) (4XHXw - 6 x 2 x w + 3 x H (XH-

1)2(XH- XW)(XW-

I)2XH

( 4 x Z x w __ 6xHx~v 2 * - x ~ + 3x~xZw)Li2(1 - x u )

(xH- 1)2(xH+

xw)(Xw-

1)2X~v

2 x 2 ( 6 - x~v - 3xH + X w x H ) L i 2 ( l - 1 / x w )

(xH- 1)2(xH-xw)(xw- l) 2 xN(3x 2 + 4xHxw - x~)Lia(1 - 1/xw) (XH -- 1)2(XH -- X w ) ( X w -- I)2X~v

+ +

2 '~ "~ " (4XHXw -- 6X2HXW + 3XHX~v -- X~v)L12( 1 - XH/XW)

( x H - 1)2(xH-xw)(xw- l)2xH x ~ ( 4 x w - 6x~v - xH + 3xHx~v)Li2( 1 -- x H / x w ) (XH-

I)2(XH-

Xw)(Xw-

I)2X2w

58

J. Urban et al./Nuclear Physics B 523 (1998) 40-58

6 x H L i 2 ( 1 -- x w ) - (a~--~w)~w---~)

2) .

(A.26)

We h a v e used the capital W, q~ a n d H to label the W - b o s o m the c h a r g e d w o u l d - b e G o l d s t o n e b o s o n a n d the p h y s i c a l H i g g s b o s o n , respectively, i.e. W ~ d e n o t e s a d i a g r a m with o n e i n t e r n a l W - b o s o n a n d o n e internal u n p h y s i c a l scalar.

References [ I ] A.J. Buras, M. Jamin and RH. Weisz, Nucl. Phys. B 347 (1990) 491. [21 BABAR Collaboration, Technical Design Report for the BABAR Detector (1995). [ 31 The Belle Collaboration, Letter of intent for a study of CP violation in B meson decays, KEK report No. 94-2 (1994). [4] J.E Gunion, H.E. Haber, G. Kane and S. Dawson, The Higgs Hunter's Guide (Addison Wesley, New York, 1990). [5J G. Burdman, Phys. Rev. I)52 (1995) 6400. 16l H.E. Haber, Introductory Low-Energy Supersymmetry, Proc. of the 1992 Theoretical Advanced Study Institute in Particle Physics (World Scientific, Singapore, 1993) p. 589. [7] T. lnami and C.N. Lira, Prog. Theor. Phys. 65 (1981) 297. 18l W. Killian and T. Mannel, Phys. Lett. B 381 (199.3) 382. 191 C,Q. Geng and John N. Ng, Phys. Rev. D 38 (1988) 2857. 110 S.L. Glashow and E.E. Jenkins, Phys. Lett. B 196 (1987) 233. [ 11 EJ. Gihnan and M.B. Wise, Phys. Lett. B 9,3 (1980) 129. [12 R.D. Field, Applications of Perturbative QCD (Addison-Wesley, New York, 1989). 113 A,J. Buras and P.H. Weisz, Nucl. Phys. B 3_33 (1990) 66, [ 14 S. Herrlich and U. Nierste, Phys. Rev. D 52 (1995) 6505. [15 W.A. Bardeen, A.J. Burns, D.W. Duke and T. Muta, Phys, Rev. D 18 (1978) 3998. 116 J. Collins, Renormalization (Cambridge Univ. Press, Cambridge, 1984). [ 17 S. Woffram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1993). [18 A. Heck, Introduction to Maple (Springer, New York, 199.3). [ 19] G. Altarelli, G. Curci, G. Martinelli and R. Petmrca, Phys. Lett. 13 99 (1981) 141. [201 G. Altarelli, G. Curci, G. Martinelli and R. Petrarca, Nucl. Phys. B 187 (1981) 46/. 1211 A.J. Buras, TUM-HEP-259/96, hep-ph/9610461. [221 M,H. Schune, talk presented at the 7th Int. Syrup. on Heavy Flavour Physics, Santa Barbara, July 1997. 12.3 ] A.J. Buras, talk presented at the 7th Int. Symp. on Heavy Flavour Physics, Santa Barbara, July 1997. 1241 Particle Data Book, Phys. Rev. D 54 (1996) I.