Physics Letters B 312 (1993) 143-147 North-Holland PHYSICS L E T T E R S B

Leading order electroweak corrections to the mixing amplitude

Panka j Agrawal , S i m o n Fo r t i n ~ a n d J o h n N. N g TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada V6T 2A3

Received 16 January 1992; revised manuscript received 15 October 1992 Editor: M. Dine

We perform a two-loop calculation to determine the leading order electroweak corrections to the B-B mixing amplitude. In this way, we find out the leading order dependence of this amplitude on the mass of the Higgs boson, Mn. This dependence is essen- tially In2(M,v). The interplay between the top quark and the Higgs boson gives rise to the term m4tlnE(MH) in the effective Lagrangian. These corrections may lead to a few percent correction to the mixing parameter.

Despite the successes of the Standard Model (SM), one of its crucial elements, viz., the Higgs sector re- mains largely unconstrained and unexplored. Only very recently, LEPI has established a lower bound on the mass o f the Higgs boson, M,v, o f O ( 55 ) GeV [ 1 ]. This bound is expected to be raised to O(Mz) when LEPII becomes operational, and no such scalars were found there. Still, unlike the other undetermined pa- rameters of the SM, there is no significant upper bound on the mass of the Higgs boson. There are stronger limits on the other undetermined mass pa- rameter o f the SM, i.e., the mass of the top quark, mr. The CDF Collaboration at the Tevatron sets rot> 91 GeV [2]. An upper bound on mt comes from the consistency of the one-loop radiative corrections to thep parameter. This is possible because o f the power dependence on m, (i.e., ~c m 2) of these corrections. It is generally agreed that if perturbation theory holds, then m r < O ( 2 0 0 ) GeV [3]. There are also con- straints on the undetermined Cabibbo-Kobayash i - Maskawa (CKM) parameters via the study of one- loop processes in rare K and B meson decays [4] . There has been attempts to put similar indirect bounds on Mn. At the one-loop level, none of the physical quantities, such as the p parameter, that have been calculated has a strong dependence on Mn. This dependence is at best logarithmic [ 5 ]. This is in ac-

Current address: Laboratoire de Physique Nucleaire, Univer- sity of Montreal, Montreal, PQ, Canada H3C 3J7.

cordance to the common lore known as the "screen- ing theorem", which states that all one-loop radiative corrections to observables do not have a power de- pendence on MH [ 6 ]. These plus the studies on per- turbative unitarity set 3//, < O ( 1 ) TeV.

Beyond one-loop calculations, one might expect a power dependence on Mu. In the limit that Mn is larger than any other mass in the theory, a two-loop calculation of the/9 parameter revealed a M 2 depen- dence [ 7 ]. However, being second order corrections, such terms are accompanied by an additional suppression factor due to coupling constants, i.e., such two-loop corrections are proportional to (g2/ 167r 2 ) 2 M 2 . So whatever one gains from the stronger dependence is lost, and the numerical significance o f such a dependence is minuscule. In view of such a situation, one should look for processes that are for- bidden at tree level. In this way, we might expect to get rid o f the suppression factor o f g2/167t 2 ( ~ l 3-3-~ )- Such forbidden processes include mixing parameters for K mesons and B mesons, and flavor changing neutral current (FCNC) decay modes o f kaons, B- mesons. We have carried out a two-loop calculation to determine the leading dependence on Mn for the cases of the B-B. Our investigation can be carried over to the K-/£ mixing amplitude quite straightfor- wardly. Apart from the parameters, mt and Mn, there are two other CKM parameters that are yet to be de- termined. The bounds on these parameters are de- rived by considering mixing and the rare decays of

0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved. 143

Volume 312, number 1,2 PHYSICS LETTERS B 5 August 1993

kaons and B-mesons [4,8 ]. If electroweak ~ correc- tions are large, they might also affect the bound on these unknown CKM parameters.

Apart from the above phenomenological consid- erations, the computation of such corrections is also interesting from the field theoretical point of view. The basic question is: what kind of functional depen- dence on MH should be expected for various observ- ables at a given order in the perturbative expansion? Various calculations, and intuitive arguments, have given rise to the conjecture known as "screening theorem" [6]. However, the discussion of this con- jecture in the literature - that goes beyond oneqoop - usually neglects the possible existence of heavy fer- mions, like the top quark [ 10 ]. Our investigations can provide another step towards a better formula- tion of precise statements of the "screening theo- rem", when there are heavy fermions in the theory.

The B-B mixing is described in the SM at the one- loop level through the box diagrams displayed in fig. 1, and the result can be summarized in terms of an effective Lagrangian

5°E~ = (VbtVTa)2(~--2w)(dyULb)(dyULb)B(x,),

(1)

where

~1 The QCD corrections to these processes have also been con- sidered and found to be of the order often percent [9 ].

Fig. 1. Scattering type one-loop diagrams contributing to the B- B mixing amplitude. There are similar annihilation type diagrams.

B ( x , ) =Bl (xt) + B2(xt , Yn) • (2)

Here Bt is the one-loop contribution, and B2 is the two-loop contribution. We have to compute these two

2 2 functions. In the above x , = m t / M w and y n = 2 2 M u / m t .

The one-loop calculation gives [ 11 ]

g4 1 Xt Bl (Xj) = -- (4n.)2 8 (Xt -- 1 )2

x~ ln(xt)). ; < ( 4 - l l x t + x 2 + 6 (3)

In this expression g = e / s i n ( O w ) , where e is the electric charge and 0w is the weak mixing angle. Bz(xt ,

Yt/) is then the electroweak correction to this func- tion Bt (x~).

The propagators, vertices, and the diagrams that contribute at the two-loop level depend on the gauge choice. We have performed the calculation in the 't Hooft-Feynman gauge. Below, we shall denote the unphysical charged scalar by ~ +, and the unphysical neutral scalar by Z. Broadly speaking, the contribut- ing two-loop diagrams can be classified in nine dif- ferent categories [ 12 ]:

( 1 ) Diagrams arising from the neutral bosons, 7, Z °, H and Z exchanged between the internal quark lines of the one-loop diagrams of fig. 1.

(2) Diagrams arising from the neutral boson 7, Z°, H and Z exchanged between the internal boson lines of the one-loop diagrams of fig. 1.

( 3 ) Diagrams arising from the neutral boson Y, Z °, H and Z exchanged between the internal boson lines and external quark lines of the one-loop diagrams of fig. 1.

(4) Diagrams arising from the neutral boson 7, Z °, H and Z exchanged between the internal quark lines and external quark lines of the one-loop diagrams of fig. 1.

(5) Vertex correction diagrams. (6) The diagrams with self-energy insertions to in-

ternal boson or fermion lines. (7) The diagrams with three charged bosons, e.g.,

the fourth diagram of fig. 2. (8) The diagrams with a four-boson vertex (ex-

cluding self-energy insertions). (9) The di-penguins. These diagrams are pro-

144

Volume 312, number 1,2 PHYSICS LETTERS B 5 August 1993

d -~

I i °I

d m d -~

it. b-- - " d

iI::I d -~

b b d

Fig. 2. Important scattering type two-loop diagrams contributing to the leading order corrections to the B-/1 mixing amplitude. There are similar annihilation type diagrams. The dark blobs de- note self-energy insertions or vertex corrections, and the cross denotes counter-term interactions.

duced through the fusion of two one-loop penguin diagrams.

Without any approximation, or further considera- tions, there are more than 10 000 diagrams that will contribute to the mixing amplitude. The computa- tion of all the diagrams is a formidable task. How- ever, it is not necessary, nor desirable, to do the com- plete calculation. Our main interest is in the large Mn behavior. Remembering that the interesting range of parameters is Mw< rot< MH, w e can neglect the ex- ternal four-momenta which is a good approximation for B and K mesons. (The result o feq . (3) is valid only in this approximation. ) Furthermore, in view of the interest in the MI-t dependence, at least as a first step, we can restrict ourselves to a leading order cal- culation. By definition, the leading order terms at two loop are of 0 (g6x2, g6XtYn), or higher. We note that the leading order term at one loop is of O(g4x,) with no Yn dependence. Restricting to the leading order introduces enormous simplifications. Now we have

to compute only about 100 diagrams (the actual number depends on the way one counts).

It turns out that in our leading order calculation only some diagrams from the categories 1, 2, 5, 7, 8 and 9 contribute. The most important o f these dia- grams are displayed in fig. 2. All these diagrams, after partial fraction, can be written in terms of seven dif- ferent integrals that can be easily computed using the techniques of ref. [7 ]. The result due to these dia- grams can be expressed in terms of logarithms and Spence functions that are complicated functions ofxt and Yn. To extract the leading order contributions these functions are expanded for Mw< rnt<Mm i.e., large xt and YH. Different functions are expanded to different order to get the contribution to the leading order terms. In our computation, we have made ex- tensive use of the MATHEMATICA system [ 13 ]. Its symbolic manipulation capabilities are very helpful in extracting leading order two-loop contributions, and are indispensable for a project of this scope.

A number of contributing diagrams have ultra-vi- olet divergences. One has to use an appropriate re- gularization procedure, and an appropriate subtrac- tion scheme to handle the infinite parts of these diagrams. The finite parts o f the renormalization constants, in terms of which counterterms in the ac- tion are given, are fixed by a set o f renormalization conditions. We use the dimensional regularization procedure, and on-shell renormalization conditions to fix the finite parts. The set o f on-shell renormali- zation conditions are described in ref. [ 14]. These counterterms give rise to diagrams which also con- tribute to the leading order. It is an important check that these diagrams cancel the divergences of the ul- tra-violet divergent diagrams.

As an example of the result of the computation, we give the contribution of the first diagram of fig. 2 to Bz(xt, y~,) to the leading order. Denoting this contri- bution by B~(xt, YH), we get

B~(x,, yi~)= (393~-~6 n'i) x2

× [ 15 - 14 n a - 12 ln2(y,q) + 2 4 ln(yn) ] . (4)

This diagram gives the leading order dependence on Mn. There are two couplings in the diagram that are proportional to M 2, and one Higgs boson propaga-

145

Volume 3 ! 2, number 1,2 PHYSICS LETTERS B 5 August 1993

tor. So, naively, one would have expected this finite d iagram to be propor t ional to M 2. However, it turns out that due to the nature of couplings, this potent ia l leading dependence drops out.

After summing up all diagrams, we get, to leading order,

g6 1 2

B2(xt, YH) = -- (4/t)4 384X/

cos2(Ow)'~ X [ ~ 2 ( 1 0 6 7 - 36rr2- 432 ~ j

- 10 In (yn) + 3 ln2(y~ / ) ] . (5)

We see that the leading dependence on M,~ is loga- r i thm squared. One can trace the origin of this term to a type 2 diagram, the first d iagram in fig. 2. The contr ibut ion of this d iagram is given in eq. (4) . No other d iagram contr ibutes to this leading behavior. We note the following features of our results. No power dependence on M , is found in our approxi- mation. Furthermore the result ofeq. (5) exhibits two Mn dependent pieces with opposi te signs. The faster growing piece as a function of M u has a smaller coef- ficient. This leads to a mutual cancellat ion of the MH dependence in the range 50 GeV < Mn< 1 TeV. This is displayed in fig. 3 where we have plot ted the func- t ion R = IB2/B~I which is a measure of the relative impor tance of the leading order correction to the am- pli tude. We also note that the two-loop contr ibut ion tends to reduce the value of the mixing parameter .

2.0

£;d

%

1.5

1.0

0.5

0 . 0 i , -1

3 0 o 8 o 0 ' lOOO 900 7o0 6oo M (CeV/c 2)

Fig. 3. R as a function of Mu. The upper line is for mr=200 GeV. The lower line is for mr= 150 GeV.

In conclusion, we find that the two-loop electro- weak correct ions to the mixing ampl i tude are at best a few percent of the one-loop contr ibut ion, and are of opposi te signs. The smallness of the corrections is a result of accidental cancellat ion among the terms and the fact that sin2(0w) is close to a quarter. Because they are of opposi te sign, these corrections tend to reduce the one-loop value. Fur thermore , at two-loop level, the ampl i tude pr imar i ly depends on Mn as In 2 (YH). Because of the nature of the M n dependent terms, the ampl i tude does not vary significantly with M , . Therefore, the M , dependence of the mixing ampl i tudes is not phenomenological ly significant. However, such investigations can provide a bet ter in- sight in the theoretical structure of models with spon- taneously symmetry breaking in general, and the S tandard Model in part icular, when there are several mass scales present. They can also be used to check the accuracy of effective field theories where one heavy mass is integrated out first [ 15,16 ].

We thank our many colleagues for useful discus- sions. P.A. would especially like to thank R. Balden for his extensive help in using MATHEMATICA. This work was suppor ted in part by the Natura l Science and Engineering Research Council of Canada.

References

[I]S.L. Wu, Plenary talk presented at the PF91 meeting (Vancouver, Canada ); ALEPH Collab., D. Decamp et al., CERN preprint CERN- PPE/91-149; DELPHI Collab., P. Abreu et al., CERN preprint CERN- PPE/91-132; OPAL Collab., M. Akrawy et al., Phys. Lett. B 253 ( 1991 ) 511; L3 Collab., B. Adeva et al., Phys. Lett. B 257 ( 1991 ) 450.

[2] F. Abe et al., Phys. Rev. Lett. 68 (1992) 447. [3] V. Amaldi el al., Phys. Rev. D 36 (1987) 1385;

G. Costa et al., Nucl. Phys. B 297 (1988) 244. [4] For a recent review see J.N. Ng, Proc. PASCOS II Symp.

(Northeastern University, Boston, MA, 1991 ). [5] See, e.g., J.F. Gunion et al., The Higgs hunter's guide

(Addison-Wesley, Reading, MA, 1990). [6] M. Veltman, Acta Phys. Polon. B 8 (1977) 475. [7]J. Van der Bij and M. Veltman, Nucl. Phys. B 44 (1983)

189.

146

Volume 312, number 1,2 PHYSICS LETTERS B 5 August 1993

[8] G. B61anger and C.Q. Geng, Phys. Rev. D 43 ( 1991 ) 140; C.O. Dib et al., Phys. Rev. D41 (1990) 1522; C.S. Kim et al., Phys. Rev. D 42 (1990) 96.

[9] F. Gilman and M. Wise, Phys. Rev. D 20 (1979) 2392. [ 10] M. Einhorn and J. Wudka, Phys. Rev. D 39 (1989) 2758. [ 11 ] See, e.g., T. Inami and C.S. Lim, Prog. Theor. Phys. 65

(1981) 297, 1772(E). [ 12] P. Agrawal, S. Fortin and J.N. Ng, in preparation. [ 13 ] S. Wolfram, MATHEMATICA (Addison-Wesley, Reading,

MA, 1991).

[ 14] For a review see K.I. Aoki et al., Suppl. Prog. Theor. Phys. 73 (1982) 1; see also A. Sirlin, Phys. Rev. D 22 (1980) 971; G. Passarino and M. Veltman, Nucl. Phys. B 160 (1979) 151; F. Jegerlehner, PSI preprint PSI-PR-91-08.

[ 15 ] H. Georgi, Nucl. Phys. B 363 ( 1991 ) 301. [ 16] P. Yao et al., Phys. Rev. D 44 ( 1991 ) 2139.

147