Next-to-leading QCD corrections to B → Xsγ in supersymmetry

ELSEVIER

Nuclear Physics B 534 (1998) 3-20

Next-to-leading QCD corrections to B supersymmetry

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in

M. Ciuchini a, G. Degrassi b, E Gambino c, G.E Giudice d,1 a Dipartimento di Fisica, Universitgt di Roma Tre and INFN, Sezione di Roma llI, Via della Vasca Navale, 84, 1-00146 Rome, Italy b Dipartimento di Fisica, Universitgt di Padova, Sezione INFN di Padova, Via F. Marzolo 8, 1-35131 Padua, Italy c Technische Universitiit Miinchen, Physik Department, D-85748 Garching, Germany d Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Received 17 June 1998; accepted 6 July 1998

Abstract We compute the QCD next-to-leading order matching conditions of the (chromo)-magnetic operators relevant for B ---+Xs7 in supersymmetric models with minimal flavour violation. The calculation is performed under the assumption that the charginos and one stop are lighter than all other squarks and the gluino. In the parameter region where a light charged Higgs boson is consistent with measurements of BR(B ---+ Xgy), we find sizeable corrections to the Wilson coefficients. As a consequence, there is a significant reduction of the stop-chargino mass region where the supersymmetric contribution has a large destructive interference with the charged-Higgs boson contribution. © 1998 Elsevier Science B.V.

1. Introduction The inclusive decay rate for B ~ X s y has first been measured by CLEO with the result B R ( B ---+ X s y ) = (2.32 ± 0.57star ± 0.35syst) x 10 - 4 [1]. Recently, a preliminary new result based on about 30% more data has been presented by the collaboration, B R ( B --+ X s y ) = (2.50±0.47stat±0.39syst) x 10 - 4 [2]. The same process has also been measured by A L E P H at LEP, with the result BR ( B --+ X s y ) = (3.11-4-0.80stati0.72syst) x 10 - 4 [ 3 ]. There has been significant theoretical effort in refining the prediction of B R ( B --+ X~y) in the Standard Model ( S M ) . Calculations are now available for the next-to-leading 1On leave of absence from INFN, Sez. di Padova, Italy. 0550-3213/98/$ - see frontmatter (~) 1998 Elsevier Science B.V. All rights reserved. PIIS0550-3213(98)00516-1

4

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

order (NLO) corrections to the anomalous dimensions [4], the matrix elements [5], and the matching conditions of the Wilson coefficients [6,7], for the leading nonperturbative effects [8], and f o r the QED corrections [9,10]. In particular the QCD NLO corrections are important since they reduce the large scale dependence of the leading result [ 11], which amount to a 30% uncertainty in the leading order (LO) theoretical prediction. Recently, Kagan and Neubert [ 10] have argued that, at the NLO, the contributions to B R ( B -+ Xsy) from the different Wilson coefficients exhibit a larger scale dependence than the total result, signaling that the theoretical uncertainty may have been underestimated in previous literature [ 12,7]. Nevertheless, even considering this effect, they find that the theoretical error can be evaluated to be about 10%. Combining the different theoretical studies, the recent complete analysis in Ref. [ 10] gives the SM prediction B R ( B --+ Xsy) = (3.29 i 0.33) x 10 -4 × BR(B ~ Xcef,)/lO.5%. While the theoretical prediction is made for the total rate of BR(B --+ Xsy), the experimental data refer only to events with photon energies between 2.2 and 2.7 GeV. The extrapolation from the data to the total rate introduces further theoretical uncertainties from the calculation of the photon energy spectrum [ 13]. This aspect has recently been emphasized by Kagan and Neubert [ 10], who have recomputed the spectrum and extrapolated from the CLEO high-energy photon data the total rate BR(B ---, X~y) = (2.66 ± a ~t; exp-0.48 -I-0.43th) x 10 -4. This is compatible with the SM within one standard deviation. To summarize, although the SM prediction is still higher than the CLEO result, the discrepancy seems no longer statistically significant. We should however wait for the forthcoming CLEO analysis and future studies at LEP to further clarify the situation. Meanwhile, we use the conservative CLEO upper limit of BR(B --+ X~y) < 4.2 x 10 -4 at 95% C.L. in our numerical analysis. The inclusive decay B --+ X~y is a particularly interesting probe of physics beyond the SM, since it is determined by a flavour-violating loop diagram. Because of this welcome sensitivity on new physics, it is important to determine the predicted rate with sufficient accuracy in a variety of models. In the case of two-Higgs doublet models, results for the next-to-leading order (NLO) QCD corrections have been presented in Refs. [7,14,15]. In these models, a new contribution arises from charged-Higgs exchange which always increases the SM prediction for BR(B --+ XsY). As a consequence, strong limits on the charged-Higgs mass can be derived. These limits are quite dependent on the treatment of theoretical errors and on the amount of discrepancy between theoretical and experimental results [7,15]. Here we want to extend our previous analysis of QCD NLO effects in BR(B X~y) to the case of supersymmetry. Refining the theoretical calculation in supersymmetry [ 16,17] (for reviews see Ref. [ 18] ) is important because BR(B --+ Xsy) provides a very stringent constraint on the parameter space of the model. Moreover, in supersymmetry it is possible to evade the strong limits on the charged Higgs mass, since the chargino contribution can interfere destructively and reduce BR(B --+ Xsy) [ 17]. Our result allows us to give a more reliable estimate of the model parameters necessary to achieve this cancellation. .

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M. Ciuchini et aL/Nuclear Physics B 534 (1998) 3-20

We have not tried to perform a complete analysis in the most general supersymmetric model, an effort, we feel, that is penalized by the fact that the final result will be very involved and will depend upon many unknown parameters. Instead, we try to focus on what we believe is the most interesting part of the parameter space for B --+ X s y . Specifically, we concentrate on the case in which the flavour violation is completely dictated by the Cabibbo-Kobayashi-Maskawa (CKM) angles (insuring predictability) and in which the charginos and a scalar partner of the top are light (insuring a sizeable new contribution). The paper is organized as follows. In Section 2 we define our assumptions on the supersymmetric parameters and discuss the underlying hypotheses. Section 3 contains the analytical results for the supersymmetric corrections to the relevant Wilson coefficients at the matching scale. A numerical analysis of BR(B --+ Xsy) in supersymmetry at the NLO is presented in Section 4.

2. Minimal flavour violation and heavy squark-gluino effective theory In this section we describe our main theoretical assumptions on the supersymmetric model: (i) the minimal flavour violation, and (ii) the heavy squark-gluino effective theory that describes chargino, stop, and the SM degrees of freedom with other supersymmetric particles integrated out. In the supersymmetric extension of the SM with general soft-breaking terms, there are a variety of new sources of flavour violation. The supersymmetry-breaking squark masses and trilinear terms lead to a mismatch in flavour space between quark and squark mass eigenstates. Flavour violation is then described by a large number of mixing angles, which cannot be determined theoretically. All we know is that present measurements and limits on various flavour-violating processes provide quite stringent bounds on these mixing angles (see, e.g., Ref. [19]). A more predictive case is what we will call m i n i m a l f l a v o u r violation, in which all flavour transitions occur only in the charged-current sector and are determined by the known CKM mixing angles. This is indeed the case in several theoretical schemes in which the communication of the original supersymmetry breaking to the observable particles occurs via flavour-independent interactions. In many of these schemes (especially when supersymmetry breaking occurs at low-energy scales) the departure from the minimal flavour violation hypothesis, caused by quantum effects, is rather Small. Our assumption can then be justified in gauge-mediated models [20] (for a review, see Ref. [21 ] ) and in certain classes of supergravity theories. At any rate, it plausibly corresponds to the unavoidable flavour violation, present in any supersymmetric model. Within this class of models, we focus on the special case of interest for B --+ X s y , in which one stop is considerably lighter than the other squarks. In order to preserve the successful fit of the electroweak precision measurements, we assume that the light stop is predominantly right-handed [22]. This assumption can also be theoretically justified by the observation that the renormalization-group evolution of the squark mass parameters

M. Ciuchini et a l . / N u c l e a r Physics B 534 (1998) 3 - 2 0

indeed pushes the right-handed stop mass to smaller values. Therefore, we concentrate on the case in which the supersymmetry-breaking parameters in the up-squark sector are flavour diagonal (but not necessarily universal) in the basis in which the corresponding up-type quark mass matrix is diagonal. The flavour-violating stop interactions arise only from charged-current effects and are completely determined by the CKM angles. The stop mass matrix is given by 2

mt=

sin 20w) cos2flM2z rot(At - / z c o t / 3 )

. . .t2 ~- - I,~ / 1 -m 2 -{-m tL

2

mt(At - / x cot/3) ) m-2 + m 2 + ~ sin 20w cos 2/3M2z " tR

(1) Here m~R, 2 mTL, 2 and At are supersymmetry-breaking parameters,/z is the Higgs mixing mass and tan/3 is the ratio between the two Higgs vacuum expectation values. We also assume that the charginos are lighter than gluinos and heavy squarks. These approximations considerably simplify the calculation and the final expressions, and they are appropriate to identify the leading supersymmetric contribution. To implement this scenario we assume the following mass hierarchy

tx~ "~ O(m~,m#,mT1) >> tzw ~ O(Mw, MH~,mt, mx±,mr2) >> mb >> AQCD- (2) Here and in the following/zw is the mass scale of the charged Higgs ( H i ) , the charginos (X+), and of the lighter stop (t2), to be identified with the ordinary electroweak scale. The scale /z~, characteristic of all other strongly interacting supersymmetric particles (squarks and gluinos), is assumed to be larger, say of the order of a TeV. We compute the QCD NLO corrections to the Wilson coefficients keeping only the first order in an expansion in I~w/tz~. As usual, the presence of different mass scales allows us to use a stack of effective theories, obtained by integrating out of the theory, at each matching scale, the heavy degrees of freedom. The effective theory just below the scale/z~ is particularly simple. The only dimension-five operators involving supersymmetric particles are the chargino (chromo)-magnetic dipole moment and the operator f t [~ t'2, obtained by integrating out the gluino. However, an explicit calculation shows that these operators do not contribute to the NLO matching conditions of magnetic-dipole operators at the scale/zw. Therefore, to our end, the inclusion of the leading tzw/tz~ corrections does not require any new operator other than renormalizable ones. The next step is the running of the intermediate effective Hamiltonian between/z~ and /zw. Although the relevant anomalous dimensions are known [23], log resummation is not really needed given the smallness of the relevant log terms, as (/Zw)/4¢r log(/z~//z 2 ) ,-~ 0.01-0.05 for typical values of the supersymmetric masses. Non-resummed NLO effects of heavy supersymmetric particles, including 1//*~ corrections, are simply given by the one-loop Feynman diagrams, containing these particles in the loop, which contribute to renormalizable operators in the intermediate effective theory. In other words, we need to compute only corrections involving gluinos and heavy squarks to the coupling constants appearing in the vertices involving the

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

7

chargino, x~-bt2, the W vector boson, W-gtq t, the charged physical, H-fTq', and unphysical, qS-cTq/, scalars. Heavy O(/z~) particle effects renormalize the masses of O(#w) particles, but these effects are reabsorbed through the definition of renormalized masses. We use on-shell masses for squarks. Quark masses are also defined on-shell, except corrections not involving supersymmetric particles, which are subtracted in the MS scheme. This definition simplifies the insertion of supersymmetric contributions into existing RGevolution formulae, where MS running quark masses are commonly used. Because of our assumption that the light stop is mainly right-handed, the stop mixing angle 0r is small. For instance, taking the supersymmetry-breaking parameter At in Eq. (1) of the order of/x~, yields 0T ~ O(/xw/tx~). In this framework, 1//x~ corrections multiplied by sin Or factors should be regarded as of higher order. Results for the renormalized vertices in the intermediate effective theory are collected in Appendix A. Notice that the x~-b72 vertex renormalization is infinite. This is not surprising, since supersymmetry and the GIM cancellation are spoiled in the intermediate effective theory, where the heavy squarks and gluinos have been integrated out. This divergence cancels out in the matching of the magnetic operators at the scale/zw against a corresponding divergence generated by the insertion of the chargino vertex into twoloop diagrams. This cancellation actually provides a check for the results of the two-loop calculation. However, the presence of the divergence calls for a regularization. We choose the naive dimensional regularization (NDR), to be consistent with the calculation of the anomalous dimension matrix [4]. However, it is known that NDR breaks supersymmetry. In particular, the renormalizations in NDR of the gauge boson and gaugino interactions with matter, as well as the Higgs boson and higgsino interactions, are different and manifestly violate supersymmetry. Supersymmetric Ward identities are restored with appropriate shifts of the gauge and Yukawa couplings in the Xab72 vertex [24], denoted as ~Tr and r/g in Appendix A. These shifts correspond to the difference of using NDR versus dimensional reduction (DR) [ 25 ], a regularization that preserves supersymmetry. The formulae in Appendix A are given in terms of few functions of the ratios m#/m~.2 2 Notice however that, whenever mT2/m 2 2~ appears, only terms up to O(mh/m~) should be retained in the corresponding functions to be consistent with the operator product expansion. Finally¢ we also keep the leading contribution in the sbottom mixing angle, since the left-right mixing term (Ab --/x tan/3) mb becomes important for large values of tan/3.

3. Wilson coefficients

This section contains the result for the NLO supersymmetric contributions to the eft Wilson coefficients Ci (tZw). Following the discussion of the previous section, the calculation of the NLO corrections to C7eft (/zw) and Cseft (/Xw) can be divided in two parts: (i) the contribution of the heavy particles at the scale / ~ that will appear as renormalization of the coupling constants in the LO diagrams. (ii) The contribution of the intermediate-scale particles that requires the computation of the two-loop gluonic

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20 corrections to the LO supersymmetric diagrams involving the charginos and the light stop. Concerning the latter, two strategies are at hand. One can match matrix elements of operators belonging to a basis obtained enforcing the equations of motion, a procedure that however requires an asymptotic expansion of the relevant diagrams in the external momenta. Alternatively, one can use a larger off-shell operator basis and perform the matching on off-shell matrix elements. In this case, one can use the freedom of the offshell status to choose a suitable kinematical configuration such that the various Feynman diagrams can be evaluated using ordinary Taylor expansions in the external momenta. This second strategy, already successfully applied by us in Ref. [7] to the calculation of the QCD corrections to the matching conditions of the AB = 1 magnetic and chromomagnetic operators in the SM and in two-Higgs doublet models, has also been employed in this calculation. Concerning the technical details we refer to Ref. [7]. In the supersymmetric model under consideration we can organize the Wilson coefficients of the operators entering the effective Hamiltonian in the following way: C?ff(l.£W) : c i(O)eff ( ].£w ) ~- ~ H c i(O)eff ( ~ W ) --~ ~ S c i(O)eff ( ].LW )

Ols(['LW) [Ci(1)eff([l~W) -~ ~Hci(1)eff(]-LW) + ~Sci(1)eff(].£w) ] 47r

(3)

3Hci(k)eff(l.~W)

where C(k)eff(lzw ) represents the SM contribution (k = 0, 1), the additional terms present in a two-Higgs doublet model, while 8Sc}k)e~(izw) contains the contribution from supersymmetric particles. Explicit expressions for Ci(k)eff(t*w) and 8Hci(k)eff(tZW) Can be found in Sections 4 and 5 of Ref. [7]. 2 Concerning the remaining contributions, the LO 8Sci(°)e~(IZw) terms are given by [ 16]

~Sci(°)eff(ld~w) = 0,

i = 1 . . . . . 6,

8S,-,(0)eff, , [2M~vr,2F(1)r" , Ui2 M w f z ~ ( 3 ) ( Z j ) t"7,S t["LW) ~" ~ [-3~--ff Vjl 7,8 ~'"J) -}- V/~--~OSt~ mX j vjlr7,8 j=l,2 2.2

M2 w F(1) 5 t l j - - ~ -_ 5 7,8 (Ylj)

l-]j2 M w tlj cos 0TF7(38) ( Ylj ) V~COSfl mxj

2-2 M w2 F O ) r " .~ ~f2j-~-~-2 7,8 tyaj) T2

(-]]2 M • - - wt 2 j s i n ,o_p(3) (Y2j) 1 v~cosflmx j vt-7,8 j ,

(4)

• 7,8 are defined in Eqs. (29), (30) of Ref. [7] and where the functions ~(x) F43)(x)_

5-7x x (3x-2) 6(x--1) 2÷ 3~x---iF 1 +x

F8(3) (x) - 2 ( x -

1) 2

x

(- x- --- - -1) ~

lnx, lnx.

We have defined 2 In Ref. [7] ~Hc/(k)effare indicated as ~C/(k)eff.

(5)

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20 tlj =Vii cos0r - tan 0TYj,

r~2 , zj= m2xj

9

t2j = Vjl sin 0T + Yj,

mE,k Ykj = m2zj ,

(6)

~2__COS 0r rn, (~w) Mw '

(7)

Yj = v ~ s i n f l

where tht(tZw) is the top-quark running mass at the scale/~w. The stop eigenstates tl = cos Or tL + sin 0r t'R and t2 = - sin or ?L + cos or ~R have mass eigenvalues mr, and mr2 and we have taken all other squarks to be degenerate with mass r~. The two matrices O and fz diagonalize the chargino mass matrix according to ( M is the weak gaugino mass)

( M

O

M Tsi. )

MwV/-~ cos fi

(8)

and are assumed to be real. Notice that in Eq. (4) and henceforth the scalar quark masses are understood as on-shell. The last ingredients needed for a complete N L O calculation in our supersymmetric model a r e ~Sci(1)eff([ZW). For the current-current and penguins operators we have

~Sci(1)eff([.Lw) = 0 ,

i = 1,2,3,5,6,

~Sc(41)eff(tzw)= ~

~ t2jEx(Y2j) j=l,2

(9) (I0)

with [ 26 ] x(ll

Ex(x) =

- 7 x + 2 x 2) 1 8 ( x - 1) 3

x 3(x-

1) 4 lnx.

(11)

The O ( a s ) corrections to the coefficient of the (chromo-)magnetic operator can be divided into four pieces ~S.--,(1)effr ~ c,Xr,(1)eff/ -~ ~,Wr,(1)eff~ -~ -- ~,~blr,(1)eff~ -~ t~7,8 ~.]-~W)= o t..7,8 I.]J~W)q- 0 1_.7,8 I.tZW) ~- 0 t.-7,8 ~.]~W) + ~'q~2t~ (l)eff,; o t.7, s ~/~w)(1)eft

.

.

(12)

.

Here 6xc.~, s represents the charglno contribution while (6 w, 8~i, 6~2)C7(,~8)eff take into account the renormalization effects due to the O(/x~) heavy particles in the W, physical and unphysical charged scalar couplings, respectively. We find

~7

t#wJ = ~ /~2j-~-2 j=l,2 L t2

(y2j) + d7 (y2j) In rn2xj - -~

x(Y2j)

(/j2 sin Or Mw (t2jG~,2(y2j) + t2jAf,2(y2j ) In 4 ~/~ cos fl mxj mxj 7]t?(3) ( y 2 j ) ~ 8 U~v - 4y.#./..s. 3 / - -dYj-z-f-t2j(Rs+Rb) Fv~l)(y2j)

~xc~'~ff(~w) =~xcT~'~ff(~) ( 7 ~ 8 , - g4e x ( y ~ ) ~ - ~1G ( y ~ ) ) •

,

(13)

(14)

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M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

6xc(81)eff(/xw)

Eq. (14) means that can be obtained from the r.h.s, of Eq. (13) by replacing the index 7 with the index 8 in the various functions and by substituting the term-4Ex(Y2j with-~Ex(Y2j). In Eqs. (13), (14) we have used

)

m2 ~ (A})s)+ A~J,) - 1 +31nm2j. ] x(85-347x+526xe) 4x2 (-8+13x+6x2) (1)

GX'l(x)=-~

-~ -

-

8F(1)(x, ~ 7 ~ ~

243 (1 _ x) 3

+

- ~ x ---1 ~,i

Li2

1-

4 x (20 - 126x + 144 x 2 + 39 x 3) 2 x 2 (21x - 10) 81 ( X _ 1)4 Inx+ 9 ( x - - 1) 4 ln2x,

(A}'ls)-[-A(bl'-t-A(b2'--2)--169-(X-~(3--7X)X

G~'2(x)=--4F(3)(x)

4(3x-5) 4 9(x--1) z

4 ( 4 - 3 0 x - t - 4 0 x 2) 9(x-l) 3 lnx-

T.L12 (1--

(15)

1)

16 ( 1 - - 3 x ) x 9(x--i7 lnZx' (16)

G~"(x)=--8F(')(x)(A},'s+) A O ) - 1 + 31n m~ ~ x(1210-437x-1427xe)-x2(49+46x÷9xZ-i 4 -2-~x---l )Li2 ~ (1 _ 1 ) 648 (x - 1) 3

x ( 8 5 - 6 0 3 x - 3 8 7 x 2 + 7 8 x 3) lnx 108 (x -- 1) 4 4-(3)"t x )" ( A (1) + A(bl) + A(b2) - 2) '2(x)=-U8

a

61 - 39x 12(x-1) 2 Ri

(2-) = 3 In ~/x2 + A -t,~

13 X 2

3 (X -- 1) 4

ln2x,

(17)

4 x (3 + 4X) Li2 (1 _ 1 ) 3 (x - 1) 3

( 7 - 6 0 x - 14x 2) lnx 6(x-1) 3 1, -

14X in2x ' 3 (x - 1) 3

(i8) (19)

(x) =

(20)

a~'l(X) = - - ~ f ~ l ) ( x ) ,

Ax'2(x) = -~-~F8(3) (x).

(21)

The various functions zl appearing in Eqs, (15)-(19) contain the effect of the renormalization of the chargino-stop-quark vertex due to the O(/X~) heavy particles. Their explicit expressions are given in Appendix A. In the same equations the terms not proportional to the one-loop functions ~O,3) * 7,8 represent the /x-independent part of the contributions coming from two-loop squark-chargino diagrams. The/zw dependence in Eqs. (13) and (14) satisfies the relation

M. Ciuchini et aL/Nuclear Physics B 534 (1998) 3-20 1 m-

OCi (°)

~70m,~

1

11

8

Av 2 Z ' Y } ? )effC(0) /=1

= "= [ m~2 ~,,2j~i (Y2j) -

(Y2))

Oj2sinOgMw (t2jAX,2(y2j)_4YjFi(3)(Y2j)) } ,

(22)

+ .v/~COS-------~ mxj which ensures that physical observables are independent o f / z w to O(o~s). In Eq. (22) y~' = 8 is the LO anomalous dimension of the top mass. Finally, we report the effects of (9(/zg) heavy particles in diagrams involving W, physical and unphysical charged scalars exchanges. As previously discussed, in our approximation they are introduced as a renormalization of the relevant couplings and the corresponding contributions are given by 4

23

4

4

r3WcT(l)eff(Ixw) = ~ (W~ + WT) G7w (tw) - ~-~ (Wb + WS) , 8W r--~(1)eft.

t~8

-~

~ w , = 5 (Wtb + Wt) Gw (tw) - -~ (W~ + WS) ,

4 8~I CT(8)eff(/zw) t a n 2 -/ 93 ,

4

(H]+Hbt) F'(1)7,8(th) + -~ (Ht + Hb) r(2)" 7,8 (th) ,

~b2 ~..,(1)eft / ~ 4 ~7,8 I']'I'W) = "9 (Ut "q- ub) "K'(1)7,8(tw) -- -34 (Ut

+ Ub) ~r(2)7,8(tw) .

(23) (24) (25) (26)

Here tw = rht(tzw)2/M~v, th = rht(IZw)2/M2:~,

G7W(x) =

23 - 67x + 50x 2 x ( 2 - 7x + 6x 2) 3 6 ( x - 1) 3 q6 ( x - 1) 4 lnx,

GW(x) =

4-5x-5x 2 x(1 - 2x) 1 2 ( x - 1) 3 + 2 ( x 1) 4 lnx,

(27)

the functions F 7,8 (2) are defined in Eqs. (54), (55) of Ref. [7], and the renormalization functions W, H, U are given in Appendix A.

4. Numerical analysis In this section we present a numerical analysis of BR(B ~ Xs3/), with special attention to the comparison between LO and NLO results in the supersymmetric model. We focus on the situation in which our approximation is adequate, i.e. light stop and charginos, and show how the strong bounds on the charged Higgs mass can be relaxed. We use the same numerical inputs and conventions as in Ref. [7], except for the photon-energy cut off E 7 > (1 - ~)mb/2, which is now chosen to be ~ = 0.90 (see discussion in Ref. [ 10] ). In our numerical analysis we also include the leading logarithmic electromagnetic effects as computed in Refs. [9,10]. They amount to a decrease of the

12

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

calculated branching ratio of about 7-8%. We neglect all other two-loop electroweak effects which appear to be very small [27]. Light supersymmetric particles can affect the measured values of the B°-/~ ° mixing A M s and of the CP-violating e r parameter and, ultimately, the extraction of the CKM angles, which enter the calculation of BR(B -~ X s y ) . Within our assumptions, it is not difficult to compute this effect. The dominant supersymmetrie contributions to AMB and eK come from box diagrams with chargino/light stop and charged Higgs/top quark exchange. Under the assumption of minimal flavour violation, these contributions have the same structure of CKM angles as the top-quark box diagram in the SM. Since the dominant effects on AMB and ex in the SM indeed come from the top quark, the CKMangle dependence is the same for all contributions, and the effect of supersymmetry is parametrized by dsusv R = 1+ - ASM

(28)

Here ASM is the amplitude of the top-quark box diagram, and AsusY is the amplitude of the chargino/light stop and charged Higgs/top quark box diagrams, whose analytical expressions are given in Appendix B. The crucial point is that R is independent of the CKM angles. In the whole supersymmetric parameter space, R turns out to be larger than one. In terms of the Wolfenstein parameters 7/ and p, the CKM structure of AMB and eK in supersymmetry is AMB ~ IY,*dY,bl2R N [(1 -- p)2 + 723 R,

(29)

eK ~ IIm(Vt~tVts)2IR "~ r/( 1 - p ) R .

(30)

The new physics contribution can be effectively reabsorbed in the Wolfenstein parameters ( 1 - p ) V ~ and ~Tv/-R. This means that the "true" values of r/ and p are related to the values r/SM and PSM extracted from the usual SM fit by (31)

T]SM

~7= v/-~ , p=1

1 - PSM

(32)

We have checked that Eqs. (31), (32) give an extremely good approximation of a complete numerical fit. The combination of the CKM angles entering the evaluation of BR(B ---+X s y ) is X=

Vt*sVtb = 1 + ( 2 p - 1 ) A 2 = 1 +

[ 2 1--~(1--PSM)

]

A2,

(33)

where A = IVusl does not depend on the supersymmetric masses. Therefore, in the presence of supersymmetry, the value of X is related to the usual SM input Xsu by

X = XsM + ( A2 + I -- XsM ) (1-- ~ R ) .

(34)

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20 20

10

I

i

I

I

I

13 I

L Oco~'~pl~te ..........

L O d e c o u p l e d ...... NLO ..... 2 H D M - L O .......... 2 H D M - N L O ...... =============================================================================================================================================

0

rP

...............................................................

1

I

I

0.4

0.6

1

I

0.8 1 heavy masses (TeV)

I

I

1.2

1.4

Fig. 1. Dependence of the prediction for the B ---+Xsy branching ratio on the heavy mass scale /~ for the following choice of parameters: tariff = 1, MH:i: = mT2 = mx2 = 100 GeV, rex1 = 300 GeV, 0r = -~-/10, Ab = A t , all heavy particle masses equal to /z~; the lighter chargino is predominantly higgsino. The value corresponding to "LOdecoupled" is the result of the LO calculation in the limit in which the heavy masses decouple. The two upper lines are the LO and NLO predictions in the two-Higgs doublet model (2HDM). This is the value o f X to be used in a supersymmetric analysis o f B R ( B ~ X s y ) . It is known that the c h a r g i n o / s t o p contribution to the Wilson coefficient C7 can interfere with the SM and charged Higgs contributions either constructively or destructively, depending on the choice o f parameters. The case of destructive interference is particularly interesting, since it is possible to make a light charged Higgs consistent with the measurement of B R ( B ~ XsT). For illustration, we choose the supersymmetric parameters corresponding to a strong cancellation between the chargino and chargedHiggs contributions, and plot in Fig. 1 the predicted B R ( B ~ XsT) as a function o f /z~. The L O result in supersymmetry is compared with the result obtained in the limit /z~ --~ oo (denoted by LOdecoupled in Fig. 1). This shows how the asymptotic behaviour is reached as /z~ grows. The N L O result is also shown in Fig. 1. In this case, we cannot define an asymptotic behaviour, as the N L O branching ratio hardly decreases with /z~ and actually starts rising again f o r / z ~ > 0.7 TeV. This non-decoupling effect is not surprising [28]. It appears because, at the NLO, the chargino vertex contains terms proportional to log(/z~//zw). Indeed, the chargino coupling constant is related to the ordinary gauge and Yukawa couplings only by supersymmetry. In the effective theory below Iz~, supersymmetry is explicitly broken and the renormalization o f the chargino vertex develops a logarithm of the large supersymmetry-breaking scale. Notice the important effect o f the N L O corrections calculated here. For the supersymmetric parameters o f Fig. 1 and for/xg, = 1 TeV, the L O branching ratio is 1.9 x 10 -4, while the N L O result is 3.7 x 10 -4. Had we neglected in the N L O analysis our new supersymmetric contributions to the matching conditions, the result would have been 2.4 x 10 -4. The large shift induced by the N L O corrections to C ~ ( t Z w ) is mainly due by the large non-decoupling logarithms o f tz~/txw. Finally, in Fig. 1 we also show the L O and N L O

14

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

Table 1 Wilson coefficients evaluated at the scale Mw in the Standard Model (SM), the two-Higgs doublet model (2HDM) and supersymmetry (SUSY) at LO and at NLO for tan/3 = 1, MH:k = mT2 = rnx2 = 100 GeV, mx~ = 300 GeV, 0r = -~'/10, Ab = At; all other squarks and the gluino are degenerate with mass of 1 TeV. The lighter chargino is predominantly higgsino

LO SM NLO SM LO 2HDM NLO 2HDM LO SUSY NLO SUSY

C7ff ( Mw )

c~ff ( Mw )

--0.198 -0.220 -0.529 -0.493 -0.143 -0.229

--0.098 -0.119 -0.336 -0.326 -0.141 -0.183

results obtained in a two-Higgs doublet model with charged-Higgs mass equal to the supersymmetric case. The light charged Higgs (taken with a mass of 100 G e V in Fig. 1) is clearly incompatible with the measurement of B R ( B ---+ X s y ) , in the absence of an appropriate chargino contribution. In order to assess the impact o f the different contributions, it is also interesting to consider the N L O Wilson coefficients C7~ at the weak scale /Xw = M w in the supersymmetric configuration chosen in Fig. 1 with all heavy masses set to 1 TeV, and compare them with the corresponding quantities in the SM and the two-Higgs doublet model. The results are shown in Table 1. Notice that, in the supersymmetric case, the Q C D corrections to C7 ff amount to about 60%. Such a large correction is clearly related to the fact that the LO approximate cancellation among different contributions is partially spoiled at the N L O level. Indeed, in the case under consideration, the N L O effects increase the SM contribution to by almost 10% and decrease the charged-Higgs and the chargino contributions by about 20% and 30%, respectively. This situation o f partial cancellation and enhanced sensitivity to N L O correction is actually the most interesting phenomenologically. It is under this condition that a light chargedHiggs mass is still allowed and that our calculation is essential. We now want to investigate the region of parameter space in which we can significantly relax the bound on the charged Higgs mass from B R ( B ---, X s y ) . We have scanned over the relevant supersymmetric parameters, assuming a stop mixing angle less than or/10 in absolute value, consistently with our assumption o f a mainly righthanded light stop. In Fig. 2 we show, for tan/3 = 2 and 4, the m a x i m u m values o f the lighter chargino and lighter stop masses for which a charged Higgs of 100 GeV is consistent with the 95% C.L. CLEO limit of 4.2 × 10 -4. We have chosen this value to give a conservative estimate. With improved experimental results and revised analyses of the photon energy spectrum, this bound may become significantly more restrictive. The effect o f the N L O corrections is to make the upper bounds on the chargino and stop masses quite more stringent. In the same figure we also present the results obtained using the renormalization group evolution and the SM and charged-Higgs contributions to the Wilson coefficients at the NLO, while the purely supersymmetric effects on CT~(tzw)

IC~f~(Mw)l

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

320 z....................................

'

............. 280

kO,tar~.b:::2 ........... NLO runningltanb=2 . . . . .

'11211

".......... ' ................. ---...

NLO,tanb=2

" ..................... . ........

.

.

NLO

" ....

..... .

240

15

.........

.

.........

LO,tanb=4 ..........

~J i'~f~i~~},~a ~~!?::,4 ....... NLO,tanb=4

.........i;i ?i':i£':il.

........

~............

E

~o

200

160

"..

"--

. . . . . . "'.

""

120

"

"""

..... "-',.. I

I

120

160

"-2.\\ "'~ I

I

"~\

200 240 light chargino mass [GeV]

"'-.. I

I

280

320

"'"

Fig. 2. Upper bounds on the lighter chargino and stop masses from the C L E O 95% C L limit on B R ( B ---+ Xs~') in the case MH± = 100 GeV. We have taken 1071 < ~r/10, I~l < 500 GeV, Ab = At, and set all heavy masses to 1 TeV. For tan/3 = 2 and 4 w e show the results of the L O and NLO calculations. The result of neglecting the new N L O supersymmetric contributions to the Wilson coefficients is labelled as "NLO running".

are evaluated at the LO. This is to show the impact of the calculation presented here. As seen in the figure, the effect of the QCD corrections to the supersymmetric contribution eft to CT,s(tZw) is very significant.

5. Conclusions In this paper we have computed the QCD NLO corrections to the matching conditions of the Wilson coefficients relevant for BR(B ~ Xsy). We have used some theoretical assumptions to simplify the result and to concentrate on the most relevant part of the supersymmetric parameter space. We have assumed minimal flavour violation or, in other words, that the flavour-violating interactions in supersymmetry are primarily in the charged-current sector and are determined by the CKM angles. This is often a good approximation in a variety of models. At worst, our result represents an unavoidable contribution to be added to new effects coming from other sources of flavour violation. This assumption allow us also to limit the number of unknown parameters. We have focused on the case in which the purely supersymmetric contribution is sizeable and can compensate the effect of a light charged-Higgs boson. For this reason, we have assumed that charginos and a mainly right-handed stop are considerably lighter than the other squarks and the gluino. This assumption has allowed us to use an effective theory, in which the heavy particles are integrated out, retaining only the first term in an expansion in the ratio between light and heavy masses. In Section 3 we have presented the analytic results for the NLO matching conditions of the Wilson coefficients in supersymmetry, under the approximations stated above. We have also performed a numerical investigation of the supersymmetric parameter region

16

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

in which there is a sizeable cancellation among various different terms. In this region the impact of the NLO corrections to the supersymmetric contribution to the Wilson coefficients is very significant because of two conspiring effects. On one side there is a large renormalization of the one-loop supersymmetric contribution, mostly coming from logarithms of the ratio between the high and the intermediate mass scales. On the other side, the effect of the large NLO corrections to the supersymmetric contribution is greatly enhanced whenever there is an approximate cancellation at the LO.

Acknowledgements We would like to thank A. Buras, G. Martinelli, M. Misiak, M. Neubert and L. Silvestrini for useful discussions. Right before submitting this paper, we learned from M. Misiak and J. Urban that they have reproduced our results in the limit of infinite gluino mass.

Appendix A In this appendix we present the results for the renormalized vertices in the effective theory at scales between /z~ and /xw. In deriving the various corrections we exploited the fact that, in our framework, the light stop is mainly right-handed and therefore we can neglect terms O(sin OHzw/IZ~). However, in the formulae below we do not expand functions of mT2/m 2 ~2 and we keep the explicit sin 2 0r terms, because we feel it will be easier for the reader to understand the origin of the different terms. A.1. W The renormalized d u W vertex is (-i)-~

V~,ay~ a_

l + -~-~ Wff ,

where W~ = 1 (cos 2 0 ~ W [ x l , w l ] + sin 20~W[xl,w2]) 2 2 2 2 with xl = m&/m~, wi = m~,/rn~ and

W[x,y]

x + y - 2xy (x-- 1)(y--l)

+ x 3 - 2 x y + x2y l n x + 2 x y - x y 2 " y3 In y. ( y - 1 ) 2 ( x - y) (x--1)2(x--y)

For equal masses W [ x , x] = O. A.2. Unphysical scalar The renormalized d t ~b+ vertex is

M. Ciuchini et aL/Nuclear Physics B 534 (1998) 3-20

17

°is Ud) --rata+ (1 + °is U d~] ( --i) - ~ M w Vtd [md a- ( l + -3--~ 37 t / l '

where 1 Ua = ~ (H1 [x~] - cos 20TH1 [Ul ] -- sin 20rH1 [u2]) -

/ztan/3H2[Xl, x2] m~

Aa + ~ tan/3 (cos 20r//2 [ ul, x2 ] + sin 20r//2 [ u2, x2 ] ) ,

+2

m~

1 (H~ [xl] - cos 20rH1 [Ul]

U

2 Aa

- -2

--

sin 20rH1 [/*2]) -- 2 Au - ~ c ° t / 3 H 2 [Ul,U2]

mg

+ 2 A~ + / x cot/3 (cos 2 0r//2 [u2, Xl ] + sin 2 0r H2 [Ul, xl ] ) m~ with xz _-

2 2 2 2 m&/m~, ui = mJm~ and

2 x -- X 2 + - - . lnx (1 - x ) 2 x y = lnx + (1 - x ) ( x - y) (1 - y ) ( y -

Ha [x] -

H2[x,y]

1 1 -x

x)

lny.

A.3. Physical scalar We write the renormalized vertex d t h + as

ig

Vtd [tan/3m~ta_ (l + ~-~rcrH~) +cot/3mta + ( 1 + 3 ~ .ees Ha)] as t j],

_ _ where

Ha=

1

+2

(H1 [Xl] -- cos 2 07H1 [Ul] - sin 20rH1 [u2]) -

2 Ad

--

I'l'tanflH2[Xl, x2]

mg

ad -- tz cot/3 (cos 2 0T//2 [ ul, xz ] + sin 2 0T//2 [ u2, x2 ]) , mg

1 - - ~ (H1 [Xl] -- cos 2 07H1 [Ul] - sin 2 Ofill [u2])

- 2 Au - / x c°t/3H2 [ul, u2] mg, + 2 Au - / x tan/3 (cos 2 0r H2 [ u2, xl ] + sin 2 0r//2 [ ul, xl ]) • m~

A.4. Chargino Unlike the case of the W and of physical and unphysical scalars, in the chargino sector the O ( a s ) corrections due to gluon and gluino exchanges are not separately finite. Therefore, as shown below, the renormalization of the chargino vertex due to the O ( ~ ) heavy particles will contain 1/E poles (E = (4 - n)/2, with n the dimension

18

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20

of the space-time). The cancellation of these poles against similar terms coming from two-loop diagrams in which a gluon is present provides a check of our calculation. We write the corrected 2adta vertex as O(s (--ig)Vtd(-- sm " O~Ua2 ~ v/~M-7"co md s fla+ (1 +~-~Cd)

+

[cos OTVa2V~Mw sin fl (1 + 3¢r mt

\

°:Sca'~ + sin OTl.'~,a-

t,dJ

(1+ 3¢r t,dJ']J } (A.1)

with (/2 is the 't Hoofl mass) 3in {m2~ A(dl) 2 t/22 ] + -}- A(2) q- Tly,

3 C d - 2e

\

Ca3

/

3 in (m2'~ + A ( 1 ) A ( 2 ) + r/y, 2 ~ /22 ] t,a + t,d

t,d- 2e

%

I

cb 3 3 (m_~'~ .(1) t,d = --2-fie -~- 2 In ~k/22 ] + at'd @ ~g.

(A.2)

The a(i) ~t(,d) functions in Eq. (A.2), also appearing in Eqs. (15)-(19), are given by d(dj) _

3 4

dO ) _

1 2H1 [x2],

3

t,d

1

2H1 [Xl ]

4

( - cot0T m.--~--2 rnt m~ /I H3[X2] + -~Hl[xl] 1 A(a2) = ~5 + ,,1 + ~1 H l[X2] _2Ad -- tz tan fl H2[ Xl,X2], m~ A(2) 5 1 t'd=2-[-n3[xl]-l--2nl[Ul]+2

1H

l[U2] --

2Au-#eotflHz[Ul,U2] ' m~

where H3[x] =

2x 1--X

lnx.

In Eq. (A.2) the factors r/r = - 3 / 2 and r/g = - 1 / 2 are induced by the fact that the MS renormalization does not preserve supersymmetry, as discussed in the text.

Appendix B

This appendix contains the expression of the parameter R defined in Eq. (28), which can be derived from the results of Ref. [ 16]. The SM contribution is

M. Ciuchini et al./Nuclear Physics B 534 (1998) 3-20 ZISM =

19

tW 3 -- 1 2 t w 2 ÷ 1 5 t W - - 4 + 6 tW2 ln tw 4 (tW -- 1) 3

(B.1)

with tw = rht(Izw)2/M 2. T h e s u p e r s y m m e t r i c contribution can be split into c h a r g e d H i g g s and c h a r g i n o contributions: (B.2)

ASUSY= AH q- ~,

,4H : cot4 fl ( th 2 -- 1 -- 2th In th ) th + 2cot213tw [F'(tw, hw) + l G'(tw, h w ) l 4 ( t h - 1) 3 (B.3) with th = rh,(tzw)Z/M2± and hw = M H 2 ± / M 2w. The functions F' and G' are g i v e n by

Ft(x,y) =

(x 2-

G'(x,y) = (x-y~(x

ylny (x-y)2(y-

y) lnx

(x-y)Z(x

- 1) 2 -

1)

1-

~ +

1)

1 (x-y)(x-

y)]

x ....... - y

lnx

1)'

y lny +

(x

-

y)Z(y

_

1)

(B.4) C o n s i s t e n t l y with our assumptions, w e keep o n l y the light stop contributions to zi:

M 4 t22it~j G'(Y2j, xij), ~=~-~ 2-----5i,j=1,2 mt mxj

(B.5)

w h e r e Y2j is the variable defined in Eq. ( 7 ) and x 0 is the ratio o f the squared masses o f two charginos.

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