Three-loop QCD corrections to B_s -> mu^+ mu^-
aa r X i v : . [ h e p - ph ] N ov SFB/CPP-13-83TTP13-034
Three-loop QCD corrections to B s → µ + µ − Thomas Hermann ( a ) , Miko laj Misiak ( b,c ) and Matthias Steinhauser ( a ) (a) Institut f¨ur Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany(b) Institute of Theoretical Physics, University of Warsaw,Ho˙za 69, PL-00-681 Warsaw, Poland(c) Theory Division, CERN, CH-1211 Geneva 23, Switzerland Abstract
The decay B s → µ + µ − in the Standard Model is generated by the well-known W -box and Z -penguin diagrams that give rise to an effective quark-lepton operator Q A at low energies. We compute QCD corrections of order α s to its Wilson coeffi-cient C A . It requires performing three-loop matching between the full and effectivetheories. Including the new corrections makes C A more stable with respect to thematching scale µ at which the top-quark mass and α s are renormalized. The cor-responding uncertainty in | C A | gets reduced from around 1 .
8% to less than 0 . B s ( d ) → ℓ + ℓ − decay modes. Introduction
The decay B s → µ + µ − is well known as a probe of physics beyond the Standard Model(SM). Recently, it has attracted a lot of attention since the LHCb and the CMS experi-ments at the CERN LHC have provided first measurements of its branching ratio [1–3].Their current results for the average time-integrated branching ratio read B ( B s → µ + µ − ) = (cid:0) . +1 . − . (cid:1) × − , LHCb [2] , B ( B s → µ + µ − ) = (cid:0) . +1 . − . (cid:1) × − , CMS [3] , (1)which leads to the weighted average [4] B ( B s → µ + µ − ) = ( 2 . ± . × − . (2)Previous upper limits can be found in Refs. [5–9]. Although the experimental uncertaintiesare still quite large, they are expected to get significantly reduced within the next fewyears.As far as the theory side is concerned, the B s meson decay into two muons is quite clean.In fact, the only relevant quantity that needs to be calculated at the leading order in α em and cannot be determined within perturbation theory is the leptonic decay constant f B s .Its square enters the branching ratio as a multiplicative factor. Recent progress in thedetermination of f B s from lattice calculations [10–15] gives a motivation for improving theperturbative ingredients, in particular the two-loop electroweak [16] and the three-loopQCD corrections.Evaluation of the latter corrections is the main purpose of the present paper. Renormal-ization scale dependence of the truncated perturbation series is going to be significantlyreduced. In our case, it refers to the branching ratio dependence on the scale µ at whichthe top-quark mass and α s are renormalized. At the two-loop order, the correspondinguncertainty amounts to around 1 . L eff = L QCD × QED (leptons and five light quarks) + N X n C n Q n + h . c . , (3)with N = V ∗ tb V ts G F M W π , (4)and the operators Q A = (¯ bγ α γ s )(¯ µγ α γ µ ) ,Q S = (¯ bγ s )(¯ µµ ) , P = (¯ bγ s )(¯ µγ µ ) . (5)In the SM, the operator Q A alone is sufficient because contributions from Q S and Q P to the branching ratio are suppressed by M B s /M W with respect to that from Q A . Inbeyond-SM theories, the Wilson coefficients C S and C P can get enhanced, especially foran extended Higgs sector (see, e.g., Refs. [17, 18]). Note that Q V = (¯ bγ α γ s )(¯ µγ α µ ) doesnot contribute at the leading order in α em due to the electromagnetic current conservation.Using Eq. (3), the following result for the average time integrated branching ratio can bederived B ( B s → µ + µ − ) = | N | M B s f B s π Γ sH β h | rC A − uC P | F P + | uβC S | F S i + O ( α em ) , (6)where Γ sH stands for the total width of the heavier mass eigenstate in the B s ¯ B s system.The quantities r , β and u are given by r = 2 m µ M B s , β = √ − r , u = M B s m b + m s . (7)In the absence of beyond-SM sources of CP-violation, we have F P = 1 and F S = 1 − ∆Γ s / Γ sL , where Γ sL is the lighter eigenstate width, and ∆Γ s = Γ sL − Γ sH . In a generic case,from the results in Refs. [19, 20] one derives F P = 1 − ∆Γ s Γ sL sin (cid:20) φ NP s + arg( rC A − uC P ) (cid:21) ,F S = 1 − ∆Γ s Γ sL cos (cid:20) φ NP s + arg C S (cid:21) , (8)where φ NP s describes the CP-violating “new physics” contribution to B s ¯ B s mixing, i.e. φ c ¯ css ≃ arg[( V ∗ ts V tb ) ] + φ NP s (see Sec. 2.2 of Ref. [21]).In the SM, the branching ratio of B s → µ + µ − is proportional to the square of the Wil-son coefficient C A which can be computed within perturbation theory. The calculationamounts to matching the amplitude for s → b µ + µ − in the full SM to the one of the ef-fective theory defined in Eq. (3). At the matching scale µ , the W and Z bosons togetherwith the top quark are integrated out simultaneously.Barring higher-order electroweak (EW) corrections, the perturbative expansion of C A reads C A = C (0) A + α s π C (1) A + (cid:16) α s π (cid:17) C (2) A + ... , (9)where α s ≡ α s ( µ ) in the MS scheme with five active quark flavours. No other definition of α s is going to be used throughout the paper. The one-loop term C (0) A has been calculated More precisely, we shall match the ¯ bs ¯ µµ one-light-particle-irreducible (1LPI) Green’s functions atvanishing external momenta. C (1) A has been found in Refs. [23–26]. In this work, we compute the three-loop QCD correction C (2) A .Let us note that C ( n ) A are µ -dependent, but C A itself is not, up to higher-order QEDeffects. It follows from the fact that the quark current in Q A is classically conservedin the limit of vanishing quark masses, while the chiral anomaly plays no role here, aswe work at the leading order in flavour-changing interactions. Once the perturbationseries on the r.h.s. of Eq. (9) is truncated, a residual µ -dependence arises. Our presentcalculation aims at making this dependence practically negligible.At each loop order, we shall split the coefficients C ( n ) A into contributions originating fromthe W -boson box and the Z -boson penguin diagrams (see Figs. 1 and 4) C ( n ) A = C W, ( n ) A + C Z, ( n ) A , (10)which are separately finite but gauge-dependent with respect to the EW gauge fixing.Here, we use the background field version of the ’t Hooft-Feynman gauge for the elec-troweak bosons, and the usual ’t Hooft-Feynman gauge for the gluons. Most of theresults have also been cross-checked using the general R ξ gauge for the gluons.For the top quark mass renormalization, we shall always use the MS scheme in the fullSM, i.e. m t ≡ m t ( µ ). The ratio m t /M W will enter our results via the following threevariables: x = m t M W , w = 1 − x , y = 1 √ x . (11)The ratio x is the only parameter on which the coefficients C ( n ) A depend, apart from thelogarithms ln( µ /M W ) or ln( µ /m t ). For the Z -penguins, this is true after taking theleading-order EW relations between M Z , M W and sin θ W into account.Our paper is organized as follows: in the next two sections, we evaluate the matchingcoefficient C A up to three loops. Calculations of the W -boxes and the Z -penguins arediscussed in Sections 2 and 3, respectively. Section 4 is devoted to a numerical analysis andexamining the size of the evaluated three loop QCD corrections. We conclude in Section 5.Logarithmically enhanced QED corrections to C A are summarized in the Appendix. W -boson boxes Sample Feynman diagrams contributing to C W, ( n ) A at one-, two- and three-loop order areshown in Fig. 1. The up- and charm-quark contributions differ only by the correspondingCabibbo-Kobayashi-Maskawa (CKM) factors because we neglect masses of these quarks.4 a ) ( b ) ( c ) s bl − l + W Wνu, c, t s bl − l + W Wνu, c, t s bl − l + W Wνu, c, t
Figure 1: Sample W -boson box diagrams contributing to C A .Consequently, it is possible to write C W, ( n ) A in terms of the top- and charm-quark contri-butions C W, ( n ) A = C W,t, ( n ) A − C W,c, ( n ) A , (12)where unitarity of the CKM matrix has been applied.To obtain C W,t, ( n ) A and C W,c, ( n ) A , we compute off-shell 1LPI amplitudes both in the fulltheory and in the effective theory, and require that they agree at the scale µ up to termssuppressed by heavy masses. In fact, on the full-theory side, all the external momenta canbe set to zero, which leads to vacuum integrals up to three loops. On the other hand, in theeffective theory, all loop corrections vanish in dimensional regularization after setting theexternal momenta and light quark masses to zero because the loop integrals are scalelessin this limit. Thus, we are only left with tree contributions.There are basically two approaches to perform the matching. In the first one, the match-ing is performed in d = 4 − ǫ dimensions setting all the light masses strictly to zero. Asa consequence, one generates spurious infrared divergences both in the full and effectivetheories. Such divergences cancel while extracting C A . However, due to the presence ofadditional poles in ǫ at intermediate steps, one has to introduce the so-called evanescentoperators in the effective Lagrangian, which complicates the calculations. In an alterna-tive matching procedure, finite light quark masses are introduced to obtain infrared andultraviolet finite results, which allows for a matching in four dimensions. In the latter case,no evanescent operators matter. In the following, we describe both matching proceduresin more detail. d dimensions The evanescent operator which enters the effective Lagrangian when the matching isperformed in d dimensions reads [25] Q EA = (¯ bγ α γ α γ α γ s )(¯ µγ α γ α γ α γ µ ) − Q A . (13)5 a ) ( b ) l − l + s b l − l + s b Figure 2: Sample one- and two-loop Feynman diagrams needed for determination of therenormalization constants Z NN , Z NE , Z EN and Z EE . Squares represent the operators Q A and Q EA .Note that this operator vanishes in d = 4 dimensions, and thus the limit d → d dimensions.Before performing the matching, we have to replace the combination C A Q A + C EA Q EA bythe corresponding renormalized expression that can be written as [25] C A Q A + C EA Q EA → Z ψ (cid:0) C A Z NN Q A + C A Z NE Q EA + C EA Z EN Q A + C EA Z EE Q EA (cid:1) , (14)where Z ψ is the MS quark wave function renormalization constant. Loop correctionsto Z ψ , Z NN , Z EE and Z NE contain no finite parts but at most poles in ǫ . As far as Z EN is concerned, we require that amplitudes proportional to C EA vanish for d →
4. Inconsequence, Z EN may contain both pole parts and (uniquely defined) finite terms. Forour purpose, the renormalization constants are needed up to two loops.The renormalization constants Z NN , Z NE , Z EN and Z EE are computed from the diagramslike those in Fig. 2, with insertions of Q A and Q EA . Since we are only interested inultraviolet poles of momentum integrals, all the masses can be set to zero, and an externalmomentum q flowing through the quark lines is introduced. Our results read Z NN = 1 ,Z NE = 0 ,Z EN = α s π
32 + (cid:16) α s π (cid:17) (cid:20) ǫ (cid:18) −
176 + 323 n f (cid:19) + 11923 − n f (cid:21) + O ( α s ) ,Z EE = 1 + O ( α s ) , (15)where n f = 5 denotes the number of active quark flavours. The results for Z NN and Z NE are true to all orders in QCD due to the (already mentioned) quark current con-servation for massless quarks. Concerning Z EN , we confirm the one-loop result from In our conventions, n -loop integrals are normalized with e µ nǫ , where e µ ≡ µ e γ / (4 π ) and γ denotesthe Euler-Mascheroni constant. Thus, no ln 4 π or γ appear in the MS renormalization constants. Z EE does not matter for ourcalculation, and thus we have left its two-loop part unevaluated.In the first step of our matching calculation, we determine the s → bµ + µ − transitionamplitude in the full theory, where the Dirac structure of each Feynman diagram is pro-jected onto Q A and Q EA (see, e.g., Ref. [27]). This gives us the unrenormalized amplitudes,which we denote by C WA, bare and C EA, bare , respectively. In this step, vacuum diagrams up tothree loops with two different mass scales have to be computed. Although some classesof Feynman diagrams of this type have been studied in the literature (see, e.g., Ref. [28]),we have decided to perform expansions in various limits, which leads to handy resultsfor the matching coefficients. Actually, we follow the same strategy as in Refs. [29, 30],namely, we expand in the limits M W ≪ m t and M W ≈ m t , i.e. y ≪ w ≪
1, whereterms up to order y and w are evaluated, respectively. A simple combination of thetwo expansions provides an approximation to the three-loop contribution, which for allpractical purposes is equivalent to an exact result.The actual calculation has been performed with the help of QGRAF [31] to generate theFeynman diagrams, q2e and exp [32] for the asymptotic expansions [33] and
MATAD [34],written in
Form [35], for evaluation of the three-loop diagrams. We have performed ourcalculation for an arbitrary gauge parameter in QCD, and have checked that it drops outin our final result for the matching coefficient.For renormalization of the full-theory contributions, we need the one-loop renormalizationconstant for the QCD gauge coupling Z SM g = 1 + α s π (cid:18) − ǫ + 13 ǫ N ǫ (cid:19) + O ( α s ) . (16)Here, N ǫ = ( µ /m t ) ǫ e γǫ Γ(1 + ǫ ) makes the renormalized α s in the full SM equal to theMS-renormalized α s in the five-flavour effective theory, to all orders in ǫ . As far as thetop quark mass is concerned, its two-loop MS renormalization constant Z m t in the fullSM is expressed in terms of the above-defined α s , which gives Z m t = 1 − ǫ α s π + (cid:16) α s π (cid:17) (cid:18) ǫ − ǫ − ǫ N ǫ (cid:19) + O ( α s ) . (17)Furthermore, for the wave-function renormalization, only the difference between the renor-malization constants in the full and effective theories has to be taken into account (seeSection 4 of Ref. [29]): ∆ Z ψ = (cid:16) α s π (cid:17) N ǫ (cid:18) ǫ − (cid:19) + O ( α s , ǫ ) . (18)At this point, all the ingredients are available to perform the matching according to thefollowing equations ( Q = c, t ): C E,QA = C E,Q, (0) A, bare + α s π (cid:16) C E,Q, (1) A, bare + δ tQ ∆ T E,t, (1) (cid:17) + O (cid:0) α s (cid:1) , W,QA = (1 + ∆ Z ψ ) X n =0 (cid:16) α s π (cid:17) n h(cid:0) Z SM g (cid:1) n C W,Q, ( n ) A, bare + δ tQ ∆ T W,t, ( n ) i − Z EN C E,QA + O (cid:0) α s (cid:1) , (19)where ∆ T E,t, (1) and ∆ T W,t, ( n ) denote contributions from the top-quark mass countertermswhich can be written as∆ T E,t, (1) = (cid:16) C E,t, (0) A, bare (cid:12)(cid:12) m bare t → Z mt m t (cid:17) α s , ∆ T W,t, (0) = 0 , ∆ T W,t, (1) = (cid:16) C W,t, (0) A, bare (cid:12)(cid:12) m bare t → Z mt m t (cid:17) α s , ∆ T W,t, (2) = (cid:16) C W,t, (0) A, bare (cid:12)(cid:12) m bare t → Z mt m t + α s π C W,t, (1) A, bare (cid:12)(cid:12) m bare t → Z mt m t (cid:17) α s . (20)Here, the following notation has been used: “ m bare t → Z m t m t ” means that the baretop quark mass is replaced by the renormalized one times the renormalization constant.Afterwards, we expand in α s and take the coefficient at [ α s / (4 π )] n ( n = 1 , C E,t, (0) A = 164 (cid:18) µ m t (cid:19) ǫ (cid:20) x − − x ln x ( x − + ǫ (cid:18) x − − ( x + 2) ln x + x ln x ( x − (cid:19)(cid:21) + O ( ǫ ) ,C E,t, (1) A = 7 − x x − + 7 x + 9 x x − ln x + x x − Li (cid:18) − x (cid:19) + ln (cid:18) µ m t (cid:19) (cid:20) − x x − + x + x x − ln x (cid:21) + O ( ǫ ) ,C E,c, (0) A = − (cid:18) µ M W (cid:19) ǫ (2 + 3 ǫ ) + O ( ǫ ) ,C E,c, (1) A = 724 + O ( ǫ ) . (21)The results for C W,tA and C W,cA will be given in Subsection 2.4.
In order to have a cross check of the results for C WA from the previous subsection, wehave performed the matching also for infrared finite quantities, which can be done in fourdimensions avoiding evanescent operators [25]. No spurious infrared divergences arisewhen small but non-vanishing masses are introduced for the strange and bottom quarks.8n the full theory, this leads to Feynman diagrams with up to four different mass scales.We evaluate them using asymptotic expansions in the limit m t , M W ≫ m b ≫ m s . (22)In addition, we use either m t ≫ M W or m t ≈ M W , as in the previous subsection. All theexternal momenta are still set to zero. Asymptotic expansions are conveniently performedwith the help of exp [32].On the effective-theory side, the loop corrections do not vanish any more due to the finitequark masses. We compute the necessary one- and two-loop Feynman integrals in thelimit m b ≫ m s . (23)After renormalization of the two-loop expression on the effective-theory side and thethree-loop result on the full-theory side, the finite parts are matched for ǫ →
0. Afterthe matching, it is possible to take the limit m s → m b →
0. This way, we obtainthe same results as in the previous calculation where the infrared divergences have beenregulated using dimensional regularization.Although we only had to compute the leading non-vanishing contributions in the lightquark masses, the calculational effort has been significantly higher than for the matchingin d dimensions described in the previous subsection. Thus, we have applied the methodwith light masses only to cross check the first two (three) terms in the expansion in y ( w ),using a general R ξ gauge though. At the one- and two-loop orders, we have confirmed the results with full dependence on x from Ref. [25], and evaluated in addition terms up to O ( ǫ ) and O ( ǫ ), respectively. Forcompleteness, we present the results for ǫ → C W,t, (0) A ( µ ) = 18( x − − x x − ln x ,C W,t, (1) A ( µ ) = − x x − + 17 x − x x − ln x + x ( x − Li (cid:18) − x (cid:19) + ln (cid:18) µ m t (cid:19) (cid:20) − x ( x − + x + x ( x − ln x (cid:21) ,C W,c, (0) A ( µ ) = − ,C W,c, (1) A ( µ ) = − . (24)Analytic expressions including O ( ǫ ) terms can be downloaded from [36].9ith the help of the exact two-loop result, we can extract the full x -dependence in frontof the ln µ terms at the three-loop level. We find C W,t, (2) A ( µ ) = C W,t, (2) A ( µ = m t ) + ln (cid:18) µ m t (cid:19) (cid:20)
69 + 1292 x − x x − − x + 105 x − x x − ln x − x + x x − Li (cid:18) − x (cid:19)(cid:21) + ln (cid:18) µ m t (cid:19) (cid:20) x + 11 x x − − x + 96 x − x x − ln x (cid:21) ,C W,c, (2) A ( µ ) = C W,c, (2) A ( µ = M W ) −
236 ln (cid:18) µ M W (cid:19) . (25)We have chosen µ = m t and µ = M W as default scales for the top and charm sectors,respectively. Analytical results for all the coefficients can be downloaded from [36]. In thefollowing, we present the results in a compact numerical form. For our two expansions,the coefficient in the charm sector reads C W,c, (2) A ( µ = M W ) = − . − . y + 0 . y ln y − . y ln y + 0 . y − . y ln y + 0 . y ln y − . y + 0 . y ln y − . y + 0 . y ln y − . · − y +2 . · − y ln y − . · − y + 2 . · − y ln y + O (cid:0) y (cid:1) , (26) C W,c, (2) A ( µ = M W ) = − .
403 + 0 . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w + O (cid:0) w (cid:1) . (27)The corresponding coefficient in the top sector is given by C W,t, (2) A ( µ = m t ) = 2 . y + 6 . y ln y − . y − . y ln y − . y − . y ln y + 35 . y + 15 . y ln y + 103 . y +149 . y ln y + 207 . y + 454 . y ln y + O (cid:0) y (cid:1) , (28) C W,t, (2) A ( µ = m t ) = − . − . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w + O (cid:0) w (cid:1) . (29)10 = M W - - - - - - = M W (cid:144) m t C A W , c H L Μ = m t - - - = M W (cid:144) m t C A W , t H L Figure 3: C W, (2) A as a function of y = M W /m t for the charm (left) and top quark sector(right). The (blue) dashed lines are obtained in the limit y ≪
1, and the (grey) solid linefor w = 1 − y ≪
1. Thinner lines contain less terms in the expansions. The physicalregion for y is indicated by the (yellow) vertical band.In Fig. 3, the results from Eqs. (26)–(29) are shown as functions of y = M W /m t . Thedashed and solid lines correspond to the y → y = √ − w → y and w . They canbe used to test convergence of the expansions, as it is expected that good agreement withthe unknown exact result is achieved up to the point where two successive orders almostcoincide.In the case of C W,c, (2) A (left panel of Fig. 3), there is a significant overlap of the expansionsaround the two limits in the region from y ≈ . y ≈ .
4. The agreement over such alarge range arises probably due to the relatively simple dependence of C W,c, (2) A on the topquark mass: m t only occurs through one-loop corrections to the gluon propagator. Notealso that the numerical effect of the top quark mass is moderate: C W,c, (2) A changes only byaround 1.5% between the m t → ∞ limit and the physical value of m t .Also in the case of C W,t, (2) A (right panel of Fig. 3) one observes an overlap of the expansionsfor y → y → y ≈ .
35. This feature allows us to use the expression inEq. (28) for y ≤ .
35, and the one in Eq. (29) for y > .
35. Due to the convergenceproperties (cf. thin lines) these expressions are excellent approximations to the exactresult in the respective regions. In particular, for the physical region of y , it is sufficientto use the expansion around m t = M W .For practical applications, it is useful to have short formulae which approximate C A nearthe physical value of y . In the range 0 . < y < .
7, the fits C W,c, (2) A ( µ = M W ) ≃ − . y − . y − . ,C W,t, (2) A ( µ = m t ) ≃ . y − . y + 0 .
189 (30)are accurate to better than 1% in the corresponding quantities.11 a ) ( b ) ( c ) l − l + s bW Zu, c, t u, c, t l − l + s bu, c, tZW W l − l + s bZW W Figure 4: Sample Z -boson penguin diagrams contributing to C A . Z -boson penguins The second type of contribution to C A arises from the so-called Z -boson penguins. Sam-ple diagrams at the one-, two- and three-loop orders are shown in Fig. 4. In contrastto the W -box diagrams, there is no contribution from evanescent operators to C Z, ( n ) A .However, flavour non-diagonal loop contributions to the light quark kinetic terms requireintroduction of an EW counterterm which already appears at the one-loop level. Thecorresponding counterterm Lagrangian (see Eq. (13) of Ref. [30] ) can be written in thefollowing form L ewcounter = i G F M W √ π V ∗ tb V ts (cid:0) Z t ,sb − Z c ,sb (cid:1) ¯ b L Ds L , (31)where D µ is the covariant derivative involving the neutral gauge boson fields ( Z , γ , g ).While only the one-loop contributions to Z Q ,sb were needed in the ¯ B → X s γ case [29], nowalso the two- and three-loop corrections of order α s and α s matter. The two-loop oneswere also necessary in Refs. [23–27]. Perturbative expansions of Z Q ,sb are convenientlywritten as Z c ,sb = X n =0 (cid:18) µ M W (cid:19) ( n +1) ǫ (cid:16) α s π (cid:17) n Z c, ( n )2 ,sb ,Z t ,sb = X n =0 (cid:18) µ m t (cid:19) ( n +1) ǫ (cid:16) α s π (cid:17) n Z t, ( n )2 ,sb . (32)For determination of Z Q ,sb , a two-point function with incoming strange quark and outgoingbottom quark has to be considered. Sample diagrams at one, two and three loops areshown in Fig. 5. We refrain from explicitly listing the results but refer to [36] for computer-readable expressions. Flavour non-diagonal renormalization of the mass terms does not matter in the present calculationbecause we can treat the bottom quark as massless. a ) ( b ) ( c ) Wu, c, ts b Ws b Ws b
Figure 5: Sample Feynman diagrams contributing to Z Q ,sb . ( a ) ( b ) ( c ) Z s btl − l + Z s btl − l + Z s btl − l + Figure 6: Sample two-loop counterterm diagrams to the Z -penguin contribution. Alto-gether, there are five such diagrams.The counterterm Z Q ,sb is either inserted in the tree-level amplitude or in two-loop diagramscontaining a closed top quark loop on the gluon propagator, as shown in Fig. 6. Insertionsof the counterterm into other loop contributions lead to massless tadpoles which vanishin dimensional regularization. There is a class of Feynman diagrams which require special attention, namely those con-taining a closed triangle quark loop (see Fig. 7). For these contributions, a naive treatmentof γ as anticommuting is not possible, and a more careful investigation is necessary. Wehave followed two approaches which are described below. Similarly to the anomaly can-cellation in the SM, contributions with the up, down, strange and charm quarks runningin the triangle loop cancel pairwise within each family. Thus, only the top and bottomquarks need to be considered, as the top is the only massive quark in our calculation.In our first approach, we adopt the prescription from Ref. [37] and replace the axial-vectorcoupling in the triangle loop as follows γ µ γ → i ε µνρσ ( γ ν γ ρ γ σ − γ σ γ ρ γ ν ) . (33)13 a ) ( b ) l − l + s bZW l − l + s bZ Figure 7: Sample Feynman diagrams containing a closed triangle fermion loop that con-tribute to C Z, (2) A . The counterterm contribution in the right diagram comes from Eq. (31).In a next step, we pull out the ε tensor and take the trace of the loop diagram in d dimensions. In the resulting object, we perform the replacements i ε µνρσ γ ν γ ρ γ σ γ ⊗ γ µ γ → γ µ ⊗ γ µ γ i ε µνρσ γ ν γ ρ γ σ ⊗ γ µ γ → γ µ γ ⊗ γ µ γ , (34)and proceed from now on in the same way as with the other diagrams contributing to C ZA .In the second approach, we do not take the trace in the triangle loop at all, but only usethe cyclicity property for traces and anticommutation relations for the γ matrices (notfor γ ) in order to put γ to the end of each product under the trace. Afterwards, weperform the tensor loop integration, and use again anticommutation relations to bringthe resulting expressions to the form γ ν γ ρ γ σ γ ⊗ γ µ γ Tr ( γ ν γ ρ γ σ γ µ γ ) ,γ ν γ ρ γ σ ⊗ γ µ γ Tr ( γ ν γ ρ γ σ γ µ γ ) , (35)where only the axial-vector part of the ( Z boson)-lepton coupling has been taken intoaccount. In a next step, we add and subtract 24 γ µ ⊗ γ µ γ to the first, and 24 γ µ γ ⊗ γ µ γ to the second structure in Eq. (35). This way, we obtain the Wilson coefficients for thetrace evanescent operators [38] Q E = γ ν γ ρ γ σ γ ⊗ γ µ γ Tr ( γ ν γ ρ γ σ γ µ γ ) + 24 γ µ ⊗ γ µ γ ,Q E = γ ν γ ρ γ σ ⊗ γ µ γ Tr ( γ ν γ ρ γ σ γ µ γ ) + 24 γ µ γ ⊗ γ µ γ (36)and a contribution to C A . Actually, the latter is given by ( −
24) times the prefactor ofthe second structure in Eq. (35).The two methods, which lead to identical results for C ZA , have been applied both to thethree-loop diagrams themselves and to the counterterm contributions (cf. Fig. 7).14 .3 Matching formula In analogy to Eq. (12) we can write C Z, ( n ) A = C Z,t, ( n ) A − C Z,c, ( n ) A + δ n, (cid:16) C Z,t, tria .A − C Z,c, tria .A (cid:17) , (37)where C Z,Q, tria .A are the contributions from the triangle diagrams described in the previoussubsection.The calculation of C Z, ( n ) A proceeds along the same lines as for the W -box contribution. Inparticular, we set all the external momenta to zero, and expand the Feynman integrals inthe full theory both for m t ≫ M W and m t ≈ M W . Furthermore, we renormalize the top-quark mass, α s and the wave function in analogy to the W -box case. As before, all loopcorrections vanish in the effective theory, which finally leads to the following matchingequation for C Z,QA ( Q = c, t ) C Z,QA = (1 + ∆ Z ψ ) X n =0 (cid:16) α s π (cid:17) n h(cid:0) Z SM g (cid:1) n C Z,Q, ( n ) A, bare + δ tQ ∆ T Z,t, ( n ) + K Q, ( n ) i + ˜ K Q + O (cid:0) α s (cid:1) , (38)with top-quark mass counterterms∆ T Z,t, (0) = 0 , ∆ T Z,t, (1) = (cid:16) C Z,t, (0) A, bare (cid:12)(cid:12) m bare t → Z mt m t (cid:17) α s , ∆ T Z,t, (2) = (cid:16) C Z,t, (0) A, bare (cid:12)(cid:12) m bare t → Z mt m t + α s π C Z,t, (1) A, bare (cid:12)(cid:12) m bare t → Z mt m t (cid:17) α s . (39) K Q, ( n ) denote tree-level contributions from the EW counterterm (31) which take a simpleform K t, ( n ) = (cid:18) −
116 + sin θ W (cid:19) (cid:18) µ m t (cid:19) ( n +1) ǫ Z t, ( n )2 ,sb ,K c, ( n ) = (cid:18) −
116 + sin θ W (cid:19) (cid:18) µ M W (cid:19) ( n +1) ǫ Z c, ( n )2 ,sb . (40)Finally, ˜ K Q stands for the counterterm contributions from two-loop diagrams like thosein Fig. 6.We observe that after inserting explicit results on the right-hand side of Eq. (38), allthe terms proportional to sin θ W cancel out, and C Z,tA becomes independent of the weakmixing angle. This can be understood by recalling similarities between the Z boson andthe photon couplings to other particles in the background field gauge, as well as thestructure of the counterterm in Eq. (31). Once the quark kinetic terms in the effective15heory are imposed to be flavour diagonal, the same must be true for dimension-four quark-photon couplings. In effect, the counterterm in Eq. (31) automatically renormalizes awayall the zero-momentum quark-( Z boson) interactions that come with sin θ W .Another interesting thing to note is that C Z,c, ( n ) A = 0 at each loop order in the backgroundfield version of the ’t Hooft-Feynman gauge, which means that only the triangle contri-butions are non-vanishing in the charm sector. One of the ways to understand this fact isagain by considering diagrams where the Z boson (together with the muons) is replacedby an external photon. With our calculation, we have confirmed the one- and two-loop results from Ref. [23]which are given by (for ǫ → C Z,t, (0) A ( µ ) = − x + x x −
1) + 2 x + 3 x x − ln x ,C Z,t, (1) A ( µ ) = 29 x + 7 x + 4 x x − − x + 14 x + 3 x x − ln x − x + x x − Li (cid:18) − x (cid:19) + ln (cid:18) µ m t (cid:19) (cid:20) x + x + x x − − x + 4 x ( x − ln x (cid:21) . (41)Furthermore, similarly to C WA , we obtain exact dependence on M W and m t for the µ -dependent terms which read C Z,t, (2) A ( µ ) = C Z,t, (2) A ( µ = m t ) + ln (cid:18) µ m t (cid:19) (cid:20) x + 4 x + 95 x − x x − Li (cid:18) − x (cid:19) + 1468 x + 1578 x − x − x x − ln x − x + 1031 x + 582 x − x x − (cid:21) + ln (cid:18) µ m t (cid:19) (cid:20) x + 315 x − x x − ln x − x + 257 x + 72 x − x x − (cid:21) . (42)For the generic three-loop contributions, terms up to order y and w have been evalu-ated, as in the W -box case in Section 2. In a numerical form, they read C Z,t, (2) A ( µ = m t ) = 0 . y + 2 .
139 + 28 . y + 33 . y ln y + 28 . y + 97 . y ln y − . y + 106 . y ln y − . y − . y ln y − . y − . y ln y − . y − . y ln y + O (cid:0) y (cid:1) ,C Z,t, (2) A ( µ = m t ) = − .
934 + 0 . w + 0 . w + 0 . w + 0 . w + 0 . w +0 . w + 0 . w + 0 . w + 0 . w + 0 . w . w + 0 . w + 0 . w + 0 . w + 0 . w +0 . w + O (cid:0) w (cid:1) . (43)For the fermion triangle contributions, all the ln µ contributions have cancelled out aftermatching. We find C Z,t, tria .A = − . y − . − . y − . y ln y − . y − . y ln y − . y − . y ln y − . y − . y ln y − . y − . y ln y − . y − . y ln y + O (cid:0) y (cid:1) ,C Z,t, tria .A = − . − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w + O (cid:0) w (cid:1) ,C Z,c, tria .A = − .
250 + 1 .
500 ln y − . y + 0 . y ln y − . y ln y + 0 . y − . y ln y + 0 . y ln y − . y + 0 . y ln y − . y + 0 . y ln y − . y + 0 . y ln y − . · − y + 1 . · − y ln y + O (cid:0) y (cid:1) ,C Z,c, tria .A = − . − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w − . w + O (cid:0) w (cid:1) . (44)In the limit of large top quark mass, the coefficient C Z, (2) A grows as m t , which has itsorigin in the Yukawa interaction of the charged pseudo-goldstones with the top quark.For this reason, we plot in Fig. 8 the combination y C Z, (2) A where sums of the resultsfrom Eqs. (43) and (44) are shown as dashed ( M W ≪ m t ) and solid lines ( M W ≈ m t ).Note that after multiplication by y , the latter is expanded in w = 1 − y . Again,one observes that the two approximations coincide for y ≈ .
4, which suggests that acombination of the two expansions covers the whole range between y = 0 and y = 1. Inthe physical region, the expansion around M W = m t provides an excellent approximation.It is interesting to note that the fermion triangle contribution is more than an order ofmagnitude larger than C Z,t, (2) A . This is particularly true for the physical value of y wherewe have y C Z,t, (2) A ≈ − . . < y < . C Z, (2) A ( µ = m t ) ≃ . y − . y + 57 . y − . . (45)17 = m t - - - - - - = M W (cid:144) m t y C A Z H L Figure 8: y C Z, (2) A as a function of y = M W /m t . The (blue) dashed lines are obtained inthe limit y ≪ w = 1 − y ≪
1. Thinner lines contain lessterms in the expansions. The physical region for y is indicated by the (yellow) verticalband. In this section, we shall discuss numerical effects of our three-loop QCD corrections. The B ( B s → µ + µ − ) branching ratio in the SM is proportional to | C A | (cf. Eq. (6) with C S = C P = 0 and F P = 1). Here, we shall consider | C A | only. Evaluation of thebranching ratio itself is relegated to a parallel article [39] where also the new two-loopEW corrections [16] are included.The relevant parameters are as follows. For the gauge boson masses, we take M Z =91 . M W = 80 .
358 GeV (calculated from G F , M Z and α em ). For thestrong coupling, α s ( M Z ) = 0 . M t = 173 . m t should be understood as renormalized on-shell withrespect to the EW interactions. As far as QCD is concerned, we use a three-loop relationfor converting M t to m t ( m t ), which gives m t ( m t ) ≃ . m t ( µ ) at other values of µ .Fig. 9 shows the matching scale dependence of | C A | . The dotted, dashed and solid curvesshow the leading order (LO), next-to-leading-order (NLO) and next-to-next-to-leading-order (NNLO) results, respectively. In the current case, they correspond to one-, two-and three-loop matching calculations.The range of the plot corresponds roughly to µ ∈ (cid:2) M W , m t (cid:3) , which might be consideredreasonable given that both the W -boson and the top quark are decoupled simultaneously.However, our Wilson coefficient has a trivial RGE (at the LO in EW interactions) butit is quite sensitive to m t . In consequence, the main reason for its µ -dependence here is18 NLONLOLO
50 100 150 200 250 3000.210.220.230.240.25 Μ in GeV È C A È Figure 9: Matching scale dependence of | C A | and the LO, NLO and NNLO in QCD butat the LO in EW interactions. The top quark mass is renormalized on shell with respectto the EW interactions, and at µ in MS with respect to QCD.the top-quark mass renormalization. Thus, for estimating uncertainties due to truncationof the QCD perturbation series at each order, we shall use a more narrow range µ ∈ (cid:2) m t , m t (cid:3) .One observes in Fig. 9 that the prediction for | C A | has already improved a lot afterincluding the NLO QCD corrections. The variation at the NLO level amounts to around1 .
8% only, for µ ∈ (cid:2) m t , m t (cid:3) . Once the new three-loop corrections are taken intoaccount, the uncertainty gets reduced to less than 0 . µ = m t the NLO correctionis moderate (2 . µ = m t has been anticipated to be an optimal scale in the past [24], there has been no convincingtheoretical argument for such a choice. Our explicit three-loop calculation has beenactually necessary to suppress the QCD matching uncertainties in | C A | to the currentsub-percent level.For µ = 160 GeV, our final result for C A is well approximated by the fit C A = 0 . (cid:18) M t . (cid:19) . (cid:18) α s ( M Z )0 . (cid:19) − . + O ( α em ) , (46)which is accurate to better than 0 .
1% for α s ( M Z ) ∈ [0 . , .
13] and M t ∈ [170 , O ( α em ) term in Eq. (46) stands for both the NLO EW matchingcorrections at µ = µ , as well as effects of the evolution of C A down to µ = µ b ∼ m b ,according to the RGE. Once the QCD logarithms get resummed, the latter effects behavenot only like O ( α em ), but also like O ( α em /α s ) and O ( α em /α s ), which means that they arepotentially more important than the NNLO QCD corrections evaluated here. However,the actual numerical impact of the O ( α em /α s ) and O ( α em /α s ) terms on the decay rate19mounts to around − .
5% only [41], which has been checked using anomalous dimensionmatrices from Refs. [42, 43]. The necessary expressions are given in the Appendix.As far as the NLO EW matching corrections are concerned, they have been known fora long time only in the m t ≫ M W limit [44]. A complete calculation of these two-loopcorrections has recently been finalized [16]. Their numerical effect depends on the schemeused at the LO. A detailed discussion of this issue is presented in Ref. [16]. Let us onlymention that the semi-perfect stabilization of µ -dependence in Fig. 9 at the NNLO inQCD takes place only because we have renormalized m t and M W on shell with respectto the EW interactions. If we used MS at µ for the EW renormalization of m t and M W ,then acceptable stability would be observed only after including the very two-loop EWcorrections. We have evaluated the NNLO QCD corrections to the Wilson coefficient C A that param-etrizes the B s → µ + µ − branching ratio in the SM. For this purpose, three-loop matchingbetween the SM and the relevant effective theory has been performed. Tadpole integralsdepending on m t and M W have been evaluated with the help of expansions starting fromthe limits m t ≈ M W and m t ≫ M W , which for all practical purposes is equivalent toan exact calculation. When masses of the light quarks and their momenta are set tozero, care has to be taken about the so-called evanescent operators, similarly to the NLOcase [25]. Such operators have also been helpful in dealing with diagrams where γ waspresent under traces.Our results for the renormalized matching coefficients C WA and C ZA can be downloadedin a computer-readable form from [36]. Including the new corrections makes C A morestable with respect to the matching scale µ at which the top-quark mass and α s arerenormalized. Apart from B s → µ + µ − , our calculation is directly applicable to all the B s ( d ) → ℓ + ℓ − decay modes, and it matters for other processes mediated by Z -penguinsand W -boxes, e.g., ¯ B → X s ν ¯ ν , K → πν ¯ ν , or short-distance contributions to K L → µ + µ − .However, it is only B s → µ + µ − for which the three-loop accuracy is relevant at present.An updated SM prediction for B ( B s → µ + µ − ) is presented in a parallel article [39] wherealso the new two-loop EW corrections [16] are included. Acknowledgements
We thank Alexander Kurz for providing to us his
FORM routine to compute tensor tadpoleintegrals to three loops. We are grateful to Christoph Bobeth, Martin Gorbahn andEmmanuel Stamou for helpful discussions and comments on the manuscript. This workwas supported by the DFG through the SFB/TR 9 “Computational Particle Physics”20nd the Graduiertenkolleg “Elementarteilchenphysik bei h¨ochster Energie und h¨ochsterPr¨azision”. M.M. acknowledges partial support by the National Science Centre (Poland)research project, decision DEC-2011/01/B/ST2/00438.
Appendix: Logarithmically enhanced QED corrections
In this appendix, we present explicit expressions for the logarithmically-enhanced QEDcorrections to C A . Beyond the LO in α em , its perturbative expansion at the matchingscale µ reads C A ( µ ) = C sA + α em ( µ )4 π C e, (1) A ( µ ) + O ( α em , α em α s ) , (47)where C sA stands for the scale-independent O ( α em ) contribution as given in Eq. (9). Usingthe RGE from Refs. [42, 43] one obtains the following result at the scale µ b C A ( µ b ) = C sA + α em ( µ b ) α s ( µ b ) F sin θ W + α em ( µ b ) α s ( µ b ) (cid:2) F + F sin θ W (cid:3) + α em G + O (cid:18) α em α s , α em α s (cid:19) , (48)where G includes all the NLO EW corrections that are not logarithmically enhanced. Thequantities F , , depend on η = α s ( µ ) α s ( µ b ) and x = m t M W . We find F = X i =1 p i η a i ,F = 3( η − η Y ( x ) ,F = 3( η − η V ( x ) + z ln ηη + X i =1 η a i (cid:20) q i + η r i + η E ( x ) s i + η t i ln (cid:18) µ M W (cid:19)(cid:21) , (49)with z ≃ . Y ( x ), V ( x ) and E ( x ) originate from the one-loop SM matching conditions [22] for variousoperators in the effective theory. They read Y ( x ) = 3 x x − ln x + x − x x − ,V ( x ) = − x + 6 x + 63 x − x + 818( x − ln x + − x + 163 x − x + 108 x x − , a i − − − . − . − . . p i − . − . − . . . . . − . q i . − . . − . − . − . r i . . − . − . . − . − . . s i . . . . − . . t i . . − . − . . − . − . − . Powers and coefficients in Eq. (49). E ( x ) = − x + 16 x − − x ) ln x + x + 11 x − x x − . (50)The coefficients in Table 1 satisfy the following identities: X i =1 p i a i = X i =1 p i = X i =1 ( q i + r i ) = X i =1 s i = X i =1 t i = 0 . (51)With the help of them one can easily check that the terms in Eq. (48) proportional to F , , are finite in the limit α s →
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