Paride Paradisi Dipartimento di Fisica dell Università di Padova and INFN, Italy

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1 INDIRECT DETERMINATIONS OF THE TOP QUARK MASS Parie Paraisi Dipartimento i Fisica ell Università i Paova an INFN, Italy Abstract We give a complete analysis of inirect eterminations of the top quark mass in the Stanar Moel using flavour an electroweak ata. Although present ata give only a poor etermination, we show how future theoretical an experimental progress in flavour physics can lea to an accuracy in M t well below 2 GeV. We revisit eterminations of M t from electroweak ata, showing how an improve measurement of the W mass leas to an accuracy at the level of 1 GeV. 1 Introuction The top quark mass (M t ) is a key input parameter of the Stanar Moel (SM). Since the top quark is the heaviest particle in the SM, its Yukawa coupling y t is sizeable an plays a crucial role in etermining the preictions of the theory at the quantum level. A precise etermination of M t is crucial for stability of 56

2 the electroweak vacuum [1] or to establish the viability of scenarios of Higgs inflation [2]. Therefore, a more precise etermination of M t will a important information to our knowlege of particle physics an cosmology. The most precise quote value of the top-quark pole mass comes from the combination of LHC an Tevatron measurements (M t ) pole = ± 0.76 GeV [3]. A theoretical concern about the extraction of M t from ata is that the pole top mass is not a physical observable. This means that its experimental etermination is one through the measurement of other physical observables (final-state invariant masses, kinematic istributions, total rates) that are especially sensitive to M t. These measurements are compare to the results of theoretical calculations, which are expresse in terms of M t in a well-efine renormalisation scheme. In the context of haron colliers, the extraction of M t suffers from a variety of effects linke to haronization that are not fully accountable by perturbative QCD calculations, like boun-state effects of the t t pairs, parton showering, an other non-perturbative corrections (see [4] for a thorough iscussion). In practice, the extraction of M t relies on moelling base on Monte-Carlo generators, an this is why [5] refers to M t as Monte-Carlo mass. These consierations justify the search for alternative strategies to etermine M t. Given that the top is the only quark associate to a sizeable Yukawa coupling, loop effects in the SM are potentially very sensitive to M t. Our goal is to ientify all processes that receive quantum corrections enhance by powers of M t (in the limit M t M W ) an infer M t from their measurements. In ref. [6] we have propose to use the comparison between experimental ata an theoretical preictions of flavour processes as a way to extract the top quark mass, uner the assumption that the SM is vali up to very short istance scales. The use of flavour ata for an inirect etermination of M t is fairly robust from the theoretical point of view, since it relies on controllable SM calculations, in which non-perturbative effects are restricte to a few wellknown haronic parameters, now uner careful scrutiny by lattice calculations. The current status of the extraction of the top mass from the fit of flavour ata is (M t ) flavour = (173.4 ± 7.8) GeV [6]. The uncertainty of this extraction is too large to be competitive with the irect measurements. However, taking into account foreseeable progress in perturbative an lattice calculations, on one sie, an experimental measurements, on the other sie, our projection for the future is that the error can be brought to about 1.7 GeV [6]. With the aim of an inirect etermination of the top mass, in ref. [6] we have also reconsiere global fits of electroweak observables, fining (M t ) EW = (177.0 ± 2.6) GeV. We fin that the etermination of M t is ominate by the measurement of M W. A reuction of the error in the measurement of M W to about 8 MeV, as foreseeable at the LHC [7], can bring own the uncertainty on M t to 1.2 GeV. 57

3 2 M t epenence of observables in the heavy-top limit The large top Yukawa coupling offers the possibility of reconstructing M t from SM quantum effects. In orer to ientify the physical observables that are most sensitive to the top mass at the one-loop level, we have evelope here a systematic proceure to extract the leaing M t epenence preicte by the SM [6]. We work in the heavy-top limit [8], in which the masses of the W an Z bosons are neglecte with respect to M t. This is achieve by consiering a gauge-less theory with massive quarks, the Higgs boson h, an 3 Golstone bosons χ (relate by the equivalence theorem to the longituinal components of the W an Z), where the only quark interaction is ( ) L = y t t R H T Vti il + h.c., H = 1 ( ) e i σ χ v 0. (1) t L 2 v + h Here y t is the top Yukawa coupling, V is the CKM matrix, H is the Higgs oublet, v = 246 GeV is the symmetry breaking scale an σ are the Golstone bosons. The next step is to integrate out the top quark. The top-less effective theory will contain a set of effective operators whose coefficients escribe the leaing top-mass epenence in the large M t limit, as illustrate in Fig Extracting M t from flavour ata When searching for new physics, it is customary to etermine the four inepenent CKM parameters from tree-level observables, which are presume to be well escribe by the SM, an then use this etermination to preict loop processes, which are expecte to hie new effects beyon the SM. In ref. [6], we took a ifferent perspective: we have assume the SM to be exactly vali an we have extracte M t from flavour processes. We have then fixe the four CKM parameters from the most precise measurements that o not epen on M t, even if they arise at loop level: V us, V cb, γ, β. (2) In principle, the extraction of (M t ) flavour woul require a global fit of all flavour observables in which the CKM parameters an the top mass are allowe to float inepenently. However, in practice, our proceure of fixing the CKM parameters in eq. (2) from processes that are insensitive to M t an then etermine M t from the remaining observables is perfectly aequate an leas to results ientical to those from a global fit. Actually, as shown in fig. 2, the etermination of M t is ominate by m Bs, which epens on the CKM parameters only through the combination V ts Vtb which is equal to V cb, up to a epenence on the angles γ an β suppresse by two powers of λ. This means that essentially V cb alone rives the error on the etermination of M t attributable to CKM 58

4 Ρ y t 2 A Z bb, Bs Μ Μ, ΠΝΝ y t 3 Mt t t Χ Χ Χ t g hzz, g hww, g ZWW y t 3 Mt A hh hh y t 4 h, Χ h, Χ h, Χ h t h h h Figure 1: Feynman iagrams illustrating the effective operators generate by integrating out the top quark. Also shown is the power counting estimate of their sensitivity to the top mass. Dashe lines enote the Higgs boson (h) or the Golstones (χ); soli lines enote the quarks. elements, while the less precisely known parameters γ an β play only a minor role. Moreover, B s µ + µ will soon become an equally important process for the etermination of M t an its CKM epenence, as in the case of m Bs, is given by V ts Vtb. So our conclusion that V cb is the most important CKM parameter for M t extraction is likely to hol true even after future theoretical an experimental improvements. Let us turn now to iscuss our forecast for the future of M t eterminations from flavour processes focusing on m Bs an B s µ + µ. m Bs The mass ifferences of the Bs, 0 B s, 0 systems in the SM can be written as [17] m Bq = G2 F 6π 2 m B q M 2 W ˆB Bq f 2 B q η B S 0 (x t ) V tq V tb 2, q =, s, (3) where η B accounts for NLO QCD corrections. The LO loop function S 0 (x t ) epens on x t = 2yt 2 /g2, 2 where g 2 is the coupling of the SM gauge group 59

5 Observable Now (2015) Error 2020 Error 2025 M W (GeV) (15) [9] 8 [7] 5 [10] sin 2 θ W (13)[9] 13 [7] 1.3 [7] αem(m 1 Z ) (13)[9] α s (M Z ) (7) [9] 7 [7] 7 [7] m Bs (MeV) (2) [9] m Bs (ps 1 ) (21)[11] τh s (ps) 1.607(10) [11] V cb (11) [12] 4 [13, 14] 3 [13, 14] B(B s µ + µ ) (7) [11] 3 [13, 14] 1.3 [13, 14] η B 0.55(1) [15] 0.5 [16] 0.2 [16] f Bs (MeV) 226(5) [12] 2 [13] 1 [13] ˆB Bs 1.33(6) [12] 2 [13] 0.7 [13] Table 1: Present values an future uncertainties for the most relevant quantities of our analysis. In the preictions for future errors we use the symbol when no significant improvement is expecte. SU(2) L an y t is the top-yukawa coupling. From eq. (3) we obtain the following value for m Bs m Bs = 16.9 ± 1.4 ps ( ) ˆBBs f 2 ( Bs 261 MeV M t GeV ) 1.52 ( Vts Vtb ) 2 ( ) ηb (4) Matching this expression with the measurement of m Bs reporte in table 1, we fin (M t ) mbs = (179.3 ± 9.7) GeV. (5) Assuming the expecte improvements by about 2020 an 2025, see table 1, we have [6] δ(m t ) mbs { ± 3.6 GeV (2020) ± 2.1 GeV (2025). (6) in goo agreement with our numerical results. 60

6 B s µ + µ The SM preiction for BR(B s µ + µ ), which accounts for NNLO QCD an NLO electroweak corrections [18] reas BR(B s µ + µ ) = (3.33 ± 0.05) 10 9 R tα R s, (7) where R tα an R s are given by R tα = R s = ( ) 0.18 ( ) 3.06 αs (M Z ) M t, (8) GeV ( ) 2 ( ) 2 ( fbs Vcb Vts Vtb /V ) 2 cb τh s 226 MeV ps, (9) where the uncertainty comes mostly from V cb an f Bs. Comparing the experimental result for BR(B s µ + µ ) quote in table 1 with eq. (7), we en up with the following preiction for M t [6] (M t ) Bs µµ = (163.8 ± 14.7) GeV. (10) On the other han, assuming the expecte improvements of table 1, we have [6] δ(m t ) Bs µµ { ± 5.3 GeV (2020) ± 2.4 GeV (2025). (11) 4 Extracting M t from electroweak precision ata Electroweak observables epen on the top mass (an on the Higgs mass) only through the ε 1, ε 2, ε 3 parameters that escribe corrections to the treelevel propagators of the weak gauge bosons, an through the ε b parameter that escribes corrections to the Zb b vertex [19]. The measurement of M W plays the key role, since we fin δm t /M t = 69 δm W /M W [6]. This means that measuring M W with a precision of 8 MeV (as foreseeable after combination of the full LHC ataset [7]) can lea to a etermination of M t within about 1.2 GeV [6]. 5 Conclusions In ref. [6], we have analyse inirect eterminations of the top quark mass M t by means of flavour an electroweak observables. Among flavour processes, m Bs an B s µ + µ provie the best probe of M t. m Bs an B s µ + µ require only V cb as CKM input an, combine 61

7 m B ± 15.9 m Bs ± 9.8 B s Μ Μ ± 14.1 K Π ΝΝ ± Ε K ± 22.9 Global flavor fit ± 7.8 irect ± Pole top mass M t in GeV Figure 2: Summary of present an future eterminations of M t from flavour ata. For future projections, we have fixe the central value of M t to the present irect measurement. with a etermination of V cb an the lattice parameters ˆB 1/2 f Bs an f Bs, are sufficient to extract a fairly accurate estimate of the M t etermination from flavour physics. Our results are summarise in fig. 2: at present flavour ata etermine M t = (173.4 ± 7.8) GeV an we have estimate that the uncertainty on M t can be brought own to 3 GeV by 2020 an to 1.7 GeV by 2025 [6]. On the other han, electroweak ata, at present, etermine M t = (177.0± 2.6) GeV [6]. A more precise measurement of M W is the key player for an improve etermination of M t from electroweak observables. As experiments at the LHC are expecte to reuce the uncertainty on M W to about 8 MeV [7], it is foreseeable that electroweak physics will etermine M t with a precision of about 1.2 GeV [6]. In the future, a global fit of all inirect eterminations of M t, from both electroweak an flavour ata, will provie significant information. Even if inirect measurements o not surpass irect eterminations in precision, the comparison between inirect an irect analyses will carry essential information, especially in view of the theoretical ambiguities in the extraction of M t from collier experiments. References 1. G. Degrassi et al., JHEP 1208 (2012)

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