Eric Laenen Theory overview of top mass and couplings
LFC15, ECT* Trento, Sept. 2015 Outline
✦ Top mass ‣ Highlight of a few key issues ‣ Dwell on some recent beautiful results ‣ Implications for experiment
✦ Top couplings ‣ Rough status of what is known ‣ and what we might eventually know
Reviews e.g.: Snowmass (2013) write-up A. Juste et al, arXiv: 1310.0799 S. Moch et al (2014) MITP workshop, arXiv:1405.481
2
The last of the mass problems? I. Newton (1687)
‣ We thought we had solved it in the 17th century Gravity holds ‣ (i) resistance force and (ii) gravitational coupling universe together
‣ New insight in 1905: condensed energy A. Einstein (1905) ‣ Non-trivial for proton K. Wilson; Durr et al (2008)
‣ Yet newer insight: coupling to condensate R, Brout, F. Englert, P. Higgs, Kibble, Hagen, Guralnik (1964 -2012)
‣ Finally ‣ Mass of confined particle? Conceptually solved, but practically subtle Does top make the universe fall apart? 4 State of the Vacuum
‣ Top quark dominant in loop corrections that make the Higgs 4-pt coupling evolve. Full two-loop analysis:
0.10 Buttazzo et al (July 2013) 0.000 3s bands in 180 0.08 6 10 7 8 l 8 10 10 Mt = 173.4 ± 0.7 GeV gray b 200 Instability 109 a3 MZInstability= 0.1184 ± 0.0007 red 0.06 l -0.005 Mh = 125.7 ± 0.3 GeV blue 178 H L quartic 1010 H L H L 19 0.04 16 11 Higgs coupling H 14L 10
GeV 150 12 the Non -0.010 176 12 10 GeV 10 in 0.02 9 of 13 quartic t stability - 8 in 10 - t
M 7 Mt = 171.4 GeV Meta perturbativity 6 M 1016
Higgs 0.00 3s bands in
5 function as M = 0.1205 174 4 Z -0.015 Mt = 173.4 ± 0.7 GeV gray mass 100 LI =10 GeV mass 1,2,3 s
Beta a M = 0.1184 ± 0.0007 red -0.02 as MZ = 0.1163 3 Z Meta-stability Mh = 125.7 ± 0.3 GeV blue pole pole 19 MtH = 175.3L GeV 172 H L 10 -0.04 Top H L H L Top H L -0.020 102 10504 106 108 Stability1010 1012 1014 1016 1018 1020 102 104 106 108 1010 1012 1014 10H 16 10L 18 1020 RGE scale m in GeV 170RGE scale m in GeV 1018 1014 Stability ‣ Depends on1.5 precise0 top quark mass 0.8 168 0 50 1s bands in100 150 200 M = 173.4 ± 0.7 GeV gray 17 120 122 124 126 128 130 132 m t 10 h 0.6 ‣ within 3001.0 MeVm or soa3 M Z Higgs= 0.1184pole± 0.0007massredM in GeV Higgs pole mass M in GeV W h h Mh = 125.7 ± 0.3 GeV blue
ê H L L 0.4
‣ W But no practicalm worriesH L about universeH L expiring m h ê 0.5 m H L h t
m Figure 3: Left: SM phase diagram in terms of Higgs and top pole masses. The plane is
contribution 0.2 and
t ê
divided into regions of absolute stability,top meta-stability, instability of the SM vacuum, and non- m H ê
0.0 l h b m ê
perturbativity of the Higgs quartic coupling.l 50.0 The top Yukawa coupling becomes non-perturbative b 3s bands in for Mt > 230 GeV. The dotted contour-linesM = show173.4 ± 0.7 theGeV instabilitygray scale ⇤I in GeV assuming -0.5 t -0.2 a M = 0.1184 ± 0.0007 red ↵3(MZ )=0.1184. Right: Zoom in the regions ofZ the preferred experimental range of Mh and Mt Mh = 125.7 ± 0.3 GeV blue (the grey areas denote the allowed region at 1, 2, and 3 ).H TheL three boundary lines correspond -1.0 -0.4 H L H L to102 1-10 4 variations106 108 1010 10 of12 ↵1014(M1016 )=01018 10.201184 0.1000072 104, and106 10 the8 10 grading10H 1012L 1014 of1016 the1018 colours1020 indicates the size 3 Z ± of the theoreticalRGE scale m error.in GeV RGE scale m in GeV
Figure 2: Upper: RG evolution of (left) and of (right) varying Mt, ↵3(MZ ), Mh by 3 . LowerThe quantity: Same as above,e↵ can with be extracted more “physical” from normalisations. the e↵ective potential The Higgs at quartic two loops coupling [107] and is explicitly ± is comparedgiven with in appendix the top YukawaC. and weak gauge coupling through the ratios sign( ) 4 /y | | t and sign( ) 8 /g , which correspond to the ratios of running masses m /m and m /m , | | 2 h t ph W respectively (left). The Higgs quartic -function is shown in units of its top contribution, (top p contribution)4.3= The3y4/ SM8⇡2 (right phase). The diagram grey shadings in cover terms values of of the Higgs RG scale and above top the masses t Planck mass M 1.2 1019 GeV, and above the reduced Planck mass M¯ = M /p8⇡. The twoPl most⇡ important⇥ parameters that determine the variousPl Pl EW phases of the SM are the Higgs and top-quark masses. In fig. 3 we update the phase diagram given in ref. [4] with our improved calculation of the evolution of the Higgs quartic coupling. The regions of stability,
metastability, and instability of the EW vacuum are shown both for a broad range of Mh and Mt, and after zooming into the region corresponding to the measured values. The uncertainty 17 from ↵3 and from theoretical errors are indicated by the dashed lines and the colour shading along the borders. Also shown are contour lines of the instability scale ⇤I .
As previously noticed in ref. [4], the measured values of Mh and Mt appear to be rather special, in the sense that they place the SM vacuum in a near-critical condition, at the border
between stability and metastability. In the neighbourhood of the measured values of Mh and Mt, the stability condition is well approximated by ↵ (M ) 0.1184 M > 129.6GeV+2.0(M 173.35 GeV) 0.5GeV 3 Z 0.3GeV . (59) h t 0.0007 ± The quoted uncertainty comes only from higher order perturbative corrections. Other non-
18 3
Consistent effective potential and the top mass
Andreassen, Frost, Schwartz.. ‣ Effective potentialsFIG. are 2. not Gauge gauge dependence invariant of the instability scale ⇤I ,defined FIG. 1. Gauge dependence of the absolute stability bound by V (⇤I ) = 0, at 1-loop in the traditional approach. There pole with mt =173.34 GeV. ‣ but their extremais noare known gauge way invariant, to make and this scalescale gauge-invariant. invariant 3 ‣ find that stability bound is gauge-dependent in perturbation theory is satisfied. For values of mh and mt close to the observed Using Eq.129.9 (9) we find that for absolute stability at SM values, there are two solutions to this equation: the 1-loop, traditional method 129.8 Higgs
(LO) NLO, the HiggsM pole mass must satisfy lower µX is where V has a maximum, and the higher LO, consistent method 129.7 µX where the minimum occurs. In the SM these scales pole pole m 129.6 2 loops, traditional method (Landaum gauge) 173.34 GeV are h > (129.40 0.58) + 2.26 ( t ) GeV 129.5 ± 1.12 GeV NLO, consistent method (10) max 10 129.4
µ =2.46 10 GeV (6) Absolute stability bound on X ⇥ This bound is around 275 MeV lower than the bound 129.3 pole min 30 0 50 100 150 200 µX =3.43 10 GeV. (7) from the traditional approach in Landau gauge (mh > ⇥ ξt 129.67 GeV). The 0.58 is pertubative and ↵s uncer-FIG. 2. Gauge dependence of the instability scale ⇤I ,defined FIG. 1. Gauge dependence± of the absolute stability bound by V (⇤I ) = 0, at 1-loop in the traditional approach. There pole tainty [2]. Sincepole the Higgs mass is known better than These numbers and results which follow use m = with m =173.34 GeV. ‣ Consistenth treatment to ordert ħ combines, at LO, tree-level with isone-loop no known way to make this scale gauge-invariant. (125.14 0.24) GeV, combined from [27, 28]. the top mass, it perhaps makes more sense to write the ± bound asis satisfied. For values of mh and mt close to the observed For the potential at the next-to-leading‣ orderFind (NLO), SM values, there are two solutions to this equation: the Using Eq. (9) we find that for absolute stability at 2 (LO) NLO, the Higgs pole mass must satisfy one contribution comes from the ~ terms in the 1-loop mpole lower µX is where V has a maximum,mpole and the125 higher.14 GeV t µX where the minimum occurs. In theh SM these scales pole pole potential with scaling: < (171.22 0.28) + 0.12 ( m) mt 173.34 GeV ~ are h > (129.40 0.58) + 2.26 ( ) ⇠ GeV ± 0.24 GeV GeV ± 1.12 GeV (11) (10) ‣ stability bound on top pole mass: µ171.2max =2 .46 1010 GeV (6) 4 2 2 X ⇥ This bound is around 275 MeV lower than the bound 1 h (⇠Bg + ⇠W g ) Fig. 1 compares the gauge-dependence of the bound pole (1,NLO) 2 1 2 µmin =3.43 1030 GeV. (7) from the traditional approach in Landau gauge (m > V (h)= 2 ⇠Bg1 ln 4 3 at 1-loop to the LO,X NLO and⇥ 2-loop bounds. For this h 256⇡ 4µ 129.67 GeV). The 0.58 is pertubative and ↵s uncer- ✓ ◆ pole ± plot weThese have numbers taken and the results U(1) which and follow SU(2) use mR⇠ gauge= tainty pa- [2]. Since the Higgs mass is known better than 3h12⇠2 g4(⇠ g2 + ⇠ g2) h 2 W 2 B 1 W 2 4 rameters(125 equal.14 0 to.24)⇠ GeV,when combinedµ6 = fromm [27,and 28]. includedthe their top mass, it perhaps makes more sense to write the +⇠W g2 ln 12 9 h (8) ± t t bound as 64µ RGE evolutionFor the potential [29]. All at the bounds next-to-leading include order 2-loop (NLO), thresh- ✓ ◆ one contribution comes from the ~2 terms in the 1-loop pole pole mt mh 125.14 GeV olds andpotential 3-loop with running. ~ scaling: We find that the bound at < (171.22 0.28) + 0.12 ( ) (2,NLO) 0 ⇠ GeV ± 0.24 GeV Another contribution V (h) comes from the and pole (11) LO is mh > 129.69 GeV which is nearly identical to 1 h4(⇠ g2 + ⇠ g2) Fig. 1 compares the gauge-dependence of the bound ln terms in 2-loop potential. In Landau gauge, these the LandauV (1, gaugeNLO)(h)= 1-loop ⇠ boundg2 ln inB the1 traditionalW 2 3 ap- (2) 256⇡2 B 1 4µ4 at 1-loop to the LO, NLO and 2-loop bounds. For this 4 pole ✓ ◆ terms are h /4 times what is written as e↵ in Eq. proach, m > 1293.7012 2 GeV.4 2 We do2 not plot the gauge-plot we have taken the U(1) and SU(2) R⇠ gauge pa- h 2 h ⇠W g2(⇠Bg1 + ⇠W g2) 4 +⇠ g ln 9 h (8) rameters equal to ⇠t when µ = mt and included their (C.4) of the published version of [2]. Finally, there is dependence ofW 2 the 2-loop64 boundµ12 since we have not com- (n>2,NLO) ✓ ◆ RGE evolution [29]. All bounds include 2-loop thresh- the contribution, V (h) from 3-loop and higher puted the gauge-dependent 2-loop potential or the daisyolds and 3-loop running. We find that the bound at Another contribution V (2,NLO)(h) comes from the 0 and pole order graphs proportional to inverse powers of .Includ- contribution. That the bound seems to asymptoteLO to is amh > 129.69 GeV which is nearly identical to ln terms in 2-loop potential. In Landau gauge, these the Landau gauge 1-loop bound in the traditional ap- ing all these terms, the potential at each extremum will finite valueterms in are unitaryh4/4 times gauge what is (⇠ written= as)maybedueto (2) in Eq. pole e↵ proach, mh > 129.70 GeV. We do not plot the gauge- be gauge-invariant. Conveniently, the higher-loop-order (C.4) of the published version of [2].1 Finally, there is dependence of the 2-loop bound since we have not com- much (but not all) of the(n>2, gauge-dependenceNLO) being in the graphs contributing at NLO vanish in Landau gauge 4 the contribution, V (h) from 3-loop and higher puted the gauge-dependent 2-loop potential or the daisy e prefactororder graphs in Eq. proportional (3) which to inverse drops powers out of .Includ- of the V contribution.=0 That the bound seems to asymptote to a (⇠B = ⇠W = 0). Thus the gauge-invariant NLO value condition.ing all these terms, the potential at each extremum will finite value in unitary gauge (⇠ = )maybedueto 1 of the potential at the minimum is simply Fig. 2be shows gauge-invariant. the gauge Conveniently, dependence the higher-loop-order of the instabilitymuch (but not all) of the gauge-dependence being in the graphs contributing at NLO vanish in Landau gauge e4 prefactor in Eq. (3) which drops out of the V =0 scale ⇤I ,definedby(⇠B = ⇠W = 0).V Thus(⇤I the) = gauge-invariant 0 [1, 2], and NLO its value Landau-condition. NLO (LO) (2,NLO) of the potential at the minimum is simply Vmin = V (µX )+V (µX ) (9) gauge value at 2-loops, including 3-loop resummationFig. in 2 shows the gauge dependence of the instability scale ⇤I ,definedbyV (⇤I ) = 0 [1, 2], and its Landau- both cases. SinceNLO the instability(LO) scale(2,NLO) is a field value, it is Vmin = V (µX )+V (µX ) (9) gauge value at 2-loops, including 3-loop resummation in To derive this, we consistently truncated to (~2) and not obviously physical. We know of no way to computeboth cases. Since the instability scale is a field value, it is d (LO) O To derive this, we consistently truncated to (~2) and not obviously physical. We know of no way to compute used dh V = 0 at h = µX . Note that this is the RG- it in a consistentd (LO) and gauge-invariant manner.O used dh V = 0 at h = µX . Note that this is the RG- it in a consistent and gauge-invariant manner. improved e↵ective potential: the resummation is implicit Fig. 3improved shows e the↵ective value potential: of theVmax resummationcomputed is implicit by variousFig. 3 shows the value of Vmax computed by various in the solution for µ . At NNLO, an infinite number of approaches. We find approximately exponential depen- in the solution for µX . At NNLO, an infinite number of approaches. We find approximatelyX exponential depen- loops are relevant, even in Landau gauge [13]. dence of Vmax (and also Vmin) on ⇠t in the traditional loops are relevant, even in Landau gauge [13]. dence of Vmax (and also Vmin) on ⇠t in the traditional Consistent effective potential and the top mass
‣ Together with testing for new physics in a consistent way they find 4 Andreassen, Frost, Schwartz.. 180 Rapid instability 10 years t=10 178
Metastability
176
10
top 174 =10
M LNP
172 12 =10 LNP Absolute stability 14 170 =10 Planck-sensitive FIG. 3. Gauge dependence of the SM potential at its maxi- LNP 16 pole pole =10 NP mum with mh =125.14 GeV and mt =173.34 GeV. L 19 =10 LNP 168 H L 120 122 124 126 128 130 132
MHiggs approach at 1-loop. Decent fits are (12)
1-loop, trad. 1/4 9 0.02⇠ +0.0003⇠2 FIG. 4. Boundaries of absolute stability (lower band, NLO) V (2.50 10 GeV)e t t max ⇡ ⇥ and metastability (upper line, LO). The thickness of the 1/4 1-loop, trad. 29 0.001⇠ 0.0001⇠2 lower boundary indicates perturbative and ↵s uncertainty. V (3.08 10 GeV)e t t min ⇡ ⇥ The theoretical uncertainty of the metastability boundary is ⇣ ⌘ unknown. The elliptical contours7 are 68%, 95% and 99% pole The consistent gauge-invariant values at NLO are confidence bands on the Higgs and top masses: mh = pole (125.14 0.23) GeV and mt =(173.34 1.12) GeV. Dotted 1/4 ± ± V NLO =2.88 109 GeV (13) lines are scales in GeV at which Vmin can be lifted positive by max ⇥ new physics. NLO 1/4 29 Vmin =2.40 10 GeV ⇥ disappear. Thus the stability of the Standard Model can Note that V corresponds to an energy density well be modified by new physics at the scale 1012 GeV. min above the Planck scale. Thus, the potential at the mini- If we vary the Higgs and top masses in the Standard mum will surely be e↵ected by quantum gravity and pos- Model, we can compute the boundary of absolute stabil- sible new physics not included in our calculation. Previ- ity. This bound is shown in Figs. 4 and 5. The dotted ous analyses have defined stability to be Planck-sensitive lines show where Vmin becomes positive when in the pres- if the instability scale ⇤ >M [1, 2]. As we have ob- ence of for the indicated value of ⇤ . Unexpectedly, I Pl O6 NP served, the instability scale is gauge dependent, so this we find that three independent conditions (1) that Vmin is not a consistent criterion. An alternative criterion is goes to zero, (2) that Eq. (5) have no solution, and (3) 1 6 that new operator, such as 6 ⇤2 h be comparable that Vmin goes positive when ⇤NP = MPl all give nearly O ⌘ NP pole pole to V when h = h . Although and V are gauge- identical boundaries in the mh /mt plane. Know- min h i O6 min invariant, the value of 6 at the field value h where the ing that quantum gravity is relevant at MPl, we should minimum occurs is gaugeO dependent, so this condition therefore be cautious about giving too strong of an in- is also unsatisfactory. A consistent and satisfactory cri- terpretation of the perturbative absolute stability bound terion was explained in [13]: the new operator must be in the SM. We also show in this plot the metastability added to the classical theory and its e↵ect on Vmin eval- bound, that the lifetime of our vacuum be larger than uated. the age of the universe. At lowest order this translates to 1 1 ( ) < 14.53 + 0.153 ln[R GeV] for all R [30]. Since Adding 6 to the potential, we find that the the po- R tential is stillO negative at its minimum in the SM even (µ) is gauge invariant, so is this criterion. Although for for operators with very large coe cients. For example, the Standard Model this approximation is probably suf- 19 ficient, it has not been demonstrated that the bound can taking ⇤NP = MPl =1.22 10 GeV, we find that min 17 ⇥ 17 4 be systematically improved in a guage-invariant way [31]. µX =6.0 10 GeV and Vmin = (1.1 10 GeV) . Comparing⇥ to Eq. (13) we see that the energy⇥ of the true In this paper, we have only discussed a single physical vacuum is very Planck-sensitive. feature of the e↵ective action: the value of the e↵ective More generally, a good fit is given by potential at its extrema. There is of course much more content in the e↵ective action, especially when tempera- 4 12 Vmin = (0.01 ⇤NP) , ⇤NP & 10 GeV (14) ture dependence is included. Unfortunately, many uses of the e↵ective action involve evaluating it for particu- 12 When ⇤NP < 3.6 10 GeV, Vmin becomes positive and lar field configurations, a procedure that has repeatedly for ⇤ < 3.1 10⇥ 12 GeV the maximum and minimum been shown to be gauge-dependent. For example, the NP ⇥ Implications of a large mass
✦ Top decays before it hadronizes fully ‣ the only “bare”(=undressed by QCD) quark ‣ gives us access to its spin (i.e. LH and RH couplings)
✦ For QCD interactions of the top, the natural scale to put in the running QCD coupling is mt. ‣ good for perturbative approach (m ) 0.1 s t ✓ but not always good enough
8 Virtual top
✦ Virtual top make other things really happen
‣ in a loop integral a fixed mass scale always occurs in the result ‣ even more if there is no particle with (roughly) equal mass to compensate h t t h WWZZ + b t
✦ Express the W mass in terms of 3 fundamental weak parameter, with loop
3 GF 2 corrections rtop = m 8 2 2 tan2 t ⇤ 1 w M 2 = W 2 3 GF ⇥2GF sin ⇥w 1 r(mt, mH ) r = m2 2 ln(m /m ) 5/6 Higgs 2 2 W H Z 8 2 tan w 9 Top Mass History
The top mass measurement was a huge successVirtual for Tevatron Physics!top mass 250
200
150
From the EW Fits pp colliders limit 100 e+e- colliders limit CDF Run1
Top Mass (GeV/c2) Top 1995 Discovery! D0 Run1 50 Run1 World Average D0 Run2 average result CDF Run2 best measurement 0 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 Year The summary of EW data/ (up to 1995) is from: hep-ph/9704332 (Chris Quigg) June , 2013 TOP2013 G.Velev 4
m world comb. 1 68% and 95% CL contours t ± σ mt = 173.34 GeV fit w/o M and m measurements σ = 0.76 GeV [GeV] 80.5 W t
W = 0.76 0.50 GeV fit w/o M , m and M measurements σ ⊕ theo W t H M direct MW and mt measurements Now impressive consistency 80.45 between top, Higgs, W mass 80.4
MW world comb. ± 1σ 80.35 MW = 80.385 ± 0.015 GeV
80.3 Jul ’14 80.25 = 50 GeV = 125.14 GeV = 300 GeV = 600 GeV G fitter SM M H M H M H M H
140 150 160 170 180 190
mt [GeV] 10 Top mass
✦ Electron mass definition is“easy”: defined by pole in full propagator
✓ If particle momentum satisfies pole condition (p2=m2), can propagate to ∞
- ⇒ there is no real ambiguity what electron “pole” mass is
✦ But: quarks are confined, so physical on-shell quarks cannot exist
✓ Leads to non-perturbative ambiguity of few hundred MeV
- (revealed by all-order pQCD!)
11 Heavy quark mass, definition(s)
1 = p/ m0 ⌃(p, m0)
↵ 1 m s +finitestu↵ 0 ⇡ ✏ ↵ 1 m = m 1+ s + z To make finite, substitute 0 R ⇡ ✏ finite ✓ ◆
Mass definitions differ in the choice of 1 c = Pole mass: pretend quarks are free and long- p/ m0 ⌃(p, m0) p/ M
MSbar mass: treat mass as a coupling zfinite =0
12 Pole mass issues
✦ Most natural definition for a free (stable) particle (electron, Z-boson) Kronfeld ‣ gauge invariant and IR safe to all orders
✦ But quarks are confined, so pole mass has intrinsic uncertainty of order ΛQCD ‣ Full QCD has no pole at the top quark mass
✓ Finite width of top does not “screen” this Smith, Willenbrock
‣ Reproduced in perturbation theory Bigi, Shifman, Uraltsev, Vainshtein, Beneke, Braun, Smith, Willenbrock
⌃(m, m) ↵n+1 nn! ⇡ s 0 n X Renormalon behaviour → order ΛQCD uncertainty 13 Heavy quark mass schemes
✦ Various definitions other than the pole and MSbar schemes have been made
✦ PS (potential subtracted) mass ‣ Substract from the pole mass the IR part of the ttbar Coulomb potential
✓ The two parts have the same IR sensitivity 3 PS 1 d q Beneke m = M 2 3 V (q) q <µ (2⇡) Z| | f
Beneke, Kiyo, Schuller; ✓ V known to 3-loop Smirnov2, Steinhauser; Anzai, Kiyo, Zumino ✦ 1S mass 3 ‣ Half the perturbative mass of (fictitious) 1 S1 state
1S 1 pt Hoang, Teubner m = M + 2 E1
14 Some mpole observations
✦ Perturbative (“asymptotic”) expansion of pole mass
m = m (1 + 0.047 + 0.01 + 0.003 + ...) Melnikov, van Ritbergen pole MS ‣ -> uncertainty about 500 MeV (or less) ‣ Uncertainty in pole mass about 300 MeV ‣ resultant uncertainty in MSbar mass smaller than m (3 loop) m (2 loop) MS MS ✓ → NNNNLO?
15 3
RS′ masses has been suggested in which (in the case of the the charm, bottom or top quark case, before combining bottom quark) the subtraction term is parameterized in the uncertainties from the numerical integration of the (3) terms of αs . In this paper we will adopt the prescription master integrals. Note that the coefficients up to three of Ref. [11]. Similarly to the PS mass also for the RS mass loops are known analytically [18, 19]. We refrain from a subtraction scale has to be specified which we again listing the corresponding results but refer to Eq. (13) of choose as µf =2GeVforbottomandµf =20GeVfor3 Ref. [46]. Analytical results are also available for the log- top. arithmic four-loop contributions since they can easily be RS′ masses has been suggested in which (in the case of the Forthe charm, the computation bottom or top of quark the scalar case, before and vector combining part of obtained using renormalization group methods. In the bottom quark) the subtraction term is parameterized in thethe fermion uncertainties propagator from thewe use numerical an automated integration setup of the which following we restrict ourselves to compact numerical res- (3) master integrals. Note that the coefficients up to three 4 terms of αs . In this paper we will adopt the prescription generates all contributing amplitudes, processes them ults. At four loops we obtain for the coefficient of (αs/π) loops are known analytically [18, 19]. We refrain from of Ref. [11]. Similarly to the PS mass also for the RS mass with FORM3 [35] and provides scalar functions involving a subtraction scale has to be specified which we again listing the corresponding results but refer to Eq. (13) of (4) 2 several million different integrals encoded in functions zm = −1744.8 ± 21.5 − 703.48 lOS − 122.97 lOS choose as µf =2GeVforbottomandµf =20GeVfor Ref. [46]. Analytical results are also available for the log- nl=3 top. witharithmic 14 di four-loopfferent indices contributions which since belong they to can 100 easily diff beerent ! 3 4 ! − 14.234 lOS − 0.75043 lOS , For the computation of the scalar and vector part of integralobtained families. using renormalization group methods. In the ! the fermion propagator we use an automated setup which Thefollowing Laporta we restrict algorithm ourselves [36] to is compact applied numerical to each res- family (4) 2 4 zm = −1267.0 ± 21.5 − 500.23 lOS − 83.390 lOS generates all contributing amplitudes, processes them usingults.FIRE5 At four[37] loops and we obtaincrusher for the[38] coe whichfficient are of ( writtenαs/π) in nl=4 with FORM3 [35] and provides scalar functions involving C++ tsort ! 3 4 .Thenweusethecode(4) [39], which is part2 of the ! − 9.9563 lOS − 0.514033 lOS , several million different integrals encoded in functions zmFIRE = −1744.8 ± 21.5 − 703.48 lOS − 122.97 lOS ! latest nl=3 version, to reveal relations between primary (4) 2 with 14 different indices which belong to 100 different ! zm = −859.96 ± 21.5 − 328.94 lOS − 50.856 lOS master integrals following3 recipes of [40]4 and end up with n =5 integral families. ! − 14.234 lOS − 0.75043 lOS , l 386 four-loop! massive on-shell propagator integrals, i.e. ! 3 4 The Laporta algorithm [36] is applied to each family (4) 2 ! − 6.4922 l − 0.33203 l , (11) zm2 =2 −1267.0 ± 21.5 − 500.23 lOS − 83.390 lOS ! OS OS using FIRE5 [37] and crusher [38] which are written in with p =nl=4M . ! 3 4 2 2 C++.Thenweusethecodetsort [39], which is part of the We have! performed− 9.9563 thelOS calculation− 0.514033 lOS allowing, for a gen- with lOS = ln(µ /M ). We obtain the µ-independent ! 2 latest FIRE version, to reveal relations between primary eral gauge(4) parameter ξ keeping terms up to order2 ξ in coefficients with an accuracy of 1.2% for nl =3,1.7% for zm = −859.96 ± 21.5 − 328.94 lOS − 50.856 lOS master integrals following recipes of [40] and end up with the expressionnl=5 we give to the reduction routines. We have nl =4)and2.5% for nl = 5. In the numerical results 386 four-loop massive on-shell propagator integrals, i.e. ! 3 4 checked! that ξ −drops6.4922 outlOS after− 0.33203 masslOS renormalization, (11) but discussed below we will assume a relative uncertainty of with p2 = M 2. ! before inserting the2 master2 integrals. 3% for all values of nl. We have performed the calculation allowing for a gen- with lOS = ln(µ /M ). We obtain the µ-independent 2 For some master integrals, analytic results could be de- For convenience we also show the four-loop results for eral gauge parameter ξ keeping terms up to order ξ in coefficients with an accuracy of 1.2% for nl =3,1.7% for rived using a straightforward loop-by-loop integration for cm which read the expression we give to the reduction routines. We have nl =4)and2.5% for nl = 5. In the numerical results general space-time dimension. We also used analytical checked that ξ drops out after mass renormalization but discussed below we will assume a relative uncertainty of (4) 2 before inserting the master integrals. results3% for obtained all values for of n non-triviall. four-loop on-shell mas- cm =1691.2 ± 21.5+828.43 lMS +189.65 lMS nl=3 For some master integrals, analytic results could be de- ter integralsFor convenience computed we also in showour earlier the four-loop paper results Ref. [41]. for In ! ! +36.688 l3 +4.8124 l4 , rived using a straightforward loop-by-loop integration for somecm which other read cases one- and two-fold Mellin-Barnes repres- ! MS MS general space-time dimension. We also used analytical entationsMSbar(4) can be derived vs which pole allow for mass a high-precision2 at 4 loopc(4) =1224.0 ± 21.5+601.98 l +134.10 l2 m results obtained for non-trivial four-loop on-shell mas- cm =1691.2 ± 21.5+828.43 lMS +189.65 lMS MS MS numericn evaluation,l=3 at least up to 20 digits. For some Marquard,nl=4 Smirnov, Smirnov, Steinhauser ter integrals computed in our earlier paper Ref. [41]. In ! ! of the master! integrals,+36.688 l we3 +4 applied.8124 l threefold4 , MB repres- ! +28.846 l3 +3.9648 l4 , some other cases one- and two-fold Mellin-Barnes repres- ! MS MS MS MS ✦ entations which enabled us to obtain a precision of eight ! entations can be derived which allow for a high-precisionImportant progress:(4) 4-loop relations between top2 quark masses(4) 2 cm =1224.0 ± 21.5+601.98 lMS +134.10 lMS c =827.37 ± 21.5+408.88 l +86.574 l digits. nl=4 m MS MS numeric evaluation, at least up to 20 digits. For some nl=5 ! M 3= cm(µ)m4 (µ) of the master integrals, we applied threefold MB repres- For factorizable! +28 integrals,.846 l +3 we.9648 obtainedl , analytic results ! 3 4 ! MS MS ! +22.023 l +3.2227 l , (12) entations which enabled us to obtain a precision of eight from known two- and three-loop results. In particular, ! MS MS c(4) =827.37 ± 21.5+408.88 l +86.574 l2 digits. we usedm Ref. [42] where the expansionMS in ϵ =(4MS− d)/2 2 2 nl=5 with l = ln(µ /m ). In the remaining part of this For factorizable integrals, we obtained analytic results ! MS has been! performed+22.023 upl to3 the+3.2227 orderl4 typical, to four-loop(12) Letter we will concentrate on the top and bottom quark from known two- and three-loop results. In particular, calculations.! (d is the space-timeMS dimensionMS used to com- we used Ref. [42] where the expansion in ϵ =(4− d)/2 mass. putewith thel momentum= ln(µ2/m integrals.)2). In the remaining part of this has been performed up to the order typical to four-loop MS As an application of the new results in Eqs. (11) WeLetter computed we will concentrate the remaining on the 332 top integrals and bottom numerically quark calculations. (d is the space-time dimension used to com- mass. and (12) we study the relations between the various pute the momentum integrals.) ‣ Use of variouswith the specialized help of FIESTA codes[43–45]. (FORM,FIESTA FIRE,returns FIESTA for each,..), many of the (master) loop As an application of the new results in Eqs. (11) threshold masses and the MS mass. We use the follow- ϵ coefficient a numerical result and the corresponding un- We computed the remaining 332 integrals numericallyintegrals doneand (12) numerically. we study the relations between the various ing input values for the strong coupling constant and the with the help of FIESTA [43–45]. FIESTA returns for each certaintythreshold from masses the and numerical the MS integration. mass. We use When the follow- inserting bottom and top quark masses: ϵ coefficient a numerical result and the corresponding un- theing master input valuesintegrals for thewe keep strong track coupling of all constant uncertainties and the and certainty from the numerical integration. When‣ This inserting is also sufficient, together with N3LO Coulomb potential,(5) for 4-loop relations to PS combinebottom them and top quadratically quark masses: in the final expression. We αs (MZ )=0.1185 [47] ,mb(mb)=4.163 GeV [6] , the master integrals we keep track of all uncertaintiesand and1S massesinterpret the resulting uncertainty as a standard devi- combine them quadratically in the final expression. We (5) Mt =173.34 GeV [48] . (13) ationαs and(MZ multiply)=0.1185 it [47] by,m fiveb in(mb the)=4 final.163 GeVresult [6] for, the interpret the resulting uncertainty as a standard devi- M =173.34 GeV [48] . 2 (13)3 4 M =relationm 1+0 betweent.4244 the↵sMS+0 and.8345 OS quark↵s +2 mass..375 This↵s + is in(8.49 0.25) ↵s)+... (5) ation and multiply it by five in the final result for the α±s with four and six active flavours is obtained from αs relation between the MS and OS quark mass. This is in agreement with adding the uncertainties from the(5) indi- αs⇣with four and six active flavours is obtained from αs where for the decoupling⌘ scale we choose twice the heavy agreement with adding the uncertainties✦ fromResult the indi- vidualwhere contributions for the decoupling linearly. scale we choose twice the heavy quark mass [46, 49]. vidual contributions linearly. Wequark are mass now [46, in 49]. the position to present numerical res- Let us have a closer look to the relation between the We are now in the position to present numerical res- ultsLet for usz have(µ) which a closer have look been to the obtained relation between by setting the the M = 163m.643 + 7.557 + 1.617 + 0.501 + 0.195 0.OS005 andGeVMS top quark mass. For µ = mt we have ults for zm(µ) which have been obtained by setting the numberOS and ofMS colours top quark to three mass. ( ForNc =3)andthenumberofµ = mt we have ± number of colours to three (Nc =3)andthenumberof 2 3 massless quarks (nl) to either 3, 4 or 5,2 corresponding3 to Mt = mt 1+0.4244 αs +0.8345 αs +2.375 αs massless quarks (nl) to either 3, 4 or 5,✦ correspondingNumerically: to niceMt progression!!= mt 1+0.4244 α sNo+0. 8345signα sof+2 an.375 impendingαs renormalon " "
16 4
4 PS 1S RS +(8.49 ± 0.25) αs input m = m = m = #loops 4.483 4.670 4.365 =163.643 + 7.557 +! 1.617 + 0.501 +0.195 ± 0.005 GeV , (14) 1 4.266 4.308 4.210 2 4.191 4.190 4.172 (6) with αs ≡ αs (mt)=0.1088. Note that the four-loop 3 4.161 4.154 4.158 term still gives a contribution of about 200 MeV which 4 4.163 4.163 4.163 is not negligible even with nowadays uncertainties from 4(×1.03) 4.159 4.159 4.159 TEVATRON and LHC [48]. The corresponding results for the bottom quark read Table II. mb(mb)inGeVcomputedfromthePS,1SandRS quark mass using one- to four-loop accuracy. The numbers M = m 1+0.4244 α +0.9401 α2 +3.045 α3 b b s s s in the last line are obtained by increasing the four-loop coef- 4 +(12" .57 ± 0.38) αs ficient in Eq. (12) by 3%. =4.163 + 0.401 + 0.201! + 0.148 +0.138 ± 0.004 GeV . (15)
(5) The results for mb(mb)computedfromthePS,1Sand Here αs ≡ αs (mb)=0.2268. Note that the four-loop RS threshold masses are shown in Tab. II. The three-loop corrections in Eq. (15) are almost as large as the three- corrections provide still sizable effects of up to 40 MeV loop term. On the other hand, the perturbative series which reduces to at most 9MeVat four loops. The uncer- for the case of the top quark has a reasonable behaviour: tainty in the four-loop MS-OS relation induces an error the three-loop coefficient is by a factor three smaller than of 4MeV. Thus we arrive at a final error of {4, 6, 5} MeV the two-loop one and the four-loop term is again smal- for the conversion from the {1S,PS,RS} mass. This is ler by a factor 2.5. This suggests that with the help of not negligible, though in general much smaller than other Eq. (14) the top quark mass can be determined with an uncertainties involved in the quark mass extraction (see, uncertainty below 200 MeV. e.g., Refs. [50], [34] and [51] for recent determinations of In practice it often happens that in a first step a mb(mb) where in intermediate steps the 1S, RS and PS threshold quark mass is extracted from comparisons has been used, respectively). of higher order calculations and experimental measure- The results of Tabs. I and II can be used, in com- ments.Impact Afterwards theon threshold MSbar mass is converted mass to the bination with similar calculations for different values of Marquard, Smirnov, Smirnov, Steinhauser MS quark mass. In Tabs. I and II we show the results αs(MZ ) and threshold masses, to construct the following for the scale invariant MS quark mass mq(mq)(q = b, t) approximation formulae ✦ Study how a differentusing one-threshold to four-loop mass accuracy measurement for the conversion. leads to MSbar mass mt(mt) PS =163.643 ± 0.023 + 0.074∆α − 0.095∆ , input mPS = m1S = mRS = GeV s mt #loops 171.792 172.227 171.215 mt(mt) 1S =163.643 ± 0.007 + 0.069∆α − 0.096∆ , 1 165.097 165.045 164.847 GeV s mt 2 163.943 163.861 163.853 m (m ) t t =163.643 ± 0.011 + 0.067∆ − 0.095∆RS , 3 163.687 163.651 163.663 GeV αs mt 4 163.643 163.643 163.643 mb(mb) PS =4.163 ± 0.004 + 0.007∆α − 0.018∆ , 4(×1.03) 163.637 163.637 163.637 GeV s mb mb(mb) 1S =4.163 ± 0.006 + 0.008∆α − 0.019∆ , Table I. mt(mt)inGeVcomputedfromthePS,1SandRS GeV s mb quark mass using one- to four-loop accuracy. The numbers m (m ) ✦ in the last line are obtained by increasing the four-loop coef- b b RS 3-loop still gives 200-250 MeV shifts =4.163 ± 0.005 + 0.004∆αs − 0.018∆mb (16) ficient in Eq. (12) by 3%. GeV ✦ PS 4-loop only gives further {44,8,20} MeV shifts with ∆αs =(0.1185 − αs(MZ ))/0.001, ∆mt = (171.792 GeV − mPS)/0.1, ∆1S =(172.227 GeV − In the case of the top quark (cf. Tab. I) the three- t mt ‣ final remaining uncertainty estimate {23,7,11} MeV m1S)/0.1, ∆RS =(171.215 GeV − mRS)/0.1, ∆PS = loop corrections amount to about 200-250 MeV which re- t mt t mb (4.483 GeV−mPS)/0.02, ∆1S =(4.670 GeV−m1S)/0.02, duces to {44, 8, 20} MeV at four loops for the {PS,1S,RS} b mb b RS RS quark mass. A 3% uncertainty in the MS-OS relation ∆mb =(4.365 GeV − mb )/0.02. induces a shift of 6MeVin mt(mt) which is in gen- Let us finally compare in Tab. III our result for (4) eral small as compared to the four-loop contribution. the four-loop coefficient cm to predictions obtained on Let us estimate the final uncertainty from the conver- the basis of different assumptions. In general good sion to the MS mass from the quadratic combination agreement is found, in particular with the results from of the 6MeVwith half of17 the four-loop contribution Refs. [34, 55, 56] which are all based on renormalon can- (i.e. {44, 8, 20}×1/2MeV). This leads to {23, 7, 11} MeV cellation. For example, in Ref. [56], the four-loop coef- which should be added in quadrature to the remaining ficient is extracted from the requirement of perturbative uncertainties of the threshold mass. stability of the combination 2mpole + VQCD where VQCD Top threshold mass
✦ Scan the ttbar threshold at linear collider by varying beam energy. The opening of the top channel leads to “smooth” theta-function
✦ Distribution can be measured very precisely. with calculation using Schrodinger equation and appropiate short-distance mass
✦ Also sensitive to top quark width, allows good measurement
✦ Calculation non-relativistic effective field theory. Two small parameters: αs and v.
✦ Choice of top quark mass scheme matters.. Threshold mass, good oang Pole mass, bad
18 N3LO for ttbar S-wave threshold production at e+e- collider
✦ Now finally the full N3LO cross section, including the last non-logarithmic terms, is known 3 ‣ Heroic effort, and it was worth it! QCD calculation under control Beneke, Kiyo, Marquard, Penin, Piclum, Steinhauser as measured by the residual scale dependence is high- lighted in Fig. 2, which shows R(µ) normalized to a ref- erence prediction defined at µ = 80 GeV. The width of the shaded band corresponds to an uncertainty of about 3% with some dependence on the center-of-mass energy √±s. The figure also shows the sensitivity to the top-quark width. The two solid lines refer to the cross section with Γt changed by 100 MeV to 1.43 and 1.23 GeV, respec- tively, computed± with µ = 80 GeV and normalized to the reference prediction. Decreasing the width implies a sharper peak, i.e. an enhancement in the peak region, and a suppression towards the non-resonant region below the peak. A few GeV above the peak the cross section is largely insensitive to the width. Increasing the width leads to the opposite effects. This pattern is clearly seen in Fig. 2, which also demonstrates that a 100 MeV de- FIG. 1. Scale dependence of the cross section near thresh- ± 3 viation from the width predicted in the Standard Model ‣ Dramatic scale reductionold. TheN2LO NLO, → NNLO N3LO. and N NegativeLO result is correction shown in blue, beyond red leads the to peak. a cross section change near and below the peak and black, respectively. The renormalization scale is varied far larger than the uncertainty from scale variation. between 50 and 350 GeV. ‣ QCD uncertainty on top quark mass can go below 50 MeV. We now turn to the question to what accuracy the top quark mass can be determined. Even if we focus ✓ But are also non-QCDthe effects total cross to sectionstudy: is EW, shown Higgs, as a function Beamstrahlung, of the center- non-resonantonly on the theoreticalterms.. accuracy, a rigorous analysis re- of-mass energy √s. The previous NLO and NNLO pre- quires accounting for the specifics of the energy points dictions are also shown for comparison to the new N3LO of the threshold scan and the correlations. However, a result (black, solid). The bands are obtained by varia- good indication is already provided by looking at the po- sition and height of the resonance peak. Fig. 3 shows this tion of the renormalization scale in the specified range. 3 After the inclusion of the third-order19 corrections one ob- information at LO, NLO, NNLO and N LO, where the serves a dramatic stabilization of the perturbative predic- outer error bar reflects the uncertainty due to the renor- tion, in particular in and below the peak region. In fact, malization scale and αs variation, added in quadrature, the N3LO curve is entirely contained within the NNLO and the inner error bar only takes the αs uncertainty one. This is different above the peak position where a into account. The central point refers to the value at clear negative correction is observed when going from the reference scale µ = 80 GeV. There is a relatively big NNLO to N3LO. For example, 3 GeV above the peak jump from LO to NLO of about 310 MeV, approximately this amounts to 8%. This arises from the large negative 150 MeV from NLO to NNLO, which reduces to only − 64 MeV from NNLO to N3LO. Furthermore, the NNLO three-loop correction to the matching coefficient cv [22]. 3 The theoretical precision of the third-order QCD result and N LO uncertainty bars show a significant overlap.
FIG. 2. Scale dependence (hatched area) of the N3LO cross FIG. 3. Position and height of the cross section peak at LO, section relative to the reference prediction. Overlaid are pre- NLO, NNLO and N3LO. The unbounded range of the LO dictions for two different values of Γt,againnormalizedtothe error bars to the right and up are due to the fact that the reference prediction. See text for details. peak disappears for large values of the renormalization scale. Mass by proxy
‣ Of course, one does not need to reconstruct the topS. Moch quark / Nuclear and Particle from Physics Proceedingsits decays. 261–262 (2015) 130–139 Needs to 135
solve implicit equation 450 σ - pp -->tt [pb] at LHC8 exp th 400 ( Q )= (mt, Q ) NNLO 350 NLO { } { } LO 300 250 ‣ using an observable σ that is optimally sensitive to mt. 200 150 100 50 ‣ Adjust mt to fit data best. 150 160 170 180 190
mt(pole)/GeV
‣ When extracting ttbar cross section, IR sensitive region is minuteFigure fraction 7: Same as fig. 6 forof the top-quarktotal mass in the on-shell scheme pole pole mt at the scale µ = mt . (Figure from ref. [26]).
result. σ , pole Figure 5: Ellipses for the 1 uncertainties in the [MH mt ] plane 280 with Higgs mass M = 125.6 0.4 GeV and α (M ) = 0.1187 σ - [pb] at LHC8 H ± s Z pp -->tt confronted with the areas in which the SM vacuum is absolutely sta- 260 m (m ) = 162 GeV ‣ ble, meta-stable and unstable up to the Planck scale. (Figure from 240 t t Pole mass should be fine here; can interpretref. [47]). “mtop” in MC as pole mass,220 with small error + - 200 (unlike e e ) 450 σ - 180 pp -->tt [pb] at LHC8 400 The160 impact of Coulomb corrections NNLO 350 NLO (which140 first appear at NLO) is confined to LO 300 120 values100 of v that contribute very little to the 250 total cross section 200 1 LO µ 150 /mt(mt) 100 NLO Figure[Mangano 8: The scale dependence at of the TOP2013 LO, NLO and NNLO QCD] 50 (vmtop In theory there is no difference between theory and practice; in practice there is P. Uwer, talk at SM@LHC, quoting Yogi Berra ✦ What do experiments do? ‣ Template method: compare observables in data with MC templates generated with different masses ‣ Matrix element method: build event likelihood for full (LO) top quark matrix element, with full kinematics ‣ “Ideogram”, and other methods 21 Mlb in leptonic top-quark decays [R. Chierici, A. Dierlamm CMS Note 2006/058], Karchilava Interesting idea : infer top mass from correlation with e/mu and J/Psi invariant mass 22 What top mass is measured? ✦ Most involve MC’s that are LO, so they could never tell the difference between different mass definitions. ✦ So what mass do hadron colliders determine? ‣ Pole mass? “Pythia” mass? ✓ Typically the path from data to a value for m involves a Monte Carlo, itself driven by a mass parameter. ✓ Path goes via (shower) cuts, efficiencies, hadronization models etc 23 Hoang, Stewart Monte Carlo top mass Moch et al ✦ MC mass does not depend on observables. Related to softHeavy radiation Quark Mass in thefor MC that MC. ✦ Hadronization affects the MC mass value ‣ Has aspects of top (or B)-meson mass! MC MSR MSR mt = mt + ‣ Use methods from B-meson physics to extract field theory mass. But uncertainty order 1 GeV. CERN Theory Seminar, May 21, 2014 ✦ To relate to field theory mass, would need mass-sensitive observable, that can also be computed in MC ‣ Calculation beyond LO (LL) Hoang, Stewart ✓ controlled errors. Factorize observable to control top mass in each factor. ‣ Hadron level observables ✦ E.g. massive thrust, DIS for massive quarks, ttbar at high pT Hoang et al 24 Proxy mass: determing the MSbar mass ✦ How to determine the MSbar mass? ‣ Problem: on-shell condition of final state top leads tot the pole mass 1 Im = ⇡ (p2 M 2) p2 M 2 + i" ✦ By proxy ‣ compute cross section using pole mass tt(M,↵s) ‣ replace pole mass by MSbar mass ‣ Now fit to data, extract MSbar mass Langenfeld, Moch, Uwer 25 MSbar mass extraction Langenfeld, Moch, Uwer ✦ Accuracy limited by mt sensitivity and PDF uncertainties ✦ Other proposals for mass-sensitive observables: ‣ (moments of) the invariant mass distribution Frederix, Maltoni ‣ tt+1 jet rate Alioli, Fernandez, Fuster, Irles, Moch, Uwer 26 S. Moch / Nuclear and Particle Physics Proceedings 261–262 (2015) 130–139 135 450 σ - pp -->tt [pb] at LHC8 400 NNLO 350 NLO LO 300 250 200 150 100 50 150 160 170 180 190 mt(pole)/GeV Figure 7: Same as fig. 6 for the top-quark mass in the on-shell scheme pole µ = pole MSbar massmt at the scaleextractionmt . (Figure from ref. [26]). σ , pole Figure 5: Ellipses for the 1 uncertainties in the [MH mt ] plane 280 with Higgs mass M = 125.6 0.4 GeV and α (M ) = 0.1187 σ - [pb] at LHC8 H ± s Z pp -->tt confronted with the areas in which the SM vacuum is absolutely sta- 260 m (m ) = 162 GeV ✦ ble, meta-stable and unstable up to the Planck scale. (Figure from 240 t t Moch In spiteref. of [47]). MLM’s argumentS. Moch / Nuclear MSbar and Particle Physicshas Proceedings better220 261–262 progression (2015) 130–139 135 200 450 σ - 418050 pp -->tt [pb] at LHC8 σ - [pb] at LHC8 400 160 pp -->tt NNLO 400 140 NNLO 350 NLO 350 NLO LO 120 LO 300 300 100 250 250 200 200 1 µ 150 150 /mt(mt) 100 100 Figure 8: The scale dependence of the LO, NLO and NNLO QCD 50 predictions50 for the tt¯ total cross section at the LHC ( √s = 8 TeV) 140 150 160 170 180 150 160 170 180 190 for a top-quark mass mt(mt) = 162 GeV in the MS scheme with mt(mt)/GeV the ABM12 PDFs and the choicemt(pole)/GeVµ = µr = µ f . The vertical bars indicate the size of the scale variation in the standard range µ/m (m ) [1/2, 2]. (Figure from ref. [26]). Figure 6: The LO, NLO and NNLO QCD predictions for the tt¯ total Figuret 7:t Same∈ as fig. 6 for the top-quark mass in the on-shell scheme √ = pole pole ‣ andcross better section atscale the LHC ( dependences 8 TeV) as a function of the top- mt at the scale µ = mt . (Figure from ref. [26]). quark mass in the MS scheme mt(mt) at the scale µ = mt(mt) with the ABM12 PDFs. (Figure from ref. [26]). pole Figure 5: Ellipses for the 1σ uncertainties in the [M , m ] plane esting to observe, that the point of minimal sensitiv- ✦ H t 280 with Higgs mass MH = 125.6 0.4 GeV and αs(MZ ) = 0.1187 ity where σLO σNLO σNNLOσ is - located[pb] at LHC8 at scales Same holds for the distribution± invariant mass mtt. pp -->tt confronted with the areas in which the SM vacuum is absolutely sta- 260µ = ≃ ≃ (mt(mt)), i.e., it coincidesm with(m ) = the 162 natural GeV hard muchble, meta-stable improved and when unstable using up to the theMS Planck mass scale. in contrast (Figure from to 240 O t t ref. [47]). scale of the process for the MS mass in fig. 8, whereas ✦ pole 220 pole From athe correlated pole mass mt . fit including the LHC andit resides Tevatron at fairly low scales, ttbarµ mcrosst /4 45section, GeV for to also gluon 200 ≃ ≃ These findings are illustrated in figs. 6 9. The the- the pole mass predictions in fig. 9. 450 − ¯ PDF and αs. σ - 180 tt Alekhin, Bluemlein, Moch ory predictions for inclusive top-quarkpp -->tt [pb] pair at LHC8 production For the distribution in the invariant mass m of the 400 160 with the MS and the pole mass are comparedNNLO in figs. 6 top quark pair the same findings can be seen in figs. 10 140 and350 7. The result in terms ofm thet(MSm masst) =mNLOt(m 162t) dis-.3 and2. 11.3 GeV For the MS mass predictions the convergence LO plays300 a much improved convergence as the higher order ±120is improved. Also the overall shape of the distribu- 100 corrections250 are successively added. The corresponding tion changes in comparison to case of the pole mass, scale dependence is shown in figs. 8 and 9 and the pre- the peak becomes more pronounced, while the posi- ‣ leading200 to the pole mass value 1 dictions with the MS mass exhibit a much better scale tion of the peak remains stable against radiative correc- µ/m (m ) stability150 of the perturbative expansion. It is also inter- tions. This is essential for precisiont t determinations of 100 M = 171.2 2.4 Figure0.7 8: The GeV scale dependence of the LO, NLO and NNLO QCD 50 ± ±predictions for the tt¯ total cross section at the LHC ( √s = 8 TeV) 140 150 160 170 180 for a top-quark27 mass mt(mt) = 162 GeV in the MS scheme with mt(mt)/GeV the ABM12 PDFs and the choice µ = µr = µ f . The vertical bars indicate the size of the scale variation in the standard range µ/m (m ) [1/2, 2]. (Figure from ref. [26]). Figure 6: The LO, NLO and NNLO QCD predictions for the tt¯ total t t ∈ cross section at the LHC ( √s = 8 TeV) as a function of the top- quark mass in the MS scheme mt(mt) at the scale µ = mt(mt) with the ABM12 PDFs. (Figure from ref. [26]). esting to observe, that the point of minimal sensitiv- ity where σ σ σ is located at scales LO ≃ NLO ≃ NNLO µ = (m (m )), i.e., it coincides with the natural hard O t t much improved when using the MS mass in contrast to scale of the process for the MS mass in fig. 8, whereas pole the pole mass m . it resides at fairly low scales, µ mpole/4 45 GeV for t ≃ t ≃ These findings are illustrated in figs. 6 9. The the- the pole mass predictions in fig. 9. − ory predictions for inclusive top-quark pair production For the distribution in the invariant mass mtt¯ of the with the MS and the pole mass are compared in figs. 6 top quark pair the same findings can be seen in figs. 10 and 7. The result in terms of the MS mass mt(mt) dis- and 11. For the MS mass predictions the convergence plays a much improved convergence as the higher order is improved. Also the overall shape of the distribu- corrections are successively added. The corresponding tion changes in comparison to case of the pole mass, scale dependence is shown in figs. 8 and 9 and the pre- the peak becomes more pronounced, while the posi- dictions with the MS mass exhibit a much better scale tion of the peak remains stable against radiative correc- stability of the perturbative expansion. It is also inter- tions. This is essential for precision determinations of Some other LHC mass proxies Frixione, Mitov ✦ In dilepton channel, can use shapes of various observables sensitive to top mass ‣ study with NLO+PS+MadSpin Label Observable ‣ + single-inclusive or mildly correlated (1,4,5) stable under above effects 1 pT (ℓ ) + − 2 pT (ℓ ℓ ) ✓ 2,3 not -> be careful with using NNLO with stable tops 3 M(ℓ+ℓ−) 4 E(ℓ+)+E(ℓ−) ‣ about 0.8 GeV theory error in studied scenario, with aMC@NLO + − 5 pT (ℓ )+pT (ℓ ) Table 1: The set of observables used in this paper, and their labelling conventions. Owing to this inclusiveness, the observable is minimally sensitive to the modelling of • Biswas,long-distance Melnikov, Schulze effects. This feature increases the reliability of the theoretical predic- tions. 84 The set of observables considered in this paper and their labelling conventions are given ] 82 • NLO study for ⟩ − − 76 + + spectively, of theBl charged-lepton pair; finally, E(ℓ )+E(ℓ )andpT (ℓ )+pT (ℓ )arethe • m 1.5 GeV uncertainty. Partons showers do in general quite ⟨ scalar sums of the74 energies and transverse momenta of the two charged leptons, respec- well in estimating uncertainties Corcella, Mescia tively. We point out72 that the latter two sums are computed event-by-event; in other words, observables #4 and170 #5 are 172not constructed 174 176 a-posteriori 178 180 given the single-inclusive energy mt [ GeV ] and transverse momentumNLO distributions of the leptons. Figure 2: Result of the linear fit to ⟨mBl⟩ is shown, with all kinematic cuts on the final state particles applied. See text for details.The extraction of the top quark mass utilises the sensitivityoftheshapes of kinematic distributions to the value of m .Itiscumbersometoworkdirectlywithdifferential distri- 28 t butions. Instead, we utilise their lower Mellin moments, whose precise definition is given 3. Dilepton channel in sect. 2.1. The idea of the method proposed in this paper is topredictthemt depen- Indence the previous of the Section moments, we saw and that then top to quark extract decays the value to final of statmtesby with comparing identified the hadrons predicted and providemeasured an interesting values way of to those determine moments. the top The quark procedure mass. In is this detailedSectionin we sect. study 2.2. inclusive final states. We focus on the case where the top and the anti-top quarks decay semileptonically, e.g. t WThe+b usel+ ofνb moments. We study for the the kinematic extraction distribution of the top of mass an invariant has been mass suggested of a b-jet previously and ain lepton,→ the context and→ the distributions of the so-called of theJ/ sumΨ method of energies [11], of or the in two connectionleptons and with the variablestwo b-jets. supposed We employto minimise the NLO the QCD dependence corrections on to topthe quark jet-energy pair production scale [12, and13]. decay,To our as knowledge,computed in the most Ref. [13]. Throughout this Section, the center-of-mass energy of proton collisions is 14 TeV. up-to-date theoretical treatment of this technique is in ref. [14]. All these papers consider We begin by summarizing the kinematic cuts that are employed to identify dilepton tt¯events [4]. l l Leptonsonly are the required first momentto be central (of variousη < 2.5 distributions); and have large transverse in the case momentumof mt extractionp⊥ > 25 GeV. from different | | miss Thereobservables, should be missing the energy results in are the either event, E not⊥ combined> 40 GeV. [14], The jet or transverse limited to momentum two observables cut [13]. is p⊥,jThese> 25 GeV. choices We mayemploy lead the tok⊥ issues,jet algorithm as we withshallR discuss=0.4. in sects.2.3,3.2.2,and3.2.4.Inthe 3.1. Invariantcase of the mass dilepton of a lepton channel, and a b ref.-jet [14] also employs one of theobservablesconsideredinthis + − It ispaper pointed (E out(ℓ in)+ Ref.E(ℓ [4])); that owing an average to the value different of the choices invariant made masssquaredofa for cuts, jetb algorithm-jet and a ,collider 2 leptonenergy,mlb and and an average PDFs, value we have of the refrained the angle frombetween making the lepton a direct and the compb-jetarison in the withW boson those results. rest frame,We also can be point used out to construct that in an ref. estimator [14] the of the simultaneous top quark mass. variatio Thenofthefactorisationand estimator reads renormalisation scales has been adopted,2 whichm2 leads to smaller scale uncertainties than M 2 = m2 + ⟨ lb⟩ . (39) those we find in this paperest (whereW the1 twocos scalesθ are varied independently, see sect. 3). −⟨ lb⟩ To see thatFinally, this is a we good remark estimator, that we other note discrete that for theparameters top quark ofdecay kinema computedtic distributions, at leading such as order inmedians perturbative and maxima, QCD and without might also any restrictions be used for on a the top final massstate extractiMest equalson. We to m havet chosen to 2 2 2 2 mt mW mW 2 2 mlb = − (1 cos θlb ) , cos θlb = 2 2 Mest = mt . (40) ⟨ ⟩ 2 −⟨ ⟩ ⟨ ⟩ mt +2mW ⇒ In reality Mest is not equal to mt for a variety of reasons–5– including i) kinematic cuts required to identify the dilepton events; ii) effects of higher order QCD corrections; iii) impossibility to choose the “correct” pair of a lepton and a b-jet and iv) the experimental issues with b-jet misidentification 2 and the jet energy resolution. The computation reported in Ref. [13] allows us to calculate Mest within the framework of perturbative QCD, accounting for the points i)-iii) exactly. We point out that the computation of NLO QCD corrections to pp tt¯ process reported in [13] includes exact spin correlations, one-loop effects in top quark decays→ and allows arbitrary constraints to be imposed on top quark decay products. These features are crucial for reproducing experimental procedures. Indeed, experimentally, it is not possible to determine the charge of the b-jet. Hence, it is unclear which of the two b-jets should be combined with the chosen, definite- sign, lepton. For the purpose of mlb reconstruction, one pairs the lepton with the b-jet that gives the smallest m value [4]. The parameter cos θ in Eq. (39) is not measured and should be lb ⟨ lb⟩ 12 To p co u p l i ng s –Johnny Appleseed Top SM couplings ✦ to W boson: flavor mixing, lefthanded Exp. tested? ‣ gW ∼ 0.45 g V (t¯ µq )W + √ tq L L µ ✦ to Z boson: parity violating 2 ‣ gZ ∼ 0.14 g 8 t¯ (1 sin2 ⇥ ) µ µ 5 t Z 4 cos ⇥ 3 w µ ? ✦ to photon: vectorlike, has charge 2/3 w ⇥ ‣ et ∼ 2/3 µ et t¯ tAµ √? ✦ to gluon: vectorlike, non-trivial in color ‣ gs 1.12 ji ∼ SU(3) ¯ a gs Ta tj µtiAµ √ ✦ to Higgs: Yukawa type ⇥ ‣ yt ∼ 1 yt htt¯ √? Top physics: check structure and strength of all these couplings 30 tt+W,Z,γ ✦ Photon Kardos, Trocsanyi ‣ NLO + PS calculation ‣ dominated by gluon fusion ‣ Control sample/background for ttH, H→γγ ‣ Z Garzelli, Kardos, Papadopoulos, ‣ NLO + PS calculation Trocsanyi ‣ not yet “seen” ‣ W ‣ NLO + PS calculation Garzelli, Kardos, ‣ ttW at LHC has little sensitivity to tWb coupling Papadopoulos, Trocsanyi ✓ Use single top production here 31 Topquarkspincorrelationsathadroncolliders 15 ( ) ( ) 1 ⇤ ⇤ completeness relation for the top quark spinors will now be ut u¯t = 2 (pt + mt)(1 5St). Following exactly the same steps the corresponding amplitude is 4 ( ) + 2 2 2 Mw M(t ⇤ be ⇤ ) = 64G V m p◆ (1 cos ⇥ + )(p p ) (2.19) ↵| ⇧ e | f | tb| (q2 M 2 )2 + M 2 2 t| e¯| e b · n w w w Summarizing this process and using a more general notation it follows that ( ) 1 ( ) ⇥ for t ⇥ , the corresponding vector appearing in the squared amplitude isp ˜t 2 (pt mtS) and • ( ) + 2 ⌅ M(t ⇥ be ⇤e) (1 + cos ⇥e+ ) and ↵| ⇧ | ⌥ ( ) 1 ( ) ⇤ for t ⇤ , the corresponding vector appearing in the squared amplitude isp ˜t 2 (pt + mtS) and • ( ) + 2 ⌅ M(t ⇤ be ⇤ ) (1 cos ⇥ + ). ↵| ⇧ e | ⌥ e The top quark momentum is decomposed into a sum of two auxiliary momenta. ( ) ( ) pt =˜pt⇥ +˜pt⇤ Top self-analyzes(2.20) its spin p◆ Ep◆ Using the definition of he covariant spin vector Sµ =(| |, ), it is easy to check that it is normalized ✦ m m p◆ µ 100%| |correlation of charged lepton with top spin such that SµS = 1. Therefore the introduced auxiliary momenta are massless. Furthermore in the ( ) top quark rest frame, with pt(mt, 0), the spatial parts‣ ofpTop ˜t⇤ andself-analyzes S are parallel, its whilespin the spatial parts ( ) ofp ˜t⇥ and S are antiparallel (fig. 7C). Returning to the process in discussion (fig. 6), from‣ Charged equations leptons (2.18), (2.19),easy to one measure can derive the normalized decay rate as a function of the angle ⇥. ✦ For spin-up top the polar angle distribution is d ( ) + 2 d d ln f 1 1 ( ) M(t ⇥ be ⇤e) 1 ( ) 1 = (1 + f cos ⇥f ) ⇥ ⇥ + = ( ) ↵| + 2 ⇧ (| ) + 2 = (1 + cos ⇥e ) d cos ⇥f 2 Total d(cos ⇥e+ ) M(t ⇥ be ⇤e) + M(t ⇤ be ⇤e) ⌃ T d(cos ⇥e+ ) 2 ↵| ⇧ | ↵| ⇧ | (2.21) ✦ Due to chiral structure of tWb coupling The degree of correlation of the decay product to the spin appears in equation (2.21) in the coe⌅cient of the cos ⇥e+ . Therefore one can conclude that the angle of the emission of the charged lepton is maximally correlated to the top quark spin. In other words, a plot of the normalized decay rate with respect to ⇥ cos ⇥e+ (eq. 2.21) would be a straight line with slope 4 . The preferred positron emission axis is the ( ) spatial part ofp ˜t⇤ (in this case cos ⇥e+ =1 maximum decay rate). Using equation (2.17) and the corresponding⇧ one for top quark spin down, as well as equation (2.20), one can derive the amplitude for the unpolarized semi leptonic top quark decay10. 2 1 ( ) + 2 ( ) + 2 M = M(t ⇥ be ⇤ ) + M(t ⇤ be ⇤ ) = ↵| | 2 ↵| ⇧ e | ↵| ⇧ e | 4 ⇥ 1 2 2 Mw ( ) ( ) ⇥ ⇤ = 128Gf Vtb 2 2 2 2 2 (˜pt pe¯)(pb pn)+(˜pt pe¯)(pb pn) 2 | | (q Mw) + Mw w · · · · ⌃ ⇤ ⇧ ⌃⌅ M 4 32 M 2 = 64G2 V 2 w (p p )(p p ) (2.22) ⌃↵| | f | tb| (q2 M 2 )2 + M 2 2 t · e¯ b · n w w w By comparing equations (2.7) and (2.22), it is clear that the top and c quark decay amplitudes di⇥er only to the fact that in the top quark decay the intermediate boson W + can be real, as expected. One may notice that in both equations (2.21) and (2.22) the mixed terms in the total amplitude are neglected. The accurate decomposition of the amplitude to spin up and down top quark is 2 1 ( ) + ( ) + 2 1 ( ) + 2 ( ) + 2 M = M(t ⇥ be ⇤ )+M(t ⇤ be ⇤ ) = M(t ⇥ be ⇤ ) + M(t ⇤ be ⇤ ) + ↵| | 2 ↵| ⇧ e ⇧ e | 2{↵| ⇧ e | ↵| ⇧ e | ( ) + ( ) + ⇥( ) + ( ) + + M(t ⇥ be ⇤ )M (t ⇤ be ⇤ ) + M(t ⇤ be ⇤ )M (t ⇥ be ⇤ ) (2.23) ↵ ⇧ e ⇧ e ↵ ⇧ e ⇧ e } 10At this point one must average over initial spins. Spin correlations for single top in MC@NLO Frixione, EL, Motylinski, Webber ✦ Top is produced polarized by EW interaction ‣ 100% correlation between top spin and charged lepton direction θ ✦ Angle of lepton with appropriate axis is different per channel ✦ Method included “a posteriori”. Also used in POWHEG Aioli, Nason, Oleari, Re Beam direction Hardest, non-b jet Robust correlation in NLO event generation (Method to infer inclusive cross section?) 33 ttH ATLAS Simulation ATLAS Simulation s = 14 TeV: ∫Ldt=300 fb-1 ; ∫Ldt=3000 fb-1 s = 14 TeV: ∫Ldt=300 fb-1 ; ∫Ldt=3000 fb-1 ✦ Should become very interesting for the new run ∫Ldt=300 fb-1 extrapolated from 7+8 TeV ∫Ldt=300 fb-1 extrapolated from 7+8 TeV H→µµ ΓZ / Γg ttH,H→µµ ✦ Γt / Γg σtth (14) ≃ 4.6 x σtth (8) VBF,H→ττ H→ ZZ Γτ / Γµ ✦ VBF,H→ WW Γ / Γ NLO calculations for signal Beenakker, Dittmaier, Kraemer, Plumper, Spira, Zerwas µ Z H→ WW Dawson, Orr, Reina, Wackeroth Γ / Γ VH,H→γγ τ Z ‣ ttH,H→γγ plus PS Frederix, Frixione, Hirschi, Maltoni, Pittau, Torrieli ΓW / ΓZ Garzelli, Kardos, Papadopoulos, Trocsanyi VBF,H→γγ Γγ / ΓZ ✓ and spin correlations Hartanto, Jaeger, Reina, Wackeroth H→γγ (+j) Γ •Γ / Γ H→γγ g Z H 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 � ≈ 2.6 × �∆µ ∆(Γ /ΓY) ∆(κ /κY) ℎ ℎ X ~ 2 X µ Γ /Γ κ /κ ‣ plus EW Frixione, Hirschi, Pagani, Shao, Zaro X Y X Y (a) (b) ✦ 1 1 and e.g. ttbb backgrounds to NLO(+PS) Figure 4: Summary of Higgs analysis sensitivities wth 300 fb and 3000 fb at ps = 14 TeV for a SM Higgs boson with a mass of 125 GeV. Left: Uncertainty on the signal strength. For the H ⌧⌧ channels ! the thin brown bars show the expected precision reached from extrapolating all tau-tau channels studied 1 1 in the current 7 TeV and 8 TeV analysis to 300 fb , instead of using the dedicated studies at 300 fb and 3000 fb 1 that are based only in the VBF H ⌧⌧ channels. Right: Uncertainty on ratios of partial ! decay width fitted to all channels. The hashed areas indicate the increase of the estimated error due to current theory systematic uncertainties. 1 1 uncertaintiesv included. The right-hand figure compares the 300 fb and 3000 fb results with no theory uncertainties included. Backgrounds difficult, but expect 4.1.1 Sensitivity to the Higgs self-coupling An important feature of the Standard Model Higgs boson is its self-coupling. The tri-linear self- 10% accuracy in Yukawa coupling by 2030 coupling HHH can be measured through an interference e↵ect in Higgs boson pair production. At hadron colliders, the dominant production mechanism is gluon-gluon fusion. At ps = 14 TeV, the production cross section of a pair of 125 GeV Higgs bosons is estimated at NLO to be2 +18% 34 15%(QCD scale) 3%(PDFs) fb. Figure 6 shows the three contributing diagrams in which the 34 Δ� /� ≈ 10% ± 2The cross section is calculated using the HPAIR package [15]. Theoretical uncertainties are provided by Michael Spira in private communication. 8 Conclusions ✦ Major progress at N3LO and N4LO for mass definition and threshold scan. ✦ Steady progress on proxy masses, and understanding MC masses. ✦ Top mass may be the last, but not the easiest theoretical problem to solve ‣ Goes to the heart of data-theory comparison ✦ With new LHC run, EW couplings of top will be tested ✦ Theory seems ready for Yukawa coupling tests 35