NNLO QCD Corrections to Differential Top-Quark Pair Production with The

NNLO QCD corrections to diﬀerential top-quark pair production with the MS mass Stefano Catani INFN Firenze

based on JHEP (2020) 027 [ 2005.00557 ] ( see also JHEP (2019) 100 [ 1906.06535 ] )

in collaboration with Simone Devoto, Massimiliano Grazzini, Stefan Kallweit, Javier Mazzitelli

Top 2020, Durham, September 14, 2020 Outline

Relation between pole mass and MS mass some general features Top-quark production at the LHC unequal role of pole mass and MS mass

Cross sections for tt¯ on-shell production from pole mass and MS mass

QCD results with MS mass total and single-diﬀerential cross sections up to NNLO

Eﬀects due to the MS running mass a ﬁrst study: invariant-mass distribution of tt¯ pair

Summary

2 POLE vs. MS MASS top-quark mass: fundamental parameter of SM to be properly deﬁned by renormalization of related UV divergences

pole mass Mt : pole of renormalized propagator (“customary” mass for physical particle)

MS mass mt(μm) : “subtract” UV divergences in dimensional regularization (more abstract deﬁnition) diﬀerent renormalization schemes are perturbatively related:

we speciﬁcally use mass relation at NNLO (k ≤ 3) coeﬃcients d(k) known for k ≤ 4

MS mass depends on arbitrary renormalization scale μm (similarly to QCD coupling αS(μR) ) and scale dependence is perturbatively computable [Renormalization Group (RG) evolution]

∞ k+1 coefficients known for d ln m (μ ) α (μ ) γk k ≤ 4 t m = − γ S m 2 ∑ k ( ) d ln μm k=0 π we specifically use RG evolution at NNLO (k ≤ 2) Note: scale dependence of mass d ln m (μ) 1 d ln α (μ) MS t ∼ S at LO much slower than αS d ln μ 2 d ln μ 3 MS mass mt(μm) can be specified by: its value at a reference scale + RG evolution (no special physical meaning; customary reference scale: m¯ t somehow analogous to reference scale MZ for αS(μR))

a scale of the order of the mass itself (“intrinsic” deﬁnition) mt(m¯ t) = m¯ t

typical values at NNLO Mt = 173 GeV ⟷ m¯ t = 164 GeV ( �( GeV) variations w.r.t. LO, NLO) ( ∼ 10 GeV diﬀerence)

(1) [ Note: at scale μm = m¯ t /2 mt(μm) = Mt + �(1 GeV) , simply because to d ∼ 0 ]

Two main consequences of scale dependence of MS mass

perturbative QCD predictions unavoidably depend on μm (in addition to renormalization scale μR from αS(μR) and factorization scale μF from PDFs)

μm can possibly be set to a scale very diﬀerent from Mt ∼ m¯ t to embody (“resum”) higher-order corrections running mass eﬀects

4 TOP QUARK at the LHC indirect studies/sensitivity :

top quark enters as virtual (highly oﬀ-shell) particle [ e.g., Higgs boson production by gluon-gluon fusion through top-quark loop ]

pole and MS masses can be introduced on equal footing

5 TOP QUARK at the LHC direct studies/sensitivity: top quark (its decay products) is (are) directly observed in the ﬁnal state

based on deﬁnite physical picture top quark is “physical”, though unstable, particle

with deﬁnite pole mass Mt ( ∼ 173 GeV) and small decay width Γt ( ∼ 1.4 GeV) then data on top-quark production extracted from quasi-resonant behavior (around pole mass) of its decay products no data without the concept of pole mass pole mass has primary role [ MS mass has (somehow) an auxiliary role ] * diﬀerence pole vs. MS mass can be much larger than width Γt theory counterpart: after integration over top-quark decay products and in narrow-width limit

compute cross section for production of on-shell top quark with pole mass Mt

[ Mt is not only a parameter of the Lagrangian but also a key kinematical parameter of the phase space (of the underlying physical picture) ]

6 ON-SHELL CROSS SECTION for tt¯ PRODUCTION: from pole to MS mass

Start from on-shell cross section σ(Mt, X) with pole mass Mt (total σ or diﬀerential dσ/dX)

e.g. up to NNLO

Perform all-order replacement Mt → mt(μm) and deﬁne MS scheme cross section σ¯ by ALL-ORDER (formal) EXACT EQUALITY

MS scheme Pole scheme Note: mass and kinematic variable(s) X are treated as independent variables

Express Mt in terms of { mt(μm) and αS(μR) } and expand σ in αS at ﬁxed mt(μm) e.g. up to NNLO

7 Explicit expressions* at LO, NLO and NNLO

NLO

NNLO

result depends on renormalization coeﬃcients d(k) , (k) n (k) pertubative terms σ of on-shell cross section and their mass derivatives ∂mσ

WARNING : mass derivatives can be very sizeable thus spoiling the perturbative convergence 0f MS cross section σ¯ see backup slides (e.g., invariant mass of tt¯ pair close to its threshold region )

* same perturbative formulae used by Langenfeld-Moch-Uwer (2009), Dowling-Moch (2014) and applied to total cross section up to NNLO and single-differential distributions up to NLO 8 within this formulation, pole scheme and MS scheme results are formally equivalent to all orders in αS but different if expanded * at fixed orders

* αS expansion at ﬁxed Mt ( in σ ) or mt(μm) ( in σ¯ ) our general expectations

at low orders, σ and σ¯ can give consistent (within scale uncertainties) results [ diﬀerences can be larger for observables close to kinematical thresholds for tt¯ on-shell production ]

at higher orders, σ and σ¯ can be quantitatively very similar equivalent perturbative description then

for observables at high scales X ≫ mtop ( e.g., top quark at large pT or tt¯ pair at high invariant mass ) investigate eﬀects of running MS mass

mt(μm) with μm ∼ X (k) Note: at such scales μm the coeﬃcients d (μm) are sizeable

our main motivation for using MS mass

9 LHC RESULTS up to NNLO two independent NNLO fully diﬀerential calculations of tt¯ on-shell production with pole mass Czakon, Fiedler, Mitov (2016) Devoto, Grazzini,Kallweit, Mazzitelli + S.C. (2019)

n (k) we use our calculation by numerically computing mass derivatives ∂mσ̂ (m) on a bin-by-bin basis (X bins)

3 auxiliary scales μi = {μR, μF, μm} and independent scale variations by a factor of two around central μ0 :

μi = ξiμ0, ξi = {1/2,1,2} with constraints μi /μj ≤ 2 15-point scale variation in MS scheme ( customary 7-point in pole scheme with 2 auxiliary scales )

we compare pole scheme and MS scheme by setting

pole scheme: Mt = 173.3 GeV and use μ0 = Mt

MS scheme: mt = 163.7 GeV and use μ0 = mt (varying μm with 0.5 < μm /μ0 < 2 ⟶ 155 GeV < mt(μm) < 173 GeV )

we use NNPDF31 and s = 13 TeV

10 TOTAL CROSS SECTION

(a) (b) (c) comparison pole scheme ( μ0 = Mt ) and MS scheme ( μ0 = mt ) order-by-order consistency of the results and very similar at NNLO

MS typically higher at central scale and with smaller uncertainties at NLO and NNLO

[ μR (a) and μm (b) dependences have similar size but opposite sign (cancellations (c)) ] MS results have faster apparent convergence * NLO * ﬁrst noticed by = 1.52 (pole), 1.32 (MS) Langenfeld-Moch-Uwer (2009) LO NNLO = 1.09 (pole), 1.01 (MS) NLO

11 pole vs. MS scheme: slower/faster apparent convergence central scales

μ0 = Mt vs. μ0 = mt : we do not have a physical interpretation but we do have a technical explanation

(valid in any scheme with renormalized mass mren. < Mt ) moreover : the apparent convergence strongly depends on the choice of

central value μ0 of auxiliary scales

Slower: MS scheme ( μ0,m = mt /2) and Faster: MS scheme ( μ0,m = mt ) and

pole scheme ( μ0 = Mt ) pole scheme ( μ0 = Mt /2 )* behave similarly behave similarly

* scale suggested by Czakon-Deymes-Mitov (2017) 12 DIFFERENTIAL CROSS SECTIONS

MS pole ratio MS/pole(Mt)

ﬁrst results at NNLO transverse-momentum p distribution of t quark : T NNLO ratio comparison pole scheme ( μ0 = Mt ) vs. MS scheme ( μ0 = mt ) overall features similar to those for total cross sections

at NNLO (see ratio MS/pole): shape diﬀerences are quite small and within scale uncert.

the results in the two schemes behave similarly at (suﬃciently) high order

13 similar comments apply to other diﬀerential cross sections : rapidity of t quark or tt¯ pair

invariant-mass mtt¯ distribution of tt¯ pair at high mtt¯

exception : invariant-mass distribution of tt¯ pair close to its threshold region

overall observations

results in pole and MS schemes become increasingly similar at high orders

NNLO results: precise QCD predictions in both schemes rapidity of t quark (antiquark)

pp tt¯ @ 13 TeV, µ0 = mt = 163.7 GeV pp tt¯ @ 13 TeV, µ0 = Mt = 173.3 GeV pp tt¯ @ 13 TeV 600 ! 600 ! 3.5 ! LO LO LO NLO NLO NLO 500 500 3.0 MS NNLO pole NNLO ratio NNLO 400 400 2.5 [pb] [pb]

pole /pole( ) | | MS Mt av av t t /d æ y 300 y 300

| | 2.0 MS d æ d æ /d 200 d æ /d 200 1.5

100 100 1.0 0 0 MS pole 1.0 1.0 NNLO pole 1.1

0.8 0.8 /d æ 1.0

0.6 0.6 NNLO d æ ratio to0 NNLO .4 ratio to0 NNLO .4 0.9 0 1 2 0 1 2 0 1 2 y y ytav | tav | | tav | | | rapidity of top-quark pair NNLO ratio pp tt¯ @ 13 TeV pp tt¯ @ 13 TeV, µ0 = mt = 163.7 GeV pp tt¯ @ 13 TeV, µ0 = Mt = 173.3 GeV 3.5 ! 700 ! 700 ! LO LO LO NLO 3.0 600 NLO 600 NLO NNLO MS NNLO pole NNLO ratio 500 500 2.5 pole [pb] [pb] | | 400 400 ¯ ¯ t t /pole( ) /d æ t t 2.0 MS Mt y y | | 300 300 MS d æ 1.5 d æ /d d æ /d 200 200 1.0 100 100

0 0 MS pole NNLO 1.0 1.0 pole 1.1 /d æ 0.8 0.8 1.0

0.6 0.6 NNLO d æ 0.9 ratio to NNLO 0.4 ratio to0 NNLO .4 0 1 2 0 1 2 0 1 2 ytt¯ y ¯ y ¯ | | | tt| | tt| 15 INVARIANT-MASS DISTRIBUTION of tt PAIR

CMS ( 1909.09193 ) 35.9 fb-1 (13 TeV) recent CMS study (2020) 350 [pb] t t Data unfolded to parton level m 300 precise measurement of cross section: Δ

mtt t t 4 bins over region ∼ 380 – 1000 GeV 250 NLO predictions in MS scheme µ = µ = m / dm r t t f t σ d 200 ABMP16_5_nlo PDF set m¯ use NLO calculation with t m (m ) = 162 GeV 150 t t FIXED mass mt(m ) = 164 GeV MS m¯ t t m (m ) = 166 GeV t t (i.e. μm = m¯ t in all bins ) 100 and ﬁt value of to data in each bin m¯ t 50

400 600 800 1000 1200 1400 1600 1800 2000 m [GeV] our conclusions : tt data/NLO consistency with a single common (i.e., bin-independent within errors)

value of m¯ t

can we study eﬀects due to running MS mass mt(μ) ? this unavoidably requires calculation with

RUNNING (bin-dependent) value of μm ( i.e., mt(μm) )

16 mtt¯ DISTRIBUTION: EFFECTS OF RUNNING MS MASS

we investigate QCD results in MS scheme with two diﬀerent options for central scale μ0 (i) FIXED mass : set μ0 = m¯ t ( for μm, μR, μF ) [ NNLO extension of CMS NLO calculation ]

(ii) RUNNING mass : set μ0 ≃ mtt¯ /2 ( for μm, μR, μF ) ( i.e. mt(mtt¯ /2) is bin-dependent and it varies by about 10 GeV : from mt ∼ 160 GeV in 1-st. bin → to mt ∼ 150 GeV in 4-th. bin )

set up : ABMP16 PDFs ( as in CMS study of mtt¯ distribution ) ; m¯ t = 161.6 GeV ( as measured at NNLO by CMS study of total cross section ) [ it corresponds to Mt = 170.8 GeV ]

* Aside comment

high (multi TeV) mtt¯ region : two very different scales, Mt and mtt¯ → resummation of soft/collinear effects [e.g., Ahrens et al. (2010), Ferroglia et al. (2012), Czakon et al. (2018) ] could be combined with running mass effects

17 pp tt¯ @ 13 TeV, mt = 161.6 GeV, µ0 = mt pp tt¯ @ 13 TeV, mt = 161.6 GeV, µ0 = µk/2 350 ! 350 ! NLO NLO NNLO 300 NNLO 300 μ0 ≃ mtt¯ /2 FIXED CMS RUNNING CMS 250 250

200 200 [pb] [pb] æ æ 150 150

100 100

50 50

0 0 1.2 1.2 dashed lines: results with 1.0 1.0 μR = μF ≃ mtt¯ /2 ratio to data ratio to0 data .8 0.8 but keeping μm = m¯t 500 1000 1500 2000 500 1000 1500 2000 mtt¯[GeV] mtt¯[GeV]

comparison FIXED vs. RUNNING (including 15-point scale variations)

practically (“by deﬁnition”) no theory diﬀerences at low mtt¯

diﬀerences at high mtt¯ are “small” and mainly driven by running of αS and PDFs very similar/consistent (within scale uncertainties) results at NNLO

our conclusions : NNLO corrections lead to reduced th. uncert. and to improved agreement with data

[ moreover : pole scheme calculation with Mt = 170.8 GeV can do a similar job ] no signiﬁcant sensitivity to running mass eﬀects 18 Summary On-shell top-quark production: reformulation of QCD calculation from pole to MS mass

tt¯ production at the LHC: first NNLO results for single-differential cross sections by using MS mass [ extension to multi-differential and/or fiducial cross section is straightforward (feasible) ]

QCD comparison pole vs. MS schemes (at ﬁxed MS mass: mt(μm) with μm ∼ m¯ t ) including perturbative uncertainties ( 15-point scale variations in MS scheme ) consistent order-by-order results and increasingly similar results at high order at NNLO: precise QCD predictions in terms of MS mass relevant for ensuing studies with MS mass

Eﬀects due to the running of MS mass

first study of running mass effects ( mt(μm) with μm ∼ mtt¯ /2 ) for invariant-mass distribution of tt¯ pair in region up to mtt¯ ∼ 1 TeV no significant sensitivity to running mass effects further studies of running mass effects feasible and warranted

19 Backup ON-SHELL CROSS SECTION for tt¯ PRODUCTION: from pole to MS mass deﬁnitely unphysical some obvious unphysical features if Mt − mt(μm) is large w.r.t. Γt e.g., consider invariant-mass mtt¯ of tt¯ pair it has a physical threshold at minimum value min. mtt¯ = 2Mt

physical threshold fulﬁlled order-by-order in αS within pole scheme

within MS scheme :

min. at LO mtt¯ = 2mt(μm) arbitrary dependence on μm ; deﬁnitely unphysical

if Mt − mt(μm) is large w.r.t. Γt

n (l) k near threshold: ∂mσ̂ very large very large N LO corrections ( badly convergent αS expansion )

21 invariant-mass distribution of tt¯ pair ¯ pp tt¯ @ 13 TeV, µ0 = mt = 163.7 GeV pp tt¯ @ 13 TeV, µ0 = Mt = 173.3 GeV pp tt @ 13 TeV ! ! 3.5 ! 4 LO 4 LO LO NLO NLO pole NLO 3.0 MS NNLO NNLO ratio NNLO 3 3 2.5 pole MS/pole(Mt) [pb/GeV] [pb/GeV] /d æ ¯ ¯ t t t t 2 2 2.0 MS d æ 1.5 d æ /dm d æ /dm 1 1 NNLO 1.0 0 0 ratio MS pole 1.0 1.0 NNLO pole 1.1

0.8 0.8 /d æ 1.0 0.6 0.6 NNLO ratio to NNLO ratio to NNLO 0.4 0.4 d æ 0.9 300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000 mtt¯[GeV] mtt¯[GeV] mtt¯[GeV]

comparison pole ( μ0 = Mt ) vs. MS ( μ0 = mt ) : similar to other distributions but exception region close to threshold ( 1st. bin: 300-360 GeV, 2nd. bin: 360-400 GeV )

low-m region : tt¯ MS results have larger uncertainty (dominated by μm variations) and larger radiative corrections

consequence of unphysical order-by-order “identiﬁcation” Mt → mt(μm) [ mis-behaviour partly alleviated at high orders and/or using wide bin size ]

sufficiently close to threshold : no point in using MS mass use pole scheme (possibly refined by resummation of Coulomb-type effects * )

22 * see talk by Li Lin Yang CMS 35.9 fb-1 (13 TeV) ) ref

µ 1.05 ( ABMP16_5_nlo PDF set t µ = 476 GeV CMS Coll. (2020) ref

) / m µ = µ 0 ref µ ( t 1

1909.09193 m

0.95

NLO extraction from differential 0.9 σtt Reference scale µ ref One-loop RGE, n = 5, α (m ) = 0.1191 0.85 f s Z

400 500 600 700 800 900 1000 µ [GeV] CMS Supplementary arXiv:1909.09193 35.9 fb-1 (13 TeV)

170 ABMP16_5_nlo PDF set

) [GeV] µ = m = 163 GeV 0 t µ ( t 160 m

150

140

130 NLO extraction from differential σtt NLO extraction from inclusive σtt 120 One-loop RGE, n = 5, α (m ) = 0.1191 f s Z 110 200 400 600 800 1000 µ [GeV] 23