The tt Total Cross Section at Hadron Colliders at NNLL Sebastian - PowerPoint PPT Presentation

1 The tt Total Cross Section at Hadron Colliders at NNLL Sebastian Klein in collaboration with M. Beneke, P. Falgari and C. Schwinn Sebastian Klein HP2.3rd, Florence 16.09.2010

2 1. Introduction • We consider the inclusive cross section of pair production of top quarks at the LHC (near) threshold ij(qq , qg , gg) → tt + X � 1 σ tt ( m t , s ) = t /s dx L ( x, µ )ˆ σ tt ( xs, m t , µ ) 4 m 2 � 1 − 4 m 2 β = t / ( xs ) • With the rediscovery of the top quark at the LHC, precision studies of its properties will be performed = ⇒ Accurate theoretical prediction of the cross section phenomenologically important and currently of much interest. • Top is expected to couple strongly with the fields responsible for electroweak symmetry breaking = ⇒ likely to play a key role in new discoveries.

3 • Partonic cross section known exactly to NLO. [Nason, Dawson and Ellis, 1988; Beenakker, Kuijf, van Neerven and Smith, 1989; Czakon and Mitov, 2008.] • Enhancement of partonic cross section near threshold β → 0 � 1 + α s � � � α s � 2 � � � σ (0) σ NLO sing σ NNLO sing + O ( β 0 ) + O ( β 0 ) σ tt ˆ = ˆ ˆ + ˆ + ... , tt 4 π tt 4 π tt c a ln 2 β + b ln β σ NLO sing ˆ = + , tt β � �� � ���� ”Threshold logarithms” ”Coulomb singularity” � 1 β 2 ; ln (0 , 1 , 2) ( β ) � σ NNLO sing σ NNLO sing ; ln (1 , 2 , 3 , 4) ( β ) ˆ = ˆ . tt tt β [Beneke, Czakon, Falgari, Mitov and Schwinn, 2009.] • Threshold logarithms: soft gluon exchange between initial-initial, initial-final and final-final state particles. Resummation in Mellin–space e.g. by [Sterman, 1987;Catani, Trentadue, 1989; Kidonakis, Sterman, 1997;Bonciani et.al., 1998;...] • Coulomb corrections: of slowly moving particles [Fadin, Khoze 1987; Strassler, 1990; NRQCD; ...] static interactions

4 • Enhanced terms can spoil convergence of perturbative series = ⇒ Resummation – Generally observed to reduce dependence on factorization scale. – Allows to predict classes of higher order corrections – Accelerated convergence of perturbative series • Recent applications – total top quark cross section [Moch,Uwer, 2008; Cacciari et. al., 2008; Kidonakis, Vogt, 2008.] – tt invariant mass distribution [Kiyo, K¨ uhn, Moch, Steinhauser, Uwer, 2009; Ahrens, Ferroglia, Neubert, Yang, 2009/10.] – squark, gluino production [Kulesza, Motyka 2008/09; Langenfeld, Moch, 2009; Beenakker et.al. 2009/10; Beneke, Falgari, Schwinn, 2010.] – Bound–state effects on kinematical distributions of top quarks at hadron colliders [Sumino, Yokoya, 2010.] • Key idea: factorization into hard, soft and Coulomb functions = ⇒ joint NNLL resummation of soft and Coulomb gluons. • Effective-theory prediction of pair production near threshold [Beneke, Falgari, Schwinn 2009/10.] using SCET + P(NRQCD) : valid for arbitrary color representations.

5 • Parametric representation of the partonic cross section near threshold: ∞ � α s � k � � � σ (0) σ tt ˆ ∝ ˆ exp ln βg 0 ( α s ln β ) + ln βg 1 ( α s ln β ) + α s ln βg 2 ( α s ln β ) tt β � �� � � �� � � �� � k =0 (LL) (NLL) (NNLL) � � 1(LL , NLL); α s , β (NNLL); α 2 s , α s β, β 2 (NNNLL); ... . . . × . • Counting: α s /β, α s ln β ∝ 1. • Fixed order expansion contains all terms of the form β 2 , ln 2 β � 1 � 1 � � β , ln 2 β , ln 4 β ; α 2 LL : α s ; ...., s β � ln β � β , ln 3 β α s ln β ; α 2 NLL : ; ...., s � 1 � α s { 1 , β ln 2 , 1 β } ; α 2 β , ln 2 , 1 β, β ln 4 , 3 β NNLL : ; ...., s • Non–relativistic log summation must be added separately - relevant from NNLL.

6 Why threshold expansion • Strictly valid for high masses 2 m t → s had . • Certainly not for tops at LHC7. Invariant mass distribution peaks at 380 GeV, corresponding to β ≈ 0 . 4, but the average β is larger. • Assume that threshold expansion provides a good approximation for the integrals over all β . Works reasonably well for gg at LO and NLO, less well for qq and probably better at NNLO, because the average is dominated by smaller β as the order increases. σ (0) • multiplying with the exact tree ˆ improves the approximation. tt Tev. LHC7 LHC14 NLOapprox 400 gg LHC14 � β � gg, NLO 0.41 0.49 0.53 NLO 300 LO 5.25 101.9 562.9 200 NLO 6.50 149.9 842.2 NLOsing 100 NLO sing 6.76 138.8 751.2 0 NLO approx 7.45 159.0 867.6 0 0.2 0.4 0.6 0.8 1 Β MSTW2008nnlo PDFs.

2. NNLL Resummation 7 • Apply NNLL soft resummation and coulomb resummation to total cross section � 2 m t � − 2 η � σ NNLL s R α ˆ = H i ( m t , µ h ) U i ( m t , µ h , µ S , µ f ) ˜ ( ∂ η , µ S ) i tt µ S i,R α � w � ∞ 2 , µ C ) , ( E = √ xs − m t ) . × exp( − 2 γ E η ) dw � 2 η J R α ( E − w Γ(2 η ) w µ S 0 • Different contributions: – Hard function H i depends on the specific process, evaluated at hard scale µ h . s ln m β ) translates via a Laplace transform – Process–independent soft function W R α ( ∝ α n i s R α into ˜ ( ∂ η , µ S ). Evaluated at soft scale µ S . i – U i evolution function from solving the RG equations of the hard and soft functions. (for DY: [Becher, Neubert,Xu, 2007.] .) – Potential function J R α encodes Coulomb effects, evaluated at Coulomb–scale µ C . • Formula valid except for non-Coulomb corrections at O ( α 2 s ), which are added separately. • New: – full LO and NLO Coulomb effects to all orders, above and below threshold. – full NNLL soft resummation (Note: full NNLL soft resummation for invariant mass distribution in [Ahrens, Ferroglia, Neubert, Yang, 2010.] ).

8 Hard and Soft Function • NLL needs H i at tree level, NNLL needs H i at NLO. Known from [Czakon, Mitov, 2008] . • NNLL needs soft function at NLO. It is given by � � � � s R α ( ρ, µ S ) = 1 + α s ( µ S ) ρ 2 + ζ 2 ˜ ( C r + C r ′ ) − 2 C R α ( ρ − 2) . 4 π • The evolution function is given by � − 2 a Γ ( µ h ,µ S ) � µ 2 � 4 m 2 � η � t h U i ( M, µ h , µ f , µ s ) = × exp 4( S ( µ h , µ f ) − S ( µ S , µ f )) µ 2 µ 2 S h � i ( µ h , µ S ) + 2 a φ,r ( µ S , µ f ) + 2 a φ,r ′ ( µ S , µ f ) − 2 a V , where a ( µ 1 , µ 2 ) , S ( µ 1 , µ 2 ) denote integrated anomalous dimensions. i , γ r , γ R α • Resummation controlled by cusp and soft anomalous dimensions Γ r cusp , γ V H,s .

9 Coulomb effects • For NNLL: J R α needed at NLO. Resummation of Coulomb effects well understood from PNRQCD and quarkonia physics. The LO Coulomb function reads �� ��� − m 2 − E � 1 � − 4 m t E � − 1 � D R α α s t J R α = 2 π Im − D R α α s 2 ln 2 + γ E + ψ 1 + . � µ 2 m t 2 − E/m t • Above threshold, E > 0, the potential function evaluates to the Sommerfeldt factor m 2 1 t D R α α s J R α = . � � � 2 exp πD R α α s m t /E − 1 • For an attractive potential, D R α < 0, there is a sum of bound states below threshold: � � n δ ( E + E n ) , E n = m t α 2 s D 2 ∞ m t D R α α s � R α J R α = − 2 . 4 n 2 2 n n =1 (see also: [Fadin, Kohze 1987; Kiyo et.al. 2009; Hagiwara, Yokoya 2009] .) • Non–Coulomb corrections can be derived from the non–Coulomb potential � � σ (0) σ NC tt α 2 ˆ = ˆ s ln β − 2 D R α (1 + v spin ) + D R α C A . tt [Beneke, Signer, Smirnov 1999; Pineda, Signer, 2006; Beneke, Czakon, Falgari, Mitov, Schwinn,2009.] .

10 Scale Choice • We use m t = 173 . 1 GeV and set µ f = µ R = m t . • Identify hard scale and factorization scale: µ h ∝ µ f . = ⇒ No large logs of the hard scale (ln( µ h /µ f )). • The form of the approximate NLO corrections implies that µ S ≈ 8 m t β 2 . However, this choice might lead to an ill–defined convolution with the parton luminosity [Becher, Neubert, Pecjak, 2007; Becher, Neubert, Xu, 2008.] = ⇒ Choose µ S such that one–loop corrections to the hadronic cross section are minimized. This guarantees well-behaved perturbative expansion at the low scale µ S . • The choice µ C ∝ m t β is required to sum correctly all NNLL terms. Additionally, the relevant scale for the bound state effects is set by the inverse Bohr radius of the tt bound state and we set µ C = max { 2 m t β, C F m t α S ( µ C ) } . • Note: only the choice µ S ∝ m t β 2 , µ C ∝ m t β reproduces correctly the threshold expansion in β

3. Preliminary Results 11 • We match the resummed cross section onto the full NLO result [Zerwas et.al., 1996; Langenfeld, Moch, 2009.] � σ N(N)LL σ N(N)LL σ approx σ NLO � ˆ ∝ ˆ − ˆ + ˆ . � tt tt tt tt α 1 , 2 s • For NLL resummation, we previously considered σ NNLO approx σ NNLO sing σ NLO ˆ = ˆ + ˆ , tt tt tt � σ NNLO approx +NLL σ NNLO approx σ NLL σ NLL � ˆ = ˆ − ˆ + ˆ . � tt tt tt tt α 2 s σ NNLO sing σ NC (Note: ˆ included in ˆ .) tt tt • A natural choice for NNLL resummation would be � σ NNLL( α s ) σ NNLL σ NNLL σ NC σ NLO � ˆ = ˆ − ˆ α s + ˆ + ˆ . � tt tt tt tt tt • Due to the limitations in choosing the soft scale running, we consider as our best approximation: � σ NNLL( α 2 s ) σ NNLO approx σ NNLL σ NNLL � σ tt ≡ ˆ ˆ = ˆ − ˆ + ˆ . � tt tt tt tt α 2 s