Threshold resummation for pair production of coloured heavy - PowerPoint PPT Presentation

Threshold resummation for pair production of coloured heavy particles at hadron colliders Pietro Falgari Institute for Particle Physics Phenomenology, Durham In collaboration with: M. Beneke, C. Schwinn PHENO 2009 Symposium, 12th May 2009 Madison Pietro Falgari (IPPP Durham) May 2009, Madison 1 / 10

Pair-production of heavy particles at hadron colliders H , H ′ ≡ t , ˜ ′ ( R ′ ) + X p i ( r i ) + p j ( r j ) → H ( R ) H q , ˜ g , ... Partonic cross section for pair-production of heavy particles at hadron colliders contains terms m 2 , ¯ kinematically enhanced in the partonic threshold region ˆ s ∼ 4 ¯ m ≡ ( m H + m H ′ ) / 2. Coulomb singularities : ∼ α n s /β n , β = � 1 − 4 ¯ m 2 / ˆ s ⇔ Coulomb interactions of slowly-moving particles s ln 2 n β 2 ⇔ soft-gluon exchange Threshold logarithms : ∼ α n Small coupling but effectively “non-perturbative” dynamics ⇒ Must be resummed to all orders when the partonic threshold region dominates the total cross section! Absolute normalisation of the total cross section Generally observed to reduce factorisation-scale dependence Pietro Falgari (IPPP Durham) May 2009, Madison 2 / 10

Moment-space VS Momentum-space resummation The theoretical basis for resummation is the factorisation of hard and soft dynamics in the threshold region (more generally for Q 2 ∼ ˆ s , even if Q 2 � = 4 ¯ m 2 ) ˆ σ = H ⊗ S Resummation traditionally performed in Mellin-moment space: s ln 2 n β ⇒ α n s ln 2 n N α n H ⊗ S ⇒ H ( N ) S ( N ) Threshold logs exponentiated by solving evolution equations for H ( N ) and S ( N ) . Requires numerical inversion of the Mellin transform and prescription to deal with Landau poles in the integrand Pietro Falgari (IPPP Durham) May 2009, Madison 3 / 10

Moment-space VS Momentum-space resummation The theoretical basis for resummation is the factorisation of hard and soft dynamics in the threshold region (more generally for Q 2 ∼ ˆ s , even if Q 2 � = 4 ¯ m 2 ) ˆ σ = H ⊗ S Resummation traditionally performed in Mellin-moment space: s ln 2 n β ⇒ α n s ln 2 n N α n H ⊗ S ⇒ H ( N ) S ( N ) Threshold logs exponentiated by solving evolution equations for H ( N ) and S ( N ) . Requires numerical inversion of the Mellin transform and prescription to deal with Landau poles in the integrand In this talk : apply formalism proposed by [ Neubert and Becher ’06 ] to resummation of the total cross section for p i p j → HH ′ + X . Based on effective-field theory description of the process (SCET+NRQCD) Threshold resummation performed directly in momentum space Pietro Falgari (IPPP Durham) May 2009, Madison 3 / 10

Cross-section factorisation near threshold s ∼ ( m H + m H ′ ) 2 Extra factorisation of the cross section near the true partonic threshold ˆ 1 pp ′ C ( ℓ ′ , i ′ ) ∗ � � C ( ℓ, i ) σ pp ′ (ˆ ˆ s , µ ) = pp ′ 2 ˆ sN pp ′ J C ∗ ⊗ W C i , i ′ ,ℓ,ℓ ′ R α � d ω J ( ℓ,ℓ ′ ) ( E − ω 2 ) W R α × ii ′ ( ω, µ ) R α Hard coefficients C ( ℓ, i ) encoding the short-distance structure of the production process pp ′ Process-independent soft function W R α ii ′ (expectation value of soft Wilson lines) { a } c ( i ′ ) ∗ dz 0 � { k } c ( i ) W R α ii ′ ( ω, µ ) = P R α e i ω z 0 / 2 � 0 | T [ S † n , ib 1 S † n , a 2 j S n , a 1 i S † v , k 1 a 3 S † n , jb 2 S v , b 4 , k 4 S v , b 3 , k 3 ]( z ) T [ S ¯ v , k 2 a 4 ]( 0 ) | 0 � { b } ¯ 4 π Potential function J ( ℓ,ℓ ′ ) encoding Coulomb interactions R α Contrary to the conventional approach there is a set of soft functions W R α ii ′ ! (corresponding to irreducible representations of R ⊗ R ′ = � α R α ) Pietro Falgari (IPPP Durham) May 2009, Madison 4 / 10

Resummation of threshold logarithms Factorisation-scale independence of the total cross section translates into evolution equation for the soft function W R α ij � ω � � d 1 � � d ln µ W R α ( ω, µ ) Γ R α W R α ( ω ′ , µ ) + W R α ( ω ′ , µ )Γ † = − d ω R α ω − ω ′ 0 [ µ ] − γ S R α W R α ( ω, µ ) − W R α ( ω, µ ) γ S † R α Γ R α controls resummation of double logs, γ S , R α resums single logs. W R α is matrix in colour space ⇒ in general mixing of different colour structures. With suitable choice of colour basis can be diagonalised to all orders in α s (at least for cases of phenomenological interest at Tevatron/LHC: t ¯ t , squarks, gluinos, etc...) Pietro Falgari (IPPP Durham) May 2009, Madison 5 / 10

ii ′ and J ( ℓ,ℓ ′ ) Resummed expressions for W R α R α Threshold logarithms are resummed by renormalising W R α at a soft scale µ s and evolving it to the common factorisation scale µ . W R α , res exp [ − 4 S R α ( µ s , µ ) + 2 a S , R α ( ω, µ ) = ( µ s , µ )] ii i i � ω � 2 η θ ( ω ) e − 2 γ E η s R α × ˜ ii ( ∂ η , µ s ) µ s ω Γ( 2 η ) µ s must be chosen such that the fixed-order perturbative expansions of W R α ii ( ω, µ s ) is well behaved. Pietro Falgari (IPPP Durham) May 2009, Madison 6 / 10

ii ′ and J ( ℓ,ℓ ′ ) Resummed expressions for W R α R α Threshold logarithms are resummed by renormalising W R α at a soft scale µ s and evolving it to the common factorisation scale µ . W R α , res exp [ − 4 S R α ( µ s , µ ) + 2 a S , R α ( ω, µ ) = ( µ s , µ )] ii i i � ω � 2 η θ ( ω ) e − 2 γ E η s R α × ˜ ii ( ∂ η , µ s ) µ s ω Γ( 2 η ) µ s must be chosen such that the fixed-order perturbative expansions of W R α ii ( ω, µ s ) is well behaved. Resummation of Coulomb corrections is well known from quarkonia physics. J ( ℓ,ℓ ′ ) related to zero-distance Green function of − � ∇ 2 / ( 2 m red ) − α s ( − C R α ) / r : R α �� − ( 2 m red ) 2 � 1 � � E − 8 m red E J ( ℓ,ℓ ′ ) ( E ) ∝ Im − 2 m red + α s ( − C R α ) 2 ln R α 4 π µ 2 � � � � − 1 α s ( − C R α ) 2 + γ E + ψ 1 − � − E / ( 2 m red ) 2 Pietro Falgari (IPPP Durham) May 2009, Madison 6 / 10

Squark-antisquark production at the LHC In the rest of this talk: q ¯ PP → ˜ q + X ˜ Perform NLL resummation of soft-gluon corrections + Coulomb singularities: Two-loop cusp anomalous dimension Γ and QCD β -function One-loop soft anomalous dimension γ S Tree-level fixed-order soft functions W R α ii ′ The effective-theory resummed cross section is matched onto the full NLO result [ Zerwas et al., ’96 ] σ match σ NLL σ NLL σ NLO � � ˆ (ˆ s , µ f ) = ˆ pp ′ (ˆ s , µ f ) − ˆ pp ′ (ˆ s , µ f ) | NLO + ˆ pp ′ (ˆ s , µ f ) pp ′ Full NLO result computed using fitted scaling functions provided by [ Langenfeld, Moch, ’09 ] Pietro Falgari (IPPP Durham) May 2009, Madison 7 / 10

q ¯ NLL corrections to ˜ ˜ q production [PRELIMINARY] K NLL − 1 = σ match σ NLO − 1 K NLL 0.10 r � 1.25 Use MSTW2008 PDFs µ set to m ˜ 0.08 q Soft � Coulomb � S � xC1 r ≡ m ˜ g / m ˜ q = 1 . 25 0.06 µ s = ¯ µ s , where ¯ µ s chosen such that one-loop soft corrections Soft � Coulomb 0.04 are minimised Coulomb 0.02 Soft m q � � TeV � 0.5 1.0 1.5 2.0 Pietro Falgari (IPPP Durham) May 2009, Madison 8 / 10

q ¯ NLL corrections to ˜ ˜ q production [PRELIMINARY] K NLL − 1 = σ match σ NLO − 1 K NLL 0.10 r � 1.25 Use MSTW2008 PDFs µ set to m ˜ 0.08 q Soft � Coulomb � S � xC1 r ≡ m ˜ g / m ˜ q = 1 . 25 0.06 µ s = ¯ µ s , where ¯ µ s chosen such that one-loop soft corrections Soft � Coulomb 0.04 are minimised Coulomb [Kulesza ’08, Talk given at IPPP 0.02 Durham ] Soft m q � � TeV � 0.5 1.0 1.5 2.0 Pietro Falgari (IPPP Durham) May 2009, Madison 8 / 10

Factorisation-scale dependence Resummation of threshold logarithms sensibly reduces the factorisation-scale dependence of the cross section m ˜ q = 500 GeV q = 1 TeV m ˜ Σ � pb � Σ � pb � 35 0.7 LO LO NLO 30 NLO 0.6 NLL NLL 25 0.5 20 0.4 15 0.3 10 0.2 5 0.1 Μ Μ 0 ���������� � 0.0 ���������� � m q 0.1 0.2 0.5 1.0 2.0 5.0 m q 0.1 0.2 0.5 1.0 2.0 5.0 Green band obtained by varying ¯ µ s / 2 < µ s < 2 ¯ µ s Pietro Falgari (IPPP Durham) May 2009, Madison 9 / 10

Conclusions and Outlook Momentum-space resummation based on effective-theory framework works well and is in good agreement with analogous results in moment space For squark-antisquark production resummation effects beyond NLO amount to 3 − 10 % in the range 0 . 2 − 2 TeV Even for small squark masses resummation dramatically improves factorisation-scale dependence of the cross section Formalism can be applied to arbitrary final states (squark-squark, squark-gluino, gluino-gluino, etc...) Pietro Falgari (IPPP Durham) May 2009, Madison 10 / 10