Analytic resummation for TMD observables Varun Vaidya 1 1 Los Alamos National Lab In collaboration with D. Kang and C. Lee DPF 2017 Varun Vaidya Analytic resummation for TMD observables DPF 2017 1 / 21
Factorization in SCET P+P → H+X, P+P → l + + l − +X. Motivation Resum large logs of qT/Q by setting renormalization scales in momentum space Obtain, for the first time, an analytic expression for resummed cross section. Varun Vaidya Analytic resummation for TMD observables DPF 2017 2 / 21
Factorization Transverse momentum cross section d σ T dy ∝ H ( µ � d 2 � q Ts d 2 � q T 1 d 2 � Q ) × q T 2 S ( � q Ts , µ, ν ) × dq 2 q T 2 , µ, ν, Q ) δ 2 ( � f ⊥ q T 1 , µ, ν, Q ) f ⊥ 1 ( x 1 , � 2 ( x 2 , � q T − � q Ts − � q T 1 − � q T 2 ) The function f ⊥ along with the soft function S forms the TMDPDF . i RG equations in two scales, µ, ν . RG equations in momentum space are convolutions of distribution functions and hard to solve directly. ν d q Ti , ν ) = γ i d ν G i ( � ν ( � q Ti ) ⊗ G i ( � q Ti , ν ) Varun Vaidya Analytic resummation for TMD observables DPF 2017 3 / 21
Factorization b space formulation d σ T dy ∝ H ( µ � bdbJ 0 ( bq T ) S ( b , µ, ν ) f ⊥ 1 ( x 1 , b , µ, ν, Q ) f ⊥ Q ) 2 ( x 2 , b , µ, ν, Q ) dq 2 RG equations in b space are simple µ d d µ F i ( µ, ν, b ) = γ i F i ∈ ( H , S , f ⊥ µ F i ( µ, ν, b ) , i ) ν d d ν G i ( µ, ν, b ) = γ i G i ∈ ( S , f ⊥ ν G i ( µ, ν, b ) , i ) � γ i � γ i µ = ν = 0 F i G i Varun Vaidya Analytic resummation for TMD observables DPF 2017 4 / 21
Resummation schemes Figure: Choice of resummation path. b space resummation: Default choice of µ = ν = 1 / b ,1007.2351 De Florian et.al., 1503:00005 V. Vaidya et. al Momentum space resummation:Both µ , ν in momentum space, distributional scale setting, 1611.08610 Tackmann et.al., 1604.02191 P. Monni et. al. Hybrid: µ in momentum space( 1007.4005, 1109.6027 Becher et.al.) Varun Vaidya Analytic resummation for TMD observables DPF 2017 5 / 21
Scale choice in momentum space Can we choose scales in momentum space for µ and ν ? Naively, we expect, µ H , ν H ∼ Q , µ L , ν L ∼ qT . Assume a power counting α s log( Q /µ L ) , log( µ L b 0 ) , α s log( Q /ν L ) , log( ν L b 0 ) ∼ 1 Attempt at NLL → running Soft function in ν a a b 0 = be − γ E / 2 d σ � ∝ U NLL ( H , µ L ) dbbJ 0 ( bq t ) U S ( ν H , ν L , µ L ) H dq 2 t � � ν H ν L d ln ν ( γ (0) S ) U NLL = ( H , µ L ) dbbJ 0 ( bq t ) e ν H � ν H � � α s − Γ 0 π log log( µ L b 0 ) U NLL = ( H , µ L ) dbbJ 0 ( bq t ) e ν L H Cross section is singular due to divergence at small b. Varun Vaidya Analytic resummation for TMD observables DPF 2017 6 / 21
Scale choice in momentum space Resum all logarithms of the form α s log 2 ( µ L b 0 ) A choice for ν in b space → include sub-leading terms µ n n = 1 � 1 − α ( µ L ) β 0 2 π log( ν H � L ν L = , ) b 1 − n 2 µ L 0 Soft exponent at NLL → Quadratic in log( µ L b 0 ) α ( µ L ) log( U NLL ( ν H , ν L , µ L )) = − 2Γ 0 × S 2 π � log( ν H ) log( µ L b 0 ) + 1 2 log 2 ( µ L b 0 ) + α ( µ L ) β 0 4 π log 2 ( µ L b 0 ) log( ν H � ) µ L µ L Varun Vaidya Analytic resummation for TMD observables DPF 2017 7 / 21
Scale choice in momentum space A choice for µ L in momentum space A choice that justifies the power counting log ( µ L b 0 ) ∼ 1 Scale shifted away from q T due to the scale Q in b space exponent. Varun Vaidya Analytic resummation for TMD observables DPF 2017 8 / 21
Analytical expression for cross section Mellin-Barnes representation of Bessel function Polynomial integral representation for Bessel function is needed � c + i ∞ � 2 t 1 dt Γ[ − t ] � 1 J 0 ( z ) = 2 z 2 π i Γ[1 + t ] c − i ∞ b space integral U S = C 1 Exp [ − A log 2 ( Ub )] � ∞ I b = dbbJ 0 ( bq T ) U S No Landau pole 0 � i ∞ � ∞ dt Γ[ − t ] dbb ( bq T 2 ) 2 t Exp [ − A log 2 ( Ub )] = C 1 Γ[1 + t ] 0 − i ∞ Varun Vaidya Analytic resummation for TMD observables DPF 2017 9 / 21
Analytical expression for cross section � c + i ∞ 2 C 1 1 dt Γ[ − t ] Γ[1 + t ] Exp [ 1 A ( t − t 0 ) 2 ] I = √ iq 2 4 π A c − i ∞ T = − 1 + A log(2 U / q T ) → saddle point t 0 Path of steepest descent is parallel to the imaginary axis α s 1 / Γ (0) Suppression controlled by 1/A ∼ 4 π cusp π t = c + ix , Γ( z )Γ(1 − z ) = sin( π z ) . , � ∞ I = 2 C 1 1 dx Γ[ − c − ix ] 2 sin[ π ( c + ix )] Exp [ − 1 A ( x − i ( c − t 0 )) 2 ] √ q 2 4 π A −∞ T Varun Vaidya Analytic resummation for TMD observables DPF 2017 10 / 21
Analytical expression for cross section An expansion for Γ[1 − ix ] 2 in weighted Hermite polynomials � ∞ ∞ � c 2 n H 2 n ( α x ) e − a 0 x 2 + i γ E Γ(1 − ix ) 2 = d 2 n +1 H 2 n +1 ( β x ) e − b 0 x 2 � � β n =0 n =0 Figure: Expansion in Hermite polynomials Varun Vaidya Analytic resummation for TMD observables DPF 2017 11 / 21
Analytical expression for cross section Expression for resummed Soft function � � 2 C � ∞ c 2 n H 2 n ( α, a 0 ) + i γ E = n =0 Im β d 2 n +1 H 2 n +1 ( β, b 0 ) I b π q 2 T H n to all orders H n ( α, a 0 ) = H 0 ( α, a 0 ) ( − 1) n n ! (1 + a 0 A ) n Floor [ n / 2] 1 1 � m � [ A ( α 2 − a 0 ) − 1](1 + a 0 A ) � (2 α z 0 ) n − 2 m × m ! ( n − 2 m )! m =0 − A ( L − i π/ 2)2 1 with H 0 ( α, a 0 ) = e √ 1 + a 0 A , z 0 = iA ( L − i π/ 2) 1+ a 0 A Varun Vaidya Analytic resummation for TMD observables DPF 2017 12 / 21
Analytical expression for cross section Fixed order terms I b acts as a generating function for residual fixed order logs � bdbJ 0 ( bq T ) log 2 n ( Ub ) Exp [ − A log 2 ( Ub )] I even = C 1 ( − 1) n d n = dA n I b ( A , L ) � bdbJ 0 ( bq T ) log 2 n +1 ( Ub ) Exp [ − A log 2 ( Ub )] I odd = C 1 ( − 1) n d n ( − 1) d = dLI b ( A , L ) dA n 2 A Varun Vaidya Analytic resummation for TMD observables DPF 2017 13 / 21
Comparison with b space resummation Figure: comparison of nnll cross section in two schemes Difference of the order of sub-leading terms. More reliable perturbative error estimation in the absence of Landau pole. Varun Vaidya Analytic resummation for TMD observables DPF 2017 14 / 21
Matching to fixed order Implement profiles in µ and ν to turn off resummation S = S (1 − z ( q T )) Q z ( q T ) S ∈ µ, ν L Soft exponent scales as (1 − z ( q T )) � Q � U S = Exp [(1 − z ) γ ν S log ] ν L This is equivalent to A → A(1-z) in I b (A,L) Varun Vaidya Analytic resummation for TMD observables DPF 2017 15 / 21
Summary Implementation momentum space resummation for transverse spectra of gauge bosons Rapidity choice in impact parameter space Virtuality choice in momentum space. Analytical expression for cross section across the entire range of q T obtained for the first time. Numerical accuracy controlled by the accuracy of the expansion for Γ[ − t ] process independent function Γ[1+ t ] Outlook Promising approach for other observables with similar factorization structure. Non-perturbative effects need to be included for low Q as well as the low q T regime. Varun Vaidya Analytic resummation for TMD observables DPF 2017 16 / 21
Backup Varun Vaidya Analytic resummation for TMD observables DPF 2017 17 / 21
Analytical expression for cross section What choice do we make for c? Obvious choice c = t 0 ? c depends on A and hence on the details of the process. For percent level accuracy, we need info about F ( x ) = Γ[ − c − ix ] Γ[1+ c + ix ] out � to x l ∼ 2 A log(10) Worst case scenario A ∼ 0.5 = ⇒ x l ∼ 1 . 5 A Taylor series expansion around the saddle point is not enough. Choose c = -1, the saddle point in the limit A → 0 for all observables and use a more suitable basis for expanding F(x) Varun Vaidya Analytic resummation for TMD observables DPF 2017 18 / 21
Analytical expression for cross section Guidelines for choosing a basis for expansion Fixed order cross section � µ L e − γ E + F ′′ [0] α ( µ L ) 2 � � � I O ( α s ) − 2Γ (0) F ′ [0] log = cusp exact q 2 4 π q T 4 T To correctly reproduce the fixed order cross section upto α n s , we need 2 n th derivative of the expansion to match the exact function F(x) We need the expansion in a basis to be accurate upto x ∼ 1 . 5 The basis functions for the expansion should be chosen so as to yield a rapidly converging and analytical result. Varun Vaidya Analytic resummation for TMD observables DPF 2017 19 / 21
(A more accurate) Analytical expression for cross section An expansion for F(x)=Γ[ − 1 − ix ] / Γ[ ix ] A general basis x n e α x 2 + β x for expansion ˆ F R ( x ) = g 1 ( Exp [ − g 2 x 2 ] − cos[ g 3 x ]) + g 4 x 2 Exp [ − g 5 x 2 ] F I ( x ) = f 1 sin[ f 2 x 2 ] + f 3 sinh( f 4 x ) ˆ Figure: (Expansion for real and imaginary parts of f(t), c is chosen to be -1 Varun Vaidya Analytic resummation for TMD observables DPF 2017 20 / 21
Numerical results Easily extended to NNLL, b space exponent kept quadratic in log( µ b ) Figure: Resummation in momentum space. Excellent convergence for both the Higgs and Drell-Yan spectrum No arbitrary b space cut-off while estimating perturbative errors. Varun Vaidya Analytic resummation for TMD observables DPF 2017 21 / 21
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