Transverse momentum distributions and the determination of the W mass Andrea Signori Loop Fest XVIII Fermilab August 13 th , 2019 1
TMDs 2
TMD PDFs extraction of a parton whose momentum has longitudinal and transverse components with respect to the parent hadron momentum richer structure than collinear PDFs probe hadron momentum courtesy A. Bacchetta 3
Motivations Nucleon tomography in momentum space : to understand how hadrons are built in terms of the elementary degrees of freedom of QCD High-energy phenomenology : to improve our understanding of high-energy scattering experiments and their potential to explore BSM physics 4
Quark TMD PDFs Φ ij ( k, P ; S, T ) ⇠ F.T. h PST | ¯ ψ j (0) U [0 , ξ ] ψ i ( ξ ) | PST i | LF U L T i σ i + γ 5 γ + γ + γ 5 Quarks k T P xP h ⊥ U f 1 1 extraction of a quark not collinear with the proton h ⊥ L g 1 1 L f ⊥ h 1 , h ⊥ T g 1 T 1 T 1 T encode all the possible h ⊥ LL f 1 LL spin-spin and spin-momentum 1 LL correlations g 1 LT h 1 LT , h ⊥ LT f 1 LT Sivers TMD PDF between the proton 1 LT and its constituents unpolarized TMD PDF g 1 T T h 1 T T , h ⊥ TT f 1 T T 1 T T similar table for gluons and for fragmentation functions bold : also collinear red : time-reversal odd (universality properties) 5
TMD factorization at work Scimemi, Vladimirov [Eur.Phys.J. C78 2018 89] + Scimemi, Vladimirov, Bertone (1902.08474) Schematically : d σ ∼ H f 1 ( x a , k T a , Q ) f 1 ( x b , k T b , Q ) δ (2) ( q T − k T a − k T b ) + O ( q T /Q ) + O ( m/Q ) dq T Low transverse momentum (TMD) region q T ⌧ Q 6
TMD factorization at work Scimemi, Vladimirov [Eur.Phys.J. C78 2018 89] + Scimemi, Vladimirov, Bertone (1902.08474) Schematically : d σ ∼ H f 1 ( x a , k T a , Q ) f 1 ( x b , k T b , Q ) δ (2) ( q T − k T a − k T b ) + O ( q T /Q ) + O ( m/Q ) dq T Low transverse momentum (TMD) region q T ⌧ Q Matching to fixed-order calculations in coll. factorization 7
TMD factorization at work �� Bacchetta, Delcarro, Pisano, Radici, AS (1703.10157) : 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � 〉 = ����� 〈 � 〉 = ����� unpolarized TMD fit including SIDIS , Drell-Yan fixed-target, Z production � ����������������� � � SIDIS @ Compass f a 1 ( x a , k 2 aT , Q 2 ) ⊗ f b 1 ( x b , k 2 bT , Q 2 ) � pp : �� 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� � 〈 � 〉 = ���� ( ������ = � ) ����������������� 〈 � 〉 = ���� ( ������ = � ) 〈 � 〉 = ���� ( ������ = � ) � 〈 � 〉 = ���� ( ������ = � ) 〈 � 〉 = ���� ( ������ = � ) 〈 � 〉 = ���� ( ������ = � ) � 〈 � 〉 = ���� ( ������ = � ) f a 1 ( x a , k 2 aT , Q 2 ) ⊗ D a → h ( z a , P 2 T , Q 2 ) ep : 1 � �� ��� ��� ��� 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � � �� [ ��� ] 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� � ����������������� � � � �� 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � 〉 = ����� 〈 � 〉 = ���� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� � ����������������� � � � ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� 8 � �� [ ��� ] � �� [ ��� ] � �� [ ��� ] � �� [ ��� ] � �� [ ��� ] � �� [ ��� ]
TMD factorization at work �� Bacchetta, Delcarro, Pisano, Radici, AS (1703.10157) : 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � 〉 = ����� 〈 � 〉 = ����� unpolarized TMD fit including SIDIS , Drell-Yan fixed-target, Z production � ����������������� � � SIDIS @ Compass f a 1 ( x a , k 2 aT , Q 2 ) ⊗ f b 1 ( x b , k 2 bT , Q 2 ) � pp : �� 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� � 〈 � 〉 = ���� ( ������ = � ) ����������������� 〈 � 〉 = ���� ( ������ = � ) 〈 � 〉 = ���� ( ������ = � ) � 〈 � 〉 = ���� ( ������ = � ) 〈 � 〉 = ���� ( ������ = � ) 〈 � 〉 = ���� ( ������ = � ) � 〈 � 〉 = ���� ( ������ = � ) f a 1 ( x a , k 2 aT , Q 2 ) ⊗ D a → h ( z a , P 2 T , Q 2 ) ep : 1 � �� ��� ��� ��� 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � � �� [ ��� ] 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� � ����������������� � e + e − : D a → h 1 1 T , Q 2 ) ⊗ D b → h 2 ( z 1 , P 2 ( z 2 , P 2 2 T , Q 2 ) 1 1 � � �� 〈 � � 〉 = ��� ��� � 〈 � � 〉 = ��� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � � 〉 = �� ��� � 〈 � 〉 = ����� 〈 � 〉 = ���� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� 〈 � 〉 = ����� � Data not available yet! ����������������� � Needed for independent analyses of TMD FFs � � ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� 9 � �� [ ��� ] � �� [ ��� ] � �� [ ��� ] � �� [ ��� ] � �� [ ��� ] � �� [ ��� ]
Structure of a TMD PDF 1 ( x, b 2 1 ( x, b 2 f a T , µ f , ζ f ) = f a T , µ i , ζ i ) b T , Fourier conjugate of k T ⇢ Z µ f �� dµ α s ( µ ) , ζ f evolution in mu two “evolution scales” × exp µ γ F µ 2 µ i → µ f µ i ◆ − K ( b T ,µ i ) ✓ ζ f evolution in zeta × ζ i → ζ f ζ i Input TMD distribution can be expanded at low b T on the collinear distributions f a X 1 ( x, b 2 C a/b ( x, b 2 T , µ i , ζ i ) = T , µ i , ζ i ) ⊗ f b ( x, µ i ) b ζ i = µ 2 i = 4 e − 2 γ E /b 2 T ≡ µ 2 A sensible choice is to set the b initial and final scale as: ζ f = µ 2 f = Q 2 10
Structure of a TMD PDF 1 ( x, b 2 1 ( x, b 2 f a T , µ f , ζ f ) = f a T , µ i , ζ i ) b T , Fourier conjugate of k T ⇢ Z µ f �� dµ α s ( µ ) , ζ f evolution in mu two “evolution scales” × exp µ γ F µ 2 µ i → µ f µ i − g K ( b T , { λ } ) ◆ − K ( b T ,µ i ) ✓ ζ f evolution in zeta × ζ i → ζ f ζ i need corrections at large bT Input TMD distribution can be expanded at low b T on the collinear distributions f a X 1 ( x, b 2 C a/b ( x, b 2 F a T , µ i , ζ i ) = T , µ i , ζ i ) ⊗ f b ( x, µ i ) NP ( x, b T ; { λ } ) b ζ i = µ 2 i = 4 e − 2 γ E /b 2 T ≡ µ 2 A sensible choice is to set the b initial and final scale as: ζ f = µ 2 f = Q 2 11
Structure of a TMD PDF 1 ( x, b 2 1 ( x, b 2 f a T , µ f , ζ f ) = f a T , µ i , ζ i ) b T , Fourier conjugate of k T ⇢ Z µ f �� dµ α s ( µ ) , ζ f evolution in mu two “evolution scales” × exp µ γ F µ 2 µ i → µ f µ i − g K ( b T , { λ } ) ◆ − K ( b T ,µ i ) ✓ ζ f evolution in zeta × ζ i → ζ f ζ i Non-perturbative structures Input TMD distribution can be expanded at low b T on the collinear distributions f a X 1 ( x, b 2 C a/b ( x, b 2 F a f b ( x, µ i ) T , µ i , ζ i ) = T , µ i , ζ i ) ⊗ f b ( x, µ i ) NP ( x, b T ; { λ } ) b ζ i = µ 2 i = 4 e − 2 γ E /b 2 T ≡ µ 2 A sensible choice is to set the b initial and final scale as: ζ f = µ 2 f = Q 2 12
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