} m 6 = 0 F 1 , F 2 in large N c limit in SU ( N c ) at 4 - loop F 1 - PowerPoint PPT Presentation

H IGH E NERGY B EHA VIOUR OF F ORM F ACTORS Taushif Ahmed Johannes Gutenberg University Mainz Germany Skype Seminar IIT Hyderabad May 10, 2018 With Johannes Henn & Matthias Steinhauser Ref: JHEP 1706 (2017) 125

G OAL & M OTIV ATION • Infrared divergences: important quantities • Consider: QCD corrections to photon - quark vertex q 2 γ ∗ V µ ( q 1 , q 2 ) = ¯ v ( q 2 ) Γ µ ( q 1 , q 2 ) u ( q 1 ) q q 1 • V ertex function: characterised by two scalar form factors F 1 , F 2  � i F 1 ( q 2 ) γ µ − 2 mF 2 ( q 2 ) σ µ ν q ν Γ µ ( q 1 , q 2 ) = Q q • Consider: Form factors of massive quarks • Important quantities: is building block for variety of observables F 1 e − e + e.g. Xsection of hadron production in annihilation & derived quantities like forward - backward asymmetry • Also consider: the massless scenario F 1 2

G OAL & M OTIV ATION • State - of - the - art results at 3 - loop [Henn, Smirnov, Smirnov, Steinhauser ’ 16] } m 6 = 0 F 1 , F 2 in large N c limit in SU ( N c ) at 4 - loop F 1 m = 0 [Henn, Smirnov, Smirnov, Steinhauser, Lee ’ 16] [Manteuffel, Schabinger ’ 16] • Next steps: compute the full results for general N c underway by several groups • W e address: What can we say about next order? indeed, IR poles can be predicted ( partially ) by exploiting RG evolution of FF 1 / ✏ 2 F 1 at 4 - loop in large N c and high energy limit upto m 6 = 0 RESULTS 1 / ✏ 3 m = 0 F 1 at 5 - loop in large N c and high energy limit upto • W e also obtain process independent functions relating massive & massless amplitudes in high - energy limit at 3 & 4 - loops RESULTS GOAL Exploit RG evolution of FF 3

P LAN O F T HE T ALK RG evolution: massive • Cute technique to solve RG evolution: massless Process independent functions Conclusions

RG E QUATION : M ASSIVE [Sudakov ’ 56; Mueller ’ 79; Collins ’ 80; Sen ’ 81] • FF satisfies KG eqn in dimensional reg. [Magnea, Sterman ’ 90] [Gluza, Mitov, Moch, Riemann ’ 07, ’ 09] a s , Q 2 µ 2 , m 2 a s , m 2 , µ 2 a s , Q 2 , µ 2 ✓ ◆  ✓ ◆ ✓ ◆� d = 1 d ln µ 2 ln ˜ ˜ + ˜ R R F µ 2 , ✏ K µ 2 , ✏ G µ 2 , ✏ ˆ ˆ ˆ − µ 2 µ 2 2 R R QCD factorisation, gauge & RG invariance • The form factor Q 2 = − q 2 = − ( p 1 + p 2 ) 2 ˆ F = Ce ln ˜ a s F d = 4 − 2 ✏ a s ≡ ˆ α s / 4 π ˆ ˆ : scale to keep dimensionless µ a s Matching coe ffi cient µ R : renormalisation scale • Goal: Solve the RG • Strategy: Use bare coupling instead of renormalised one ˆ a s a s [Ravindran ’ 06: For Massless] 5

S OL VING RG E QUATION : M ASSIVE RG invariance of FF wrt µ R a s , m 2 , µ 2 a s , Q 2 , µ 2 ✓ ◆ ✓ ◆ d d ˜ ˜ R R µ 2 � � �� K µ 2 , ✏ G µ 2 , ✏ = − A a s ˆ = − ˆ R d ln µ 2 µ 2 d ln µ 2 µ 2 R R R R Cusp anomalous dimension µ 2 R a s , m 2 , µ 2 dµ 2 ✓ ◆ Z ˜ R m 2 � R µ 2 � � � � � �� K µ 2 , ✏ = K a s , ✏ A a s ˆ − R µ 2 µ 2 R R m 2 µ 2 R a s , Q 2 , µ 2 dµ 2 ✓ ◆ Z ˜ R R Q 2 � µ 2 � � � � � �� G µ 2 , ✏ = G a s , ✏ A a s ˆ + R µ 2 µ 2 R R Q 2 Boundary terms 6

S OL VING RG E QUATION : M ASSIVE ln ˜ Initial goal: Solve for in powers of bare F ˆ a s Need all quantities in powers of ˆ a s Expand B ∈ { K, G, A } ∞ X a k λ 2 �� λ 2 � � � � λ ∈ { m, Q, µ R } B B k a s ≡ s k =1 Renormalisation constant � ✓ µ 2 ◆ − ✏ a s = a s ( µ 2 µ 2 � ˆ R ) Z a s R µ 2 R Use ◆ − k ✏ ∞ ✓ λ 2 ˆ X Z − 1 a s ( λ 2 ) = 1 + Z − 1 , ( k ) a k ˆ s a s µ 2 k =1 functions of � i , ✏ ˆ Expansion of in powers of B a s 7

S OL VING RG E QUATION : M ASSIVE Soln of in powers of B ˆ a s ◆ − k ✏ ∞ ✓ λ 2 X λ 2 �� a k ˆ � � B B k = ˆ a s s µ 2 k =1 ˆ B 1 = B 1 , B 2 = B 2 + B 1 ˆ ˆ Z − 1 , (1) , a s with Z − 1 , (1) Z − 1 , (2) B 3 = B 3 + 2 B 2 ˆ ˆ + B 1 ˆ , a s a s ⌘ 2 n ⇣ o B 4 = B 4 + 3 B 3 ˆ ˆ Z − 1 , (1) Z − 1 , (1) ˆ + 2 ˆ Z − 1 , (2) + B 1 ˆ Z − 1 , (3) + B 2 a s a s a s a s and so on… The integral becomes a polynomial integral trivial µ 2 ! − k ✏ ! − k ✏ # R " ∞ dµ 2 � 2 µ 2 Z 1 X a k ˆ R µ 2 R � � �� A a s A k = ˆ − R s µ 2 µ 2 µ 2 k ✏ R k =1 � 2 8

U N - RENORMALISED S OLUTION : M ASSIVE Solution of KG in powers of bare ˆ a s ◆ − k ✏ ˆ ◆ − k ✏ ˆ "✓ Q 2 # ∞ a s , Q 2 µ 2 , m 2 ✓ m 2 ✓ ◆ Q m ln ˜ X ˜ ˜ a k ˆ = ˆ k ( ✏ ) + k ( ✏ ) F µ 2 , ✏ L L s µ 2 µ 2 k =1 k ( ✏ ) = − 1 Q  G k + 1 � ˆ Renormalised Solution ˜ ˆ ˆ L A k , 2 k ✏ k ✏ � ✓ µ 2 with ◆ − ✏ m k ( ✏ ) = − 1  K k − 1 � ˆ a s = a s ( µ 2 µ 2 ˜ ˆ ˆ � ˆ R ) Z a s L A k R µ 2 2 k ✏ k ✏ R ∞ h i X s ( Q 2 ) ˜ s ( m 2 ) ˜ L Q a k k + a k L m = k k =1 a s ( µ 2 To obtain the renormalised solution in powers of general R ) a s ( µ 2 R ) use d - dimensional evolution of ∞ d X � k a k +2 µ 2 µ 2 µ 2 � � � � � � = − ✏ a s a s Solved iteratively − R R s R d ln µ 2 R k =0 9

R ENORMALISED S OLUTION : M ASSIVE Renormalised Solution ∞ ln ˜ R ) ˜ X a k s ( µ 2 F = L k k =1 µ 2 R = m 2 at one loop For ( !) ! ( !) L 2 L 1 = 1 − 1 + L G 1 − A 1 L G 1 − A 1 L ˜ G 1 + K 1 − A 1 L − ✏ 2 2 2 4 3 ✏ ( !) ( !) ( !) L 3 L 4 L 5 G 1 − A 1 L G 1 − A 1 L G 1 − A 1 L + ✏ 2 − ✏ 3 + ✏ 4 + O ( ✏ 5 ) 12 4 48 5 240 6 At two loop ( !) ( !) ! L 2 = 1 − 1 1 � 0 + L G 2 − A 2 L ˜ G 1 + K 1 − A 1 L G 2 + K 2 − A 2 L ✏ 2 4 4 2 2 ✏ ! ( ! !) − � 0 L 2 L 2 − � 0 L 3 G 1 − A 1 L G 2 − A 2 L G 1 − A 1 L − ✏ 4 3 2 3 4 4 ( ! !) ( ! L 3 − 7 � 0 L 4 L 4 G 2 − A 2 L G 1 − A 1 L G 2 − A 2 L + ✏ 2 − ✏ 3 3 4 48 5 6 5 !) − � 0 L 5 G 1 − A 1 L + O ( ✏ 4 ) and so on… 16 6 L = log( Q 2 /m 2 ) 10

N EW R ESULTS : M ASSIVE • Conformal theory : all order result β i = 0 ! ∞ ( − ✏ k ) l − 1 L l G k + � 0 l K k − A k L ˜ X L k = 2 l ! l + 1 l =0 • Form Factor e ln ˜ m 2 � F � � � consistent with literature up to 3 - loop F = C a s , ✏ [Gluza, Mitov, Moch, Riemann ’ 07, ’ 09] • State - of - the - art results at 3 - loop in large N c F 1 , F 2 [Henn, Smirnov, Smirnov, Steinhauser ’ 16] • New results in 1704.07846 F 1 at 4 - loop in large N c and high energy limit 1 upto ✏ 2 m 2 /q 2 F 2 is suppressed by in high energy limit 11

to new! new! , new! D ETERMINING U NKNOWN C ONSTANTS : M ASSIVE N c limit Determining unknown constants G, K, C in large Comparing with explicit computations O ( ✏ 2 ) to to G 2 G 1 O ( ✏ ) [Gluza, Mitov, Moch, Riemann ’ 07 ’ 09] O ( ✏ 0 ) G 3 F 1 at 3-loop [Henn, Smirnov, Smirnov, Steinhauser ’ 16] K 1 , K 2 [Gluza, Mitov, Moch, Riemann ’ 09] K 3 , C 2 to O ( ✏ ) [Gluza, Mitov, Moch, Riemann ’ 09] C 1 to O ( ✏ 2 ) , C 3 to O ( ✏ 0 ) C 1 to O ( ✏ 4 ) , C 2 to O ( ✏ 2 ) explicit computation A 4 became available recently [Henn, Smirnov, Smirnov, Steinhauser ’ 16] [Henn, Smirnov, Smirnov, Steinhauser, Lee ’ 16] 12

C OMMENTS : M ASSIVE • Excludes singlet contributions Obey similar • Excludes closed heavy - quark loops exponentiation [K ü hn, Moch, Penin, Smirnov ’ 01] [Feucht, K ü hn, Moch ’ 03] N c Sub - leading in large limit Hence, we have not considerer these 13

M ASSLESS S CENARIO

new! RG E QUATION : M ASSLESS • FF satisfies KG eqn a s , Q 2 µ 2 , m 2 a s , m 2 , µ 2 a s , Q 2 , µ 2 ✓ ◆  ✓ ◆ ✓ ◆� d = 1 d ln µ 2 ln ˜ ˜ + ˜ R R F µ 2 , ✏ K µ 2 , ✏ G µ 2 , ✏ ˆ ˆ ˆ − µ 2 µ 2 2 R R [Sudakov ’ 56; Mueller ’ 79; Collins ’ 80; Sen ’ 81] Solved exactly the similar way [Ravindran ’ 06] ◆ − k ✏ ˆ ◆ − k ✏ ˆ "✓ Q 2 # ∞ a s , Q 2 µ 2 , m 2 ✓ m 2 ✓ ◆ Q m ln ˜ X ˜ ˜ a k ˆ = ˆ k ( ✏ ) + k ( ✏ ) F µ 2 , ✏ L L s µ 2 µ 2 k =1 Up to 4 - loop: present [Moch, Vermaseren, Vogt ’ 05] [Ravindran ’ 06] 5 - loop solution 15

RG E QUATION : M ASSLESS • Conformal theory : all order result β i = 0 ( ) ( ) Q k = 1 1 + 1 − 1 ˆ ˜ 2 k 2 A k 2 k G k L − ✏ 2 ✏ [Bern, Dixon, Smirnov ’ 05] • FF [TA, Banerjee, Dhani, Rana, Ravindran, Seth ’ 17] F = Ce ln ˜ F Matching coe ffi cient = 1 • State - of - the - art results at 4 - loop in large N c F [Henn, Smirnov, Smirnov, Steinhauser, Lee ’ 16] • New results in 1704.07846 at 5 - loop in large N c and high energy limit F 1 upto ✏ 3 16

new! at 4-loop to D ETERMINING U NKNOWN C ONSTANTS : M ASSLESS N c limit Determining unknown constants in large Comparing with explicit computations G 1 to O ( ✏ 6 ) , G 2 to O ( ✏ 4 ) , G 3 to O ( ✏ 2 ) [Baikov, Chetyrkin, Smirnov, Smirnov, Steinhauser ’ 09] [Gehrmann, Glover, Huber, Ikizlerli, Studerus ’ 10] O ( ✏ 0 ) G 4 F [Henn, Smirnov, Smirnov, Steinhauser, Lee ’ 16] K i = K i ( A k , β k ) do not appear in the final expressions get cancelled against similar terms arising from G