HIGH ENERGY BEHAVIOUR OF FORM FACTORS

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1 HIGH ENERGY BEHAVIOUR OF FORM FACTORS Taushif Ahmed Johannes Gutenberg University Mainz Germany Skype Seminar IIT Hyderabad May 10, 018 With Johannes Henn & Matthias Steinhauser Ref: JHEP 1706 (017) 15

2 GOAL & MOTIVATION Infrared divergences: important quantities Consider: QCD corrections to photon-quark vertex q q 1 q V µ (q 1,q )= v(q ) µ (q 1,q )u(q 1 ) Vertex function: characterised by two scalar form factors F 1,F Important quantities: Also consider: the massless scenario µ (q 1,q )=Q q applef 1 (q ) µ i m F (q ) µ q Consider: Form factors of massive quarks F 1 e e + annihilation & derived quantities like forward- e.g. Xsection of hadron production in backward asymmetry is building block for variety of observables F 1

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

4 PLAN OF THE TALK RG evolution: massive Cute technique to solve RG evolution: massless Process independent functions Conclusions

5 RG EQUATION: MASSIVE FF satisfies KG eqn in dimensional reg. [Sudakov 56; Mueller 79; Collins 80; Sen 81] [Magnea, Sterman 90] [Gluza, Mitov, Moch, Riemann 07, 09] d ln F d ln µ â s, Q µ, m µ, = 1 apple K â s, m, µ R µ, + G â s, Q, µ R µ, QCD factorisation, gauge & RG invariance The form factor F = Ce ln F Matching coefficient Q = q = (p 1 + p ) d =4 â s ˆ s /4 â s â s µ : scale to keep dimensionless µ R : renormalisation scale Goal: Solve the RG Strategy: Use bare coupling â s instead of renormalised one a s [Ravindran 06: For Massless] 5

6 SOLVING RG EQUATION: MASSIVE RG invariance of FF wrt µ R d d ln K â s, m, µ R µ, = d d ln G â s, Q, µ R µ, = A a s Cusp anomalous dimension K â s, m, µ R µ, = K a s m, Z m d A a s G â s, Q, µ R µ, = G a s Q, + Z Q d A a s Boundary terms 6

7 SOLVING RG EQUATION: MASSIVE Initial goal: Solve for ln F in powers of bare â s Need all quantities in powers of â s Expand B a s 1X a k s B k B {K, G, A} {m, Q, µ R } k=1 Renormalisation constant µ Use â s = a s ()Z as Z 1 a s ( )=1+ 1X k=1 â k s µ k Ẑ 1,(k) a s functions of i, Expansion of B in powers of â s 7

8 SOLVING RG EQUATION: MASSIVE Soln of with B in powers of ˆB 1 = B 1, â s B a s The integral becomes a polynomial integral = 1X k=1 â k s ˆB = B + B 1 Ẑ 1,(1) a s, ˆB 3 = B 3 +B Ẑ 1,(1) a s + B 1 Ẑ 1,() a s, µ k ˆBk ˆB 4 = B 4 +3B 3 Ẑ 1,(1) a s + B n Ẑ 1,(1) a s + Ẑ 1,() a s o trivial + B 1 Ẑ 1,(3) a s and so on Z d A a s = 1X k=1 â k s " 1 k µ! k µ! k # Â k 8

9 UN-RENORMALISED SOLUTION: MASSIVE Solution of KG in powers of bare ln F â s, Q µ, m µ, = 1X k=1 â k s â s " Q µ k ˆ LQ k ( )+ m µ k ˆ Lm k ( ) # Renormalised Solution â s = a s ()Z as µ with ˆ L Q k ( ) = ˆ L m k ( ) = 1 k 1 k apple Ĝ k + 1, apple k Âk 1 ˆK k k Âk = 1X h a k s(q ) L Q k + ak s(m ) L m k i k=1 To obtain the renormalised solution in powers of general a s () use d-dimensional evolution of d d ln a s = a s 1X k=0 ka k+ s a s () Solved iteratively 9

10 RENORMALISED SOLUTION: MASSIVE Renormalised Solution For = m L 1 = 1 ( + ( L 3!) 1 G 1 + K 1 A 1 L + L!) 1 G 1 at one loop A 1 L 4 ln F = G 1 3 ( L 4 48 G 1 1X k=1 a k s() L k! ( A 1 L L 4 G 1!) ( A 1 L + 4 L G 1!) A 1 L 3 A 1 L 6!) + O( 5 ) At two loop L = 1 ( 0L 4 ( + 0L L 3 3 G 1 G 1 G 1 + K 1 G A 1 L!)! ( A 1 L L 3! A L 7 0 L !) A 1 L + O( 4 ) 6 (!) G + K A L + L! A L 0L 3 A 1 L G G G 1 A 1 L 5 10!) 3 ( L 4 6 G and so on G!) A L 5! A L! L = log(q /m )

11 NEW RESULTS: MASSIVE Conformal theory i =0: all order result L k = 1X l=0 ( k) l 1 Ll l! G k + 0l K k! A k L l +1 Form Factor F = C a s m ln F, e State-of-the-art results consistent with literature up to 3-loop [Gluza, Mitov, Moch, Riemann 07, 09] F 1,F at 3-loop in large N c [Henn, Smirnov, Smirnov, Steinhauser 16] New results in F 1 at 4-loop in largen c and high energy limit upto F is suppressed by m /q 1 in high energy limit 11

12 DETERMINING UNKNOWN CONSTANTS: MASSIVE Determining unknown constants G, K, C in large N c limit Comparing with explicit computations G 1 to O( ), G to O( ) [Gluza, Mitov, Moch, Riemann 07 09] K 1,K G 3 to O( 0 ) new! F 1 at 3-loop [Henn, Smirnov, Smirnov, Steinhauser 16] [Gluza, Mitov, Moch, Riemann 09] K 3 new! C 1 to O( ), [Gluza, Mitov, Moch, Riemann 09] C to O( ) C 1 to O( 4 ), C to O( ), C O( 0 3 to ) new! explicit computation A 4 became available recently [Henn, Smirnov, Smirnov, Steinhauser 16] [Henn, Smirnov, Smirnov, Steinhauser, Lee 16] 1

13 COMMENTS: MASSIVE Excludes singlet contributions Excludes closed heavy-quark loops Obey similar exponentiation [Kühn, Moch, Penin, Smirnov 01] [Feucht, Kühn, Moch 03] Sub-leading in large N c limit Hence, we have not considerer these 13

14 MASSLESS SCENARIO

15 RG EQUATION: MASSLESS FF satisfies KG eqn d ln F d ln µ â s, Q µ, m µ, = 1 apple K â s, m, µ R µ, + G â s, Q, µ R µ, [Sudakov 56; Mueller 79; Collins 80; Sen 81] Solved exactly the similar way [Ravindran 06] ln F â s, Q µ, m µ, = 1X k=1 â k s " Q µ k ˆ LQ k ( )+ m µ k ˆ Lm k ( ) # Up to 4-loop: present [Moch, Vermaseren, Vogt 05] [Ravindran 06] 5-loop solution new! 15

16 RG EQUATION: MASSLESS Conformal theory i =0: all order result ( ) ( ˆ L Q k = 1 1 k A k + 1 FF 1 k G k ) [Bern, Dixon, Smirnov 05] [TA, Banerjee, Dhani, Rana, Ravindran, Seth 17] F = Ce ln F State-of-the-art results Matching coefficient = 1 F at 4-loop in large N c [Henn, Smirnov, Smirnov, Steinhauser, Lee 16] New results in F at 5-loop in large N c and high energy limit upto

17 DETERMINING UNKNOWN CONSTANTS: MASSLESS Determining unknown constants in large N c limit Comparing with explicit computations G 1 to O( 6 ), G O( 4 ) O( to, G 3 to ) G 4 to O( 0 ) new! F [Baikov, Chetyrkin, Smirnov, Smirnov, Steinhauser 09] [Gehrmann, Glover, Huber, Ikizlerli, Studerus 10] at 4-loop [Henn, Smirnov, Smirnov, Steinhauser, Lee 16] K i = K i (A k, k) do not appear in the final expressions get cancelled against similar terms arising from G

18 COMMENTS: MASSIVE & MASSLESS G are same for massive and massless [Mitov, Moch 07] expected! Governed by universal cusp AD Manifestly clear in our methodology G â s, Q, µ R µ, = G a s Q, + Z Q d A a s For massive K i enter only into the poles of Lk Constants and O( k ) terms can be determined from massless calculation could lead to deeper understanding of the connection between massive & massless FF

19 PROCESS INDEPENDENT FUNCTION QCD factorisation: massive amplitudes shares essential properties with the corresponding massless ones in the high-energy limit M (m) = Y i{all legs} Massive apple Z (m 0) [i] m 1/ M (0) µ Massless [Moch, Mitov 07] Universal and depends only on the external partons! Can be computed using simplest amplitudes: FF Z (m 0) [q] = F (Q,m,µ ) F (Q,µ ) Q independence is manifestly clear: governed by G, same for massive & massless FF O( 0 ) at 3-loop, upto O(1/ ) at 4-loop new! Relates dimensionally regularised amplitudes to those where the IR divergence is regularised with a small quark mass. 19

20 CONCLUSIONS RG equations governing massive & massless quark-photon FF are discussed. Elegant derivation for analytic solution is proposed key idea: use bare coupling Q dependence is governed by G & cusp AD: same for massive & massless Massive: non-trivial matching coefficient C 1 Massive: F 1 at 4-loop in large N c and high energy limit to Massless: F at 5-loop in large N c and high energy limit THANK YOU! to 1 3 0