PSEUDO SCALAR FORM FACTORS AT 3-LOOP QCD. Taushif Ahmed Institute of Mathematical Sciences, India March 22, 2016

Transcription

1 PSEUDO SCALAR FORM FACTORS AT 3-LOOP QCD Taushif Ahmed Institute of Mathematical Sciences, India March, 016

2 PROLOGUE: SM & MSSM SM Complex scalar doublet (4 DOF) 3 DOF transform into longitudinal modes of W ± & Z Neutral Higgs boson Yukawa coupling between Higgs field and fermions Masses to fermions

3 PROLOGUE: SM & MSSM SM Complex scalar doublet (4 DOF) 3 DOF transform into longitudinal modes of W ± & Z Neutral Higgs boson Yukawa coupling between Higgs field and fermions MSSM Masses to fermions Requires Higgs doublets 5 physical Higgs bosons h, H : CP even neutral charged h, H, A H ± A : CP odd 3

4 PROLOGUE: STATE OF THE ART 4 CP even Inclusive production cross section at CP odd N 3 LO QCD [Anastasiou, Duhr, Dulat, Furlan, Herzog, Mistlberger] Inclusive production cross section at NNLO QCD [Harlander, Kilgore; Anastasiou, Melnikov]

5 PROLOGUE: STATE OF THE ART 5 CP even Inclusive production cross section at CP odd N 3 LO QCD [Anastasiou, Duhr, Dulat, Furlan, Herzog, Mistlberger] Inclusive production cross section at NNLO QCD [Harlander, Kilgore; Anastasiou, Melnikov] What is next? Go beyond NNLO for CP odd! requires 1. Virtual correction at 3-loop. Real corrections at N 3 LO

6 PROLOGUE: STATE OF THE ART 6 CP even Inclusive production cross section at CP odd N 3 LO QCD [Anastasiou, Duhr, Dulat, Furlan, Herzog, Mistlberger] Inclusive production cross section at NNLO QCD [Harlander, Kilgore; Anastasiou, Melnikov] What is next? Go beyond NNLO for CP odd! requires Our GOAL 1. Virtual correction at 3-loop. Real corrections at N 3 LO

7 PLAN OF THE TALK Underlying Theory Defining Form Factors Calculation of FF Feynman Diagrams Prescription of 5 IBP & LI: Master Integrals Unrenormalised Results UV Renormalisation Coupling Const Renorm Operator Renorm Universal Structure of FF Determining Oper Renorm Axial Anomaly 7

8 UNDERLYING THEORY 8

9 LAGRANGIAN 9 Original Theory Pseudo scalar couples to quarks through Yukawa L A = i g c v A m t t 5 t + n l X i=1 m i i 5 i! g c = coupling constant, depends on specific theory v = G vev = F m t = top quark mass A = pseudo scalar field t = top quark field n l = no of light quarks = 5

10 EFFECTIVE LAGRANGIAN 10 Effective Theory [Chetyrkin, Kniehl, Steinhauser and Bardeen] g c = cot In tan in MSSM! 1 top quark loop dominates Simplifications occur if m A << m t effective theory by int out top loop L A e = Ah 1 massless QCD 8 C GO G 1 C JO J i O G (x) =G µ G a a,µ µ G µ a G a C G = a s 5 4 G 1 F cot C J = apple a s C F 3 3ln µ R m t O J (x) =@ µ µ 5 + a sc () J + C G

11 FEYNMAN RULES 11 Vanishes Different from scalar Higgs

12 1 UNDERLYING THEORY DEFINING FF

13 DEFINING FORM FACTORS 13 Form Factors Loop correction to T gg/q q!color neutral particle Our Interest QCD Pseudo-scalar GOAL gluon FF quark FF at 3-loop

14 DEFINING FORM FACTORS 14 Recall Existence of -operators O G O J Our Strategy Calculate FF for individual operators Later we will combine

15 DEFINING FORM FACTORS 15 Gluon FF Corresponding to O G [ ] =1+â s Q µ S ˆF G,(1) g +â s Q µ S ˆF G,() g +â 3 s Q µ 3 S 3 ˆF G,(3) g + F G g h h ˆM G,(0) g G,(0) ˆM g ˆM G,(n) g i G,(0) ˆM g i

16 DEFINING FORM FACTORS 16 Gluon FF Corresponding to O J F [ ] F =1+â s Q µ S ˆF J,(1) g +â s Q µ S ˆF J,() g + F J g h h ˆM G,(0) g G,(0) ˆM g ˆM J,(n+1) g J,(1) ˆM g i i

17 DEFINING FORM FACTORS 17 Quark FF Corresponding to O J [ ] =1+â s Q µ S ˆF J,(1) q +â s Q µ S ˆF J,() q +â 3 s Q µ 3 S 3 ˆF J,(3) q + F J q h h ˆM J,(0) q ˆM J,(0) q ˆM J,(n) q i ˆM J,(0) q i

18 DEFINING FORM FACTORS Quark FF F Corresponding to O G [ ] F =1+â s Q µ S ˆF G,(1) q +â s Q µ S ˆF G,() q + F G q h h ˆM J,(0) q J,(0) ˆM q ˆM G,(n+1) q G,(1) ˆM q i i 18

19 GOAL 19 Calculating F G F J and F J F G g g q q at 3-loop level

20 0 UNDERLYING THEORY DEFINING FF CALCULATION OF FF

21 FEYNMAN DIAGRAMS 1 Qgraf [P. Nogueira] 1586 ˆM G,(3) type g i 447 ˆM J,(3) g i type 44 ˆM J,(3) q i type 400 ˆM G,(3) q i type

22 5 PRESCRIPTION Color simplification in SU(N) theory } in-house Lorentz & Dirac algebra in d-dimensions codes What about 5 & " µ? inherently 4-dimensional problem of defining in d ( =4 ) dimensions Prescription 5 = i 1 4! " { 5, µ } 6= 0 " µ " µ = 4! µ [µ 1 1] [ t Hooft and Veltman] Treat in d-dimensions

23 5 PRESCRIPTION 3 Problem Chiral Ward identities fails to hold Remedy Finite renormalization of axial current is required [Larin] Will come back to this

24 IBP & LI 4 Removing unphysical DOF of gluons 1. Internal: Ghost loops. External: Polarization sum in axial gauge Results Thousands of 3-loop scalar integrals! LiteRed IBP & LI identities [Lee] [Chetyrkin,Tkachov; Gehrmann, Remeddy] Master Integrals (topologically different)

25 5 MIS Master Integrals [Gehrmann, Huber & Maitre 05; Gehrmann, Heinrich, Huber & Studerus 06; Heinrich, Huber & Maitre 08; Heinrich, Huber, Kosower & Smirnov 09; Lee, Smirnov & Smirnov 10] Results Unrenormalized 3-loop FF in power series of (d =4+ )

26 6 UNDERLYING THEORY DEFINING FF CALCULATION OF FF UV RENORMALIZATION

27 COUPLING CONS RENORM 7 Dimensional Regularization d =4+ Coupling Constant Renorm â s S = µ µ R / Z as a s Z as =1+a s apple 0 + a s apple a 3 s apple i QCD beta functions

28 OPERATOR RENORM 8 Overall Operator Renorm O G & O J requires additional renorm [O G ] R = Z GG [O G ] B + Z GJ [O J ] B [O J ] R = Z s 5Z s MS [O J] B O G mixes under renorm Finite renorm Z s 5 5 prescription

29 OPERATOR RENORM 9 t Hooft & Veltman Prescription { 5, µ } =0 identities} Chiral Ward Violated in d-dimensions Fails to restore correct renorm axial current J µ 5 µ 5 = i 1 3! "µ Introduce finite renorm const Z s µ J µ 5 [J µ 5 ] R = Zs 5Z s MS [J µ 5 ] B [O J ] R = Z s 5Z s MS [O J] B [Larin]

30 RENORM FF Corresponding to [O G ] R [O G ] R = Z GG [O G ] B + Z GJ [O J ] B S G g Z GG h ˆM G,(0) g M G g i + Z GJ h ˆM G,(0) g M J g i S G q Z GG h ˆM J,(0) q M G q i + Z GJ h ˆM J,(0) q M J q i 3-Loop F G g R SG g S G,(0) g F G q R SG q a s S G,(1) q X n=1 1X n=1 a n s a n s h i Fg G,(n) R h i Fq G,(n) R n =3 n = 30

31 RENORM FF Corresponding to [O J ] R S J g Z s 5Z s MS [O J ] R = Z s 5Z s MS [O J] B h ˆM G,(0) g M J g i S J q Z s 5Z s MS h ˆM J,(0) q M J q i 3-Loop F J g R SJ g a s S J,(1) g F J q R SJ q S J,(0) q X n=1 1X n=1 a n s a n s h i Fg J,(n) R h i Fq J,(n) R n = n =3 31

32 RENORM FF 3 Operator renorm is expressed as matrix OG O J R = ZGG Z JG Z GJ Z JJ OG O J B { Z ij with Z JG =0 Z JJ = Z s 5Z s MS

33 33 How do we determine Z ij? Universal Structure of FF

34 34 UNDERLYING THEORY DEFINING FF CALCULATION OF FF UV RENORMALIZATION UNIVERSAL STRUCTURE OF FF

35 KG QCD Factorization, Gauge & RG invariances Q d dq ln F (â s,q,µ, ) = 1 apple K (â s, µ R µ, )+G (â s, Q µ R, µ R µ, ) K G : poles in : finite in RG invariance µ R d dµ R K (â s, µ R µ, ) = µ R d dµ R G (â s, Q µ R, µ R µ, ) = A (a s(µ R)) A : cusp anomalous dimensions A g = C A A q : maximally non-abelian C F 35

36 KG: SOLUTION 36 Solution to order by order [Moch, Vogt, Vermaseren; Ravindran; Magnea] with ˆL,1 ( ) = 1 n ˆL, ( ) = 1 3 n ˆL,3 ( ) = 1 4 n ln F (â s,q,µ, ) = + 1 n A,1 o + 1 0A,1 o A,3 + 1 n o 0A,1 1X i=1 n o G,1 ( ) â i s Q µ i 1 o A, 0G,1 ( ) S i ˆL,i ( ) + 1 n 1 o G,( ) + 1 n A, A, + 4 o 3 0G,1 ( ) 1 4 o 3 1 G,1 ( ) 3 0 G, ( ) + 1 n 1 o 3 G,3( )

37 KG: SOLUTION Solution to order by order [Moch, Vogt, Vermaseren; Ravindran; Magnea] with ˆL,1 ( ) = 1 n ˆL, ( ) = 1 3 n ˆL,3 ( ) = 1 4 n ln F (â s,q,µ, ) = + 1 n A,1 o + 1 0A,1 o A,3 + 1 n o 0A,1 1X i=1 n o G,1 ( ) â i s Q µ i 1 o A, 0G,1 ( ) S i ˆL,i ( ) + 1 n 1 o G,( ) + 1 n A, A, + 4 o 3 0G,1 ( ) 1 4 o 3 1 G,1 ( ) 3 0 G, ( ) + 1 n 1 o 3 G,3( ) All the poles, except single, can be predicted from previous order 37

38 KG: SINGLE POLE ALSO PREDICTABLE 38 3-Loop results of FF [Ravindran, Smith, van Neerven; Moch et. al.; Gehrmann et. al.] G,1 ( ) = B,1,1 + f,1 + 1X k=1 k g,k,1 G, ( ) = B,, + f, 0 g,1,1 + 1 X k=1 k g,k,

39 KG: SINGLE POLE ALSO PREDICTABLE 39 3-Loop results of FF G,1 ( ) = B,1,1 + f,1 + 1X k=1 k g,k,1 G, ( ) = B,, + f, 0 g,1,1 + 1 X [Ravindran, Smith, van Neerven; Moch et. al.; Gehrmann et. al.] k=1 k g,k, B,i : collinear,,i : UV, f,i : soft

40 KG: SINGLE POLE ALSO PREDICTABLE 40 3-Loop results of FF [Ravindran, Smith, van Neerven; Moch et. al.; Gehrmann et. al.] : collinear, : UV, f,i : soft

41 KG: SINGLE POLE ALSO PREDICTABLE 41 3-Loop results of FF [Ravindran, Smith, van Neerven; Moch et. al.; Gehrmann et. al.] G, ( ) = B,, + f, 0 g,1,1 + 1 X k=1 k g,k, : collinear, : UV, : soft

42 KG: SINGLE POLE ALSO PREDICTABLE 4 3-Loop results of FF G,1 ( ) = B,1,1 + f,1 + 1X k=1 k g,k,1 G, ( ) = B,, + f, 0 g,1,1 + 1 X [Ravindran, Smith, van Neerven; Moch et. al.; Gehrmann et. al.] k=1 k g,k, B,i : collinear,,i : UV, f,i : soft Single pole can also be predicted! [Ravindran, Smith, van Neerven]

43 KG: SINGLE POLE ALSO PREDICTABLE 43 3-Loop results of FF G,1 ( ) = B,1,1 + f,1 + 1X k=1 k g,k,1 G, ( ) = B,, + f, 0 g,1,1 + 1 X [Ravindran, Smith, van Neerven; Moch et. al.; Gehrmann et. al.] k=1 k g,k, B,i : collinear,,i : UV, f,i : soft Single pole can also be predicted! [Ravindran, Smith, van Neerven] Explicit computation gives

44 44 KG: UNIVERSALITY OF f,i f g,i = C A C F f q,i [Ravindran, Smith, van Neerven] : universal Extract,i from poles of FF since A, B & f are universal A, B & f are same which appear in scalar and vector FF known up to 3-loop level

45 45 DETERMINING Z ij

46 DETERMINING Z s MS Recall F J R SJ a s S J,(1) = Zs 5Z s MS F J KG Consider unrenorm F J calculate UV anomalous dimen F J q F J g J q,i i =1,, 3 J g,i i =1, } J g,i = J q,i As expected J,1 =0 J,3 = C AC F n + C A C F n f n 3578 o n + CF n f o 7 J, = C A C F n o C F n f 44 3 n 6 o 7 o + C F n f n + C A C F 10 3 n 308 o 3 o 46

47 J DETERMINING Z s MS µ R d dµ R ln Z (a s,µ R, ) = 1X i=1 a i s i n Z s 44 o n MS =1+a s hc A C F + C F n f 3 n 44 o n 80 + CF n f + C F n 5 o f oi n + a 3 s hc o 3 AC F 7 81 n 616 o n + C A CF oi + C A C F n f Overall operator renorm const of [O J ] B Agrees with existing results: different methodology [Larin] Z s 5 =1+a s { 4C F } + a s C F [O J ] R = Z s 5Z s MS [O J] B Finite renorm can t be fixed in this way! C AC F C F n f [Larin] 47

48 DETERMINING Z ij Recall F G g R SG g S G,(0) g = Z GG F G g + Z GJ F J g hm G,(0) g M J,(1) g i hm G,(0) g M G,(0) g i F G q R SG q a s S G,(1) q = Z GGFq G hm J,(0) q h a s hm J,(0) q M G,(1) q M G,(1) q i + Z (1) i + Z GJ F J q hm J,(0) q GJ hmj,(0) q M J,(0) q M J,(0) q i i i Consider Z 1 GG [F G ] R Effectively treat as unrenorm FF However, this involves Z GJ Z GG with bare FF Requires parametrisation of anomalous dimensions Z ij in terms of Non-trivial due to operator mixing! 48

49 DETERMINING Z ij Introduce ij µ R d dµ R Z ij ik Z kj i, j, k = G, J Matrix equation Z ij = ij + a s h ij,1 i + a s h 1 n 0 ij,1 + ik,1 kj,1 o + 1 n ij, oi + a 3 s h 1 3 n ij, ik,1 kj,1 + 4 o 3 ik,1 kl,1 lj,1 + 1 n ij, ij, o 3 ik, kj,1 + 1 n oi 3 ij,3 ik,1 kj, Recall Z JJ = Z s 5Z s MS i =1, JJ = a s h C F i + a s h n C AC F + 14C F C F n f o 6C F n f i dependence is uncommon but crucial! 49

50 50 DETERMINING Z ij Recall to all order With this parametrisation of, consider Z 1 GG [F G ] R Z 1 GG [F G g ] R Z 1 GG [F G q ] R KG KG } Solve coupled linear eqns & i =1,, 3

51 DETERMINING Z ij Findings Agrees with existing by Larin New result up to 3-loop Agrees with [Zoller] 51

52 5 DETERMINING Z ij Findings Results of uniquely specifies & up to With these we compute renorm FF

53 UNDERLYING THEORY DEFINING FF CALCULATION OF FF UV RENORMALIZATION UNIVERSAL STRUCTURE OF FF AXIAL ANOMALY 53

54 AXIAL ANOMALY RELATION 54 Axial Anomaly RG Invariance Our results are in agreement with this in Crucial check

55 APPLICATIONS 55 Soft-virtual cross section at and N 3 LO Threshold resum cross section at N 3 LL [TA, Kumar, Mathews, Rana & Ravindran] [arxiv: ] For total inclusive production cross section, it is an important ingredient.

56 56 FINAL REMARKS Pseudo scalar FF at 3-loop QCD have been computed In dimensional regularization, t Hooft-Veltman prescription for 5, which requires finite renorm. By exploiting universal IR structure independent determination of operator renorm constants. An important ingredient to precision theoretical and phenomenological study. Immediate applications: N 3 LO SV & N 3 LL

57 Thank you!