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

SLIDE 1

HIGH ENERGY BEHA

VIOUR OF FORM FACTORS

Taushif Ahmed Johannes Gutenberg University Mainz Germany Skype Seminar IIT Hyderabad May 10, 2018 With Johannes Henn & Matthias Steinhauser Ref: JHEP 1706 (2017) 125

SLIDE 2

GOAL & MOTIV

ATION

• Infrared divergences: important quantities
• Consider: QCD corrections to photon-quark vertex

Γµ(q1, q2) = Qq  F1(q2)γµ − i 2mF2(q2)σµνqν

• Also consider: the massless scenario

F1

2

• Important quantities: is building block for variety of observables

F1, F2

• V

ertex function: characterised by two scalar form factors

F1

• Consider: Form factors of massive quarks

V µ(q1, q2) = ¯ v(q2)Γµ(q1, q2)u(q1)

q1

q2 q

γ∗

e.g. Xsection of hadron production in annihilation & derived quantities like forward- backward asymmetry

e−e+

SLIDE 3

GOAL & MOTIV

ATION

• State-of-the-art results

}

m 6= 0 at 3-loop

F1, F2

Nc limit in

[Henn, Smirnov, Smirnov, Steinhauser ’16]

m = 0

at 4-loop

F1

in large

[Henn, Smirnov, Smirnov, Steinhauser, Lee ’16]

• Next steps: compute the full results for general

underway by several groups

Nc

SU(Nc)

• W

e address: What can we say about next order?

indeed, IR poles can be predicted (partially) by

exploiting RG evolution of FF

F1 at 4-loop in large Nc and high energy limit upto

m 6= 0 m = 0 F1 at 5-loop in large Nc and high energy limit upto

1/✏2

1/✏3

RESULTS

• W

e also obtain process independent functions relating massive & massless amplitudes in high-energy limit at 3 & 4-loops

RESULTS

3

GOAL

Exploit RG evolution of FF

[Manteuffel, Schabinger ’16]

SLIDE 4

PLAN OF THE TALK

RG evolution: massive

• Cute technique to solve

RG evolution: massless Process independent functions Conclusions

SLIDE 5

RG EQUATION: MASSIVE

• FF satisfies KG eqn in dimensional reg.

− d d ln µ2 ln ˜ F ✓ ˆ as, Q2 µ2 , m2 µ2 , ✏ ◆ = 1 2  ˜ K ✓ ˆ as, m2 µ2

R

, µ2

R

µ2 , ✏ ◆ + ˜ G ✓ ˆ as, Q2 µ2

R

, µ2

R

µ2 , ✏ ◆

[Magnea, Sterman ’90] [Gluza, Mitov, Moch, Riemann ’07, ’09] [Ravindran ’06: For Massless] [Sudakov ’56; Mueller ’79; Collins ’80; Sen ’81] 5

• Strategy: Use bare coupling instead of renormalised one

ˆ as as

ˆ as

• Goal: Solve the RG

ˆ as ≡ ˆ αs/4π

d = 4 − 2✏

Q2 = −q2 = −(p1 + p2)2

: scale to keep dimensionless

µ

µR : renormalisation scale

Matching coefficient

F = Celn ˜

F

QCD factorisation, gauge & RG invariance

ˆ as

• The form factor

SLIDE 6

SOL

VING RG EQUATION: MASSIVE

RG invariance of FF wrt µR Cusp anomalous dimension Boundary terms ˜ K ✓ ˆ as, m2 µ2

R

, µ2

R

µ2 , ✏ ◆ = K

• as
• m2

, ✏

• −

µ2

R

Z

m2

dµ2

R

µ2

R

A

• as
• µ2

R

• ˜

G ✓ ˆ as, Q2 µ2

R

, µ2

R

µ2 , ✏ ◆ = G

• as
• Q2

, ✏

• +

µ2

R

Z

Q2

dµ2

R

µ2

R

A

• as
• µ2

R

• 6

d d ln µ2

R

˜ K ✓ ˆ as, m2 µ2

R

, µ2

R

µ2 , ✏ ◆ = − d d ln µ2

R

˜ G ✓ ˆ as, Q2 µ2

R

, µ2

R

µ2 , ✏ ◆ = −A

• as
• µ2

R

SLIDE 7

SOL

VING RG EQUATION: MASSIVE

Need all quantities in powers of ˆ

as ˆ as

Initial goal: Solve for in powers of bare

ln ˜ F

B

• as
• λ2

≡

∞

X

k=1

ak

s

• λ2

Bk

Expand

B ∈ {K, G, A} λ ∈ {m, Q, µR}

functions of i, ✏

7

Use Expansion of in powers of

B

ˆ as

ˆ as = as(µ2

R)Zas

• µ2

R

✓ µ2 µ2

R

◆−✏

Z−1

as (λ2) = 1 + ∞

X

k=1

ˆ ak

s

✓λ2 µ2 ◆−k✏ ˆ Z−1,(k)

as

Renormalisation constant

SLIDE 8

SOL

VING RG EQUATION: MASSIVE

with and so on… The integral becomes a polynomial integral

8

B

• as
• λ2

=

∞

X

k=1

ˆ ak

s

✓λ2 µ2 ◆−k✏ ˆ Bk

Soln of in powers of

B

ˆ as

ˆ B1 = B1 , ˆ B2 = B2 + B1 ˆ Z−1,(1)

as

, ˆ B3 = B3 + 2B2 ˆ Z−1,(1)

as

+ B1 ˆ Z−1,(2)

as

, ˆ B4 = B4 + 3B3 ˆ Z−1,(1)

as

+ B2 n ⇣ ˆ Z−1,(1)

as

⌘2 + 2 ˆ Z−1,(2)

as

• + B1 ˆ

Z−1,(3)

as

µ2

R

Z

2

dµ2

R

µ2

R

A

• as
• µ2

R

• =

∞

X

k=1

ˆ ak

s

1 k✏ " 2 µ2 !−k✏ − µ2

R

µ2 !−k✏# ˆ Ak

trivial

SLIDE 9

UN-RENORMALISED SOLUTION: MASSIVE

Solution of KG in powers of bare with

ˆ ˜ L

Q k (✏) = − 1

2k✏  ˆ Gk + 1 k✏ ˆ Ak

• ,

ˆ ˜ L

m k (✏) = − 1

2k✏  ˆ Kk − 1 k✏ ˆ Ak

• d

d ln µ2

R

as

• µ2

R

• = −✏as
• µ2

R

• −

∞

X

k=0

kak+2

s

• µ2

R

• =

∞

X

k=1

h ak

s(Q2) ˜

LQ

k + ak s(m2) ˜

Lm

k

i

Solved iteratively

9

ˆ as

To obtain the renormalised solution in powers of general as(µ2

R)

use d-dimensional evolution of

as(µ2

R)

ln ˜ F ✓ ˆ as, Q2 µ2 , m2 µ2 , ✏ ◆ =

∞

X

k=1

ˆ ak

s

"✓Q2 µ2 ◆−k✏ ˆ ˜ L

Q k (✏) +

✓m2 µ2 ◆−k✏ ˆ ˜ L

m k (✏)

#

ˆ as = as(µ2

R)Zas

• µ2

R

✓ µ2 µ2

R

◆−✏

Renormalised Solution

SLIDE 10

RENORMALISED SOLUTION: MASSIVE

˜ L1 = 1 ✏ ( − 1 2 G1 + K1 − A1L !) + L 2 G1 − A1 L 2 ! − ✏ ( L2 4 G1 − A1L 3 !) + ✏2 ( L3 12 G1 − A1 L 4 !) − ✏3 ( L4 48 G1 − A1L 5 !) + ✏4 ( L5 240 G1 − A1L 6 !) + O(✏5)

L = log(Q2/m2)

For at one loop

µ2

R = m2

10

ln ˜ F =

∞

X

k=1

ak

s(µ2 R) ˜

Lk

Renormalised Solution

˜ L2 = 1 ✏2 ( 4 G1 + K1 − A1L !) − 1 ✏ ( 1 4 G2 + K2 − A2L !) + L 2 G2 − A2L 2 ! − 0L2 4 G1 − A1L 3 ! − ✏ ( L2 2 G2 − A2L 3 ! − 0L3 4 G1 − A1L 4 !) + ✏2 ( L3 3 G2 − A2L 4 ! − 70L4 48 G1 − A1L 5 !) − ✏3 ( L4 6 G2 − A2L 5 ! − 0L5 16 G1 − A1L 6 !) + O(✏4)

At two loop and so on…

SLIDE 11

NEW RESULTS: MASSIVE

F = C

• as
• m2

, ✏

• eln ˜

F

• Form Factor
• State-of-the-art results

˜ Lk =

∞

X

l=0

(−✏k)l−1 Ll 2 l! Gk + 0lKk − AkL l + 1 !

• Conformal theory : all order result

βi = 0 at 3-loop in large Nc F1, F2 consistent with literature up to 3-loop

[Gluza, Mitov, Moch, Riemann ’07, ’09] [Henn, Smirnov, Smirnov, Steinhauser ’16]

• New results in 1704.07846

F1 at 4-loop in largeNc and high energy limit

is suppressed by F2

m2/q2 1 ✏2

upto

11

in high energy limit

SLIDE 12

DETERMINING UNKNOWN CONSTANTS: MASSIVE

Determining unknown constants G, K, C in large Comparing with explicit computations

Nc limit

G3

O(✏0)

to new!

[Henn, Smirnov, Smirnov, Steinhauser ’16]

G1

G2 O(✏2)

O(✏)

to ,

F1 at 3-loop

K1, K2

[Gluza, Mitov, Moch, Riemann ’09]

new!

K3

[Gluza, Mitov, Moch, Riemann ’07 ’09]

C1 to O(✏2)

, C2 to O(✏)

[Gluza, Mitov, Moch, Riemann ’09]

C1 to O(✏4) , C2 to O(✏2)

O(✏0)

, C3 to new! explicit computation

12

to

A4 became available recently

[Henn, Smirnov, Smirnov, Steinhauser ’16] [Henn, Smirnov, Smirnov, Steinhauser, Lee ’16]

SLIDE 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 limit

Nc

Hence, we have not considerer these

13

SLIDE 14

MASSLESS SCENARIO

SLIDE 15

RG EQUATION: MASSLESS

• FF satisfies KG eqn

− d d ln µ2 ln ˜ F ✓ ˆ as, Q2 µ2 , m2 µ2 , ✏ ◆ = 1 2  ˜ K ✓ ˆ as, m2 µ2

R

, µ2

R

µ2 , ✏ ◆ + ˜ G ✓ ˆ as, Q2 µ2

R

, µ2

R

µ2 , ✏ ◆

Solved exactly the similar way

[Sudakov ’56; Mueller ’79; Collins ’80; Sen ’81] [Ravindran ’06]

5-loop solution new! Up to 4-loop: present

[Moch, Vermaseren, Vogt ’05] [Ravindran ’06]

15

ln ˜ F ✓ ˆ as, Q2 µ2 , m2 µ2 , ✏ ◆ =

∞

X

k=1

ˆ ak

s

"✓Q2 µ2 ◆−k✏ ˆ ˜ L

Q k (✏) +

✓m2 µ2 ◆−k✏ ˆ ˜ L

m k (✏)

#

SLIDE 16

RG EQUATION: MASSLESS

• State-of-the-art results
• Conformal theory : all order result

βi = 0

ˆ ˜ L

Q k = 1

✏2 ( − 1 2k2 Ak ) + 1 ✏ ( − 1 2k Gk )

[Bern, Dixon, Smirnov ’05] [TA, Banerjee, Dhani, Rana, Ravindran, Seth ’17]

• FF

at 4-loop in large Nc

[Henn, Smirnov, Smirnov, Steinhauser, Lee ’16]

• New results in 1704.07846

upto

1 ✏3 F

at 5-loop in large Nc and high energy limit

F

16

F = Celn ˜

F

Matching coefficient = 1

SLIDE 17

DETERMINING UNKNOWN CONSTANTS: MASSLESS

Determining unknown constants in large Comparing with explicit computations

Nc limit G1 to O(✏6) O(✏4) O(✏2)

, G2 to , G3 to

[Baikov, Chetyrkin, Smirnov, Smirnov, Steinhauser ’09] [Gehrmann, Glover, Huber, Ikizlerli, Studerus ’10]

O(✏0)

to new! at 4-loop

G4 F

[Henn, Smirnov, Smirnov, Steinhauser, Lee ’16]

do not appear in the final expressions

Ki = Ki(Ak, βk)

get cancelled against similar terms arising from G

SLIDE 18

COMMENTS: MASSIVE & MASSLESS

G are same for massive and massless

expected! Governed by universal cusp AD Manifestly clear in our methodology

[Mitov, Moch ’07]

˜ G ✓ ˆ as, Q2 µ2

R

, µ2

R

µ2 , ✏ ◆ = G

• as
• Q2

, ✏

• +

µ2

R

Z

Q2

dµ2

R

µ2

R

A

• as
• µ2

R

• enter only into the poles of

For massiveKi

˜ Lk

Constants and O(✏k) terms can be determined from

massless calculation

could lead to deeper understanding of the connection

between massive & massless FF

SLIDE 19

PROCESS INDEPENDENT FUNCTION

M(m) = Y

i∈{all legs}

 Z(m|0)

[i]

✓m2 µ2 ◆1/2 M(0)

Massless Massive

[Moch, Mitov ’07]

Universal and depends only on the external partons!

• Can be computed using simplest amplitudes: FF

Z(m|0)

[q]

= F(Q2, m2, µ2) F(Q2, µ2)

• QCD factorisation: massive amplitudes shares essential properties with

the corresponding massless ones in the high-energy limit independence is manifestly clear: governed by G, same for

Q2

at 3-loop, upto O(✏0) O(1/✏2) at 4-loop new!

19

Relates dimensionally regularised amplitudes to those where the massive & massless FF IR divergence is regularised with a small quark mass.

SLIDE 20

CONCLUSIONS

RG equations governing massive & massless quark-photon FF are discussed. Elegant derivation for analytic solution is proposed key idea: use bare coupling dependence is governed by G & cusp AD: same for

Q2

Massive: non-trivial matching coefficient C Massive: F1 at 4-loop in large Nc

1 ✏2

to Massless:

1 ✏3

at 5-loop in large Nc

F

to

20

massive & massless and high energy limit and high energy limit

THANK YOU!