Two-loop QED correctionsto Bhabha scattering

Thomas Becher

Loopfest VI, Fermilab, April 16-18, 2007

work with Kirill Melnikov, hep-ph/soon

Overview• Bhabha scattering

• luminosity determination

• radiative corrections

• A simple relation between massive and massless scattering amplitudes

• Mass factorization for me2 << Q2.

• 2-loop QED differential cross section

Bhabha scattering• Used to measure luminosity at e+e− colliders

• Large angle scattering at low energy meson factories

• Babar, Belle, BEPC-BES, CLEO-C, Daphne, VEPP-2M, ...

• Small angle scattering at high-energy machines

• LEP, SLD, ILC, ...• Electro-weak and new physics at large angles!

L =dN

dtdΩ

∣∣measured

dσdΩ

∣∣theory precise prediction crucial

Tree level cross section

• Cross section diverges as t→0. • Even at the Z-pole, small angle scattering is

large and dominated by QED.• LEP experiments used Bhabha between

20 mrad < θ < 60 mrad for L determination

dσ

dΩ=

α2

s

[t2 + u2

2s2+

s2 + u2

2t2+

u2

st

]

2

+

Homi J. Bhabha ‘36

Current precision• State-of-the art: Monte-Carlo generators

that implement NLO and include logarithmically enhanced higher order corrections.

• Small angle scattering• MC: 0.05%• Exp: LEP 0.035%, Giga-Z 0.02%

• Large angles• MC: 0.5% accuracy.

• New: BABAYAGA@NLO: 0.1%

• Exp: Cleo-C, BaBar, Belle 1%, Daphne 0.3%Balossini, Calame, Montagna, Nicrosini, Piccini

NNLO QED status• NNLO result for θ→0 known.

• Only form factor corrections are needed

• Dominant part is included in BHLUMI MC

• Massless 2-loop virtual corrections calculated

• Ongoing work on massive NNLO• Planar master integrals

• Electron loop contribution known.

Jadach, Placzek, Richter-Was, Ward, Was

Bonciani et al. ‘04

Bern, Dixon, Ghinculov ‘01

Fadin, Kuraev, Lipatov, Merenkov & Trentadue ‘92

Czakon, Gluza, Riemann ‘06

• Terms suppressed by powers of the electron mass are negligible in all applications

• Condition is fulfilled in practice

• e.g. for at LEP • Keep lepton mass at leading power

• necessary, if isolated leptons are observed rather than “lepton jets” (this is the case for large angle scattering)

• for easier comparison with exisiting MC’s

Expansion in me2 ≪ s,|t|,|u|

θ ! 2m√s

θ ! 0.01 mrad

• Expansion of diagrams is nontrivial• interplay of different momentum regions• need loop integrals to subleading powers to

obtain leading power cross section• Can instead use known massless result:

• Photonic logarithmic terms derived from divergent part of massless result.

• Complete leading power photonic corrections inferred from massless result and known mass dependence of vector form factor.

Expansion in me2 ≪ s,|t|,|u|

Penin ‘05

Glover, Tausk and van der Bij ‘01

α2 lnm2

s

• Penin’s derivation of the result is somewhat complicated • uses photon mass as IR regulator• depends on non-renormalization of leading Sudakov log’s

• Have much simpler method to restore logarithmic mass dependence of amplitudes

• Mass effects appear as wave function renormalization on external legs of massless amplitude

• • this relation also works for QCD• Note: relation involves additional soft part for diagrams with

massive fermion loops.

“Mass from no mass”m2

e ! s, |t|

4

somewhat formal decoupling transformation. The lead-ing term of the expansion of the massive Dirac form Ffactor is thus just given by the massless form factor mul-tiplied by the square of the jet-function.

F (Q2,m) = Zj(m2)F (Q2, 0) +O(m2/Q2) . (16)

There are a number of subtleties that need mentioning:while it is easy to see that the soft contribution is scale-less, the above factorization theorem does not hold in atheory with scalar fields in the same kinematics. As waspointed out by Smirnov [14], the form factor integralsreceive contributions from nonstandard momentum re-gions which he calls ultra-collinear which induce induceQ2 dependence not associated with the hard region. Weshow in Appendix ?? that these ultra-collinear regionsonly contribute at subleading power in gauge theories.Another difficulty is that in massive theories the individ-ual collinear contributions are not regulated dimension-ally and additional analytic regulators have to be used atintermediate stages [15].

To determine the jet-function Zj(m2) to NNLO inQED, we now divide the dimensionally regularized mas-sive form factor by the massless one. The O(α2

s) resultfor expansion of the massive vector form factor in thelimit Q2 ! m2 is given in Ref. [16]. The massless resultcan be obtained from [17]. Taking the ratio we obtain

Zj = 1 +(α

π

)m−2ε

[1

2ε2+

14ε

+π2

24+ 1

+ε

(2 +

π2

48− ζ(3)

6

)+ ε2

(4− ζ(3)

12+

π4

320+

π2

12

)]

+(α

π

)2m−4ε

[1

8ε4+

18ε3

+1ε2

(1732

+π2

48

)

+1ε

(8364− π2

24+

2ζ(3)2

)+

561128

+61π2

192− 11

24ζ(3)

−π2

2ln(2)− 77π4

2880

]+O

(αε3,α2ε

). (17)

Corrections: added 1 in first line. 2ζ(3)2 → 2ζ(3)

3 infourth line. The fact that the result is independent ofQ2 is an explicit demonstration of the mass factorizationto two loop order.

IV. BHABHA SCATTERING

In this Section we apply the factorization formula es-tablished for the form factor in the previous Section toBhabha scattering. To get the scattering amplitudeM inwhich electron mass is used as a regulator of the collinearsingularities, all we need to do is to multiply M by asuitable wave function renormalization constant for eachelectron and positron leg. We write

M(pi,m) = Zj(m)n/2M(pi) +O(m2/Q2), (18)

where n = 4 is the number of external massive particlesfor the Bhabha scattering.

We can now use the two-loop result for the Bhabhascattering amplitude computed in the massless limit [2]and employ Eqs.(18,17) to obtain the scattering ampli-tude in which collinear divergences are regularized by theelectron mass and the infra-red divergences are regular-ized dimensionally.

With the massive scattering amplitude at hand,the computation of the two-loop QED corrections tothe large-angle Bhabha scattering cross-section becomesstraightforward. We write the perturbative expansion ofthe scattering cross-section as

dσ

dΩ=

α2

sZ4

j exp (α

πFsoft)

(dσ0 +

α

πs−εdσv

1

+(α

π

)2s−2ε

(dσv,1l2

2 + dσv,2l2

)+O(α3)

). (19)

In this formula, Zj is the renormalization factor given inEq.(17), Fsoft refers to the soft radiation which, in case ofQED, is known to factorize and exponentiate and dσv

1,2denote contributions to the cross-section that originatefrom virtual radiative corrections. We denote the two-loop virtual correction due to the interference of the two-loop amplitude with the tree-level amplitude by dσv,2l

2and the two-loop virtual correction due to the interfer-ence of the one-loop amplitude with itself by dσv,1l2

2 .The massless cross-sections dσv

1 and dσv,2l2 can be

found in Ref. [2], while dσv,1l2 can be obtained fromRef. [18], as we explain below.

In Ref. [18] the result for the interference of the one-loop amplitude with itself was published for quark-quarkscattering in QCD. To extract the QED piece, relevantfor Bhabha scattering, from these results, we need toanalyse the color algebra; such an analysis shows thatdσv,1l2 can be obtained from the computation of Ref. [18]by taking the N → 0 limit of the QCD result, where Nis the number of colors, and subtracting from it suit-ably weighted products of the one-loop Bhabha scatter-ing cross-section dσv

1 and the electron Dirac form fac-tors in the massless approximation. Note that the di-vergent terms in dσv,1l2 can be obtained from Catani’sdecomposition [19] of the one-loop scattering amplitudefor e+e− → e+e− and the O(α) correction to the Bhabhascattering cross-section in dimensional regularization pre-sented in Ref. [2]. For this reason we only present thefinite part of dσv,1l2; it is explicitely given in the Ap-pendix.

Finally, the description of soft radiation in Eq.(19) isencapsulated in the functions Fsoft; this function has tobe computed for massive electron-positron scattering. Itis well-known that in QED the emission of soft photonsexponentiates. The function Fsoft, Eq.(19), is determinedby the integral

Fsoft = −4π2

∫

k0≤wcut

ddk

(2π)d−12k0JµJµ, (20)M(pi)

see also Moch and Mitov hep-ph/0612149

Structure of form factor at large Q2

• Typical momentum regions / relevant scales:

• Explicit in Soft-Collinear Effective Theory • QCD fields are split into soft and collinear fields. • Hard part is absorbed into Wilson coefficient.

p1

p2

hard Q2 = (p1 − p2)2

collinear to p1: p21

collinear to p1: p22

soft: Λ2

soft =p21p

22

Q2

Form factor in dimensional regularization

off-shell on-shellmassless

on-shell massive

H≡H(Q2) same in all three cases!IR finite.

Jet and soft function scaleless!

Soft and collinear divergencies for d→ 4

Jet function J≡J(m2)Soft function scaleless!Soft divergencies for

d→ 4

H

JS

J

H H

J J

Jet-function• Determine Zj (=J2) by taking ratio of

massive to massless form factor

Q2 independent

Zj = 1 +(α

π

)m−2ε

[1

2ε2+

14ε

+π2

24+ 1 + ε

(2 +

π2

48− ζ(3)

6

)+ ε2

(4− ζ(3)

12+

π4

320+

π2

12

)]

+(α

π

)2m−4ε

[1

8ε4+

18ε3

+1ε2

(1732

+π2

48

)+

1ε

(8364− π2

24+

2ζ(3)2

)

+561128

+61π2

192− 11

24ζ(3)− π2

2ln(2)− 77π4

2880

]

agrees with Moch and Mitov hep-ph/0612149

Zj =F (Q2,m2

e, ε)F (Q2,me = 0, ε)

Fermion loop contributions

• At the leading power, diagram gets contributions from hard, collinear and soft photon exchange:

δS = α20 m−4ε

f ln(

m2e

Q2

)f(ε)

mf

meme

S = 1 + δS = 1− (4πα0)2∫

ddk

(2π)d

p1 · p2

(p1 · k)(p2 · k)k2Π(k2,m2

f )

• Multiply massless Bhabha amplitude with Zj2 and S

• Square and add soft radiation with Eγ < ω ≪ me

Massive Bhabha

7

The easiest way to compute the soft matrix element is to employ dispersion representation for thevacuum polarization function Π(k2). We obtain

δS(Q2,m2i , Ni) = Nia

20m

−4εi ln

(Q2

m2e

) (− 1

12ε2+

536ε− 7

27+O(ε)

), (21)

where a0 = α0/π e−γε(4π)ε and γ is the Euler constant. Eq.(16) provides a relation between closedlepton loop contributions to massive and massless form factors and allows a derivation of Ne,µ-dependentO(α2) corrections to Bhabha scattering from the massless results of Ref.[4].

To determine the square of the jet-function Zj to NNLO in QED, we use Eqs.(15,16) and divide theratio of dimensionally regularized massive and massless form factors by the soft matrix element Eq.(18).The expansion of the massive vector form factor in the limit Q2 " m2

e through O(α2) can be foundin Refs. [19, 20]5. The massless result can be obtained from Ref. [21]. We express the jet functionthrough the bare QED coupling constant α0. We find

Zj = 1 + a0m−2ε

[1

2ε2+

14ε

+π2

24+ 1 + ε

(2 +

π2

48− ζ(3)

6

)+ ε2

(4− ζ(3)

12+

π4

320+

π2

12

)]

+a20m

−4ε

[1

8ε4+

1ε3

(18−

Nf

24

)+

1ε2

(1732

+π2

48−

Nf

36

)+

1ε

(8364− π2

24+

2ζ(3)3

−Nf

(209432

+5π2

144

)− Nµ

6ln

m2µ

m2e

)+

561128

+61π2

192− 11

24ζ(3)− π2

2ln(2)− 77π4

2880

+Nf

(33792592

− 11π2

108+

ζ(3)36

)+ Nµ

(136

ln3 m2µ

m2e

+2572

ln2 m2µ

m2e

+(

193216

+π2

18

)ln

m2µ

m2e

−12411296

+7π2

54− ζ(3)

3

)]+O

(αε3,α2ε

). (22)

The independence of the jet function Zj of the hard scale Q2 is an explicit demonstration of themass factorization to two-loop order.

IV. BHABHA SCATTERING

We can now apply the factorization formula, established for the electron Dirac form factor in theprevious Section, to Bhabha scattering. To get the scattering amplitude M in which electron mass isused as a regulator of the collinear singularities, we only need to multiply the massless amplitude Mby the square root of the jet function Z1/2

j for each electron and positron leg and by the product ofsoft functions that account for soft exchanges in s, t and u channels. For the soft matrix element, wemay use Eq.(20) but we have to be careful with the overal sign there since, while it was derived for thespace-like electron form factor, we need to use it to describe the electron-positron interaction in s- andu- channels. We write

M(pi,me) = Z2j (m2

e)M(pi)S(s, t, u) +O(m/pi), (23)

where the soft function S(s, t, u) is given by

S(s, t, u) =

1 + 2∑

Q2

∑

i=e,µ

δS(Q2,m2i ,λQ2Ni)

(24)

5 We need the O(ε2) term in the one-loop contribution to the massive form factor which is not provided in Ref. [19].However it is simple to compute it. The corresponding result is given in Appendix A.

dσ

dΩ= exp(

α

πFsoft)× Z4

j × |S|2 × dσ

dΩ

∣∣∣∣virtual,me=0,mµ=0

massive virtual

Input

• Input for the determination of Zj

• 2-loop massless FF• 2-loop massive FF• heavy fermion contribution

• Input to calculate Bhabha scattering• 1-loop to O(ε2) • 2-loop virtual• (1-loop) x (1-loop)

• Fsoft to O(ε)

Bern, Dixon, Ghinculov ‘01

Bern, Dixon, Ghinculov ‘01

inferred from Anastasiou et al. ‘00

our own evaluation

e.g. Gehrmann, Huber, Maitre ‘05 Bernreuther et al. ‘04

Kniehl ‘89Hoang, Teubner ‘98

Result

• Full agreement with Penin for the photonic two-loop corrections.

• first independent check of his result

• Agreement with Bonciani et al. for the electron loop contribution.

• typo in their paper

• Result for the muon contribution is new.

Structure of the result:

• Collinear logarithms

• Muon mass logarithms

• Soft logarithms

3

the factorization formula to compute NNLO QED corrections to Bhabha scattering. We conclude inSection V. Some useful formulas are collected in the Appendix.

II. NOTATION

Consider the process e+(p1) + e−(p2) → e+(p3) + e−(p4) for energies and scattering angles suchthat the absolute values of all kinematic invariants (p1 + p2)2 = s, (p1 − p3)2 = t and (p1 − p4)2 = uare much larger than the electron mass squared, s, |t|, |u|# m2

e.The Bhabha scattering cross-section is computed in the perturbative expansion in the on-shell fine

structure constant3 α

dσ

dΩ=

α2

s

dσ0

dΩ

[1 +

(α

π

)δ1 +

(α

π

)2δ2 + O(α3)

], (1)

where

dσ0

dΩ=

(1− x + x2

x

)2

+ O(m2e/s), (2)

x = (1− cos θ)/2 and cos θ is the scattering angle.We compute QED corrections to Bhabha scattering taking into account electron and muon contri-

butions to photon vaccum polarization diagrams. We work in the approximation s # m2µ # m2

e. Forphenomenological applications, the contribution of τ leptons may be required; it can be obtained fromthe formulas below with an obvious modification mµ → mτ provided that the high-energy conditions# m2

τ is valid.The higher order corrections to the Bhabha scattering cross-section depend logarithmically on the

mass of the electron. Also, in order to arrive at a physically meaningful result, we need to allow forsoft radiation with the energy of each emitted photon below some value ωcut $ m. The perturbativecorrections to Bhabha scattering cross-section depend logarithmically on ωcut. The corrections sensitiveto soft and collinear physics are numerically enhanced relative to other corrections; it is thereforecustomary to separate those corrections when presenting results for perturbative coefficients. The O(α)correction in Eq.(1) is well-known [9]; in the limit of small electron mass it can be written as

δ1 =(

4Lsoft + 3 +23Nf

)ln

(s

m2e

)+ δ(0)

1 , (3)

where Lsoft = ln(2wcut/√

s) and we introduced label Nf to distinguish corrections due to closed leptonloops. In particular, including electron and muon loops Nf = Ne + Nµ where Ni is the number ofleptonic spicies of the i-th flavor. To arrive at realistic predictions for Bhabha scattering, we use Ne = 1and Nµ = 1.

The correction that is not enhanced by the logarithm of the electron mass to the center of massenergy ratio reads

δ(0)1 =

[−4 + 4 ln

x

1− x

]Lsoft −

2Nµ

3ln

m2µ

m2e− 4− 2π2

3

−2Li2(x) + 2Li2(1− x)−10Nf

9+ f(x). (4)

3 By “on-shell” we mean the fine structure constant defined through the photon propagator at zero momentumtransfer.

dσtree

dΩ

4

The function f(x) is defined as

f = (1− x + x2)−2

(13− 2

3x +

94x2 − 13

6x3 +

43x4

)π2 +

(3− 4x +

92x2 − 3

2x3

)ln(x)

+(

34x− x2

4− 3

4x3 + x4

)ln2(x) +

[−1

2x− 1

2x3 +

(2− 4x +

72x2 − x3

)ln(x)

]ln(1− x)

+(−1 +

52x− 7

2x2 +

52x3 − x4

)ln2(1− x) + Nf

(23− x

3

)(1− x + x2) ln(x)

. (5)

The second term in the perturbative expansion of the Bhabha scattering cross-section is enhancedby up to three powers of the logarithm of the electron mass. We write the NNLO correction as

δ2 = −Nf

9ln3

(s

m2e

)+ δ(2)

2 ln2

(s

m2e

)+ δ(1)

2 ln(

s

m2e

)+ δ(0)

2 , (6)

where

δ(2)2 = 8L2

soft +(

12 +83Nf

)Lsoft +

92

+ Nf

(−1

3ln

x

1− x+

5518

)+

Nµ

3ln

m2µ

m2e

+N2

f

3; (7)

δ(1)2 =

[−16 + 16 ln

x

1− x

]L2

soft +[−28− 8

3π2 + 12 ln

x

1− x− 8Li2(x) + 8Li2(1− x) (8)

+Nf

(83

lnx

1− x− 64

9

)+ 4f(x)− 8Nµ

3ln

m2µ

m2e

]Lsoft −

938− 5π2

2+ 6ζ(3)− 6Li2(x)

+6Li2(1− x) + 3f(x) + Nµ

(−1

3ln2 m2

µ

m2e

+(

23

lnx

1− x− 37

9

)ln

m2µ

m2e

)

−Nf

(83Li2(x) +

28127

)+ Nfg(x) + N2

f

(−10

9+

(2− x)3(1− x + x2)

ln(x))− 2

3NfNµ ln

m2µ

m2e.

and the function g(x) reads

g = (1− x + x2)−2

(23x4 − 5

4x2 − 1

12x3 +

1712

x− 13

)ln2(x)

+(

(1− 2x)(

23x3 − 1

2x2 +

23

)ln(1− x) +

379− 56x

9+

47x2

6− 67x3

18+

10x4

9

)ln(x)

−(

23x2 − 7

6x +

23

)(x2 − x + 1) ln2(1− x) +

(−10

3x2 +

3118

x3 +3118

x− 109− 10

9x4

)ln(1− x)

+(

1112

x2 +89x4 − 1

9+

29x− 23

18x3

)π2

. (9)

The δ(2)2 and δ(1)

2 terms were computed in [10–12]. The term δ(0)2 in Eq.(6) was computed in

Refs. [5, 7, 8]. We present the result of our computation of this term below.

III. MASS FACTORIZATION

We begin with the description of the method that we use to compute the NNLO QED correctionsto the Bhabha scattering cross-section. The key to our approach is the factorization formula thatrelates massless and massive amplitudes for a given process; such a relationship between massless and

δ2 = M2 ln2 m2µ

m2e

+ M1 lnm2

µ

m2e

+ M0

δ2 = S2 ln2

(2ω√

s

)+ S1 ln

(2ω√

s

)+ S0

Eγ < ω

Size of the two-loop corrections

• Assume MC takes care of soft radiation. • Set

s = 1GeV2

θ

103(

α π

) 2δ(

2)

total

photonic corrections

e-loop

μ-loop

0 25 50 75 100 125 150 175!2

0

2

4

6

Lsoft = ln(2Esoftγ /

√s)→ 0

Non-logarithmic corrections

• Assumes MC takes care of soft radiation and implements correct terms.

s = 1GeV2

θ

103(

α π

) 2δ(

2)

total photonic

non-logarithmic photonic

0 25 50 75 100 125 150 175!2

0

2

4

6

ln(m2e/s)

Small angle expansion

• Small angle expansion work up to large angles!• (Plot shows expansion of full result, not

comparison with Fadin et al.)

0 25 50 75 100 125 150 175!0.1

!0.05

0

0.05

0.1

θ

δ 2(θ!

1)−

δ 2δ 2

Summary• Have established a simple relation between

massless and massive amplitudes at large momentum transfers.

• Have applied it to Bhabha scattering at NNLO

• rederivation of results of Penin for photonic corrections and of Bonciani et al. for electron loops.• first independent check of these results

• new result for µ-loop contribution• Same relation can also be used for QCD

processes, such as heavy quark production.see Moch and Mitov hep-ph/0612149