- p. 1/24

Two-loop QCD amplitudes for Higgs → b+ b̄+ g

Prakash Mathews

Saha Institute of Nuclear Physics

• Higgs+1 jet

• Two loop amplitudes b+ b̄ → H + g• IBP, LI, MI

• Two loop IR structure

• Summary

with Taushif Ahmed, Maguni Mahakhud, Narayan Rana and V. Ravindran

RADCOR 2015

- p. 2/24

Run-I@LHC

• At the experimentally accessible energy scales, the Run-I@ LHC hasestablished the SM framework as the true theory of electrowe ak interactions

• The discovered new boson at 125 GeV behaves like the SM scalar and itsmass fixes the last free parameter of the Lagrangian

• Negative results from the searches for signals of new physic s tightlyconstrain many new physics scenarios, surviving parameter space is nolonger appropriate to address the physics problems they wer e intented tosolve

• Experimental program at the LHC relies heavily on the precis e theoreticalpredictions for the relevant signals and the many QCD backgr ounds

• Remarkable agreement between the predicted SM values and th e measuredcross sections spanning a broad range is a significant valida tion of thetheoretical framework

- p. 3/24

The Higgs

• Post discovery of a new particle

mH = 125.09 ± 0.24 GeV

ATLAS & CMS combined measurement in the H → γγ and H → ZZ → 4ℓchannels for

√s = 7, 8 TeV

PRL 114 (2015) 191803

• With increasing dataset, emphasis shifted to determining it s properties andtesting the consistency of the SM against the data

• Spin, Charge conjugation and Parity probed by examining the angulardistributions of the decay channels H → γγ, H → ZZ, H → WW . Datafavours a CP-even, spin-zero particle

• Strengths of the couplings with gauge bosons and fermions ex plored for anumber of benchmark models

• Results consistent with expectation of a SM Higgs boson

- p. 4/24

Precision studies

• Precise theoretical predictions of the Higgs boson observa ble will bekey to precision studies of its properties at the LHC

• New physics models look for deviation from the SM predictions andconstraining these models needs precise theoretical inputs

• Theoretical uncertainties as a result of the missing higher o rderterms in pQCD at the LHC energies are large and is comparable to t heexperimental errors

• To study the properties of the Higgs boson, differential dis tributionof the Higgs boson play an important role

• Observables with jet vetos enhance the significance of the si gnalconsiderably enabling the study the properties of Higgs cou plings

- p. 5/24

Exclusive observables

• NNLO predictions of the Higgs bosons with one jet through eff ectivegluon fusion process is available

Gehrmann, Jaquier, Glover, Koukoutsakis

Boughezal, Caola, Melnikov, Petriello, Schulze

• As the experimental accuracy improves it is important to includ e thesub-dominant contributions to the production

• Higgs through bb̄ annihilation with 1-jet is one such process and isknown only upto NLO level

• Here we present one of the ingradients for the production Hig gs+1jet in bb̄ anhillation viz. two loop QCD amplitudes

- p. 6/24

Branching fraction of SM Higgs

• Branching fraction of SM Higgs as a function of Higgs mass

- p. 6/24

Branching fraction of SM Higgs

• Branching fraction of SM Higgs as a function of Higgs mass

• Mass range of interest MH = 120 − 130 GeV, significant contribution comes fromH→ bb̄, τ+τ−, WW ∗ and ZZ∗

- p. 7/24

Interaction part of the action

b-quarks with ◦ scalar Higgs (SM) and ◦ pseudoscalar Higgs (MSSM)

SbI = −λ∫

d4x φ(x) ψb(x) ψb(x) SM

= −λ̃∫

d4x φ̃(x) ψb(x) γ5 ψb(x) MSSM

Yukawa coupling: λ =

√2

υmb λ̃ =

−λ sinαcosβ

φ̃ = h

λ cosαcosβ

φ̃ = H

λ tanβ φ̃ = A

• α: mixing angle between weak and mass eigenstates of neutralscalars

• Scale of the problem is the Higgs boson mass and b-quark mass ismuch smalller, hence both in the phase space integral and matri xelements, the b-quark mass is treated as massless like the light quarkflavours, while retaining the b-quark mass in the Yukawa coupl ing

- p. 8/24

Variable Flavour Scheme (VFS)

• In the VFS, one assumes the initial state b-quarks inside the p roton,as a result of emission of collinear b b̄ states from the gluonsintrinsically present inside the proton

• Being collinear, give large logs, which need to be resummed. Th eresummed contribution is the source for non-vanishing b and b̄ partondistribution functions inside the proton in the VFS scheme. A ctiveflavours nf = 5

• Fully inclusive cross section for Higgs production in assoc iationwith bottom quark to NNLO level accuracy is known in the VFS

Harlander, Kilgore

- p. 9/24

Higgs decay H(q) → b(p1) + b̄(p2) + g(p3)

• Mandelstam variables

s ≡ (p1 + p2)2 > 0, t ≡ (p2 + p3)2 > 0, u ≡ (p1 + p3)2 > 0

s+ t+ u = M2H ≡ Q2 > 0

• Dimensionless invariants

0 < x ≡ s/Q2 < 1, 0 < y ≡ u/Q2 < 1, 0 < z ≡ t/Q2 < 1

x+ y + z = 1

• Two-loop four-point functions with one off-shell external leg andmassless internal propagators can be expressed in terms of HPL s and2dHPLs as functions of dimensionless invariants

- p. 10/24

Analytical continuation: Higgs+1 jet production Q2 = M2H > 0

• Continuation from the Euclidean region to any physical Mink owskian regionrequires in general the analytic continuation

1 → 3 ⇐⇒ 2 → 2

• b(−p1) + b(−p2) → g(p3) + H(p4) s > 0, t < 0, u < 00 < u1 ≡ −us < 1, 0 < v1 ≡

Q2

s< 1

• b(−p2) + g(−p3) → b(p1) + H(p4) s < 0, t > 0, u < 00 < u2 ≡ −ut < 1, 0 < v2 ≡

Q2

t< 1

• b(−p1) + g(−p3) → b(p2) + H(p4) s < 0, t < 0, u > 00 < u3 ≡ − tu < 1, 0 < v3 ≡

Q2

u< 1

• Relations for HPLs and 2dHPLs needed for analytic continuati on of the2-loop, 4-point master integrals to kinematics of all 2 → 2 scatterings with oneoff-shell external leg

Gehrmann and Remiddi NPB640 (2002) 379

- p. 11/24

Amplitude |M〉 = Sµ(b, b̄; g)εµ of H(q) → b(p1)+ b̄(p2)+g(p3)

• Using equations of motion and p3 · ε = 0 the general form

Sµ(b, b̄; g) = ū(p1){

A′ p1µ + A′′ p2µ + A2 /p3γµ

}

v(p2)

• QCD Ward identities ⇒ A′/p2.p3 = −A′′ /p1.p3 ≡ A1. Amplitude reduces

Sµ(b, b̄; g) εµ = ū(p1){

A1 (p2.p3 p1µ − p1.p3 p2µ) + A2 /p3γµ}

v(p2) εµ

≡ A1 T1 + A2 T2

• Coefficients Am (m = 1, 2) can be expanded in powers of as = g2s/16π2

Am =λ

µǫR4π

√asT

aij

{

A(0)m + asA(1)m + a

2sA

(2)m + O(a3s)

}

Coefficients A(l)1,2 completely specify the amplitude order by order inperturbation theory

- p. 12/24

Projection Operators

• Using appropriate d-dimensional projection operators and summing over thespin, coefficients A1,2 can be extracted to a particular order in pQCD

Am =∑

spins

P(Am) Sµ(b, b̄; g) εµ

• Projection operators

P(A1) =2(d − 2)

s2 t u (d − 3)T1† +

1

s t u (d − 3)T2†

P(A2) =1

s t u (d − 3)T1† +

1

2 t u (d − 3)T2†

• By suitably crossing the Higgs decay (1 → 3) amplitudes can be related toHiggs + 1 jet (2 → 2) production amplitude, with the A1,2 now expressed interms of ui and vi

- p. 13/24

Calculation of unrenormalised ampliudes |M̂ (l)〉

Process H → b + b̄ + g to 2-loop

|M〉 = λ̂µǫ0

Sǫ

(

âsµǫ0

Sǫ

) 12

{

|M̂(0)〉 +(

âsµǫ0

Sǫ

)

|M̂(1)〉 +(

âsµǫ0

Sǫ

)2

|M̂(2)〉 + O(â3s)}

• Dimensional regularisation d = 4 + ǫ

• Gauge choice: ◦ External gluon (axial) ◦ Internal gluon (Feynman)

• Feynmann diagrams are generate using QGRAF

◦ Tree level 2◦ 1-loop level 13◦ 2-loop level 251

• Convert raw QGRAF symbolic output to FORM readable formatincorporating the Feynman rules (in-house FORM codes)

• The Integrals are reduced to master integrals (MI) using IBP and LI identities(mathematica packages FIRE and LiteRed)

- p. 14/24

Planar topologies of master integrals

Gehrmann and Remiddi

- p. 15/24

Non-planar topologies of master integrals

Gehrmann and Remiddi

- p. 16/24

1-loop

• Integral belongs to one of the following sets:

{D, D1, D12, D123} , {D, D2, D23, D123} , {D, D3, D31, D123}

D = k21, Di = (k1−pi)2, Dij = (k1−pi−pj)2, Dijk = (k1−pi−pj−pk)2

• Scalar products {Sij} with loop momenta k1 and external momentapi can be expressed, in terms of D’s in a set. Each set {D} form acomplete basis and are linearly independent

• At 1-loop, number of {Sij} = max number of propagators. Does nothold for 2-loop and additional auxilliary denominators need to beintrocuced

I =∫ l

∏

ℓ=1

ddkℓ{Sij}

Dn11 · · · Dnmm

NLO m ≤ 4 {Sij} = 4NNLO m ≤ 7 {Sij} = 9

- p. 17/24

Two Loop

• Nine independent scalar products involving loop momenta k1,2 and externalmomenta p1,2,3. Feynman integral contain terms belonging to one of thefollowing six sets:

{D0, D1, D2, D1;1, D2;1, D1;12, D2;12, D1;123, D2;123} ,

{D0, D1, D2, D1;1, D2;1, D1;12, D2;12, D1;123, D0;3} ,

{D0, D1, D2, D1;2, D2;2, D1;23, D2;23, D1;123, D2;123} ,

{D0, D1, D2, D1;2, D2;2, D1;23, D2;23, D1;123, D0;1} ,

{D0, D1, D2, D1;3, D2;3, D1;31, D2;31, D1;123, D2;123} ,

{D0, D1, D2, D1;3, D2;3, D1;31, D2;31, D1;123, D0;2}

D0 = (k1 − k2)2, Dα = k2α, Dα;i = (kα − pi)2, Dα;ij = (kα − pi − pj)2,

D0;i = (k1 − k2 − pi)2, Dα;ijk = (kα − pi − pj − pk)2 α = 1, 2; i = 1, 2, 3

- p. 18/24

UV divergences; d = 4 + ǫ; MS scheme

• Bare couplings related to renormalised âsµǫ0Sǫ =

asµǫ

R

Z(µ2R)

âs

µǫ0Sǫ =

as

µǫR

[

1 + as

(

1

ǫra1;1

)

+ a2s

(

1

ǫ2ra2;2 +

1

ǫra2;1

)

+ O(a3s)]

Sǫ = exp

[

ǫ

2(γE − ln 4π)

]

, ra1;1 = 2β0 , ra2;2 = 4β20 , ra2;1 = β1

β0 =

(

11

3CA −

4

3TFnf

)

, β1 =

(

34

3C2A −

20

3CATFnf − 4CF TFnf

)

λ̂

µǫ0Sǫ =

λ

µǫR

[

1 + as

(

1

ǫrλ1;1

)

+ a2s

(

1

ǫ2rλ2;2 +

1

ǫrλ2;1

)

+ O(a3s)]

rλ1;1 = 6CF , rλ2;2 =(

18C2F + 6β0CF

)

, rλ2,1 =

(

3

2C

2F +

97

6CF CA −

10

3CF TF nf

)

- p. 19/24

UV

• Using the bare couplings

|M〉 = λµǫR

(as)12

(

|M(0)〉 + as|M(1)〉 + a2s|M(2)〉 + O(a3s))

|M(0)〉 =(

1

µǫR

) 12

|M̂(0)〉

|M(1)〉 =(

1

µǫR

) 32 [

|M̂(1)〉 + µǫR(ra1

2+ rλ1

)

|M̂(0)〉]

|M(2)〉 =(

1

µǫR

) 52 [

|M̂(2)〉 + µǫR(3ra1

2+ rλ1

)

|M̂(1)〉

+µ2ǫR

(

ra22

− r2a1

8+

ra12

rλ1 + rλ2

)

|M̂(0)〉]

ra1=

( 1

ǫra1;1

)

, ra2=

( 1

ǫ2ra2;2

+1

ǫra2;1

)

, rλ1=

( 1

ǫrλ1;1

)

, rλ2=

( 1

ǫ2rλ2;2

+1

ǫrλ2;1

)

|M̂(l)〉 unrenormalised color-space vector represents the lth loop amplitude

- p. 20/24

Infrared factorisation: Catani’s formula

• IR divergent structure of amplitudes well understood. Reno rmalised amplitudes

|M(1)〉 = 2 I(1)b (ǫ) |M

(0)〉 + |M(1)fin〉

|M(2)〉 = 2 I(1)b (ǫ) |M

(1)〉 + 4 I(2)b (ǫ) |M

(0)〉 + |M(2)fin〉

• Universal substraction operators I(i)

I(1)b (ǫ) =

1

2

e−ǫ2γE

Γ(1 + ǫ2)

{

( 4

ǫ2−

3

ǫ

)

(CA − 2CF )[(

−s

µ2R

) ǫ2]

+(

−4CA

ǫ2+

3CA

2ǫ+

β0

2ǫ

)

[

(

−t

µ2R

) ǫ2+(

−u

µ2R

) ǫ2

]}

,

I(2)b (ǫ) = −

1

2I(1)b (ǫ)

[

I(1)b (ǫ) −

2β0

ǫ

]

+e

ǫ2γE Γ(1 + ǫ)

Γ(1 + ǫ2)

[

−β0

ǫ+ K

]

I(1)b (2ǫ)

+(

2H(2)q (ǫ) + H

(2)g (ǫ)

)

Catani PLB427 (1998) 161; Sterman et. al. PLB552 (2003) 48

- p. 21/24

IR

H(2)q (ǫ) =

1

ǫ

{

CACF

(

−245

432+

23

16ζ2 −

13

4ζ3

)

+ C2F

( 3

16−

3

2ζ2 + 3ζ3

)

+CFnf

( 25

216−

1

8ζ2

)

}

H(2)g (ǫ) =

1

ǫ

{

C2A

(

−5

24−

11

48ζ2 −

1

4ζ3

)

+ CAnf

(29

54+

1

24ζ2

)

−1

4CFnf −

5

54n2f

}

K =

(

67

18−

π2

6

)

CA −10

9TFnf

Born amplitude |M(0)〉 and the finite parts |M(l)fin〉, l = 1, 2 are process dependent,needs to be explicitly computed

- p. 22/24

lth loop amplitude |M(l)〉 = 4π T aij{A(l)1 T1 +A

(l)2 T2}

• Renormalised coefficients A(l)m in terms of their bare counterparts Â(l)m

A(0)m =

(

1

µǫR

) 12

Â(0)m

A(1)m =

(

1

µǫR

) 32 [

Â(1)m + µ

ǫR

(ra1

2+ rλ1

)

Â(0)m

]

A(2)m =

(

1

µǫR

) 52 [

Â(2)m + µ

ǫR

(3ra12

+ rλ1

)

Â(1)m

+µ2ǫR

(

ra2

2−

r2a1

8+

ra1

2rλ1 + rλ2

)

Â(0)m

]

• Subtracting terms proportional to universal part I(l)b , Finite parts of A(l)m

A(1)m = 2 I

(1)b (ǫ) A

(0)m + A

(1)finm

A(2)m = 2 I

(1)b (ǫ) A

(1)m + 4 I

(2)b (ǫ) A

(0)m + A

(2)finm

• IR poles structure agree exactly, providing a crucial test o f our computation

- p. 23/24

Finite parts of A(l)finm

A(l)finm =

l∑

n=0

A(0)m B(l)m;n lnn(

− Q2

µ2

)

A(0)1 = −

4i

t uand A(0)2 = i

(1

t+

1

u

)

• Coefficients B(l)m;n are given in the paper.

• Without using projectors, we independently computed 〈M(0)|M(l)〉 forl = 1, 2, as an additional check

• Corresponding coefficients for Higgs+ 1 jet production at ha drons collidersare also provided in the arXiv submission using analyticall y continued HPLsand 2d HPLs

Gehrmann and Remiddi NPB640 (2002) 379

- p. 24/24

Summary

• The potential sub dominant contribution to Higgs + 1 jet produ ctionat LHC comes from b+ b̄ → H + g

• Amplitudes for the partonic process H → b+ b̄+ g and processesrelated by crossing are presented to 2-loop level in QCD

• The crossing processes, contribute to exclusive observabl esinvolving Higgs boson and a jet

• The IR structure of the amplitudes are in accordance with theprediction of Catani upto two loop level

Two-loop QCD amplitudes for Higgs b + + gmynaranjaRun-I@LHCmynaranjaThe HiggsmynaranjaPrecision studiesmynaranjaExclusive observablesmynaranjaBranching fraction of SM HiggsmynaranjaBranching fraction of SM Higgs

mynaranjaInteraction part of the actionmynaranjaVariable Flavour Scheme (VFS)mynaranjaHiggs decay H (q) b (p1) + (p2) + g (p3) mynaranjaAnalytical continuation: Higgs+1 jet production Q2=MH2>0mynaranjaAmplitude |M"526930B = S (b,;g) of H (q) b (p1) + (p2) + g (p3) mynaranjaProjection OperatorsmynaranjaCalculation of unrenormalised ampliudes |(l) "526930B mynaranjaPlanar topologies of master integrals mynaranjaNon-planar topologies of master integralsmynaranja1-loop mynaranjaTwo Loop mynaranjaUV divergences; d=4+; MS schememynaranjaUVmynaranjaInfrared factorisation: Catani's formulamynaranjaIRmynaranjalth loop amplitude |M(l) "526930B = 4 Taij { A1(l) T1 + A2(l) T2 } mynaranjaFinite parts of Am(l)fin mynaranjaSummary