two-loop qcd amplitudes for higgs · 2015. 6. 15. · • as the experimental accuracy improves it...
TRANSCRIPT
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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
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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
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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
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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
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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
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Branching fraction of SM Higgs
• Branching fraction of SM Higgs as a function of Higgs mass
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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∗
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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
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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
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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
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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
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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
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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
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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)
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Planar topologies of master integrals
Gehrmann and Remiddi
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Non-planar topologies of master integrals
Gehrmann and Remiddi
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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
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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
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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
)
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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
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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
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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
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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
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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
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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