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Effective Field Theory for Higgs Plus Jet Production
S. Dawson, BNL September 29, 2014
S. Dawson, I. Lewis, and M. Zeng, arXiv: 1409:6299
• Dimension 5 operator contributes only to gg è h
• This operator is generated by mt è ∞ limit of SM o Dimension-‐‑7 in BSM 1/mt
2 corrections to SM dimension-‐‑5 result
• Consider here only dimension-‐‑7 operators which contribute to ggèh+jet (other operators contribute to ggè h+jet jet, eg)
• Question: Can we disentangle effects of higher dimension operators from higher order QCD?
EFT in the Strong-‐‑Higgs Sector
2
O1 = Gµ⌫,AGµ⌫,Ah
Is there a kinematic regime where EFT makes sense in strong sector?
• Gluon fusion largest Higgs production channel
Gluon Fusion of Higgs
[GeV] HM80 100 200 300 400 1000
H+X
) [pb
]
→(p
p σ
-210
-110
1
10
210= 8 TeVs
LHC
HIG
GS
XS W
G 2
012
H (NNLO+NNLL QCD + NLO EW)
→pp
qqH (NNLO QCD + NLO EW)
→pp
WH (NNLO QCD + NLO EW)
→pp
ZH (NNLO QCD +NLO EW)
→pp
ttH (NLO QCD)
→pp
[LHC Higgs cross section working group]
σgg(mh=125 GeV)=18.97 pb +7.2%-‐‑7.8% +7.5%
-‐‑6.9% (NNLO+)
scale PDF
SM predictions good to ~ ±15%
Similar uncertainties at 14 TeV and 100 TeV
3
Data Consistent with SM Hypothesis
BSM physics that contributes to gg èh highly constrained
4
• Exact cross section for gg èh computed to NLO with b and t loops
• b loops are ~-5% of SM gg èh
Gluon Fusion
[Anastasiou, Buehler, Herzog, Lazopoulos, ArXiv:1107.0683] 5
h
t, b
yb SM
σ(y
b)(pb)
mh=125 GeV
√S=7 TeV
• Assume no new Higgs bosons and CP conservation • At dimension-5, one new operator in SU(3) sector:
• C1SM computed to O(αs
3) • Effective theory corresponds to mt è∞ limit of SM:
o Doesn’t include b contributions or 2-loop EW corrections o EFT not valid in MSSM with large tan β
• For MH~2mt resonant structure not captured by EFT
Effective Theory
6
O1 = Gµ⌫,AGµ⌫,AhLeff = C1O1
Higgs and Gluon Fusion • SM gluon fusion well approximated by mt è∞ limit
L =↵s
12⇡
h
vGA
µ⌫Gµ⌫,A
NLO cross section including full b and t mass effects (solid)
LO cross section (with full b and t mass effects) times K factor in mt → ∞ limit (dotted)
[Kraemer, Laenen, Spira, hep-‐‑ph/9611272]
mh (GeV)
σ (g
gèh) pb
√S=14 TeV
7
0.9
0.925
0.95
0.975
1
1.025
1.05
1.075
1.1
100 120 140 160 180 200 220 240 260 280 300
mNNLO/mNNLOeff
MH/GeV
pp @ 14 TeV1/Mt
n, n=0,...,6
EFT for ggh works at NNLO
[Harlander, Mantler, Marzani, Ozeren, 0912.2104; Pak, Rogel, Steinhauser, 0911.4662]
n=0
8
• EFT used for NNLO K factor • Top contribution numerically verified at NNLO by
expansion in (mh2/mt
2)n
• EFT (n=0) accurate to ~ 2% for mh=126 GeV
ggèh is a 1-‐‑scale problem (mh)
n=2 n=4 n=6
mh (GeV)
σNNLO/σ
NNLO
EFT
EFT
1/mt2 expansion
converges well for gg èh
EFT for ggh with New Physics • New physics could be in ggh vertex or Yukawa
couplings
• gg è h cannot distinguish δC1 from δYt
Leff ⇠ LSM + �C1GAµ⌫G
µ⌫,Ah+ �Yttth
A(gg ! h) = ASM
✓m2
h
m2t
◆1 +
�Yt
Y SMt
�+ASM (0)
�C1
CSM1
Almost equal in Standard Model
9
*Include SM top and bopom in LSM
Ex: Top Partner Models
o Top partners allow Dirac masses and non-diagonal Yij
o Often has SM rate, despite new physics in loop
10
top top partners
Cancellation of new physics effects
L ⇠ Yijf ifjh+mif ifi
[Azatov, Galloway, 1103.5646; Low, Rapazzi, Vicci, 0907.5413]
• Top and top partner heavy è EFT works • But…. New physics effects too small to be observed
Top Partners and New Physics
11
0 0.2 0.4 0.6 0.8 1
sin !L
0.9
0.95
1
"N
NL
O/"
NN
LO
(SM
)
Max sin!L allowed by STU
MT = 1 TeV
Singlet Fermion Model#s=8 TeV, pp$H
MH
= 125 GeV
Leading piece in mh2/mt
2 expansion independent of top partner mixing
[Dawson, Furlan,1310.7593,1205.4733]
ggèh in top partner model, √S=14 TeV
L ⇠ mt
vc2Ltth+
MT
vs2LTTh
• To see new physics, perhaps h+jet is more sensitive • At LO, ggègh, qgè gh, qqè gh from top/bottom
loops • Bottom quark contribution is negligible in SM • Can also calculate using EFT for top contribution
Higgs plus Jet Production
12 [Azatov, Paul, 1309.5273; Banfi, Martin, Sanz, 1308.4771]
t,b
100 200 300 400 500pT (GeV)
1e-05
0.0001
0.001
0.01
0.1
gg, exact (top only)gg, mt A'
qg, exact (top only)qg, mt A'
All channels, exact (top only)All channels, mt A'
ppAH+jet, 3S=14 TeVCT12 NLO PDFs, µ2=pT
2+mh2, LO
Higgs plus Jet Production • Look for new
physics at high pT
• Just where EFT breaks down!
• qg important at high pT
• EFT accurate for pT < 200 GeV
Mh=126 GeV
EFT in this plot includes only dimension-‐‑5 operator
13
• Take exact 1-loop calculation and expand in powers of s/mt
2 • Adding more terms in 1/mt
2 doesn’t improve agreement with mtè∞ limit (dimension-5 EFT)
• [General feature of 2 è 2 processes]
Aside: Including more terms
14
200 400 600 800 1000
MHH
(GeV)
0
0.01
0.02
d!
/MH
H (
fb/G
eV)
ExactLow Energy Theorem
Corrections up to O(1/mt
2)
Corrections up to O(1/mt
4)
pp"HH, #S=8 TeVµ=M
HH, CT10 NLO PDFs, m
H=125 GeV
EFT
[Harlander, Neumann, Ozeren, Wiesemann, 1206.0157; Dawson, Furlan, Lewis, arXiv: 1210.6663]
ppèh jet
1/mt6
1/mt4
• EFT is expansion in 1/Λ2
o SM gg è h, expansion is mh2/mt
2
o SM gg è hg, terms of O(s/mt2, pT
2/mt2)
• To dimension-7:
• O1 is usual operator described by low energy theorems
EFT for Higgs + Jet Production
L = C1O1 +1
⇤2⌃5
i=2CiOi +O✓
1
⇤4
◆
O1 = GAµ⌫G
µ⌫,Ah
CSM1 =
↵s
12⇡v
15 * Include only operators which contribute to h jet production
• Small number of dimension-7 operators which contribute to Higgs + jet
• O4 involves 4 light fermions, contributes to Higgs +jet production starting at NLO
EFT for Higgs + Gluons
16
O2 = D�
GA
µ⌫
D�GA,µ⌫h
O3 = fABC
GA,µ
⌫
GB,⌫
�
GC,�
µ
h
O4 = D�GA
�⌫
D⇢
GA,⇢⌫h !eom
g2s
h⌃nlf
i,j=1( i
�µ
TA i
)( j
TA j
)
O5 = GA
�⌫
D⌫D⇢GA,�
⇢
h !eom
gs
h⌃nlf
i=1GA
µ⌫
Dµ i
�⌫TA i
Convenient Basis • Define: (valid for on-shell Higgs production)
• New basis, O6, O3, O4, O5
• Jacobi identities (no eom)
O6 = �D⇢D⇢(GAµ⌫G
µ⌫,A)h = m2hO1
O6 = �2O2 + 4gsO3 + 4O5
Leff = C1O1 +1
⇤2(C3O3 + C4O4 + C5O5)
C1 = C1 �m2
h
2⇤2C2
17
SM Coefficients, Ci • C1 from gg è h at 2-loops
• Match EFT calculation with exact result
CSM,pole
1 =↵s
12⇡v
⇢1 +
↵s
4⇡
5C
A
� 3CF
��
+7↵
s
m2h
1440⇡vm2t
⇢1 +
↵s
⇡
29C
A
84+
19CF
21+
3
2C
F
ln
✓m2
t
µ2R
◆��
Small coefficient of αs mh2/mt
2 term well known
[Dawson, Kauffman, PRD 49 (1994) 2298] 18
• 2-‐‑loop diagrams computed by expanding propagators in powers of 1/mt
2
• qg è qh • gg ègh
• Matching to SM gives: • [At NLO, we need αs
2 contributions from 2-loop SM]
LO QCD in EFT to dimension-‐‑7 A ⇠ (. . . )
gs
C1
t+ gs
C5
⇤2
�
19
Not a lot of free parameters
[Neill,0908.1573; Harlander, Neumann, 1308.2225]
A ⇠ [. . . ]gsC1 + [. . . ]gsC3
⇤2
CSM3 = � gs↵s
360⇡v, CSM
5 =11↵s
360⇡v
LO Result with dimension-‐‑7 EFT
20
O3 = fABCGA,µ⌫ GB,⌫
� GC,�µ h
O5 = GA�⌫D
⌫D⇢GA,�⇢ h
• Effect of SM dimension-7 operators small in gg channel, poorly behaved in qg, qq channels
EFT for Higgs + Gluons at LO
21 [Dawson, Lewis, Zeng,1409.6299; Harlander, Neumann, 1308.2225]
Deviation from exact result
EFT doesn’t work for qq
• O1 and O3 do not interfere at tree level in mhè0 limit unless 3 or more jets in final state
• LO gg ègh
Understanding LO contributions
O3 = fABCGA,µ⌫ GB,⌫
� GC,�µ h
22
O1 :M(+ + +h) ⇠ m4h
M(�++h) ⇠ non-vanishing for mh = 0
O3 :M(+ + +h) ⇠ non-vanishing for mh = 0
m(�++h) = 0
• Include dimension-5 and dimension-7 operators • Tree level:
o ggègh interactions well described by O1
o gqègh interactions become large at high pT
• Goal of current work: o Include effects of dimension-7 operators at NLO QCD
o Do calculation for arbitrary coefficients, Ci
EFT for Higgs + jet Production
23
Does including NLO contributions extend pT range of EFT?
[Dawson, Lewis, Zeng,1409.6299]
Naively, NLO corrections might be same size as BSM effects
• For arbitrary Ci o Compute 1-loop 0è gggh and 0è qqgh amplitudes with
insertion of Oi
o Renormalize Ci
o Combine with real emission contributions for each Ci
o Factorize initial state singularities into PDF definitions
• Calculation done with 2 cut-off phase space slicing o Checks: Known NLO result for Higgs+jet from O1 reproduced o pT spectrum from O1 agrees with HqT program in mt è∞ limit o Calculation is finite and independent of collinear and soft
cut-offs
Ingredients of NLO Calculation
24
1×102 1×103 1×104 1×105
R (GeV)1
2
3
4
5
R i
R3=[C3(R)/gs3(R)]/[C3(mh)/gs
3(mh)]
R5=[C5(R)/gs2(R)]/[C5(mh)/gs
2(mh)]
• C1 renormalization well known o Identical to αs renormalization at 1-loop; known to O(αs
3)
• Calculate ggh, gggh, qqgh on-shell at 1-loop, match divergences to tensor structures from Oi
• Find renormalization group running of coefficients
Aside on NLO renormalization
* C5 renormalization is new result 25
Cbare1 = (1 + �Z↵s)C1
Cbare3 = (1 +
3
2�Z↵s +
↵s
2⇡✏r�3CA)C3
Cbare5 = (1 + �Z↵s +
↵2
2⇡✏r�(
11
6C3 +
4
3CF ))C5
Use running to make connection with high scale physics
NLO Results
50 100 150 200 250 3000.01
0.1
1
10
100
pT !GeV"
d!#dp T
!fb#GeV" s "14 TeV, NLO
Total
Self
$#1%O1#O5$#1%O1#O3O1
26
Look for new physics due to O5 operator at high pT
* Plots use SM Ci to αs2log(µ2/mt
2)
pT dependent K factor
27
At NLO, result well approximated by O1 only, including constant rescaling to get mh
2/mt2 terms
pT dependent K factor
O1-‐‑O3 don’t interfere at tree level, so LO is small
Can rescale K factors for individual operators to get BSM results!
28
• SM h + jet known to NNLO for mtè∞ o (dimension-‐‑5 EFT)
• SM h+ jet known at NLO with numerical inclusion of 1/mt2
terms o (dimension 7 EFT)
• Our results can be interpreted as analytic formulae for the 1/mt
2 NLO contribution to h+jet in SM • For pT< 200 GeV, K factor almost entirely from O1
o Supports validity of usual approach of including only O1
Connection with SM h + jets
29
[Boughezal, Caola, Melnikov, Petriello, 1302.6216; Harlander, Neumann, Ozeren, Wiesemann, 1206.0157]
• Computed NLO corrections to Higgs plus jet using
dimension-7 operators in strong EFT o First analytic result for 1/mt
2 corrections in SM
• Major result: effects from operators other than O1 small o Usual approach of including only O1 makes sense
• K factors presented which can be used for arbitrary BSM models
• EFT not valid for pT > 200 GeV
Conclusion
30