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