EFT analysis of the double Higgs production in gluon fusion Aleksandr Azatov CERN XSWG HH meeting 24-02-2015 AA, R.Contino, G.Panico, M.Son arXiv: 1502.00539
When can we use EFT analysis ? NP If new physics states are heavier than the SM states as well as the typical mass scale of the process Λ > E . EFT We can integrate these states out and parametrize their effects in terms of the higher dimensional operators. The effects of new physics will appear as a � E � corrections in the series. SM Λ
Operators important for the Higgs pair production in gluon fusion Assuming that the Higgs boson is neutral under U (1) em the most generic lagrangian parametrizing the Higgs pair production in gluon fusion is L n . l . = − m t ¯ � � m 2 2 v h 3 + g 2 � � v + c 2 t h 2 v + c 2 g h 2 c t h c g h G 2 tt − c 3 h s v 2 4 π 2 2 v 2 µν where we have kept only the terms with up to two derivatives. Terms with the higher number of derivatives should appear as a � E � corrections in series. Λ Low et al; Goertz et al,1205.5444 ,1405.7040,1410.3471
Double Higgs production in gluon fusion L n . l . = − m t ¯ � � m 2 2 v h 3 + g 2 � � v + c 2 t h 2 v + c 2 g h 2 c t h c g h G 2 tt − c 3 h s v 2 4 π 2 2 v 2 µν g g g h h h t h g g g h h h g g h h t t h g g h h c t , c g will be constrained also by the single Higgs measurements c 2 t , c 2 g , c 3 can constrained only by the double Higgs measurements
Linear vs non-linear lagrangian The constraints from EWPT from LEP and single Higgs measurements at LHC 7+8 TeV indicate strongly that the Higgs boson comes as a part of the electroweak doublet. Assuming the doublet structure and keeping only the dim 6 operators the relevant lagrangian for the Higgs interactions becomes � 3 + ¯ ∆ L lin . = m 2 g 2 c u ¯ − ¯ c 6 � HH † ¯ q L H c t R + h . c . � � H † H w H † HG 2 v 2 y t c g h s v 2 2 v 2 m 2 µν L n . l . = − m t ¯ � � m 2 2 v h 3 + g 2 � � v + c 2 t h 2 v + c 2 g h 2 c t h c g h G 2 tt − c 3 h s v 2 4 π 2 2 v 2 µν � � c u , c 2 t = − 3 4 π c t = 1 − ¯ 2 ¯ c u , c g = c 2 g = ¯ c g α 2 c 3 = 1 − ¯ c 6 The doublet structure of the Higgs interactions fixes the relations between tth ( ggh ) and tthh ( gghh ) interactions
EFT limitations We must be in the regime when the effects of the dimension ( > 6) operators are not important Linear lagrangian shows what are the range of the Wilson coefficients � � m � H † H � n � c u H † H 1 + ¯ � ∂ q L H c t R � L y = y t ¯ + c n , m v 2 v v 2 n � n � v v 2 g 2 � � m we can estimate the size of c n ∼ ∗ Λ 2 Λ Expansion is valid if only v 2 g 2 ≪ 1 where g ∗ and Λ are the coupling ∗ Λ 2 constant and the mass scale of the new resonances.
Dimension-8 vs dimension-6 operators g 2 g 2 w H † HG 2 w G 2 µν | D σ H | 2 O 6 = ¯ µν , O 8 = ¯ c g s c D 0 s m 2 m 4 keeping only the largest terms growing with energy α � y 2 t + g 2 6 ( E ) + g 2 � A ∼ 8 ( E ) + ... 4 π E 2 E 4 g 2 c g 4 π g 2 c D 0 4 π 6 ( E ) ∼ ¯ v 2 , 8 ∼ ¯ m 2 W v 2 α 2 α 2 We can estimate this contributions to be 6 ( E ) ∼ g 2 ∗ E 2 8 ( E ) ∼ g 2 ∗ E 4 g 2 g 2 Λ 2 , Λ 4
Dimension-8 vs dimension-6 operators g 2 g 2 w H † HG 2 w G 2 µν | D σ H | 2 O 6 = ¯ µν , O 8 = ¯ c g s c D 0 s m 2 m 4 keeping only the terms fastest growing with energy α � y 2 t + g 2 6 ( E ) + g 2 � A ∼ 8 ( E ) + ... 4 π E 2 E 4 c g 4 π c D 0 4 π g 2 g 2 6 ( E ) ∼ ¯ v 2 , 8 ∼ ¯ m 2 W v 2 α 2 α 2 We can estimate this contributions to be 6 ( E ) ∼ g 2 ∗ E 2 � � 8 ( E ) ∼ g 2 ∗ E 4 λ 2 g 2 g 2 , Λ 2 g 2 Λ 4 ∗ In Composite PNGB higgs models dimension 6 operators have an additional suppression ⇒ thus there will be energy region where the � � λ contribution of the dim-8 will larger than dim 6, E � Λ and g ∗ comparable to SM E � Λ / √ g ∗ and at the same time within validity of EFT E < Λ
Choosing the better strategy in extracting the Higgs couplings g h t A � ∼ c 2 α s 4 π y 2 , g h t t g h m 2 A 3 ∼ c g c 3 α s h , h g h v 2 4 π g h s A 4 ∼ c 2 g α s ˆ , g h v 2 4 π g h � 2 t m 2 � log m 2 h α s 4 π y 2 , A △ ∼ c t c 3 s + i π h t g h t ˆ s ˆ g h � 2 t � log m 2 α s 4 π y 2 , A △ nl ∼ c t 2 s + i π t g h t ˆ Different contributions scale differently with the center of mass energy √ s , exclusive measurements will have better sensitivities on the Higgs couplings
Angular distributions A ( g ( p a ) g ( p b ) → h ( p c ) h ( p d )) = ( P µν 0 M 0 + P µν 2 M 2 ) ǫ µ ( p a ) ǫ ν ( p b ) M 0 , 2 are the contributions mediating J Z = 0, J z = ± 2 transitions In the SM we are dominated by the 1.0 0.0001 M 0 contribution 0.001 0.8 NP contributions coming from Dim-6 0.01 0.1 0.6 operators contribute only to M 0 cos Θ min 0.5 0.9 Dim-8 operators can contribute to the 0.4 M 2 0.2 ( η µν ∂ ρ h † ∂ ρ h − 4 ∂ µ h ∂ ν h G a µα G a α � ν 0.0 500 1000 1500 2000 m hh � GeV � c dim − 8 ∼ g 2 ∗ Λ 4 Λ = 1 . 9 TeV , g ∗ = 3
Simulation details Signal yield as a function of c i was simulated by the dedicated code, which was tested against hpair. We have decided to study only bb γγ final state due to cleaner signal and smaller background In order to take into account NLO and NNLO QCD corrections to the Higgs production we use the k -factor calculated in the infinite top mass limit k 14 = 2 . 27 (1309.6594 ) We bin our events in center of mass energy √ s And for every bin after application of the selection cuts we extract the Higgs pair production cross section as a polinomial in c i couplings. � A 1 c 4 t + A 2 c 2 2 t + A 3 c 2 t c 2 3 + A 4 c 2 g c 2 3 + A 5 c 2 2 g + A 6 c 2 t c 2 σ = σ SM t + A 7 c 3 t c 3 + A 8 c 2 t c t c 3 + A 9 c 2 t c g c 3 + A 10 c 2 t c 2 g + A 11 c 2 t c g c 3 � + A 12 c 2 t c 2 g + A 13 c t c 2 3 c g + A 14 c t c 3 c 2 g + A 15 c g c 3 c 2 g
Combining single and double Higgs measurements The bounds from single and double Higgs measurements are correlated In order to estimate the strength of the combined constraints we have constructed the approximate ”likelihood” based on the ATLAS high luminosity studies of the Higgs interaction measurements
Simulation details, yields for 14 TeV LHC 3 ab − 1 Most of the Higgses are produced at threshold. We use the traditional cut based analysis to differentiate the background from the signal.
Results In order to understand the HL-LHC prospects on measuring the various Higgs couplings we wanted to combine the information from the double and single Higgs production measurements. In order to derive approximate the LHC sensibility on the Higgs couplings we have used the ATLAS projections for the HL-LHC. 0.10 10 0.05 5 0.00 c 3 c 2 g 0 � 0.05 � 5 � 0.10 � 10 � 0.15 � 1.0 � 0.5 0.0 0.5 1.0 � 0.5 0.0 0.5 1.0 c 2 t c 2 t
Results: inclusive vs exclusive Binnning in m hh improves the sensibility. However at 14 TeV we are dominated by the low energy bins, thus the improvement is not so big. L � 3ab � 1 LHC s � 14TeV 10 5 c 3 0 � 5 � 1.0 � 0.5 0.0 0.5 1.0 c 2t
Results: inclusive vs exclusive Binnning in m hh improves the sensibility. However at 14 TeV we are dominated by the low energy bins, thus the improvement is not so big. L � 3ab � 1 LHC s � 14TeV L � 3ab � 1 s � 100TeV 10 10 8 6 5 4 c 3 c 3 2 0 0 � 2 � 5 � 4 � 1.0 � 0.5 0.0 0.5 1.0 � 1.0 � 0.5 0.0 0.5 1.0 c 2t c 2 t
Results 0.25 0.20 0.15 0.10 0.05 0.00 � 5 0 5 10 c 6 c 3 = 1 − ¯ c 6 68% credibility intervals are LHC HL-LHC FCC [ − 1 . 2 , 6 . 1] [ − 1 . 0 , 1 . 8] ∪ [3 . 5 , 5 . 1] [ − 0 . 33 , 0 . 29]
Interpreting LHC results 8 LHC HL-LHC 6 [ − 1 . 2 , 6 . 1] [ − 1 . 0 , 1 . 8] ∪ [3 . 5 , 5 . 1] 4 LHC is sensitive to the order one c 6 2 deviations in the trilinear Higgs coupling 0 � In this regime the EFT is probably not valid, unless the BSM model has some � 2 strong dynamics coupled through the � 4 Higgs portal. � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 0.3 c u
Results The single Higgs production in gluon fusion is proportional to s � 14TeV L � 3ab � 1 LHC v + g 2 − m t c t h 4 π 2 c g h s v ⇒ 0.05 σ ∝ | c t + 12 c g | 2 � � 4 π | 2 σ ∝ | 1 − ¯ c u + 12 α 2 ¯ c g c g � � 4 Π � Α 2 � 0.00 � We have large degeneracy in the Higgs couplings in ¯ c u , ¯ c g space which � 0.05 is broken only by the tth and h → γγ measurements. The double Higgs production can � 0.10 � 0.4 � 0.2 0.0 0.2 0.4 break this degeneracy due to the c u different scaling of the cross section with ¯ c u , ¯ c g
Summary If there is a mass gap between the new physics and the SM states EFT presents a coherent framework for analysing the Higgs interactions. The double Higgs production in gluon fusion is sensitive to the ttH ( H † H ), ( H † H ) 3 operators, which modify not only the HH † G 2 µν , ¯ trilinear coupling, but also the interactions between the Higgs boson and the top quark and gluons. In order to extract the Wilson coefficients the combination with single Higgs production measurements is essential. Studying energy distributions is very important in constraining the EFT operators. It looks like HL-LHC can determine the trilinear coupling with order one uncertainty.
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