RGE of d = 6 Operators in SM EFT

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1 RGE of d = 6 Operators in SM EFT Elizabeth Jenkins Department of Physics University of California, San Diego Madrid HEFT2014, September 29, 2014

2 Papers C. Grojean, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Scaling of Higgs Operators and h γγ Decay, JHEP 1304:016 (2013). E.E. Jenkins, A.V. Manohar and M. Trott, On Gauge Invariance and Minimal Coupling," JHEP 1309:063 (2013). E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and λ Dependence," JHEP 1310:087 (2013). E.E. Jenkins, A.V. Manohar and M. Trott, Naive Dimensional Analysis Counting of Gauge Theory Amplitudes and Anomalous Dimensions," Phys. Lett. B726 (2013) E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence," JHEP 1401:035 (2014). R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology," JHEP 1404:159 (2014). R. Alonso, E.E. Jenkins and A.V. Manohar, Holomorphy without Supersymmetry in the Standard Model Effective Field Theory," arxiv: [hep-ph].

3 Generalization of the SM to SM EFT Discovery of Higgs boson h with mass 126 GeV at LHC and no other experimental signal of new physics up to energies Λ 1 TeV implies that SM gives good description of nature up to high energies Λ. If no new particles below Λ, it is possible to study effect of arbitrary new physics at high energy Λ on electroweak scale observables by generalizing SM theory to SM EFT containing Lagrangian terms with d > 4. Low-energy fields of SM EFT are SM fields. L = L SM + 1 Λ d 4 C i O (d) i L = L SM + 1 Λ 2 i i C i O (6) i +

4 Generalization of the SM to SM EFT Leading higher dimension operators are d = 6. Assuming B and L conservation, there are 59 independent dimension-six operators which form complete basis of d = 6 operators. 59 operators divided into eight operator classes. 1 : X 3 2 : H 6 3 : H 4 D 2 4 : X 2 H 2 5 : ψ 2 H 3 6 : ψ 2 XH 7 : ψ 2 H 2 D 8 : ψ 4 X = G A µν, W I µν, B µν ψ = q, l, u, d, e Buchmuller & Wyler 1986 Grzadkowski, Iskrzynski, Misiak and Rosiek 2010

5 Dimension Six Operators Q G Q G Q W Q W 1 : X 3 f ABC Gµ Aν Gν Bρ Gρ Cµ ABC Aν f G µ Gν Bρ Gρ Cµ ɛ IJK Wµ Iν Wν Jρ Wρ K µ IJK Iν ɛ W µ Wν Jρ Wρ K µ 2 : H 6 3 : H4D2 Q H (H H) 3 Q H (H H) (H H) Q HD ( H D ) ( µh H D ) µh 5 : ψ 2 H 3 + h.c. Q eh (H H)( l pe r H) Q uh (H H)( q pu r H) Q dh (H H)( q pd r H) 4 : X 2 H 2 Q HG H H GµνG A Aµν Q H G H H G µνg A Aµν Q HW H H WµνW I Iµν Q H W H H W µνw I Iµν Q HB H H B µνb µν Q H B H H B µνb µν Q HWB H τ I H WµνB I µν Q H W B H τ I H W µνb I µν 6 : ψ 2 XH + h.c. Q ew ( l pσ µν e r )τ I HWµν I Q eb ( l pσ µν e r )HB µν Q ug ( q pσ µν T A u r ) H Gµν A Q uw ( q pσ µν u r )τ I H Wµν I Q ub ( q pσ µν u r ) H B µν Q dg ( q pσ µν T A d r )H Gµν A Q dw ( q pσ µν d r )τ I H Wµν I Q db ( q pσ µν d r )H B µν 7 : ψ 2 H 2 D Q (1) Hl (H i D µh)( l pγ µ l r ) Q (3) Hl (H i D I µh)( l pτ I γ µ l r ) Q He (H i D µh)(ē pγ µ e r ) Q (1) Hq (H i D µh)( q pγ µ q r ) Q (3) Hq (H i D I µh)( q pτ I γ µ q r ) Q Hu (H i D µh)(ū pγ µ u r ) Q Hd (H i D µh)( d pγ µ d r ) Q Hud + h.c. i( H D µh)(ū pγ µ d r ) Buchmuller & Wyler 1986 Grzadkowski, Iskrzynski, Misiak and Rosiek 2010

6 Dimension Six Operators 8 : ( LL)( LL) 8 : ( RR)( RR) 8 : ( LL)( RR) Q ll ( l p γ µ l r )( l s γ µ l t ) Q (1) qq ( q p γ µ q r )( q s γ µ q t ) Q (3) qq ( q p γ µ τ I q r )( q s γ µ τ I q t ) Q (1) lq ( l p γ µ l r )( q s γ µ q t ) Q (3) lq ( l p γ µ τ I l r )( q s γ µ τ I q t ) Q ee (ē p γ µ e r )(ē s γ µ e t ) Q uu (ū p γ µ u r )(ū s γ µ u t ) Q dd ( d p γ µ d r )( d s γ µ d t ) Q eu (ē p γ µ e r )(ū s γ µ u t ) Q ed (ē p γ µ e r )( d s γ µ d t ) Q (1) ud (ū p γ µ u r )( d s γ µ d t ) Q (8) ud (ū p γ µ T A u r )( d s γ µ T A d t ) Q le ( l p γ µ l r )(ē s γ µ e t ) Q lu ( l p γ µ l r )(ū s γ µ u t ) Q ld ( l p γ µ l r )( d s γ µ d t ) Q qe ( q p γ µ q r )(ē s γ µ e t ) Q (1) qu ( q p γ µ q r )(ū s γ µ u t ) Q (8) qu ( q p γ µ T A q r )(ū s γ µ T A u t ) Q (1) qd ( q p γ µ q r )( d s γ µ d t ) Q (8) qd ( q p γ µ T A q r )( d s γ µ T A d t ) 8 : ( LR)( RL) + h.c. Q ledq ( l j pe r )( d s q tj ) 8 : ( LR)( LR) + h.c. Q (1) quqd ( q pu j r )ɛ jk ( q s k d t ) Q (8) quqd ( q pt j A u r )ɛ jk ( q s k T A d t ) Q (1) lequ ( l pe j r )ɛ jk ( q s k u t ) Q (3) lequ ( l pσ j µν e r )ɛ jk ( q s k σ µν u t ) Buchmuller & Wyler 1986 Grzadkowski, Iskrzynski, Misiak and Rosiek 2010

7 Dimension Six Operators 1 : X 3 2 : H 6 3 : H 4 D 2 4 : X 2 H 2 5 : ψ 2 H 3 6 : ψ 2 XH 7 : ψ 2 H 2 D 8 : ψ 4 Operators in classes 1,2,3,4 are flavor singlet Operators in classes 5,6,7 have two flavor indices p, r Operators in class 8 have four flavor indices p, r, s, t n g = 1, 2, 3 number of d = 6 operator coefficients is large!

8 Counting of Coefficients Class N op CP-even CP-odd n g 1 3 n g ng ng ng ng ng(9ng + 7) ng(9ng 7) : (LL)(LL) 5 4 n2 g(7ng ) n2 g(n g 1)(n g + 1) : (RR)(RR) 7 8 ng(21n3 g + 2ng n g + 2) ng(21ng + 2)(ng 1)(ng + 1) : (LL)(RR) 8 4ng(n 2 g 2 + 1) ng(n 2 g 1)(n g + 1) : (LR)(RL) 1 ng ng : (LR)(LR) 4 4ng ng : All ng(107n3 g + 2ng n g + 2) ng(107n3 g + 2ng 2 67n g 2) Total (107n4 g + 2ng ng n g + 72) (107n4 g + 2ng ng 2 30n g + 48) Table : Number of CP-even and CP-odd coefficients in L (6) for ng flavors. The total number of coefficients is (107n 4 g + 2n 3 g + 135n 2 g + 60)/4, which is 76 for ng = 1 and 2499 for ng = 3.

9 SM EFT Higher dimensional operators generated at scale µ = Λ by integrating out new physics particles. Analysis model independent. L (6) = i C i (Λ)O (6) i (Λ) = i C i (m h )O (6) i (m h ) Matrix elements of operators taken at µ = m h (electroweak scale) for electroweak observables. RG evolution of dimension-six operator coefficients produces interesting mixing. Need to do RG evolution for accurate measurements of h γγ, h Z γ, gg h. We have computed full one-loop anomalous dimension matrix γ ij for dimension-six operators!

10 Structure of Anomalous Dimension Matrix Rescale Operators according to Naive Dimensional Analysis: X gx, and chirality flip operators scaled by Yukawa coupling y: ψ 2 H 3 yψ 2 H 3, ψ 2 XH gyψ 2 XH Anomalous dimension entries ( ) λ nλ ( ) y 2 ny ( ) g 2 ng γ 16π 2 16π 2 16π 2, N = n λ + n y + n g N = 1 + w i w j w operators {2} {3, 5, 7, 8} {4, 6} {1} 2 H 6 {2} H 4 D 2, yψ 2 H 3, ψ 2 H 2 D, ψ 4 {3, 5, 7, 8} g 2 X 2 H 2, gyψ 2 XH {4, 6} g 3 X 3 {1}

11 Anomalous Dimension Matrix H 6 H 4 D 2 yψ 2 H 3 ψ 2 H 2 D ψ 4 g 2 X 2 H 2 gyψ 2 XH g 3 X 3 Class NDA Weight H 6 λ, y 2, g 2 λ 2, λg 2, g 4 λy 2, y 4 λy 2, λg 2, y/// 4 0 λg 4, g 6 0 λg ///// 6 H 4 D 2 0 λ, y 2, g 2 y/// 2 y 2, g 2 0 g/// 4 y////// 2 g 2 g/// 6 yψ 2 H 3 0 λ, y 2, g 2 λ, y 2, g 2 λ, y 2, g 2 λ, y 2 g 4 g///// 2 λ, g 4, g 2 y 2 g/// 6 ψ 2 H 2 D 0 g 2, y 2 y/// 2 g 2, λ, // y 2 g 2, y 2 g/// 4 g////// 2 y 2 g/// 6 ψ g 2, y 2 g 2, y 2 0 g 2 y 2 g/// 6 g 2 X 2 H 2 0 1// 0 1// 0 λ, y 2, g 2 y 2 g 4 gyψ 2 XH // 1// 1 g 2 g 2, y 2 g 4 g 3 X // 0 g 2 Table : Form of the one-loop anomalous dimension matrix γij for dimension-six operators Qi rescaled according to naive dimensional analysis. The operators are ordered by NDA weight, rather than by operator class. The possible entries allowed by the one-loop Feynman graphs are shown. The cross-hatched entries vanish.

12 RGE RG running of SM parameters due to d = 6 operators. All corrections calculated. RG running of d = 6 operators. All corrections calculated. Formulae are many pages. RG mixing of d = 6 operators gives nontrivial implications for flavor structure of BSM physics. d = 6 operators affect interpretation of SM parameters at tree level.

13 Holomorphy Observation: 1-loop anomalous dimension matrix respects holomorphy to a large extent. Recall d = 6 Operator Classes 1 : X 3 2 : H 6 3 : H 4 D 2 4 : X 2 H 2 5 : ψ 2 H 3 6 : ψ 2 XH 7 : ψ 2 H 2 D 8 : ψ 4 Divide d = 6 Operators into Holomorphic, Antiholomorphic and Non-Holomorphic Operators X ± µν = 1 2 (X µν i X µν ), X ± µν = ±ix ± µν, X µν ɛ µναβ X αβ /2 X µν = X µν X X ± ψ L, R

14 Holomorphy Definition The holomorphic part of the Lagrangian, L h, is the Lagrangian constructed from the fields X +, R, L, but none of their hermitian conjugates.

15 Holomorphy 6 : ψ 2 XH + h.c. i 2 ɛαβµν σ µν P R = σ αβ P R Q RX = ( Lσ µν R ) X µν H = ( Lσ µν R ) X + µν H, Q RX = ( Rσ µν L ) X µν H = ( Rσ µν L ) X µν H, L d=6 C RX Q RX + C RX Q RX 6 : Q RX, Q RX Q RX, Q RX

16 Holomorphy 1 : X 3 Q X = f ABC X Aν µ X Bρ ν X Cµ ρ, Q X = f ABC X Aν µ X Bρ ν X Cµ ρ, Q X,+ 1 ( ) QX 2 iq X = f ABC X µ + Aν X ν + Bρ X ρ + Cµ Q X, 1 ( ) QX + 2 iq X = f ABC Xµ Aν Xν Bρ Xρ Cµ L d=6 C X Q X + = C C X Q X X,+ Q X,+ + C X, Q X, C X,± ( ) C X ± ic X 1 : Q X, Q Q X X,+, Q X,

17 Holomorphy 4 : X 2 H 2 Q HX = X µν X µν H H, Q H X = X µν X µν H H, Q HX,+ X + 2 H H = 1 ( X) 2 X i H H 4 Q HX, X 2 H H = 1 ( X) 2 X + i H H 4 L d=6 C HX Q HX + C H Q X H = C X HX,+ Q HX,+ + C HX, Q HX, C HX,± ( ) C HX ± ic H X 4 : Q HX, Q H Q X HX,+, Q HX,

18 Holomorphy 8 : ψ 4 = ( LR ) ( LR ), ( RL ) ( RL ) Q LRLR = ( ) ( L j R ɛ jk L k R ), Q LRLR L d=6 C LRLR Q LRLR + C LRLR Q LRLR 8 : Q LRLR, Q LRLR Q LRLR, Q LRLR Holomorphic + Antiholomorphic

19 Holomorphy 8 : ψ 4 = ( LR ) ( RL ) Q ledq = ( l j e) (dqj ), Q ledq L d=6 C ledq Q ledq + C ledq Q ledq 8 : Q ledq, Q ledq Q ledq, Q ledq Non holomorphic

20 Holomorphy 8 : ψ 4 = JJ = ( Lγ µ L ) ( Lγ µ L ), ( Rγ µ R ) ( Rγ µ R ), ( Lγ µ L ) ( Rγ µ R ) 3 : H 4 D 2 = ( ) 3 2 : H 6 = H H ( ) ( ) ( ) ( ) H H H H, H D µ H H D µ H Self Hermitian Non holomorphic

21 Holomorphy 5 : ψ 2 H 3 + h.c. ( (LR Q RH = H H) ) H, ) (RL Q RH (H = ) H H L d=6 C RH Q RH + C RH Q RH Non holomorphic

22 Holomorphy 7 : ψ 2 H 2 D ( Q HL = H i (Lγ D µ H) µ L ), ( Q HR = H i ) (Rγ D µ H µ R ) ( H ) Q Hud = i D µ H (uγ µ d), Q ( H ) (dγ Hud = i D µ H µ u ) Self Hermitian (except Q Hud ) Non holomorphic

23 Holomorphy L d=6 = L h + L h + L n = C h Q h + C h Q h + C n Q n { Q h X + 3, X +2 H 2, ( Lσ µν R ) } X + H, (LR)(LR) { Q h X 3, X 2 H 2, ( Rσ µν L ) } X H, (RL)(RL) { } Q n H 6, H 4 D 2, ψ 2 H 3, ψ 2 H 2 D, (LR)(RL), JJ Definition The holomorphic part of the Lagrangian, L h, is the Lagrangian constructed from the fields X +, R, L, but none of their hermitian conjugates.

24 Holomorphy Ċ i 16π 2 µ d dµ C i = j=h,h,n γ ij C j, i = h, h, n γ hh γ hh γ hn γ hh γ hh γ hn γ nh γ nh γ nn = γ hh γhh 0 γ nh γnh γ nn ( γhh γ hn γ nh γ nn ) ( γhh 0 = γ nh γ nn ) Ċ h = j=h γ hj C j

25 Holomorphy (X + ) 3 (X + ) 2 H 2 ψ 2 X + H (LR)(LR) (LR)(RL) JJ ψ 2 H 3 H 6 H 4 D 2 ψ 2 H 2 D (X + ) 3 h (X + ) 2 H 2 h h h ψ 2 X + H h h h h F (LR)(LR) 0 h F h F Y uy e,d Y uy e,d 0 (LR)(RL) 0 0 Y uy d, Y uy e h F 0 JJ 0 0 Y uy e,d ψ 2 H 3 0 Y u,d,e h h H 6 0 H 4 D ψ 2 H 2 D Table : Form of the one-loop anomalous dimension matrix for d = 6 operators. Y is a Yukawa coupling. The first 4 rows and columns involve holomorphic operators (and their conjugates), and the rest involve non-holomorphic operators. The RGE for the rows can depend on the C of each column, or their conjugates. Entries which must vanish by NDA are denoted by 0, those for which there is no one-loop diagram (after taking equations of motion into account) are denoted by, and those which vanish by explicit computation are denoted by 0. Entries with h are non-zero, and satisfy holomorphy, i.e. they depend on C but not C. Entries with hf satisfy holomorphy because anti-holomorphic contributions are forbidden by NDA and flavor symmetry. Entries with are non-zero, but not holomorphic. The red entries violate holomorphy.

26 Summary Complete RGE of dimension-six operators of SM EFT computed for first time. Contribution of dimension-six operators to running of SM parameters calculated for first time. RG evolution of dimension-six operators important for Higgs boson production gg h and decay h γγ and h Z γ, which occur at one loop in SM. Significant constraints on flavor structure of SM EFT. Test of MFV hypothesis. Approximate holomorphy of 1-loop anomalous dimension matrix of dimension-six operators.