The Standard Model Effective Field Theory and Dimension 6 Operators

Transcription

1 The Standard Model Effective Field Theory and Dimension 6 Operators Aneesh Manohar University of California, San Diego 17 Mar 2015 / Vienna A Manohar (UCSD) 17 Mar 2015 / Vienna 1 / 46

2 Outline The SMEFT Equations of Motion Operators Naive Dimensional Analysis µ eγ and MFV Holomorphy A Manohar (UCSD) 17 Mar 2015 / Vienna 2 / 46

3 Talk based on: Rodrigo Alonso, Elizabeth Jenkins, Michael Trott, AM 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].

4 SMEFT A model independent way to include the effects of new physics. The assumption is that there are no new particles at the electroweak scale. This is what the data indicates. Fields are the SM fields. H breaks SU(2) U(1). L = L SM + L 5 Λ + L 6 Λ Need to include the Cannot just stop at a given dimension. A Manohar (UCSD) 17 Mar 2015 / Vienna 4 / 46

5 H = ( ( ϕ + 1 v + h + iϕ 0 ) 2 ) ϕ +, ϕ 0 give mass to W, Z, and h is the Higgs scalar. H H W 2 ( 1 2 v 2 + hv + 1 ) 2 h2 W 2 Couplings of h fixed because it is part of H. An alternate approach is to assume SU(2) U(1) U(1) broken symmetry, i.e. a chiral field U, and a scalar field h, with arbitrary couplings. A Manohar (UCSD) 17 Mar 2015 / Vienna 5 / 46

6 Notation Fields are three generations of fermions L : q r, l r, R : u r, d r, e r r = 1,..., n g = 3 the scalar doublet H, and SU(3) SU(2) U(1) gauge fields. L = L SM + 1 Λ L(5) + 1 Λ 2 L(6) +... note the dots Λ is the scale of new physics, and assume Λ > v Power Counting L (6) m2 H Λ 2 L (8) m4 H Λ 4 L (6) L (6) m4 H Λ 4 m H a typical electroweak scale (v, m t, m Z ) A Manohar (UCSD) 17 Mar 2015 / Vienna 6 / 46

7 SMEFT: dim 5 L 5 = c rs ( H i l iαr ) ( H j l jβs ) ɛ αβ i : SU(2) index α : Lorentz index r : flavor index Violates lepton number L = 2. This is the operator that gives Majorana mass to the neutrino. The scale is the seesaw scale and is high. Not relevant for physics at LHC. A Manohar (UCSD) 17 Mar 2015 / Vienna 7 / 46

8 SMEFT: dim 6 Lots of operators at dimension six. 59 operators that preserve baryon number, and 4/5 that violate baryon number. Buchmuller and Wyler, Nucl.Phys. B268 (1986) 621 Grzadkowski et al. JHEP 1010 (2010) 085 Flavor indices run over n g = 3 values baryon number conserving operators, including flavor indices CP-even and 1149 CP-odd operators. Alonso, Jenkins, AM, and Trott, JHEP 1404 (2014) 159. Notation: L : q r, l r, R : u r, d r, e r r = 1,..., n g = 3 H j, Hj = ɛ ij H j X µν : G A µν, W I µν, B µν A Manohar (UCSD) 17 Mar 2015 / Vienna 8 / 46

9 Fierz Identities Write down all possible gauge invariant operators of dimension 6. Use Fierz identities: ψ 1 γ µ P L ψ 2 ψ 3 γ µ P L ψ 4 = ψ 3 γ µ P L ψ 2 ψ 1 γ µ P L ψ 4 So in the four-quark operators: p γµ q rβj q λk s γ µq tσl q λk s γµ q rβj q αi p γ µq tσl q αi q γ µ Γ q q γ µ Γ q, Γ Γ 1 1, T A T A, τ I τ I, T A τ I T A τ I Two independent contractions: Q (1) qq = q p γ µ q r q s γ µ q t 1 1 Q (3) qq = q p γ µ τ I q r q s γ µ τ I q t τ I τ I A Manohar (UCSD) 17 Mar 2015 / Vienna 9 / 46

10 not needed: Q (8) qq = q p γ µ T A q r q s γ µ T A q t T A T A Q (3,8) qq = q p γ µ τ I T A q r q s γ µ τ I T A q t τ I T A τ I T A [T A] α [T A] λ = 1 β σ 2 δα σ δβ λ 1 δβ α 2N δλ σ 1 + swap color c [τ I] i [τ I] k = 2δl i δk j δj i δk l 1 + swap weak j l 1 swap both swap weak swap color A Manohar (UCSD) 17 Mar 2015 / Vienna 10 / 46

11 SM Equations of Motion D 2 H k λv 2 H k + 2λ(H H)H k + q j Y u u ɛ jk + d Y d q k + e Y e l k = 0, for the Higgs field (mix H and fermion operators) i /D q j = Y u u H j + Y d d H j, i /D d = Y d q j H j, i /D u = Y u q j H j, i /D l j = Y e eh j, i /D e = Y e l j H j, for the fermion fields, and [D α, G αβ ] A = g 3 j A β, [Dα, W αβ ] I = g 2 j I β, Dα B αβ = g 1 j β, for the gauge fields, where [D α, F αβ ] is the covariant derivative in the adjoint representation. A Manohar (UCSD) 17 Mar 2015 / Vienna 11 / 46

12 Equations of Motion Can use classical equations of motion in a quantum field theory. No quantum corrections needed. Green s functions can change, but the S-matrix does not. The two versions of the operator do not agree diagram by diagram. Only the total contribution to the S-matrix is unchanged. A Manohar (UCSD) 17 Mar 2015 / Vienna 12 / 46

13 SMEFT Operators 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 A Manohar (UCSD) 17 Mar 2015 / Vienna 13 / 46

14 Dimension Six Operators Q G Q G 1 : X 3 f ABC Gµ Aν Gν Bρ Gρ Cµ ABC Aν f G µ Gν Bρ Gρ Cµ 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 W ɛ IJK Wµ Iν Wν Jρ Wρ K µ Q dh (H H)( q pd r H) Q W IJK Iν ɛ W µ Wν Jρ Wρ K µ 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 ) A Manohar (UCSD) 17 Mar 2015 / Vienna 14 / 46

15 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 ψ 4 JJ, (LR)(LR), (LR)(RL)

16 In the broken phase: H = [ ( ϕ + 1 v + h + ϕ 0 ) 2 ] X 2 H 2 : h γγ, h gg ψ 2 XH : l r σ µν e s F µν µ eγ ψ 2 H 3 : higgs coupling mass = 3 1 A Manohar (UCSD) 17 Mar 2015 / Vienna 16 / 46

17 Power Counting for the RGE Amplitudes and anomalous dimensions obey power counting: µ d dµ C(6) C (6) µ d dµ C(8) C (8) + [C (6)] 2 In the SM, because of the dimension two operator H H, have µ d dµ C(4) C (4) + m 2 H C(6) +... equivalently m 2 H v 2 SM parameter RG evolution affected. A Manohar (UCSD) 17 Mar 2015 / Vienna 17 / 46

18 Anomalous Dimension Matrix g 3 X 3 H 6 H 4 D 2 g 2 X 2 H 2 yψ 2 H 3 gyψ 2 XH ψ 2 H 2 D ψ g 3 X 3 1 g H 6 2 g 6 λ λ, g 2, y 2 g 4, g 2 λ, λ 2 g 6, g 4 λ λy 2, y 4 0 λy 2, y 4 0 H 4 D 2 3 g 6 0 g 2, λ, y 2 g 4 y 2 g 2 y 2 g 2, y 2 0 g 2 X 2 H 2 4 g g 2, λ, y 2 0 y yψ 2 H 3 5 g 6 0 g 2, λ, y 2 g 4 g 2, λ, y 2 g 2 λ, g 4, g 2 y 2 g 2, λ, y 2 λ, y 2 gyψ 2 XH 6 g g 2 1 g 2, y ψ 2 H 2 D 7 g 6 0 g 2, y 2 g 4 y 2 g 2 y 2 g 2, λ, y 2 y 2 ψ 4 8 g g 2 y 2 g 2, y 2 g 2, y 2 Structure of anomalous dimension matrix. Jenkins, Trott, AM: Many entries exist because of EOM A Manohar (UCSD) 17 Mar 2015 / Vienna 18 / 46

19 Naive Dimensional Analysis Georgi, AM: NPB234 (1984) 189 Jenkins, Trott, AM: Related recent work by Buchalla et al. arxiv: L = f 2 Λ 2 ( ψ f Λ ) a ( H f ) b ( yh Λ NDA weight w powers of f 2 in denominator. ) c ( ) D d ( ) gx e Λ Λ 2, Λ = 4πf L = f 2 Λ 2 ( H f ) 6 = Λ 2 (H H) 3 f 4, w = 2 γ ij ( ) λ nλ ( ) y 2 ny ( ) g 2 ng 16π 2 16π 2 16π 2, N = n λ + n y + n g N = 1 + w i w j A Manohar (UCSD) 17 Mar 2015 / Vienna 19 / 46

20 Familiar Example: b sγ Use Then O q = bγ µ P L u uγ µ P L s w = 1 O g = g 16π 2 m b b σ µν G µν P L s w = 0 µ d [ Cq dµ C g ] [ = L L + 1 L 1 L ] [ Cq C g ] where L is the number of loops of the diagram. A Manohar (UCSD) 17 Mar 2015 / Vienna 20 / 46

21 Tree-Loop Mixing (incorrect) Claim Operators can be classified into tree and loop Based on some notion of minimal coupling The mixing of tree and loop operators vanishes at one loop Bauer et al. for HQET provides a counterexample A Manohar (UCSD) 17 Mar 2015 / Vienna 21 / 46

22 γ 68 as a counterexample [ µ d dµ C eb = 1 pr 16π 2 [ µ d dµ C ew = 1 pr 16π 2 µ d dµ C ub pr = 1 16π 2 µ d dµ C uw pr = 1 16π 2 [ [ 4g 1 N c (y u + y q )C (3) 2g 2 N c C (3) lequ prst 4g 1 (y e + y l ) C (3) 2g 2 C (3) Class 6 are the magnetic dipole operators: Class 8 are four-fermion operators Q (3) lequ stpr lequ prst [Y u ] ts ] +... lequ stpr [Y u ] ts ] +... [Y e ] ts ] +... [Y e ] ts ] +..., Q eb pr = ( l p,a σ µν e r )H a B µν lequ = ( l pσ j µν e r )ɛ jk ( q s k σ µν u t ) = 4( l pe j r )ɛ jk ( q s kα u αt ) 8( l pu j αt )ɛ jk ( q s kα e r ) = 4Q (1) lequ 8( l j pu αt )ɛ jk ( q kα s e r ) A Manohar (UCSD) 17 Mar 2015 / Vienna 22 / 46

23 Full anomalous dimension matrix computed, including all Yukawa couplings, not just y t. Alonso, Jenkins, Trott, AM: JHEP 1310 (2013) 087, JHEP 1401 (2014) 035, arxiv: Group theory can be quite complicated the penguin graph gives an anomalous dimension 7 pages long. Concentrate on a small part of the result. A Manohar (UCSD) 17 Mar 2015 / Vienna 23 / 46

24 There are some big numbers: The evolution of the H 6 coefficient is µ d dµ C H = 1 16π 2 [ 108 λ C H 160 λ 2 C H + 48 λ 2 C HD ] +... Independent of normalization of C H. For m H 126 GeV, 108 λ/(16π 2 ) 0.1. A Manohar (UCSD) 17 Mar 2015 / Vienna 24 / 46

25 X 2 H 2 (Gauge-Higgs) Operators Grojean, Jenkins, Trott, AM: JHEP 1304 (2013) 016 Look at only 4 operators (since there are 59 operators) C HW c W, etc. rescaled by g i g j /(2Λ 2 ) O G = g2 3 2 Λ 2 H H G A µ νg A µ ν, O B = g2 1 2 Λ 2 H H B µ ν B µ ν, O W = g2 2 2 Λ 2 H H W a µ νw a µ ν, O WB = g 1 g 2 2 Λ 2 H τ a H W a µ νb µ ν, Also CP-odd versions, which do not interfere with the SM amplitude. When you expand them out in the broken phase, H h + v, get h γγ, h γz and gg h. A Manohar (UCSD) 17 Mar 2015 / Vienna 25 / 46

26 Considered these operators in an earlier work, and an explicit model that produced these operators. AM, M.B. Wise, PLB636 (206) 107, PRD74 (2006) An exactly solvable model that produces only the O W, O WB and O B operators and the W 3 operator: AM: PLB 726 (2013) 347 L = L SM + D µ S α D µ S α V, the usual Standard Model Lagrangian L SM, the S α, α = 1,..., N kinetic energy term, and the potential V = m 2 S S α S α + λ 1 N H H S α S α + λ 2 N H τ a H S α τ a S α + λ 3 N S α S α S β S β + λ 4 N S α τ a S α S β τ a S β A Manohar (UCSD) 17 Mar 2015 / Vienna 26 / 46

27 c W = (λ 1/λ 3 ) 48 log Λ2 3 Φ, c B = (λ 1/λ 3 )Y 2 S 12 log Λ2 3 Φ, c WB = (λ 2/λ 4 )Y S 24 log Λ2 4 Φ c W 3 = N g π 2 Φ. disproves claims that these are loop-supressed operators, and smaller than other contributions such as four-fermion operators. All terms depend on RG invariant quantities. A Manohar (UCSD) 17 Mar 2015 / Vienna 27 / 46

28 Constraint on c WB from the S parameter. c WB = 1 Λ 2 8π v 2 S. The gg h amplitude gets contribution from c G For h γγ, For h γz, c γγ = c W + c B c WB c γz = c W cot θ W c B tan θ W c WB cot 2θ W, A Manohar (UCSD) 17 Mar 2015 / Vienna 28 / 46

29 Contribution to the Higgs Decay Rate Γ(h γγ) Γ SM (h γγ) and I γ Note the 4π π2 v 2 c γγ Λ 2 I γ Similar expressions for gg h and h γz. 2 A Manohar (UCSD) 17 Mar 2015 / Vienna 29 / 46

30 Brief Experimental Summary ATLAS: µ γγ = 1.17 ± 0.27 CMS: µ γγ = 1.14 ± 0.25 Naive combination of these results (not recommended) gives If due to c γγ : µ γγ ± 0.18 v 2 Λ 2 c γγ(m h ) 0.086, ± The second solution is preferred. The first solution is when c γγ switches the sign of the standard model h γγ amplitude. The experiments are sensitive to these effects if the new physics scale Λ is near a few TeV. (4.5 TeV) A Manohar (UCSD) 17 Mar 2015 / Vienna 30 / 46

31 Magnetic Dipole Operator Ceγ = 1 C eb 1 C ew rs g 1 rs g 2 rs L = ev 2 Ceγ rs e r σ µν P R e s F µν + h.c. where r and s are flavor indices ({e e, e µ, e τ } {e, µ, τ}) and Ceγ = {Y (s) + e ( csc2 θ W + 14 )} sec2 θ W Ceγ rs rs + 2Ceγ rv [Y e Y e] vs + (2 sin θ W cos θ W ) [Y ey e ] rw C ez ( ) cos2 θ W [Y ey e ] rw Ceγ + ws e2 (12 cot 2θ W ) C ez rs cot θ W [Y ( ) e] rs CHWB + ic H W B ws + 8 ) 3 e2 [Y e] rs (C γγ + i Cγγ + e (cot 2 θ W 5 3 tan θ W + 16[Y u ] wv C (3) lequ rsvw as corrected by Signer and Pruna, arxiv:1408:3565 ) ) [Y e] rs (C γz + i CγZ A Manohar (UCSD) 17 Mar 2015 / Vienna 31 / 46

32 The current experimental limit BR(µ eγ) < from the MEG experiment implies at the low energy scale µ m µ. This bound implies v 2 me Ceγ µe TeV 2 m t C (3) m lequ TeV 2 e µett using the estimate ln(λ/m H )/(16π 2 ) 0.01 for the renormalization group evolution, and assuming that this term is the only contribution to at low energies. Ceγ µe A Manohar (UCSD) 17 Mar 2015 / Vienna 32 / 46

33 The anomalous magnetic moment of the muon is δa µ = 4m µv 2 Re Ceγ µµ which yields the limits C HWB 0.6 TeV 2, C γγ 4 TeV 2, m t Re C (3) m lequ µ 7 TeV 2, µµtt assuming that each of these is the only contribution to Ceγ µµ. The bound on the electric dipole moment of the electron translates to the limits CH W B TeV 2, C γγ TeV 2, m t Im C lequ m e TeV 2 using the recently measured upper bound d e < e cm from the ACME collaboration, again assuming each of these terms is the only contribution. eett A Manohar (UCSD) 17 Mar 2015 / Vienna 33 / 46

34 Minimal Flavor Violation MFV based on the structure of flavor violation in the SM. i SU(3) i i = Q, U, D, L, E Y D U Q Y D U D The RGE preserves MFV, since it is a symmetry But if MFV is violated by dimension-six operators, then MFV feeds into all the sectors via the RGE Allows one to test MFV in a model-independent way MFV not compatible with GUTS. Accidental symmetry of the SM. Recent polarization B modes indicate a GUT scale. A Manohar (UCSD) 17 Mar 2015 / Vienna 34 / 46

35 Empirical Observation ( ) Ceγ C γγ + i Cγγ, Ceγ no ( ) C γγ i Cγγ, Ceγ f (z) depends only on z but not z Find this after adding all graphs and using the EOM. Individual contributions are not holomorphic, and only the total respects holomorphy. Observation: 1-loop anomalous dimension matrix respects holomorphy to a large extent. Using EOM, so equivalent to computing (on-shell) S-matrix elements. A Manohar (UCSD) 17 Mar 2015 / Vienna 35 / 46

36 Holomorphy 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 Complex self-duality condition in Minkowski space. A Manohar (UCSD) 17 Mar 2015 / Vienna 36 / 46

37 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. These transform as (0, 1 2 ) or (0, 1) under the Lorentz group, i.e. only under the SU(2) R part of SU(2) L SU(2) R.

38 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 Some of the n operators are complex. A Manohar (UCSD) 17 Mar 2015 / Vienna 38 / 46

39 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 = γ hh γ hh = γ hh γ hn = γ hn only need to look at: ( γhh γ hh γ hn ) γ nh γ nh γ nn

40 0: Vanishes by NDA, i.e. NDA gives a negative loop order : There is no one-loop diagram (including from EOM) h F : Holomorphic. Nonholomorphic terms forbidden by NDA and flavor symmetry 0: Vanishes by explicit computation, after adding all contributions. Individual graphs need not vanish. h: Holomorphic, by explicit computation : Non-zero Q ew = ( l p σ µν e r )τ I HW I µν Q (1) lequ = ( l j pe r )ɛ jk ( q k s u t ) but NDA says only order y 2. Ċ ew Y e Y e Y uc (1) lequ A Manohar (UCSD) 17 Mar 2015 / Vienna 40 / 46

41 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 u Y d, Y uy e h F 0 JJ 0 0 Y u Y e,d ψ 2 H 3 0 Y u,d,e h h H 6 0 H 4 D ψ 2 H 2 D

42 The 1 1 block is holomorphic γ hh = 0 The 1 2 block vanishes except for the red terms proportional to Y u Y e or Y u Y d. L Y = q j Y d d H j q j Y u u H j l j Y e e H j + h.c. H j = ɛ ij H j ψ 2 H 3 behaves to some extent like a holomorphic operator. one entry present even if Yukawa couplings set to zero. A Manohar (UCSD) 17 Mar 2015 / Vienna 42 / 46

43 RGE of SM parameters Recall that µ d dµ C(4) m 2 H C(6) ( ) µ d dµ τ = µ d 4π dµ gx 2 i θ X 2π = 2m2 H πgx 2 C HX,+ where θ-terms are normalized as L (θ X g 2 X /32π2 )X X and X {SU(3), SU(2), U(1)}. τ is the SUSY holmorphic gauge coupling A Manohar (UCSD) 17 Mar 2015 / Vienna 43 / 46

44 Entry: Some Numerology ) Ċ H = 3g2 (g g2 2 12λ Re(C HW,+ ) ) 3g1 (g g2 2 4λ Re(C HB,+ ) ( ) 3g 1 g 2 g1 2 + g2 2 4λ Re(C HWB,+ ) +... The C HB,+ and C HWB,+ terms vanish if g g2 2 = 4λ: m 2 H = 2m2 Z = (129 GeV)2, and the C HW,+ term vanishes if g g2 2 = 12λ: If g g2 2 = 4λ: m 2 H = 2 3 m2 Z m2 W = (119 GeV)2, Ċ H = 6g 2 1 g2 2 Re(C HW,+) +... A Manohar (UCSD) 17 Mar 2015 / Vienna 44 / 46

45 No explanation for this at present. A Manohar (UCSD) 17 Mar 2015 / Vienna 45 / 46

46 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. Does it hold in a more general gauge theory? Does any of it extend beyond one loop? A Manohar (UCSD) 17 Mar 2015 / Vienna 46 / 46