The Higgs Boson Mass at 2 Loops in the Finely Tuned Split Supersymmetric Standard Model

SLAC-PUB-10702 hep-ph/0408240 September 2004

The Higgs Boson Mass at 2 loops in the Finely Tuned Split Supersymmetric Standard Model

Michael Binger Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, USA e-mail: [email protected]

Abstract

The mass of the Higgs boson in the ﬁnely tuned Split Supersymmetric Stan- dard Model is calculated. All 1 loop threshold eﬀects are included, in addition to the full RG running of the Higgs quartic coupling through 2 loops. The 2 loop corrections are very small, typically less than 1GeV. The 1 loop threshold corrections to the top yukawa coupling and the Higgs mass generally push the Higgs mass down a few GeV.

Work supported by the Department of Energy contract DE-AC02-76SF00515. 1 Introduction

The so-called gauge hierarchy problem of the standard model (SM) of particle physics has been a fruitful source of inspiration for beyond the SM physics. Most notably, a main reason for the prominence of supersymmetry was its natural solution to this problem. In recent years, additional pieces of evidence for the existence of supersymmetry have arisen from gauge coupling unification and from dark matter, although these successes have been partially offset by difficulties with FCNC and CP violation which arise from light SUSY scalars. Thus, it may be reasonable to abandon the original motivation for SUSY and consider the implications of a theory which main- tains all of the successes of the MSSM, except the hierarchy problem, and does away with many of the difficulties. This proposal, called finely tuned, or split SUSY, has recently appeared [1, 2], and some phenomenology has been discussed [3, 4, 5, 6, 7]. In split supersymmetry, a single Higgs scalar is fine tuned to be light, with the under- standing that the fine tuning will be resolved by some anthropic-like selection effects. This approach may have a natural realization within inflation and string theory[8]. Basically, an almost infinite landscape of vacua may contain a small percentage with the desired fine tuned parameters, which are necessary for life and the properties of our universe.

2 The Higgs Mass

The starting point for our analysis is the split SUSY lagrangian

2 λ 2 = m H†H (H†H) + F Hˆ †Qu + F H†Qd + F H†Le + h.c. L − 2 u d e M M M 1 BB 2 W AW A 3 gaga µHT ǫH − 2 − 2 − 2 − u d

H† e e a a f f He †e fa fa κuσ W + κ′ B Hu + κdσ W + κ′ B Hd + h.c., (1) − √2 u √2 − d f f +e f0 T f e f where Hˆ = iσ H∗ and H =(H ,H ) . − 2 The following discussion will be similar to the work of [9][10]. After electroweak symmetry breaking, the bare Higgs mass is Mh0 (µ) = λ0(µ)v0(µ) where v0 = √2 H0 . The pole mass is related to the bare mass throughq h i 2 2 Th M = M +Re Σh(Mh,µ)+3 , (2) h h0 v where Σh is the self energy and Th is the tadpole. The bare vev is related to the renormalized vev via muon decay :

2 2 Th v0(µ)= vF 1 + ΠWW (0) + E 2 2 , (3) − MH vF !

2 where v = 1/ √2G 246.22GeV, µ is the renormalization scale, and E rep- F F ≈ resents universalq vertex and box corrections to muon decay in the standard model. Finally the bare coupling is related to the MS coupling by

β λ (µ)= λ(µ)+ λ C , (4) 0 2λ UV where C = 1 γ + log4π. Putting these together, one ﬁnds UV ǫ − E

Mh = λ(µ)vF (1 + δh(µ))

q1 Th βλ δh(µ) = Πh(Mh)+ 2 + E + ΠWW (0) + CUV . (5) 2 M vF 2λ h This formula includes all 1-loop threshold and RG corrections, and the can be improved to include the leading RG 2-loop corrections. The scale µ should be chosen near Mh, although at one loop the µ (and CUV ) dependence exactly cancels from Eq.(5). The result for δh(µ) in the SM has been given in [9]. The corrections for split SUSY will be given below. The algorithm used will be as follows. In the next section, the chargino and neutralino mass eigenstates and mixing matrices will be defined, since the results are written in terms of them. Next, the top yukawa coupling will carefully be extracted from the top mass, including all one loop threshold corrections. The bottom and tau yukawas are also included, although their effect is slight. The 2 loop running of the Higgs quartic coupling will be given. Solving the coupled RG equations for g1,g2,g3, FU , FD, FL, κu, κd, κu′ , κd′ yields the required inputs to solve the λ RGE. Once this is done, all that remains is to include the finite threshold corrections in Eq.(5).

2.1 The Chargino and Neutralino mass spectrum The chargino and neutralino mass matrix diagonalization proceeds similar to the MSSM[11]. The mass matrices are given by

κuv M2 √2 X = κdv (6) √2 µ !

and ′ ′ κdv κuv M1 0 √2 √2 −κ v κ v 0 M d u 2 √2 √2 ′ − Y = κdv κdv . (7) √2 √2 0 µ ! − ′ − κuv κuv µ 0 √2 − √2 − 3 + These are diagonalized by matrices U, V in the chargino sector (χi χi−) and N in 0 the neutralino sector (χi ):

+ + 0 0 χi = Vijψj χi− = Uijψj− χi = Nijψj , (8) where the gauge eigenstates are

+ + W W − 0 0 0 T ψj = + ψj− = ψj =(B, W3, Hd , Hu) . (9) Hu ! Hd− ! f f e f f f The matrices U, V, Nfare speciﬁed byf

1 (N) (N) (N) (N) (N) N ∗Y N − = M = diag M1 , M2 , M3 , M4 1 (N{) 2 } NY †Y N − = (M ) 1 (C) (C) (C) U ∗XV − = M = diag M1 , M2 1 (C) 2 { T } VX†XV − = (M ) = U ∗XX†U (10)

2.2 The Top Quark Yukawa coupling and pole mass Since the Higgs mass is most sensitive to the top yukawa coupling, it is important to carefully extract this from the pole mass. In the canonical work of Hempﬂing and Kniehl [10], the authors found the following relation between the MS yukawa coupling in the Standard Model (SM) and the pole mass :

Mt yt(µ)= √2 (1 δt(µ)). (11) vF − The correction term, which comes from relating the bare vev to the Fermi constant in muon decay, is given by

ΠWW (0) E βyt δt(µ) = ReΠt(Mt)+ 2 + + CUV . (12) 2MW 2 2yt

In this formula, Πt represents the top self energy and E is the vertex and box corrections to muon decay, neither of which receive new contributions in split SUSY. However, the W boson self energy, ΠWW , does receive corrections, which we calculate below. The UV divergence, CUV , multiplying the top yukawa beta function βyt comes from the relation between the bare and MS yukawa coupling and is cancelled by the divergent parts of Πt, ΠWW , and E. It turns out that in the range of Higgs masses of interest, the SM contribution is approximately constant with respect to Mt and Mh and is well approximated by δSM (M , M )= 0.044 (M (GeV) 125)0.00002 + (M (GeV) 178)0.0008. t t h − − h − t − Now we turn to the split SUSY corrections.

4 0 1 ± The W ± χi χj vertex is given by igγµ(LijPL +RijPR), with Lij = √2 Ni4Vj∗2 + − − 1 − Ni2Vj∗1 and Rij = √2 Ni∗3Uj2 + Ni∗2Uj1. From this we derive the split SUSY corrections involving charginos and neutralinos :

4 2 2 2 (C,N) 2 2 2 a 16π ΠWW (0) = 2MW CUV X2(SS)+ g (LijLij∗ + RijRij∗ ) a (log 1/2) − µ2 − Xi=1 jX=1 h b2 a2b2 a2 + b2(log 1/2) + log µ2 − a2 b2 b2 i − 2 2 ab 2 a 2 b + 2(LijRij∗ + RijLij∗ ) a (log 1)+ b (log 1) ,(13) a2 b2 − µ2 − µ2 − ! − h i (C) (N) where we used the shorthand a = Mj , b = Mi and X2(SS) is given in Eq.(14). (C,N) SS ΠW W (0) The resulting correction term δt (Mt) = 2 depends on the soft gaugino mass 2MW terms, tan β, and the scalar mass scale M . Generically however δSS(M ) <0.01. S | t t | Analogous corrections to the bottom and tau yukawa couplings are numerically∼ unimportant.

2.3 The 2-loop running of the Higgs Quartic Coupling It is useful to deﬁne the following invariants involving the standard model yukawas and the new split susy yukayas :

Y2(SM) = Tr 3FU† FU +3FD† FD + FL†FL h 2 2 i 2 Y4(SM) = Tr 3(FU† FU ) + 3(FD† FD) +(FL†FL) h 3 3 3i Y6(SM) = Tr 3(FU† FU ) + 3(FD† FD) +(FL†FL) h i 17 2 9 2 2 Y (SM) = g + g +8g Tr(F † F ) G 20 1 4 2 3 U U 1 2 9 2 2 3 2 2 + g + g +8g Tr(F † F )+ (g + g )Tr(F †F ) 4 1 4 2 3 D D 4 1 2 L L 2 2 2 2 X2(SS) = 3(κu + κd)+ κu′ + κd′ 4 4 2 2 2 2 2 2 X4(SS) = 5(κu + κd)+2κuκd + 2(κuκu′ + κdκd′ ) +(κu′ + κd′ ) . (14) The running of the quartic coupling λ of the Higgs boson is governed by ∂λ 1 1 β = β(1) + β(2) + (15) λ ≡ ∂µ 16π2 λ (16π2)2 λ ··· The 1 loop beta function is given by [2] 1 27 9 9 β(1) = 12λ2 9λ g2 + g2 + g4 + g2g2 + g4 +4λY (SM) 4Y (SM) λ − 5 1 2 100 1 10 1 2 4 2 2 − 4 + 2λX (SS) X (SS). (16) 2 − 4 5 The 2 loop result is conveniently divided into two terms,

(2) (2) (2) βλ = βλ (SM ′)+ βλ (SS), (17) where SS is the new split SUSY contribution and SM ′ denotes the standard model result modiﬁed to include gauginos and higgsinos in gauge boson self energies. This is given by replacing the number of generations in the SM result :

Ng(1)=3+3/10 = 33/10 Ng(2)=3+3/2=9/2 (18)

(2) 3 2 βλ (SM ′) = 78λ 24λ Y2(SM) λY4(SM) 42λTr(FU† FU FD† FD)+20Y6(SM) − − − − 2 2 12Tr[F † F (F † F + F † F )F † F ]+10λY (SM)+54λ(g + g /5) − U U U U D D D D G 2 1 313 687 117 λ 10N (2) g4 +2N (1) g4 g2g2 − 8 − g 2 − 200 g 1 − 20 2 1 h i 2 2 2 8 2 2 2 2 64g Tr[(F † F ) +(F † F ) ] g Tr[2(F † F ) (F † F ) + 3(F †F ) ] − 3 U U D D − 5 1 U U − D D L L 3 4 2 63 2 171 2 27 2 9 2 g2Y2(SM)+ g1 g2 g1 Tr(FU† FU )+ g2 + g1 Tr(FD† FD) − 2 " 5 − 50 5 10 33 2 9 2 497 6 97 8 4 2 + g2 g1 Tr(FL†FL) + 8Ng(2) g2 + Ng(2) g2g1 5 − 2 # 8 − − 40 5 717 8 531 24 + N (1) g2g4 + N (1) g6 (19) − 200 5 g 2 1 − 1000 25 g 1 The new split SUSY yukawas contribute

(2) 2 λ 4 4 2 2 2 2 2 2 4 4 2 2 β (SS) = 12λ X (SS) 5(κ + κ )+44κ κ + 2(κ κ′ + κ κ′ )+ κ′ + κ′ 12κ′ κ′ λ − 2 − 4 u d u d u u d d u d − u d h 1 6 6 6 6 2 2 2 2 4 2 4 2 80κuκdκu′ κd′ + 47(κu + κd)+5(κu′ + κd′ )+7κuκd(κu + κd)+11(κuκu′ + κdκd′ ) − 2" i 2 2 2 2 2 2 2 2 + 21κuκd(κu′ + κd′ )+38κuκdκu′ κd′ (κu + κd)+42κuκdκu′ κd′ (κu′ + κd′ )

2 4 2 4 2 4 4 2 2 2 2 2 + 17(κuκu′ + κdκd′ + κu′ κd′ + κu′ κd′ )+19κu′ κd′ (κu + κd) # 15 1 + λ (g2 + g2)X (SS)+8g2(κ2 + κ2) 4 2 5 1 2 2 u d h i 2 4 4 2 2 2 4 3 2 2 4g 5(κ + κ )+2κ κ +(κ κ′ + κ κ′ ) g X (SS)+36(κ + κ ) − 2 u d u d u u d d − 2 4 2 u d h i h i 3 2 2 2 2 2 2 9 4 + g g 21(κ + κ ) (κ′ + κ′ ) g X (SS). (20) 10 1 2 u d − u d − 100 1 2 h i 2.4 The Higgs Self Energy and Tadpole Corrections The Higgs tapole and self energy is written in terms of the mixing matrices which appear in the Feynman rules. The interaction Lagrangian in terms of the physical

6 mass eigenstate Dirac and Majorana fermions are given by

h C C h 0 N C 0 int = ψ±(PLL + PRR )ψ± + ψ (PL(R )∗ + PRR )ψ , (21) L −√2 i ij ij j 2 i (ij) (ij) j and the mixing matrices are given by C C Rij = (Lji)∗ = κuVi2Uj1 + κdVi1Uj2 N R = (κ N κ′ N )N (κ N κ′ N )N ij u i2 − u i1 j4 − d i2 − d i1 j3 1 RN = (RN + RN ). (22) (ij) 2 ij ji (C) (N) The Higgs tadpole is determined by setting iTh = i(Th + Th ) equal to the tadpole diagrams involving charginos and neutralinos. The result is 2 (M C )2 16π2T (C) = 2√2 Re RC (M C )3 C log i +1 (23) h − ii i UV − µ2 Xi=1 h i 4 (M C )2 16π2T (N) =2 Re RN (M N )3 C log i +1 . (24) h (ii) i UV − µ2 Xi=1 h i The Higgs self energies are easily written in terms of the canonical 1 loop basis 2 functions A(m), B0(k , M1, M2)[12].

2 2 (C) 2 1 C 2 C 2 2 2 2 2 16π Σh (p ) = ( Lij + Rij ) A(Mi)+ A(Mj)+(Mi + Mj p )B0(p , Mi, Mj) " 2 | | | | − i,jX=1 C C 2 + 2Re MiMjRij(Lij)∗B0(p , Mi, Mj) (25) #

4 2 (N) 2 N 2 2 2 2 2 16π Σh (p ) = R(ij) A(Mi)+ A(Mj)+(Mi + Mj p )B0(p , Mi, Mj) "| | − i,jX=1 N N 2 + 2Re MiMjR(ij)(R(ij))∗B0(p , Mi, Mj) (26) #

Re Σh(Mh) Using Πh(Mh)= 2 , the above results contribute Mh (C) (N) SS 1 (C) (N) Th + Th (CN) δh (µ)= Πh (Mh) + Πh (Mh)+ 2 + ΠWW (0) . (27) 2 M vF h Combining this with the SM results of [9], the total threshold correction is given by SM SS δh(µ)= δh (µ)+ δh (µ). (28)

A useful check of these results is the cancellation of divergences CUV in Eq.(5). This involves repeated use of the deﬁnitions in Eq.(10), and has been explicitly veri- ﬁed.

7 3 Results for the Higgs Mass

The corrections to the Higgs mass considered in this paper are of three varieties:

Top Yukawa. The 1-loop threshold corrections to the Yukawa coupling initial • value given in Eq.(13) are ampliﬁed because yt is raised to the fourth power (1) in βλ . The size of the correction grows with the soft gaugino and higgsino masses. For soft masses less than a TeV, the downward shift in the Higgs mass is < 2GeV. ∼ 2 loop running of λ. The 2 loop correction to the beta function is numerically • small, and the SM and split SUSY contributions tend to cancel each other in (2) most of parameter space. The shift in the Higgs mass due to including βλ is less than 1GeV.

Threshold corrections. The correction given in Eq.(28) typically pushes • down the Higgs mass by 1 4GeV, the larger shift ocurring for small tan β and − small Ms. Typically, the SM contributes most of this shift, with the split SUSY corrections < 1GeV. ∼ 5 In regions of parameter space with small tan β, Ms 10 GeV, and large soft masses of order 1TeV, the corrections discussed above can∼ shift the Higgs mass down 5GeV. However, generically the corrections are < 2GeV. ∼ Of course, all of these small corrections are nearly∼ wiped out by the large uncer- tainties in the top mass M = 178 4.3GeV, which translates into an uncertainty of t ± about 10GeV in Mh.

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