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Lecture III Higgs in Supersymmetric Models

Outline

The significance of the TeV-scale—Part 2 •

The MSSM Higgs sector at tree-level •

Saving the MSSM Higgs sector—the impact of radiative corrections •

LHC searches for the heavy Higgs states of the MSSM •

Alignment without decoupling in the MSSM? •

Beyond the MSSM Higgs sector • The significance of the TeV scale—Part 2

Despite the tremendous success of the (SM) of , we know that there are some missing pieces.

Neutrinos are not massless—perhaps suggestive of a new high energy • (see-saw) scale.

Dark is not accounted for. • There is no explanation for the asymmetry of the . • There is no explanation for the inflationary period of the very early universe. • The gravitational interaction is omitted. •

Consequently, the SM is an low-energy effective theory. New high energy scales must exist where more fundamental physics resides. Indeed, gravitational effects are likely to be relevant only at the Planck scale, M =(c~/G )1/2 1019 GeV , PL N ≃ 2 which arises as follows. Consider the gravitational , GN M /r, of a particle of M evaluated at its Compton , r = ~/(Mc). If this energy is of order the rest mass, Mc2 (i.e., when M M ), then ∼ PL pair production is possible and quantum gravitational effects can no longer be ignored.

Denote scale at which new physics enters by Λ. Predictions made by the SM depend on a number of parameters that must be taken as input to the theory. These parameters are sensitive to ultraviolet physics, and since the physics at very high energies is not known, one cannot predict their values.

However, one can determine the sensitivity of these parameters to the ultra- violet scale (which one can take as the Planck scale or some other high energy scale at which new physics beyond the Standard Model enters). In the 1930s, it was already appreciated that a critical difference exists between bosons and . are logarithmically sensitive to ultraviolet physics. Ultimately, this is due to the chiral of massless fermions, which implies that

δm m ln(Λ2/m2 ) . f ∼ f f

No such symmetry exists for bosons (in the absence of ), and consequently we expect quadratic sensitivity of the squared-mass to ultraviolet physics, δm2 Λ2 . B ∼ 2 2 2 1 2 2 In the SM, mh = λv and mW = 4g v imply that

2 mh 4λ 2 = 2 , mW g which one would expect to be roughly of (1). A 125 GeV O satisfies this expectation. However, the Higgs boson is a consequence of a spontaneously broken scalar potential, 2 1 2 V (Φ) = µ (Φ†Φ) + λ(Φ†Φ) , − 2 2 1 2 where we identify µ = 2λv in terms of the expectation value v of the Higgs field. The parameter µ2 is quadratically sensitive to Λ. Hence, to obtain v = 246 GeV in a theory where v Λ requires a significant fine-tuning ≪ of the ultraviolet parameters of the fundamental theory.

Indeed, the one-loop contribution to the squared-mass parameter µ2 would be expected to be of order (g2/16π2)Λ2. Setting this quantity to be of order of v2 (to avoid an unnatural fine-tuning of the tree-level parameter and the loop contribution) yields

Λ 4πv/g (1 TeV) ≃ ∼ O

A natural theory of electroweak (EWSB) would seem to require new physics at the TeV scale to govern the EWSB dynamics. The Principle of

In 1939, Weisskopf announces in the abstract to this paper that “the self-energy of charged obeying Bose statistics is found to be quadratically divergent”….

…. and concludes that in theories of elementary bosons, new phenomena must enter at an energy scale of order m/e (e is the relevant )—the first application of naturalness. Principle of naturalness in modern times How can we understand the magnitude of the EWSB scale? In the absence of new physics beyond the SM, its natural value would be the Planck scale (or perhaps the GUT scale or seesaw scale that controls masses). The alternatives are:

Naturalness is restored by a symmetry principle—supersymmetry—which • ties the bosons to the more well-behaved fermions.

The Higgs boson is an approximate —the only other known • mechanism for keeping an elementary scalar light.

The Higgs boson is a composite scalar, with an inverse length of order the • TeV-scale.

The naturalness principle does not apply. The Higgs boson is very • unlikely, but unnatural choices for the EWSB parameters arise from other considerations (landscape?). Avoiding quadratic sensitivity to Λ with elementary scalars

A lesson from history

The self-energy in classical E&M goes like e2/a (a 0), i.e., it → is linearly divergent. In quantum theory, fluctuations of the electromagnetic fields (in the “single electron theory”) generate a quadratic divergence. If these divergences are not canceled, one would expect QED to break down at an energy of order me/e far below the Planck scale.

The linear and quadratic divergences will cancel exactly if one makes a bold hypothesis: the existence of the (with a mass equal to that of the electron but of opposite ).

Weisskopf was the first to demonstrate this cancellation in 1934...well, actually he initially got it wrong, but thanks to Furry, the correct result was presented in an erratum.

A remarkable result:

e e

e

e

e

a b

The linear and quadratic divergences of a quantum theory of elementary fermions are precisely canceled if one doubles the particle spectrum—for every fermion, introduce an anti-fermion partner of the same mass and opposite charge.

In the process, we have introduced a new CPT-symmetry that associates a fermion with its anti-particle and guarantees the equality of their masses. TeV-scale supersymmetry to the rescue

Take the SM and double the particle spectrum. Introduce supersymmetry (SUSY), which dictates that for every boson, there is a fermion of equal mass and vice versa.

Supersymmetry relates the self-energy of the boson to the self-energy of its fermionic partner. Since the latter is only logarithmically sensitive to Λ, we conclude that the quadratic sensitivity of the boson squared-mass to ultraviolet physics must exactly cancel. Naturalness is restored!

Since no exist that are degenerate in mass with the corresponding SM particle, SUSY must be a broken symmetry. Conclusion:

The effective scale of SUSY-breaking cannot be much larger than of order 1 TeV, if SUSY is responsible for the EWSB scale. The minimal supersymmetric extension of the SM

The minimal supersymmetric extension of the Standard Model (MSSM) consists of the fields of the two-Higgs-doublet extension of the Standard Model and the corresponding superpartners. A particle and its superpartner together form a supermultiplet. The corresponding field content of the supermultiplets of the MSSM and their gauge quantum numbers are shown in the following table.

The enlarged Higgs sector of the MSSM constitutes the minimal structure needed to guarantee the cancellation of anomalies from the introduction of the superpartners. Moreover, without a second Higgs doublet, one cannot generate mass for both “up”-type and “down”-type (and charged ) in a way consistent with a holomorphic superpotential.

To account for supersymmetry (SUSY) breaking, one adds the most general set of soft-SUSY-breaking terms consistent with the SM gauge symmetry. content of the MSSM Super- Super- Bosonic Fermionic multiplets field fields partners SU(3) SU(2) U(1)

/ V8 g g 8 1 0 0 0 gauge/ V W ± , W W ± , W 1 3 0 b e Vb′ B f Bf 1 1 0 slepton/ L (ν , e−) (ν, e−) 1 2 1 b L L e L − Ec e˜+ ec 1 1 2 b e Re L squark/ Qb (uL, dL) (u, d)L 3 2 1/3 c c U u∗ u 3¯ 1 4/3 b e Re L − c c D d∗ d 3¯ 1 2/3 b eR L 0 0 Higgs/ Hd (H e, H−) (H , H−) 1 2 1 b d d d d − + 0 + 0 higgsino Hbu (Hu , Hu) (Heu , He u) 1 2 1 b e e The fields of the MSSM and their SU(3) SU(2) U(1) quantum numbers are listed. The is × × 1 given in terms of the third component of the T3 and U(1) hypercharge Y by Q = T3 + 2Y . For simplicity, only one generation of quarks and leptons is exhibited. For each lepton, quark, and Higgs super- multiplet, there is a corresponding anti-particle multiplet of charge-conjugated fermions and their associated scalar partners. The L and R subscripts of the squark and slepton fields indicate the helicity of the corresponding fermionic superpartners. More on cancellation

One-loop VVA and AAA triangle diagrams with three external gauge bosons, with fermions running around the loop, can contribute to a gauge anomaly.

(Here V refers to a γµ vertex and A refers to a γµγ5 vertex.) A theory that possesses gauge anomalies is inconsistent as a quantum theory.

To cancel the gauge anomalies, we must satisfy certain group theoretical constraints. W iW jB triangle Tr(T 2Y ) = 0 , ⇐⇒ 3 BBB triangle Tr(Y 3) = 0 . ⇐⇒ Example: contributions of the fermions to Tr(Y 3)

Tr(Y 3) = 3 1 + 1 64 + 8 1 1+8=0 . SM 27 27 − 27 27 − − + 0  Suppose we only add the (Hu , Hu). The resulting anomaly factor 3 3 is Tr(Y ) = Tr(Y )SM + 2, leading to a gauge anomaly. This anomaly is e e canceled by adding a second higgsino doublet with opposite hypercharge. A connection between SUSY-breaking and EWSB

Suppose that ΛSSB is the energy scale of a fundamental theory of SUSY- breaking. The soft-SUSY-breaking parameters of the MSSM are set at ΛSSB, and these parameters evolve via RG running down to the EWSB scale. This provides an elegant mechanism of radiatively-generated EWSB.

700 ~ ~ g 600 ~qL qR ~ t 500 L ~ tR ______400 2 √ µ2+m Hd 0 0

300 m ~ 1/2 lL ~ 200 W ~ m lR 0 ~ 100 B

0

-100 Hu

-200

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 The Higgs sector of the MSSM

The Higgs sector of the MSSM is a 2HDM, whose Yukawa couplings and Higgs potential are constrained by SUSY. Instead of employing to hypercharge-one scalar doublets Φ , it is more convenient to introduce a Y = 1 doublet 1,2 − H iσ Φ∗ and a Y = +1 doublet H Φ : d ≡ 2 1 u ≡ 2 H1 Φ0 H1 Φ+ H = d = 1 ∗ , H = u = 2 . d 2 u 2 0 H ! Φ1−! Hu! Φ2 ! d − The origin of the notation originates from the Higgs Yukawa Lagrangian:

= hij(¯ui uj H2 u¯i dj H1) hij(d¯i dj H1 d¯i uj H2)+h.c.. LYukawa − u R L u − R L u − d R L d − R L d

2 Note that the neutral Higgs field Hu couples exclusively to up-type quarks 1 and the neutral Higgs field Hd couples exclusively to down-type quarks. That is, the Higgs sector of the MSSM is a Type-II 2HDM. The Higgs potential of the MSSM is:

2 2 i i 2 2 i i 2 ij i j V = m + µ H ∗H + m + µ H ∗H m ǫ H H +h.c. 1 | | d d 2 | | u u − 12 d u 1 2 2 i i j j 2 1 2 i i 2 + g + g′ H∗H H ∗H + g H ∗H ,  8 d d − u u 2 | d u|   where ǫ12 = ǫ21 = 1 and ǫ11 = ǫ22 = 0, and the sum over repeated indices − 2 is implicit. Above, µ is a supersymmetric Higgsino mass parameter and m1, 2 2 m2, m12 are soft-SUSY-breaking masses. The quartic Higgs couplings are related to the SU(2) and U(1)Y gauge couplings as a consequence of SUSY. In the Higgs basis, we can identity the coefficients of the quartic terms of the scalar potential,

1 2 2 2 1 2 2 1 2 Z = Z = (g + g′ )cos 2β, Z = Z + (g g′ ) ,Z = Z g , 1 2 4 3 5 4 − 4 5 − 2 1 2 2 2 1 2 2 Z = (g + g′ )sin 2β, Z = Z = (g + g′ )sin2β cos 2β. 5 4 7 − 6 4

Here, we have used the phase freedom to define the Higgs basis field H2 such that Z5, Z6 and Z7 are real. Minimizing the Higgs potential, the neutral components of the Higgs fields acquire vevs: 1 vd 1 0 Hd = , Hu = , h i √2 0 ! h i √2 vu !

2 2 2 2 2 2 where v v + v = 4m /g (246 GeV) and∗ ≡ d u W ≃ vu 1 tan β , 0 β 2π. ≡ vd ≤ ≤

2 2 At the scalar potential minimum, the squared-mass parameters m1, m2 and 2 m12 can be re-expressed in terms of the two Higgs vevs, vu and vd (or equivalently in terms of mZ and tan β) and the CP-odd Higgs mass mA,

2m2 2m2 sin2β = 12 = 12 , m2 + m2 + 2 µ 2 m2 1 2 | | A m2 m2 tan2 β 1m2 = µ 2 + 1 − 2 . 2 Z −| | tan2 β 1 − ∗The Higgs fields can be rephased such that the vevs are real and positive. That is, the tree-level MSSM Higgs sector conserves CP, which implies that the neutral Higgs mass eigenstates are states of definite CP. At this stage, we can already see the tension with naturalness, if the SUSY parameters are significantly larger than the scale of electroweak symmetry 2 breaking. In this case, mZ will be the difference of two large numbers,

m2 m2 tan2 β 1m2 = µ 2 + 1 − 2 , 2 Z −| | tan2 β 1 − requiring some fine-tuning of the SUSY parameters in order to produce the correct Z boson mass. In the literature, this tension is referred to as the little .

2 2 In the above equation, µ, m1 and m2 are parameters defined at the . The question of fine-tuning should really be addressed to the fundamental SUSY-breaking parameters at some high energy scale, which ultimately determine the low-energy parameters appearing the above expression. Higgs bosons of the MSSM

The five physical Higgs particles consist of a charged Higgs pair

H± = Hd± sin β + Hu± cos β, one CP-odd scalar

0 0 0 A = √2 Im Hd sin β + Im Hu cos β ,  and two CP-even scalars

h0 = (√2Re H0 v )sin α +(√2Re H0 v )cos α, − d − d u − u H0 =(√2Re H0 v )cos α +(√2Re H0 v )sin α, d − d u − u where we have now labeled the Higgs fields according to their electric charge. The angle α arises when the CP-even Higgs squared-mass matrix (in the 0 0 Hd—Hu basis) is diagonalized to obtain the physical CP-even Higgs states. Tree-level MSSM Higgs masses

The charged Higgs mass is given by

m2 = m2 + m2 , H± A W and the CP-even Higgs bosons h0 and H0 are eigenstates of the squared-mass matrix 2 2 2 2 2 2 2 mA sin β + mZ cos β (mA + mZ)sin β cos β 0 = − 2 . M (m2 + m2 )sin β cos β m2 cos2 β + m2 sin β ! − A Z A Z

The eigenvalues of 2 are the squared-masses of the two CP-even Higgs M0 scalars

m2 = 1 m2 + m2 (m2 + m2 )2 4m2 m2 cos2 2β , H,h 2 A Z ± A Z − Z A  q  and α is the angle that diagonalizes the CP-even Higgs squared-mass matrix.†

2 1 2 †Note the contrast with the SM where the Higgs mass is a free parameter, mh = 2λv . In the MSSM, all Higgs self-coupling parameters of the MSSM are related to the squares of the electroweak gauge couplings. Aside: the decoupling limit of the MSSM In the limit of m m , the expressions for the Higgs masses and mixing A ≫ Z angle simplify and one finds m2 m2 cos2 2β, h ≃ Z m2 m2 + m2 sin2 2β, H ≃ A Z m2 = m2 + m2 , H± A W 4 2 2 mZ sin 4β cos (β α) 4 . − ≃ 4mA

Two consequences are immediately apparent. First, mA mH m , up ≃ ≃ H± to corrections of (m2 /m ). Second, cos(β α) = 0 up to corrections O Z A − 2 2 of (mZ/mA). This is the decoupling limit, since at energy scales below O 0 0 approximately common mass of the heavy Higgs bosons H± H , A , the effective Higgs theory is precisely that of the SM.

In particular, we will see that in the limit of cos(β α) 0, all the h0 − → couplings to SM particles approach their SM limits. Tree-level MSSM Higgs couplings

1. Higgs couplings to pairs (V = W or Z)

g 0 = g m sin(β α) , g 0 = g m cos(β α) , h V V V V − H V V V V −

0 where g 2m /v. There are no tree-level couplings of A or H± to V V . V ≡ V 2. Higgs couplings to a single gauge boson

The couplings of V to two neutral Higgs bosons (which must have opposite 0 0 0 CP-quantum numbers) is denoted by g 0 (p p ), where φ = h or H φA Z φ − A 0 and the momenta pφ and pA point into the vertex, and

g cos(β α) g sin(β α) gh0A0Z = − , gH0A0Z = − − . 2cos θW 2cos θW 3. Summary of Higgs boson– couplings

The properties of the three-point and four-point Higgs boson-vector boson couplings are conveniently summarized by listing the couplings that are proportional to either sin(β α) or cos(β α) or are angle-independent. As − − a reminder, cos(β α) 0 in the decoupling limit. − →

cos(β α) sin(β α) angle-independent − − 0 + 0 + H W W − h W W − — H0ZZ h0ZZ — 0 0 0 0 + + ZA h ZA H ZH H− , γH H− 0 0 0 W ±H∓h W ±H∓H W ±H∓A 0 0 0 ZW ±H∓h ZW ±H∓H ZW ±H∓A 0 0 0 γW ±H∓h γW ±H∓H γW ±H∓A 0 0 + — — VVφφ, VVA A ,VVH H−

0 0 + where φ = h or H and V V = W W −, ZZ, Zγ or γγ. 4. Higgs-fermion couplings

Since the neutral Higgs couplings to fermions are flavor-diagonal, we list only the Higgs coupling to 3rd generation fermions. The couplings of the neutral

Higgs bosons to ff¯ relative to the SM value, gmf /2mW , are given by

0 0 + sin α h b¯b (or h τ τ −): = sin(β α) tan β cos(β α) , − cos β − − − cos α h0tt¯: = sin(β α)+cot β cos(β α) , sin β − − 0 0 + cos α H b¯b (or H τ τ −): = cos(β α)+tan β sin(β α) , cos β − − sin α H0tt¯: = cos(β α) cot β sin(β α) , sin β − − − 0 0 + A b¯b (or A τ τ −): γ5 tan β,

0 A tt¯: γ5 cot β,

0 where the γ5 indicates a pseudoscalar coupling. Note that the h ff¯ couplings approach their SM values in the decoupling limit, where cos(β α) 0. − → Similarly, the charged Higgs boson couplings to fermion pairs, with all particles pointing into the vertex, are given by‡ g g ¯ = m cot βP + m tan βP , H−tb t R b L √2mW g h i g + = m tan βP . H−τ ν τ L √2mW h i Especially noteworthy is the possible tan β-enhancement of certain Higgs- fermion couplings. The general expectation in MSSM models is that tan β lies in a range: m 1 < tan β < t . ∼ ∼ mb Near the upper limit of tan β, we have roughly identical values for the top and bottom Yukawa couplings, h h , since t ∼ b

√2 mb √2 mb √2 mt √2 mt hb = = , ht = = . vd v cos β vu v sin β

‡Including the full flavor structure, the CKM matrix appears in the charged Higgs couplings in the standard way for a charged-current interaction. Saving the MSSM Higgs sector—the impact of radiative corrections

The tree-level Higgs mass result to a startling prediction,

m m cos 2β m . h ≤ Z| |≤ Z This is clearly in conflict with the observed Higgs mass of 125 GeV. However, the above inequality receives quantum corrections. The Higgs mass can be shifted due to loops of particles and their superpartners (an incomplete cancellation, which would have been exact if supersymmetry were unbroken):

t t h0 h0 h0 1,2 h0 e 3g2m4 M 2 X2 X2 m2 < m2 + t ln S + t 1 t , h Z 8π2m2 m2 M 2 − 12M 2 ∼ W   t  S  S  where A µ cot β governs stop mixing and M 2 is the average squared- t ≡ t − S mass of the top-squarks t1 and t2 (which are the mass-eigenstate combinations of the interaction eigenstates, tL and tR). e e e e A quick and dirty derivation of the logarithmic correction to the Higgs mass

Consider the one-loop effective potential for a coupled to matter,

(1) Veff(Φ) = Vscalar(Φ) + V (Φ) .

If we regulate the divergence of the loop correction by a momentum cutoff Λ, then Λ2 1 M 2(Φ) 1 V (1)(Φ) = Str M 2(Φ) + Str M 4(Φ) ln i , 32π2 i 64π2 i Λ2 − 2      2 1 where Mi (Φ) are the relevant squared-mass matrices for 0, 2 and 1, in which the scalar vacuum expectation values are replaced by the corresponding scalar fields, and the supertrace is defined as

Str M 2 ( 1)J (2J + 1)C Tr M 2 , ≡ − J J XJ 1 where the sum is taken over J = 0, 2 and 1, and CJ is a counting factor. In particular, CJ = 1 for Majorana spinors and real bosons, CJ = 2 for 2 complex bosons, etc. In softly-broken SUSY theories, Str Mi (Φ) = 0.

We focus on the contributions of the and its scalar superpartners, tL and tR (“stops”). For simplicity, we ignore stop mixing, in which case,

2 2 2 2 1 2 2 1 e e m˜ = m˜ = m˜ M + h v , mt = htvu . t tL tR ≃ S 2 t u √2 Hence, 2 1 2 2 (1) 3 2 1 2 2 2 MS + 2ht vu 1 V (vd, vu)= 2 2 2 (MS + 2ht vu) ln 2 64π · · ( " Λ ! − 2# h4v4 1h2v2 1 t u ln 2 t u , − 4 " Λ ! − 2#) where we have included a color factor of 3, a factor of 2 for complex bosons and a second factor of 2 to account for both tL and tR.

Exercise: Repeat the above computation includinge thee effects of stop mixing. Including the tree-level piece, we can write

m2 m2 v V (v , v )= 1(v v ) 1 12 d d u 2 d u 2 − 2 m12 m2 ! vu! − 1 2 2 2 2 2 (1) + (g + g′ )(v v ) + V (v , v ) . 32 d − u d u

Requiring that vu is a minimum of the scalar potential, we set ∂V/∂vu = 0, which yields

(1) 2 2 vd 1 2 2 2 2 2 1 ∂V m2 = m12 8(g + g′ )(vd vu) . vu − − − vu ∂vu

2 Using the minimum condition to eliminate m2, we obtain,

2 (1) 2 (1) ∂ V 2 vd 1 2 2 2 1 ∂V ∂ V 2 = m12 + 4(g + g′ )vu + 2 . ∂vu vu − vu ∂vu ∂vu

We now insert the expression previously derived for V (1) into the above result. The end result is: 2 (1) (1) 2 ∂ V 1 ∂V 3 4 2 mt˜ 2 = 2ht vu ln 2 . ∂vu − vu ∂vu 8π mt ! In particular, notice that the factors of Λ have canceled out!

We therefore see that the effect of the top quark and top squark loops is to modify the element 2 of the CP-even Higgs squared-mass matrix, M22 3g2m4 m2 δ 2 = t ln t˜ , 22 2 2 2 2 M 8π mW sin β mt ! after writing ht = √2mt/vu and vu = (2mW /g)sin β. Diagonalizing the CP- even Higgs squared-mass matrix in the limit of m m and incorporating A ≫ Z the radiative correction obtained above, we obtain the radiatively-corrected upper bound, 2 4 2 2 2 2 3g mt mt˜ mh mZ cos 2β + 2 2 ln 2 , ≤ 8π mW mt ! which reproduces the logarithmic correction that was quoted earlier. The state-of-the-art computation includes the full one-loop result, all the significant two-loop contributions, some of the leading three-loop terms, and -group improvements. The final conclusion is that mh < 130 GeV [assuming that the top-squark mass is no heavier than a few∼ TeV], which is compatible with the observed Higgs boson mass.

Maximal mixing corresponds to choosing the MSSM Higgs parameters in such a way that mh is maximized (for a fixed tan β). This occurs for Xt/MS 2. As tan β varies, mh reaches is maximal value, (m ) 130 GeV, for tan β 1∼and m m . h max ≃ ≫ A ≫ Z Radiatively-corrected Higgs couplings

Although radiatively-corrections to couplings tend to be at the few-percent level, there is some potential for significant effects: large radiative corrections due to a tan β-enhancement (for tan β 1) • ≫ CP-violating effects induced by complex SUSY-breaking parameters that • enter in loops In the MSSM, the tree-level Higgs–quark Yukawa Lagrangian is supersymmetry-conserving and is given by Type-II structure, tree = ǫ h Hiψj ψ + ǫ h Hi ψj ψ +h.c. Lyuk − ij b d Q D ij t u Q U Two other possible dimension-four gauge-invariant non-holomorphic Higgs- quark interactions terms, the so-called wrong-Higgs interactions,

k k k k Hu∗ψDψQ and Hd ∗ψU ψQ , are not supersymmetric, and hence are absent from the tree-level Yukawa Lagrangian. Nevertheless, the wrong-Higgs interactions can be generated in the effective low-energy theory below the scale of SUSY-breaking. In particular, one-loop radiative corrections, in which supersymmetric particles (squarks, higgsinos and ) propagate inside the loop can generate the wrong-Higgs interactions.

i i Hu∗ Hu∗ i Q Qi i D U∗ ∗i Q ∗ D∗ U Q e e ψi a e ψ ψi ψe ψ ψ Qe g× e D Q e Hu× Hed D e (a) (b) One-loop diagrams contributing to the wrong-Higgs Yukawa effective operators. In (a), the cross ( ) corresponds to a factor of the × gluino mass M3. In (b), the cross corresponds to a factor of the higgsino Majorana mass parameter µ. Field labels correspond to of the corresponding particle at each vertex of the triangle.

If the superpartners are heavy, then one can derive an effective field theory description of the Higgs-quark Yukawa couplings below the scale of SUSY- breaking (MSUSY), where one has integrated out the heavy SUSY particles propagating in the loops. The resulting effective Lagrangian is:

eff i j k k = ǫ (h + δh )ψ H ψ + ∆h ψ H ∗ψ Lyuk − ij b b b d Q b b u Q i j k k +ǫij(ht + δht)ψtHuψQ + ∆htψtHd ∗ψQ +h.c.

In the limit of M m , SUSY ≫ Z 2 2αs ht ∆hb = hb µM3 (M˜ , M˜ , Mg)+ µAt (M˜ , M˜ ,µ) , 3π I b1 b2 16π2 I t1 t2   where, M3 is the Majorana gluino mass, µ is the supersymmetric Higgs-mass parameter, and b1,2 and t1,2 are the mass-eigenstate bottom squarks and top squarks, respectively. The loop integral is given by e e a2b2 ln(a2/b2)+ b2c2 ln(b2/c2)+ c2a2 ln(c2/a2) (a,b,c)= . I (a2 b2)(b2 c2)(a2 c2) − − − Note that (a,b,c) 1/max(a2,b2,c2) in the limit where at least one of the I ∼ arguments of (a,b,c) is large. Hence, the one-loop contributions to ∆h do I b not decouple when M µ A M˜ M˜ M m . 3 ∼ ∼ t ∼ b ∼ t ∼ SUSY ≫ Z Phenomenological consequences of the wrong-Higgs Yukawas

The effects of the wrong-Higgs couplings are tan β-enhanced modifications of some physical . To see this, rewrite the Higgs fields in terms of the physical mass-eigenstates (and the Goldstone bosons): 1 H1 = (v cos β + H0 cos α h0 sin α + iA0 sin β iG0 cos β) , d √2 − − 1 H2 = (v sin β + H0 sin α + h0 cos α + iA0 cos β + iG0 sin β) , u √2 2 H = H− sin β G− cos β, d − 1 + + Hu = H cos β + G sin β. For simplicity, we neglect below possible CP-violating effects due to complex couplings. Then, the b-quark mass is:

hbv δhb ∆hbtan β hbv mb = cos β 1+ + cos β(1+∆b) , √2 hb hb ≡ √2   which defines the quantity ∆b. In the limit of large tan β the term proportional to δhb can be neglected, in which case, ∆ (∆h /h )tan β . b ≃ b b Thus, ∆ is tan β–enhanced if tan β 1. As previously noted, ∆ survives b ≫ b in the limit of large MSUSY; this effect does not decouple.

The effective Yukawa Lagrangian (neglecting CP-violating effects) yields,

0 0 0 int = g 0 h qq¯+ g 0 H qq¯ ig 0 A qγ¯ 5q + ¯bg ¯tH− +h.c. . L − h qq¯ H qq¯ − A qq¯ H−tb q=t,b,τ X    

The one-loop corrections can generate measurable shifts in the decay rate for h0 b¯b: → mb sin α 1 δhb g ¯ = 1+ ∆b (1 + cot α cot β) . h◦bb − v cos β 1 + ∆ h −  b  b   At large tan β 20—50, ∆ can be as large as 0.5 in magnitude and of either ∼ b sign, leading to a significant enhancement or suppression of the Higgs decay rate to b¯b. Non-decoupling effects in h0 b¯b: a closer look → The origin of the non-decoupling effects can be understood by noting that below the scale MSUSY, the effective low-energy Higgs theory is a completely general 2HDM. Thus, it is not surprising that the wrong-Higgs couplings do not decouple in the limit of M . SUSY → ∞ However, suppose that m (M ). Then, the low-energy effective A ∼ O SUSY Higgs theory is a one-Higgs doublet model, and thus gh0b¯b must approach its SM value. Indeed in this limit,

m2 sin4β m4 cos(β α)= Z + Z , − 2m2 O m4 A  A 2m2 m4 1+cot α cot β = Z cos 2β + Z . − m2 O m4 A  A Thus the previously non-decoupling SUSY radiative corrections do decouple as expected. LHC searches for the heavy Higgs states of the MSSM

At present, the LHC search channel with the greatest reach looks for •

+ gg (H or A) τ τ − (at moderate values of tan β) , → → + gg b¯b(H or A) b¯bτ τ − (at large values of tan β) . → →

No signal above background has been observed. This provides model- dependent limits on the MSSM parameter space which rules out regions of

the mA–tan β plane.

The precision Higgs data suggests that the mixing of the observed Higgs • boson with a non-SM-like Higgs eigenstate is small. This points to the

decoupling regime which provides a lower bound on the value of mA (except in the parameter region of alignment without decoupling). β 10

tan 9 ATLAS Preliminary 8 s= 7 TeV, ∫ Ldt = 4.6•4.8 fb•1 7 s= 8 TeV, ∫ Ldt = 20.3 fb•1 6 Combined h → γγ , ZZ*, WW*, ττ , bb 5 κ κ κ Simplified MSSM [ V, u, d] 4 Exp. 95% CL Obs. 95% CL 3 2 1 0 200 300 400 500 600 700 800 900 1000

mA [GeV]

Regions of the (mA, tan β) plane excluded in a simplified MSSM model via fits to the measured rates of Higgs boson production and decays. The likelihood contours corresponding approximately to 95% CL, are indicated for the data and expectation assuming the SM Higgs sector. The light shaded and hashed regions indicate the observed and expected exclusions, respectively. Taken from ATLAS-CONF-2014-010. The infamous LHC wedge region of large m and moderate values of • A tan β will be difficult to probe at the LHC. For values of mA > 2mt, the H, A tt¯ is a challenging signal. In a limited parameter regime with → 2m < m < 2m and moderate tan β, the decay H hh may provide h H t → some opportunity for discovery.

2.3 fb -1 (13 TeV)

β 80 -1 ATLAS Preliminary, s=13 TeV, 3.2 fb 95% CL Excluded: tan CMS Observed ± 1 σ Expected mMSSM ≠ 125 ± 3 GeV 70 MSSM m mod+ scenario, M = 1 TeV h h SUSY Preliminary Expected ± 2 σ Expected 7 + 8 TeV(HIG-14-029) H/A → ττ , 95% CL limits 60 β mod+ 60 mh scenario

m = 126.3 GeV tan h 50 50 = 500 GeV H m = 300 GeV 40 40 H m = 1000 GeV H m 30 m = 126 GeV 30 h

Observed 20 Expected 20 1 σ m = 125 GeV 10 h 2 σ m = 123.5 GeV 10 h ATLAS Run-I (Obs.)

200 300 400 500 600 700 800 900 1000 1100 1200 200 400 600 800 1000 1200 1400 mA [GeV] mA (GeV) The alignment limit without decoupling in the MSSM

Recall that in the Higgs basis, the CP-even neutral Higgs squared-mass matrix is given by Z v2 Z v2 2 = 1 6 . 2 2 2 M Z6v mA + Z5v !

The mixing angle of rotation from the Higgs basis to the mass basis is α β. −

It follows that in both the decoupling limit (mH mh) and the alignment 2 ≫2 2 limit without decoupling [ Z 1 and m Z v (v )],§ | 6| ≪ H − 1 ∼ O Z2v4 cos2(β α)= 6 1 , − (m2 m2 )(m2 Z v2) ≪ H − h H − 1 Z2v4 Z v2 m2 = 6 (v2) . 1 − h m2 Z v2 ≪ O H − 1

2 2 §Note the upper bound m Z v on the mass of h is saturated in the exact alignment limit. h ≤ 1 The MSSM values of Z1 and Z6 (including the leading one-loop corrections):

3v2s4 h4 M 2 X2 X2 Z v2 = m2 c2 + β t ln S + t 1 t , 1 Z 2β 8π2 m2 M 2 − 12M 2   t  S  S  3v2s2 h4 M 2 X (X + Y ) X3Y Z v2 = s m2 c β t ln S + t t t t t . 6 − 2β Z 2β − 16π2 m2 2M 2 − 12M 4 (   t  S S )

2 where M m˜ m˜ , X A µ cot β and Y = A + µ tan β. S ≡ t1 t2 t ≡ t − t t As previously noted, m2 Z v2, is consistent with m 125 GeV for suitable h ≤ 1 h ≃ choices for mS and Xt. Exact alignment (i.e., Z6 = 0) can now be achieved due to an accidental cancellation between tree-level and loop contributions,¶

3v2s2 h4 M 2 X (X + Y ) X3Y m2 c = β t ln S + t t t t t . Z 2β 16π2 m2 2M 2 − 12M 4   t  S S  That is, Z 0 is possible for a particular choice of tan β. 6 ≃ ¶See M. Carena, H.E. Haber, I. Low, N.R. Shah and C.E.M. Wagner, Phys. Rev. D 91, 035003 (2015). ∆χ 2 alt µ -2 ln(L) alt µ HS mh scenario ( =3mQ) mh scenario ( =3mQ) 25 20 25 20 95% CL excl. 95% CL Carena et al. FeynHiggs-2.10.2 mod+ µ SusHi-1.4.1 mh ( =200GeV) HiggsSignals-1.3.0 20 15 20 15 FeynHiggs-2.10.2 SusHi-1.4.1 HiggsBounds-1.2.0 β β 15 10 15 10 tan tan

10 10 5 5

5 5 0 0 200 250 300 350 400 450 500 200 250 300 350 400 450 500

MA [GeV] MA [GeV]

Constraints from LHC Higgs searches in the alignment benchmark scenario (with µ = 3MS). + Left panel: distribution of the exclusion likelihood from the CMS φ τ τ − search and the → observed 95% CL exclusion line as obtained from HiggsBounds. 2 Right panel: likelihood distribution, ∆χHS obtained from testing the signal rates of the Higgs boson h against a combination of Higgs rate measurements from the and LHC experiments, obtained with HiggsSignals. See P. Bechtle, S. Heinemeyer, O. St˚al, T. Stefaniak and G. Weiglein, EPJC 75, 421 (2015). Likelihood analysis: allowed regions in the tan β–mA plane

2 Preferred parameter regions in the (MA, tan β) plane (left) and (MA, µAt/MS) plane (right), where M 2 = m m , in a pMSSM-8 scan. Points that do not pass the direct S t˜1 t˜2 constraints from Higgs searches from HiggsBounds and from LHC SUSY particle searches from CheckMATE are shown in gray. Applying a global likelihood analysis to the points 2 that pass the direct constraints, the color code employed is red for ∆χh < 2.3, yellow 2 for ∆χh < 5.99 and blue otherwise. The best fit point is indicated by a black star. (Taken from P. Bechtle, H.E. Haber, S. Heinemeyer, O. St˚al, T. Stefaniak, G. Weiglein and L. Zeune, in preparation.) The alignment limit of the NMSSM Higgs sector

In the MSSM, a supersymmetric Higgsino mass term µ appears, whose magnitude is similar to that of a typical SUSY-breaking mass. Although phenomenology demands it, there is no theoretical reason for these two mass scales to be connected.

This problem is addressed in the next-to-minimal supersymmetric extension of the Standard Model (NMSSM). In this model, a new singlet supermultiplet is added consisting of a complex singlet scalar field S and its fermionic superpartner. An effective µ-term appears, µ = λvs, where λ governs the coupling of the singlet scalar to the Higgs doublet fields and vs is the vev of the singlet scalar field.

The analog of the Higgs basis fields are:

h √2Re H0 v , H √2Re H0 , HS √2(Re S v ) , ≡ 1 − ≡ 2 ≡ − s G √2 Im H0 , A √2 Im H0 , AS √2 Im S. ≡ 1 ≡ 2 ≡ CP is not automatically conserved in the NMSSM Higgs sector, but we shall assume it for simplicity. In this case, the symmetric squared-mass matrix for the CP-even scalars in the Higgs basis is

2 2 Z1v Z6v √2 v C1 + (Zs1 + 2Zs5)vs     2 v 2 2 =  M A + Z5v C3 + C4 + 2(Zs3 + Zs7 + Zs8)vs  , MS  √2    2   v 2  C1 + 3(C5 + C6)vs + 4(Zs4 + 2Zs9 + 2Zs10)v   − 2v s  s 

where M 2 2µ(A + κv )/s and µ λv . Note that κ governs the A ≡ λ s 2β ≡ s self-coupling of the singlet scalar field. The coefficients of the scalar potential appear in the squared-mass matrix 2, M

1 2 1 2 ... + Z (H†H ) + ... + Z (H†H ) + Z (H†H )H†H +h.c. + ... V ∋ 2 1 1 1 2 5 1 2 6 1 1 1 2   +S†S Zs1H1†H1 + ... + (Zs3H1†H2 +h.c.)+ Zs4S†S  2 2 2  2 4 + Zs5H1†H1S + ... + Zs7H1†H2S + Zs8H2†H1S + Zs9S†S S + Zs10S +h.c. n o 3 + C1H1†H1S + ... + C3H1†H2S + C4H2†H1S + C5(S†S)S + C6S +h.c. .   Exact alignment occurs when 2 = 2 = 0. That is, M12 M13

Z6 = 0 , C1 +(Zs1 + 2Zs5)vs = 0 .

In the NMSSM, including the leading one-loop radiative corrections,

3v2s4 h4 M 2 X2 X2 Z v2 =(m2 1λ2v2)c2 + 1λ2v2 + β t ln S + t 1 t , 1 Z − 2 2β 2 8π2 m2 M 2 − 12M 2   t  S  S  3v2s2 h4 M 2 X (X + Y ) X3Y Z v2 = s (m2 1λ2v2)c β t ln S + t t t t t . 6 − 2β Z − 2 2β − 16π2 m2 2M 2 − 12M 4 (   t  S S )

Note that the squared-mass bound of the SM-like Higgs boson is now modified at tree-level by a positive quantity,

m2 (Z v2) = m2 c2 + 1λ2v2s2 . h ≤ 1 tree Z 2β 2 2β Regions of parameter space exist in which the term proportional to λ2 provides 2 a significant contribution to mh. In contrast to the MSSM, in the NMSSM one can set Z6 = 0 and obtain mh = 125 GeV, with only small contributions from the one-loop radiative corrections. This leads to a preferred choice of NMSSM parameters,

λ 0.65 , tan β 2 ∼ ∼

DOW Λ  P K= *H9  

 

L ;W=  *H9 ±  =  *H9 Λ WUHH  H P K 6 0 

 ;W=06

ΛDOW P  K = ± *H9            WDQ Β WDQ Β 2 The exact alignment limit also requires that 13 = 0. In the NMSSM, 2 M M s2 κs A 2β + 2β = 1 . 4µ2 2λ which to further correlations of the NMSSM parameter space.

ΛDOW  Κ = Λ DOW  H L = = PK6 *H9  W Β  P K  *H9 

 L  *H9 H  6

$ 

P  

 

       P$ H*H9 L Near the alignment limit, we have m m M . A ≃ H ≃ A Phenomenological Implications of the NMSSM Higgs sector

[reference: M. Carena, H.E. Haber, I. Low, N.R. Shah and C.E.M. Wagner, Phys. Rev. D 93, 035013 (2016)]

Numerous H HH and H V H channels are kinematically allowed. • → → The parameters µ and M are correlated by the alignment conditions. • A Typical values of µ 100–300 GeV yield / states that • | |∼ are approximately unmixed higgsino (with mass µ ) and singlino (with ∼ | | mass 2 κµ/λ ) states. Thus, we expect significant branching ratios of ∼ | | + 0 0 0 H, A χ χ− , χ χ , H± χ±χ . → i j → i

Imposing perturbativitye upe toe thee Planck scale impliese e that κ < 1λ. • 2 0 ∼ Consequently, the singlino is the LSP (denoted as χ1). A potential decay mode of the neutral higgsino is χ0 χ0 + h. 2 → 1 e e e