Reφ = 1 2. h ff λ. = λ f

Transcription

1 I. THE FINE-TUNING PROBLEM A. Quadratic divergence We illustrate the problem of the quadratic divergence in the Higgs sector of the SM through an explicit calculation. The example studied is that of the two point function (inverted propagator) of the Higgs scalar at vanishing external momentum, computed at the one loop level. Let φ be the SM neutral Higgs field with Reφ = 1 (h + v), (1) where v = (1/ G F ) 1/ 46 GeV. Take f to be a generic matter fermion field with a Yukawa coupling to φ via the term L f fφ = λ f fl f R φ + h.c. Thus, the fermion (tree level) mass is m f = λ f v/. = λ f h ff λ fv ff. () We now compute the one loop f f contribution to the scalar two point function: Π f hh (0) = ( 1) d 4 ( k (π) 4Tr i λ ) ( f i i λ ) f i k m f k m f = λ d 4 k k + m f f (π) 4 (k m f ) = λ d 4 [ k 1 m ] f f +. (3) (π) 4 k m f (k m f ) The first term in the final RHS of (3) is quadratically divergent. Suppose the integral is cutoff by a scale Λ which is then set equal to the Planck mass M Pl GeV. Then the one loop correction to m S would be more than 30 orders of magnitude larger than m S itself. Furthermore, the correction (3) is independent of m S. This is an indication that within the SM, m S is an unnatural parameter. Setting m S = 0 does not increase the symmetry of the theory. Of course, one could simply renormalize such quadratic divergences away in the same way that logarithmic divergences are disposed of. But, still, the residual finite correction in (3) would be of order m f λ f /(8π). Such a correction would be managably small for a SM fermion like the top quark. But any heavy fermions that couple to the SM Higgs would give contributions that have to be cancelled by unnatural amount of fine-tuning. 1

2 B. Naturalness and Supersymmetry Loops induced by scalar fields, contributing to the Higgs two point function, can also be considered. Let us construct a toy model by adding to the model of () additional complex scalar fields, fl, f R, with the following couplings to the Higgs field: L f fφ = λ f φ ( f L + f R ) + (A f φ f L f R + h.c.) = 1 λ f h ( f L + f R ) + v λ f h( f L + f R ) + h (A f fl f R + h.c.) + (4) where, in the last equality, we display only the h-dependent terms. The f-loops make the following additional contributions to the two point function: Π f hh (0) = λ d 4 k 1 1 f + (π) 4 k m fl k m fr + ( λ f v) d 4 k 1 (π) 4 (k m fl ) + 1 (k m fr ) + A f d 4 k (π) 4 1 k m fl 1 k m fr. (5) Only the first line in (5) containts a quadratic divergence. This, however, can be cancelled with that in the fermionic contribution (3), i.e. Π f f hh (0)+Πhh(0) becomes free of any quadratic divergence, provided the following coupling constant equality is obeyed: λ f = λ f. (6) Note that the inequality λ f < 0 is required in (4) to keep the Hamiltonian bounded from below. Another important point to note is that the above cancellation is the quadratic divergence is independent of the masses m fl, m fr, or the coupling strength A f. Now that the quadratic divergence has disappeared from Π f f hh (0)+Πhh(0), the remaining logarithmic ones can be cancelled by contributions from logarithmically infinite counterterms introduced in the Lagrangian density as part of the renormalization procedure. In the MS renormalization scheme, one can replace the logarithmic divergence in our loop integrals by the logarithm of the square of the renormalization scale µ. Utilizing the B 0 -function of Passarino and Veltman, we can then make the following types of replacements: d 4 ( ) k 1 1 (m iπ k m 1 k m 1 m )B 0(0, m 1, m ) ( ) ( ) m 1 1 ln m 1 µ m 1 ln m µ, (7)

3 d 4 k iπ 1 (k m ) ln m µ. (8) The consequent expression of the sum of (3) and (5) is simplified in case of equal masses, m fl = m fr m f. (9) The choices (6) and (9) as well as the substitutions (7) lead to the result: f, f Πhh (0) = iλ f 16π m f ( 1 ln m f µ ) + 4m f ln m f µ + m f 1 ln m f µ 4m f ln m f µ i A f 16π m f ln µ. (10) Thus, if along with (9) we also require the relations m f = m f, (11) A f = 0, (1) we will have Π f f hh (0) + Πhh (0) = 0. (13) Eq. (13) can be restated as follows. If the fermion f Yukawa coupling strength squared equals the quartic coupling between the Higgs and the scalars f L,R, if the masses of the femion and of the scalars are equal, and if the A f trilinear scalar coupling is zero, the entire one loop renormalization of the Higgs self energy Π hh (0) vanishes. We are now ready for the supersymmetric interpretation of the above. In an exactly supersymmetric theory, the two scalars f L,R are the left and right superpartners (sfermions) of the fermion f. Moreover, the coupling strength equality (6), the mass equalities (9) and (11), and the required null value of the (supersymmetry breaking) parameter A f (1) are all ensured by supersymmetry. Indeed, with these conditions, the vanishing of the renormalization of the Higgs self energy holds in all orders of perturbation theory as a consequence of the nonrenormalization theorem valid in supersymmetric theories. The naturalness aspect is also made clear by the introduction of a certain kind of small supersymmetry breaking, namely that the breaking is confined to m f m f and A f 0, but does not change the coupling equality (6). These are the features of softly broken supersymmetry: The only supersymmetry breaking parameters are coefficients of operators that have mass dimension less than four. 3

4 Suppose that we characterize this supersymmetry breaking in terms of two small parameters A f and δ m, with δ m = m f m f. (14) Thus δ m characterizes the mass splitting within the f f supermultiplet. With the assumption that δ m, A f m f, we can approximate ln(m f/µ ) ln(m f /µ ) + δ m /m f and rewrite Eq. (15) as [ ] f, f Πhh (0) = iλ f 4δ 16π m + (4δ m + A f ) ln m f + O(δ µ m, A f δ m ). (15) Hence the one loop renormalization of the Higgs self energy is linearly proportional to the small supersymmetry breaking parameters δ m and A f, restricting the correction to one of modest magnitude, though m f may be quite large. Thus the introduction of the superpartners f L,R with the interactions (4) has served two purposes: (1) The quadratic divergence in the scalar self-energy is cancelled; () The scalar mass is protected from large loop corrections involving heavy particles as long as the mass splitting between the heavy fermion and boson superpartners is itself of the order of the sacalar mass. This then is a toy model example of how naturalness is restored by supersymmetry in the scalar sector of the SM. C. The SM Higgs The hierarchy problem arises from the fact that there are quadratically divergent loop contributions to the Higgs mass which drive the Higgs mass to unacceptably large values unless the tree level mass parameter is finely tuned to cancel the large quantum corrections. The most significant of these divergences come from three sources. They are - in order of decreasing magnitude - one loop diagrams involving the top quark, the electroweak gauge bosons, and the Higgs itself. For the sake of concreteness (and, also, because this is the scale that will be probed by the LHC), let us assume that the SM is valid up to a cut-off scale of 10 TeV. Then, the contributions from the three diagrams are 3 8π Y t Λ ( TeV ) (16) 4

5 from the top loop, from the gauge loop, and 1 16π g Λ (700 GeV ) (17) 1 16π λλ (500 GeV ) (18) from the Higgs loop. Thus the total Higgs mass is approximately m h = m tree [ ](00 GeV ). (19) In order for this to add up to a Higgs mass of order a few hundred GeV as required in the SM, fine tuning of order one part in 100 is required. This is the hierarchy problem. Is the SM already fine tuned given that we have experimentally probed to near 1 TeV and found no NP? Setting Λ = 1 TeV in the above formulas we find that the most dangerous contribution from the top loop is only about (00 GeV ). Thus no fine tuning is required, the SM with no NP up to 1 TeV is perfectly natural, and we should not be surprised that we have not yet seen deviations from it at colliders. We can now turn the argument around and use the hierarchy problem to predict what forms of new physics exist at what scale in order to solve rge hierarchy problem. Consider for example the contribution to the Higgs mass from the top loop. Limiting the contribution to be no larger than about 10 times the Higgs mass (limiting fine tuning to less than 1 part in 10) we find a maximum cut-off for Λ = TeV. In other words, we predict the existence of new particles with masses less than TeV which cancel the quadratically divergent contribution to the Higgs mass from the top loop. In order for this cancellation to occur naturally, the new particles must be related to the top quark by a symmetry. In practice, this means that the new particles have to carry similar quantum numbers to top quarks. In supersymmetry, these new particles are of course the top squarks. The contributions from gauge loops also need to be canceled by new particles which are related to the SM SU() U(1) gauge bosons by a symmetry. The masses of these states are predicted to be at or below 5 TeV for the cancellation to be natural. Similarly, the Higgs loop requires new states related to the Higgs at 10 TeV. We see that the hierarchy problem can be used to obtain specific predictions. 5

6 II. FORMALISM Supersymmetry is a symmetry between bosons and fermions or, more precisely, it is a symmetry between states of different spin. For example, a spin-0 particle is mapped into a spin- 1 particle under a supersymmetry transformation. Thus, the generators Q α, Q α of the supersymmetry transformation must transform in the spin- 1 representations of the Lorentz group. These new fermionic generators form together with the four-momentum P m and the generators of the Lorentz transformations M mn a graded Lie algebra which features in addition to commutators also anticommutators in their defining relations. The simplest (N = 1) supersymmetry algebra reads: {Q α, Q β} = σ m α β P m, {Q α, Q β } = {Q α, Q β} = 0, [ Q α, P m ] = [Qα, P m ] = 0, [Q α, M mn ] = 1 σmnβ α Q β, [ Q α, M mn] = 1 σmn β α Q β, (0) where we used the notation and convention of ref. [1]. The particle states in a supersymmetric field theory form representations (supermultiplets) of the supersymmetry algebra (0). We do not recall the entire representation theory, but only highlight a few generic features: There is an equal number of bosonic degrees of freedom n B and fermionic degrees of freedom n F in a supermultiplet: n B = n F. (1) The masses of all states in a supermultiplet are degenerate. In particular the masses of bosons and fermions are equal 1 m B = m F. () Q has mass dimension 1 differ by 1. and thus the mass dimensions of the fields in a supermultiplet 1 This follows immediately from the fact that P is a Casimir operator of the supersymmetry algebra (0): [P, Q] = [P, M mn ] = 0. 6

7 The two irreducible multiplets which are important for constructing the supersymmetric Standard Model are the chiral multiplet and the vector multiplet, which we discuss in turn now. A. The chiral supermultiplet The chiral supermultiplet Φ contains a complex scalar field A(x) of spin 0 and mass dimension 1, a Weyl fermion ψ α (x) of spin 1 and mass dimension 3, and an auxiliary complex scalar field F(x) of spin 0 and mass dimension : Φ = (A(x), ψ α (x), F(x)). (3) Φ has off-shell four real bosonic degrees of freedom (n B = 4) and four real fermionic degrees of freedom (n F = 4) in accord with (1). The supersymmetry transformations act on the fields in the multiplet as follows: δ ξ A = ξψ, δ ξ ψ = ξf + i σ m ξ m A, δ ξ F = i ξf + i ξ σ m m ψ, (4) where we used the conventions of ref. [1]. The parameters of the transformation ξ α are constant, complex, anticommuting Grassmann parameters obeying ξ α ξ β = ξ β ξ α. (5) The transformation (4) can be thought of as generated by the operator δ ξ = ξq + ξq, (6) with Q and Q obeying (0). This can be explicitly checked by evaluating the commutators [δ ξ, δ η ] on the fields A, ψ and F. Exercise: Show that [δ ξ, δ η ] = i(ησ m ξ ξσm η) m by using (6) and (0). Exercise: Evaluate the commutator [δ ξ, δ η ] using (4) for all three fields A, ψ and F and show that this is consistent with the results of the previous exercise. The field F has the highest mass dimension of the members of the chiral multiplet and therefore is called the highest component. As a consequence it cannot transform into any 7

8 other field of the multiplet but only into their derivatives. This is not only true for the chiral multiplet (as can be seen explicitly in Eq. (4)), but holds for any supermultiplet. This fact can be used to construct Lagrangians that transform into a total derivative under supersymmetry transformations leaving the corresponding actions invariant [1]. For the chiral multiplet, a supersymmetric and renormalizable Lagrangian is given by L(A, ψ, F) = i ψ σ m m ψ m Ā m A + F F +m(af + Ā F 1 (ψψ + ψ ψ)) +Y (A F + Ā F Aψψ Ā ψ ψ), (7) where m and Y are real parameters. This action has the peculiar property that no kinetic term for F appears. Consequently, the equations of motion for F are purely algebraic: δl δ F = F + mā + Y Ā = 0, δl δf = F + ma + Y A = 0. (8) Thus F is non-dynamical, auxiliary field, which can be eliminated from the action algebraically by using equations of motion. This yields L(A, ψ, F = mā Y Ā ) = i ψ σ m m ψ m Ā m A where V (A, Ā) is the scalar potential given by V (A, Ā) = ma + Y A m (ψψ + ψ ψ) Y (Aψψ + Ā ψ ψ) V (A, Ā), (9) = m AĀ + my (AĀ + ĀA ) + Y A Ā = F F. (30) =0 δl δ F = δl δf As can be seen from (9) and (30), after elimination of F, a standard renormalizable Lagrangian for a complex scalar A and a Weyl fermion ψ emerges. However, (9) is not the most general renormalizable Lagrangian for such fields. Instead, it satisfies the following properties: L depends on only two independent parameters, the mass parameter m and the dimensionless Yukawa coupling Y. In particular, the (AĀ) coupling is not controlled by an independent parameter (as it would be in non-supersymmetric theories) but determined by the Yukawa coupling Y. 8

9 The masses for A and ψ coincide, in accordance with (). V is positive semi-definite, V 0. B. The vector supermultiplet The vector supermultiplet V contains a gauge boson v m of spin 1 and mass dimension 1, a Weyl fermion (called the gaugino) λ of spin 1 and mass dimension 3, and a real scalar field D of spin 0 and mass dimension : V = (v m (x), λ α (x), D(x)). (31) Similar to the chiral multiplet, the vector multiplet has n B = n F = 4. The vector multiplet can be used to gauge the action of the previous section. The generators T a of a compact gauge group G have to commute with the supersymmetric generators: [T a, Q α ] = [T a, Q α ] = 0. (3) Therefore all members of the chiral multiplet (A, ψ, F) have to reside in the same representation of the gauge group. Similarly, the members of the vector multiplet have to transform in the adjoint representation of G and thus they are all Lie-algebra valued fields: v m = v a m T a, λ α = λ a α T a, D = D a T a. (33) The supersymmetry transformations of the components of the vector multiplet are: δ ξ v a m = i λ a σ m ξ + i ξ σ m λ a, δ ξ λ a = iξd a + σ mn ξf a mn, δ ξ D a = ξσ m D m λa D m λ a σ m ξ. (34) The field strength of the vector bosons F a mn and the covariant derivative D m λ a are defined as follows: F a mn = mv a n nv a m gfabc v b m vc n, D m λ a = m λ a gf abc v b m λc, (35) 9

10 where f abc are the structure constants of the Lie algebra and g is the gauge coupling. A gauge invariant, renormalizable and supersymmetric Lagrangian for the vector multiplet is given by L = 1 4 F a mn F mna i λ a σ m D m λ a + 1 Da D a. (36) As before, the equation of motion for the auxiliary D-field is purely algebraic, D a = 0. A gauge invariant, renormalizable Lagrangian containing a set of chiral multiplets (A i, ψ i, F i ) coupled to vector multiplets is found to be L(A i, ψ i, F i, v a m, λa, D a ) = 1 4 F a mn F mna i λ a σ m D m λ a + 1 Da D a where the covariant derivatives are defined by D m A i D m Ā i i ψ i σ m D m ψ i + F i F i + i g(āi T a ij ψj λ a λ a T a ij Ai ψj ) +gd a Ā i T a ija j 1 W ijψ i ψ j 1 W ij ψi ψj + F i W i + F i Wi, (37) D m A i = m A i + igv a mt a ija j, The superpotential is a holomorphic function W(A): while W i and W ij in Eq. (37) are its derivatives: D m ψ i = m ψ i + igv a m T a ij ψj. (38) W(A) = 1 m ija i A j Y ijka i A j A k, (39) W i W A = m ija j + Y i ijk A j A k, W ij W A i A j = m ij + Y ijk A k. (40) By explicitly inserting (40) into (37), one finds that the m ij are the mass parameters while the Y ijk are the Yukawa couplings. Supersymmetry forces W to be a holomorphic function of the scalar fields A while renormalizability restricts W to be at most a cubic polynomial of A. The parameters m ij and Y ijk are further constrained by gauge invariance. As before, F i and D a obey algebraic equations of motion: L F = 0 = i F i + W i = 0, L F = 0 = i F i + W i = 0, L D a = 0 = Da + gāi T a ij Aj = 0. (41) 10

11 They can be used to eliminate the auxiliary fields F i and D a from the Lagrangian (37): where L (A i, ψ i, v a m, λa, F i = W i, D a = gāi T a ij Aj ) = 1 4 F a mn F mna i λ a σ m D m λ a D m A i D m Ā i i ψ i σ m D m ψ i +i g(āi Tij a ψj λ a λ a Tij a Ai ψj ) 1 W ijψ i ψ j 1 W ij ψi ψj V (A, Ā), (4) V (A, Ā) = W W i i + 1 g (Āi Tij a Aj )(Āi T a (F i F i + 1 ) Da D a = ij Aj ) L L =0, F D a =0 As before, the scalar potential V (A, Ā) is positive semi-definite. Exercise: Insert (41) into (37) and derive (4). 0. (43) [1] J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press (199). 11