Aitchison I. Supersymmetry in Particle Physics: An Elementary Introduction

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7.3 Combining chiral and gauge supermultiplets 115

So (7.44) becomes

−2qα[χ

†

· ξ

†

χ · λ + χ

†

· λ

†

χ · ξ ], (7.46)

which will cancel (7.43) if (again using χ · ξ = ξ · χ and χ

†

· λ

†

= λ

†

· χ

†

)

A = 2α. (7.47)

So far, there is nothing to prevent us from choosing α = 1, say, in (7.42) and

(7.47). However, a constraint on α arises when we consider the variation of the

− φ interaction term in (7.32), namely

−iqδ

†

∂

φ − (∂

φ)

†

φ). (7.48)

The terms in δ

yield a change

iqα[(∂

†

)(ξ

†

¯σ

λ + λ

†

¯σ

ξ)φ − (ξ

†

¯σ

λ + λ

†

¯σ

ξ)φ

†

∂

φ]. (7.49)

A similar change arises from the terms Aq[φ

†

(δ

χ) · λ + λ

†

· (δ

†

)φ] in (7.39),

namely

Aq[φ

†

(−iσ

iσ

†T

∂

φ) · λ + λ

†

· ∂

†

(−iσ

iσ

φ)]. (7.50)

The ﬁrst spinor dot product is

†

(−iσ

)(−iσ

μT

)(−iσ

)λ = iξ

†

¯σ

λ, (7.51)

using (4.30). The second spinor product is the hermitian conjugate of this, so that

(7.50) yields a change

Aqi[φ

†

(∂

φ)ξ

†

¯σ

λ − (∂

†

)φλ

†

¯σ

ξ]. (7.52)

Along with (7.49) and (7.52) we must also group the last two terms in (7.41), which

we write out again here for convenience

Bq[φ

†

φ(−αi)(ξ

†

¯σ

∂

λ − (∂

†

)¯σ

ξ)], (7.53)

and integrate by parts to yield

αiBq{[(∂

†

)φ + φ

†

∂

φ](ξ

†

¯σ

λ) − [(∂

†

)φ + φ

†

∂

φ](λ

†

¯σ

ξ)}. (7.54)

Consider now the terms involving the quantity ξ

†

¯σ

λ in (7.49), (7.52) and (7.54),

which are

iqα[(∂

†

)φ − φ

†

∂

φ] + Aqiφ

†

∂

φ + αiBq[(∂

†

)φ + φ

†

∂

φ]. (7.55)

These will all cancel if the condition (7.47) holds, and if in addition

B =−1. (7.56)

116 Vector (or gauge) supermultiplets

From (7.47) and (7.42) it now follows that

. (7.57)

We conclude that, as promised, the combined Lagrangian will not be SUSY-

invariant unless we modify the scale of the transformations of the gauge supermul-

tiplet, relative to those of the chiral supermultiplet, by a non-trivial factor, which

we choose (in agreement with what seems to be the usual convention) to be

α =−

√

. (7.58)

With this choice, the coefﬁcient A is determined to be

A =−

√

2, (7.59)

and our combined Lagrangian is ﬁxed.

We have, of course, not given a complete analysis of all the terms in the SUSY

variation of our Lagrangian, an exercise we leave to the dedicated reader, who will

ﬁnd that (with one more adjustment to the SUSY transformations) all the variations

do indeed vanish (after partial integrations in some cases, as usual). The need for

the adjustment appears when we consider the variation associated with the terms

Aq[φ

†

(δ

χ) · λ + λ

†

· (δ

†

)φ] in (7.39), which includes the term

Aq[φ

†

ξ · λF + λ

†

· ξ

†

φ]. (7.60)

This cannot be cancelled by any other variation, and we therefore have to modify the

transformation for F and F

†

so as to generate a cancelling term from the variation

of F

†

F in the Lagrangian. This requires

F =−

√

2qλ

†

· ξ

†

φ + previous transformation (7.61)

and

†

=−

√

2qξ · λφ

†

+ previous transformation, (7.62)

where we have now inserted the known value of A.

In summary then, our SUSY-invariant combined chiral and U(1) gauge super-

multiplet Lagrangian is

L = (D

φ)

†

φ) + iχ

†

¯σ

χ + F

†

F −

μν

+ iλ

†

¯σ

∂

λ +

−

√

2q[(φ

†

χ) · λ + λ

†

· (χ

†

φ)] − qφ

†

φ D. (7.63)

Note that the terms in the last line of (7.63) are interactions whose strengths are

ﬁxed by SUSY to be proportional to the gauge-coupling constant q, even though

7.3 Combining chiral and gauge supermultiplets 117

they do not have the form of ordinary gauge interactions; the terms coupling the

photino λ to the matter ﬁelds may be thought of as arising from supersymmetrizing

the usual coupling of the gauge ﬁeld to the matter ﬁelds.

The equation of motion for the ﬁeld D is

D = qφ

†

φ. (7.64)

Since no derivatives of D enter, we may (as in the W–Z case for F

and F

†

, cf.

equations (5.21) and (5.22)) eliminate the auxiliary ﬁeld D from the Lagrangian

by using (7.64). The effect of this is clearly to replace the two terms involving D

in (7.63) by the single term

−

(φ

†

φ)

. (7.65)

This is a ‘(φ

†

φ)

’ type of interaction, just as in the Higgs potential (1.4), but here

appearing with a coupling constant, which is not an unknown parameter but is

determined by the gauge coupling q. In the next section we shall see that the same

feature persists in the more realistic non-Abelian case. Since the Higgs mass is (for

a ﬁxed vev of the Higgs ﬁeld) determined by the (φ

†

φ)

coupling (see (1.3)), it

follows that there is likely to be less arbitrariness in the mass of the Higgs in the

MSSM than in the SM. We shall see in Chapter 10, when we examine the Higgs

sector of the MSSM, that this is indeed the case.

7.3.2 The non-Abelian case

Once again, we proceed in two steps. We start from the W–Z Lagrangian for a

collection of chiral supermultiplets labelled by i, and including the superpotential

terms:

∂

†

∂

+ χ

†

i¯σ

∂

+ F

†



∂W

∂φ

−

∂

∂φ

· χ

+ h.c.



(7.66)

into which we introduce the gauge couplings via the covariant derivatives

∂

→ D

= ∂

+ igA

φ)

(7.67)

∂χ

→ D

= ∂

+ igA

χ)

, (7.68)

where g and A

are the gauge coupling constant and gauge ﬁelds (for example, g

and gluon ﬁelds for QCD), and the T

are the hermitian matrices representing the

generators of the gauge group in the representation to which, for given i, φ

and χ

belong (for example, if φ

and χ

are SU(2) doublets, the T

’s would be the τ

/2,

with α running from 1 to 3). Recall that SUSY requires that φ

, χ

and F

must all

be in the same representation of the relevant gauge group. Of course, if, as is the

118 Vector (or gauge) supermultiplets

case in the SM, some matter ﬁelds interact with more than one gauge ﬁeld, then

all the gauge couplings must be included in the covariant derivatives. There is no

covariant derivative for the auxiliary ﬁelds F

, because their ordinary derivatives

do not appear in (7.66). To (7.68) we need to add the Lagrangian for the gauge

supermultiplet(s), equation (7.29), and then (in the second step) additional ‘mixed’

interactions as in (7.33).

We therefore need to construct all possible Lorentz- and gauge-invariant renor-

malizable interactions between the matter ﬁelds and the gaugino (λ

) and auxiliary

) ﬁelds, as in the U(1) case. We have the speciﬁc particle content of the SM

in mind, so we need only consider the cases in which the matter ﬁelds are either

singlets under the gauge group (for example, the R parts of quark and lepton ﬁelds),

or belong to the fundamental representation of the gauge group (that is, the triplet

for SU(3) and the doublet for SU(2)). For matter ﬁelds in singlet representations,

there is no possible gauge-invariant coupling between them and λ

or D

, which are

in the regular representation. For matter ﬁelds in the fundamental representation,

however, we can form bilinear combinations of them that transform according to

the regular representation, and these bilinears can be ‘dotted’ into λ

and D

to give

gauge singlets (i.e. gauge-invariant couplings). We must also arrange the couplings

to be Lorentz invariant, of course.

The bilinear combinations of the φ

and χ

which transform as the regular rep-

resentation are (see, for example, [7] Sections 12.1.3 and 12.2)

†

,φ

†

,χ

†

, and χ

†

, (7.69)

where, for example, T

= τ

/2 in the case of SU(2), and where the τ

, (α =

1, 2, 3) are the usual Pauli matrices used in the isospin context. These bilinears

are the obvious analogues of the ones considered in the U(1) case; in particular

they have the same dimension. Following the same reasoning, then, the allowed

additional interaction terms are

Ag[(φ

†

) · λ

+ λ

α†

· (χ

†

)] + Bg(φ

†

, (7.70)

where A and B are coefﬁcients to be determined by the requirement of SUSY-

invariance.

In fact, however, a consideration of the SUSY transformations in this case shows

that they are essentially the same as in the U(1) case (apart from straightforward

changes involving the matrices T

). The upshot is that, just as in the U(1) case, we

need to change the SUSY transformations of (7.30) by replacing ξ by −ξ/

√

2, and

by modifying the transformation of F

†

=−

√

2gφ

†

ξ · λ

+ previous transformation, (7.71)

7.3 Combining chiral and gauge supermultiplets 119

and similarly for δ

. The coefﬁcients A and B in (7.70) are then −

√

2 and −1,

respectively, as in the U(1) case, and the combined SUSY-invariant Lagrangian is

gauge + chiral

= L

gauge

(equation (7.29))

W−Z, covariantized

(equation (7.66), with ∂

→ D

as in (7.67) and (7.68))

−

√

2g[(φ

†

) · λ

+ λ

α†

· (χ

†

)] − g(φ

†

. (7.72)

We draw attention to an important consequence of the terms −

√

2g[...] in (7.72),

for the case in which the chiral multiplets (φ

,χ

) are the two Higgs supermultiplets

and H

, containing Higgs and Higgsino ﬁelds (see Table 8.1 in the next chapter).

When the scalar Higgs ﬁelds H

and H

acquire vevs, these terms will be bilinear

in the Higgsino and gaugino ﬁelds, implying that mixing will occur among these

ﬁelds as a consequence of electroweak symmetry breaking. We shall discuss this

in Section 11.2.

The equation of motion for the ﬁeld D

= g



(φ

†

), (7.73)

where the sum over i (labelling a given chiral supermultiplet) has been re-instated

explicitly. As before, we may eliminate these auxiliary ﬁelds from the Lagrangian

by using (7.73). The complete scalar potential (as in ‘L = T − V’) is then

V(φ

,φ

†

) =|W



i, j



†



†



, (7.74)

where in the summation we have recalled that more than one gauge group G will

enter, in general, given the SU(3)×SU(2)×U(1) structure of the SM, with different

couplings g

and generators T

. The ﬁrst term in (7.74) is called the ‘F-term’,

for obvious reasons; it is determined by the fermion mass terms M

and Yukawa

couplings (see (5.19)). The second term is called the ‘D-term’, and is determined

by the gauge interactions. There is no room for any other scalar potential, inde-

pendent of these parameters appearing in other parts of the Lagrangian. It is worth

emphasizing that V is a sum of squares, and is hence always greater than or equal

to zero for every ﬁeld conﬁguration. We shall see in Section 10.2 how the form of

the D-term allows an important bound to be put on the mass of the lightest Higgs

boson in the MSSM.

The MSSM

8.1 Speciﬁcation of the superpotential

We have now introduced all the interactions appearing in the MSSM, apart from

specifying the superpotential W . We already assigned the SM ﬁelds to supermul-

tiplets in Sections 3.2 and 4.4; let us begin by reviewing those assignments once

more.

All the SM fermions, i.e. the quarks and the leptons, have the property that their

L(‘χ ’) parts are SU(2)

doublets, whereas their R (‘ψ’) parts are SU(2)

singlets.

So these weak gauge group properties suggest that we should treat the L and R

parts separately, rather than together as in a Dirac 4-component spinor. The basic

‘building block’ is therefore the chiral supermultiplet, suitably ‘gauged’.

We developed chiral supermultiplets in terms of L-type spinors χ: this is clearly

ﬁne for e

−

,μ

−

, u

, d

, etc., but what about e

−

,μ

−

, etc.? These R-type particle ﬁelds

can be accommodated within the ‘L-type’ convention for chiral supermultiplets by

regarding them as the charge conjugates of L-type antiparticle ﬁelds, which we use

instead. Charge conjugation was mentioned in Section 2.3; see also Section 20.5

of [7] (but note that we are here using C

=−iγ

). If (as is often done) we denote

the ﬁeld by the particle name, then we have e

−

≡ ψ

−

, while e

≡ χ

. On the other

hand, if we regard e

−

as the charge conjugate of e

, then (compare equation (2.94))

−

≡ ψ

−

= (e

)

≡ iσ

†T

. (8.1)

To remind ourselves of how this works (see also Section 2.4), consider a Dirac mass

term for the electron:



−

)



−

)

= ψ

†

−

+ χ

†

−



iσ

†T



†

−

+ χ

†

−

iσ

†T

= χ

(−iσ

)χ

−

+ χ

†

−

iσ

†T

= χ

· χ

−

+ χ

†

−

· χ

†

. (8.2)

120

8.1 Speciﬁcation of the superpotential 121

Table 8.1 Chiral supermultiplet ﬁelds in the MSSM.

Names spin 0 spin 1/2 SU(3)

, SU(2)

, U(1)

squarks, quarks Q (

)(u

, d

) 3, 2,1/3

(× 3 families) or (χ

,χ

)

†

= (u

)

3, 1, −4/3

or χ

¯u

= ψ

†

= (d

)

3, 1,2/3

or χ

= ψ

sleptons, leptons L (˜ν

)(ν

, e

) 1, 2, −1

(× 3 families) or (χ

,χ

)

†

= (e

)

1, 1,2

or χ

¯e

= ψ

Higgs, Higgsinos H

, H

)(

) 1, 2,1

, H

−

)(

−

) 1, 2, −1

Table 8.2 Gauge supermultiplet ﬁelds in the MSSM.

Names spin 1/2 spin 1 SU(3)

, SU(2)

, U(1)

gluinos, gluons

gg 8, 1,0

winos, W bosons



, W

1, 3,0

bino, B boson

BB 1, 1,0

So it is all expressed in terms of χ’s. It is also useful to note that



−

)



−

)

=−χ

· χ

−

+ χ

†

−

· χ

†

. (8.3)

The notation for the squark and slepton ﬁelds was explained in Section 4.4, follow-

ing equation (4.97).

In Table 8.1 we list the chiral supermultiplets appearing in the MSSM (our y

is twice that of [46], following the convention of [7], Chapter 22). Note that the

‘bar’ on the ﬁelds in this table is merely a label, signifying ‘antiparticle’, not (for

example) Dirac conjugation: thus χ

−

and χ

, for instance, now become χ

and χ

¯e

The subscript

can be added to the names to signify the family index: for example,

= u

, u

= c

, u

= t

, and similarly for leptons. In Table 8.2, similarly, we

list the gauge supermultiplets of the MSSM. After electroweak symmetry breaking,

the W

and the B ﬁelds mix to produce the physical Z

and γ ﬁelds, while the

corresponding ‘s’-ﬁelds mix to produce a zino (

) degenerate with the Z

, and a

massless photino ˜γ .

122 The MSSM

So, knowing the gauge groups, the particle content, and the gauge transforma-

tion properties, all we need to do to specify any proposed model is to give the

superpotential W . The MSSM is speciﬁed by the choice

W = y

· H

− y

· H

− y

· H

+ μH

· H

. (8.4)

The ﬁelds appearing in (8.4) are the chiral superﬁelds indicated under the ‘Names’

column of Table 8.1. In this formulation, we recall from Section 6.4 that the F-

component of W is to be taken in the Lagrangian. We can alternatively think of

W as being the same function of the scalar ﬁelds in each chiral supermultiplet, as

explained in Section 6.4. In that case, the W

of (5.17) and W

of (5.8) generate

the interaction terms in the Lagrangian via (5.3). In either case, the y’s are 3 × 3

matrices in family (or generation) space, and are exactly the same Yukawa couplings

as those which enter the SM (see, for example, Section 22.7 of [7]).

In particular, the

terms in (8.4) are all invariant under the SM gauge transformations. The ‘·’ notation

means that SU(2)-invariant coupling of two doublets;

also, colour indices have

been suppressed, so that ‘

’, for example, is really

αi

, where the upstairs

α = 1, 2, 3 is a colour 3 (triplet) index, and the downstairs α is a colour

3 (antitriplet)

index. These couplings give masses to the quarks and leptons when the Higgs ﬁelds

and H

acquire vacuum expectation values: there are no ‘Lagrangian’ masses

for the fermions, since these would explicitly break the SU(2)

gauge symmetry.

In summary, then, at the cost of only one new parameter μ, we have got an

exactly supersymmetric extension of the SM. This is, of course, not the same ‘μ’

as appeared in the Higgs potential of the SM, equation (1.4). We follow a common

notation, although others are in use which avoid this possible confusion.

[In parenthesis, we note a possibly confusing aspect of the labelling adopted

for the Higgs ﬁelds. In the conventional formulation of the SM, the Higgs

ﬁeld φ =





generates mass for the ‘down’ quark, say, via a Yukawa

interaction of the form (suppressing family labels)

φd

+ h.c. (8.5)

However, we stress once again – see Section 3.2 and footnote 1 of Chapter 5, page 72 – that whereas in the SM

we can use one Higgs doublet and its charge conjugate doublet (see Section 22.6 of [7]), this is not allowed in

SUSY, because W cannot depend on both a complex scalar ﬁeld φ and its Hermitian conjugate φ

†

(which would

appear in the charge conjugate via (3.40)). By convention, the MSSM does not include Dirac-type neutrino mass

terms, neutrino masses being generally regarded as ‘beyond the SM’ physics.

To take an elementary example: consider the isospin part of the deuteron’s wavefunction. It has I = 0; i.e. it is

the SU(2)-invariant coupling of the two doublets N

(1)



(1)



, N

(2)



(2)



. This I = 0 wavefunction

is, as usual,

√

( p

(1)

(2)

− n

(1)

(2)

), which (dropping the 1/

√

2) we may write as N

(1)T

iτ

(2)

≡ N

(1)

· N

(2)

Clearly this isospin-invariant coupling is basically the same as the Lorentz-invariant spinor coupling ‘χ

(1)

· χ

(2)

’

(see (2.47)–(2.49)), which is why we use the same ‘·’ notation for both, we hope without causing confusion.

8.1 Speciﬁcation of the superpotential 123

where q





. In this case, the SU(2) dot product is simply q

†

φ,

which is plainly invariant under q

→ Uq

,φ→ U φ.Nowq

†

φ =

†

+ d

†

; so when φ

develops a vev, (8.5) contributes

φ



+ h.c. (8.6)

which is a d-quark mass. Why, then, do we label our Higgs ﬁeld





with a subscript ‘u’ rather than ‘d’? The point is that, in the SUSY version

(8.4), the SU(2) dot product involving the superﬁeld H

is taken with the

superﬁeld Q which has the quantum numbers of the quark doublet q

rather

than the antiquark doublet q

†

. If we revert for the moment to the procedure

of Section 5.1, and write W just in terms of the corresponding scalar ﬁelds,

the ﬁrst term in (8.4) is



L j

−

L j



. (8.7)

The ﬁrst term here will, via (5.8) and (5.3), generate a term in the Lagrangian

(cf. (5.45))

−

(χ

¯u

· χ

L j

+ χ

· χ

¯u

L j

+ h.c. =−y

(χ

¯u

· χ

L j

+ h.c.

(8.8)

When H

develops a (real) vacuum value v

(see Section 10.1), this will

become a Dirac-type mass term for the u-quark (cf. (8.2)):

−(m

uij

¯u

· χ

L j

+ h.c.) (8.9)

where

uij

= v

. (8.10)

Transforming to the basis which diagonalizes the mass matrices then

leads to ﬂavour mixing exactly as in the SM (see Section 22.7 of [7], for

example).]

The fermion masses are evidently proportional to the relevant y parameter, so

since the top, bottom and tau are the heaviest fermions in the SM, it is sometimes

useful to consider an approximation in which the only non-zero y’s are

= y

; y

= y

; y

= y

. (8.11)

124 The MSSM

Writing W now in terms of the scalar ﬁelds, and omitting the μ term, this gives

W ≈y





−



− y





−



− y



¯τ



˜ν

τ L

−

− ˜τ



(8.12)

The minus signs in W have been chosen so that the terms y

, y

and

¯τ

˜τ

have the correct sign to generate mass terms for the top, bottom and tau

when H

 = 0 and H

 = 0. Note that

and

¯τ

could equally well have

been written as

†

and ˜τ

†

It is worth recalling that in such a SUSY theory, in addition to the Yukawa

couplings of the SM, which couple the Higgs ﬁelds to quarks and to leptons, there

must also be similar couplings between Higgsinos, squarks and quarks, and between

Higgsinos, sleptons and leptons (i.e. we change two ordinary particles into their

superpartners). There are also scalar quartic interactions with strength proportional

to y

, as noted in Section 5.1, arising from the term ‘|W

’ in the scalar potential

(7.74). In addition, there are scalar quartic interactions proportional to the squares

of the gauge couplings g and g



coming from the ‘D-term’ in (7.74). These include

quartic Higgs couplings such as are postulated in the SM, but now appearing with

coefﬁcients that are determined in terms of the parameters g and g



already present in

the model. The important phenomenological consequences of this will be discussed

in Chapter 10.

Although there are no conventional mass terms in (8.4), there is one term which is

quadratic in the ﬁelds, the so-called ‘μ term’, which is the SU(2)-invariant coupling

of the two different Higgs superﬁeld doublets:

W (μ term) = μH

· H

= μ(H

− H

), (8.13)

where the subscripts 1 and 2 denote the isospinor component. This is the only such

bilinear coupling of the Higgs ﬁelds allowed in W , because the other possibilities

†

· H

and H

†

· H

involve hermitian conjugate ﬁelds, which would violate SUSY.

As always, we need the F-component of (8.13), which is (see (6.61))



−

+ H

−

− H



−



−



, (8.14)

and we must include also the hermitian conjugate of (8.14). The second term in

(8.14) will contribute to (off-diagonal) Higgsino mass terms. The ﬁrst term has the

general form ‘W

’ of Section 5.1, and hence (see (5.22)) it leads to the following

term in the scalar potential, involving the Higgs ﬁelds:

|μ|



+|H

−





. (8.15)

But these terms all have the (positive) sign appropriate to a standard ‘m

†

φ’

bosonic mass term, not the negative sign needed for electroweak symmetry breaking