Labelle P. Supersymmetry DeMYSTiFied

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Supersymmetry Demystified

4. A correction to the mass terms of the scalar ﬁelds. For example, we could

replace mA

/2bymA

/2 + δmA

5. It’s an interaction that breaks supersymmetry but does not disturb the cancel-

lation of quadratic divergences.

Chapter 10

1. The photino is invariant under a gauge transformation, like the abelian ﬁeld

strength F

μν

2. A new term must be added to the transformation of the auxiliary ﬁeld, namely

√

2qφ

λ · ¯χ. In addition, all the transformations of the ﬁeld of the vector

multiplet must be multplied by a factor of −1/

√

3. It is a term in the lagrangian linear in the auxiliary ﬁeld D, which is gauge-

invariant for an abelian gauge theory.

= ∂

−

(λ

)

∗

5. Off-shell, a gauge ﬁeld has three degrees of freedom. We therefore only need

one extra off-shell bosonic degree of freedom to match the four off-shell

degrees of freedom of the Weyl spinor. In the case of the chiral multiplet,

the boson was a complex scalar ﬁeld that has only two off-shell degrees of

freedom, which is why we needed to incorporate two extra bosonic degrees

of freedom off-shell.

Chapter 11

1. Zero.

2. Using the fact that θ

= θ

, we see that the expression is differentiated and

is identically zero. But even if we were only differentiating θ

we would get

zero.

3. Eight terms. A constant, three terms linear in each Grassmann variable, three

bilinear terms (that we may take to be θ

,θ

and θ

), and ﬁnally, the

trilinear combination θ

4. A total of four, counting the hermitian conjugates as independent.

5. It gives −∂

; see Section 11.6.

APPENDIX C Solutions to Quizzes

467

Chapter 12

1. There is a complex scalar ﬁeld φ, a left-chiral Weyl spinor χ , and off-shell, a

complex auxiliary ﬁeld F .

2. Because it is a total derivative, which we can ignore.

3. Five. Extracting the D term increases the dimension by 2, and left-chiral

superﬁelds have dimension 1.

4. So that the superpotential will be itself a left-chiral superﬁeld. In this way, we

know that the F term of the superpotential is a supersymmetric lagrangian.

5. It is the function of left-chiral superﬁelds that contain at least one hermitian

conjugate 

†

. In the case of the Wess-Zumino lagrangian, it is simply K =



†

, the kinetic term of the superﬁeld.

Chapter 13

1. It transforms as a left-chiral Weyl spinor.

2. It is dimensionless (so it may appear as the argument of an exponential).

3. A gauge ﬁeld A

, a left-chiral Weyl spinor λ, and a real auxiliary scalar

ﬁeld D.

4. N

− 1, which is the number of ﬁelds in the adjoint representation.

5. Two. This is so because the electron is a Dirac particle, which is equivalent

to two Weyl spinors (which we may take to be the left-chiral electron and

positron states).

Chapter 14

1. Because the auxiliary ﬁelds D

are not gauge-invariant, so we may not write

a Fayet-Illiopooulos term.

2. It implies that some scalar superpartners of the known fermions would be

lighter than these fermions, which is ruled out experimentally.

3. The minimum of the potential V (φ

) must be equal to zero when at least one

of the scalar ﬁelds is nonzero.

4. In F-type SUSY breaking and in D-type SUSY breaking when the sum of the

U (1) charges of the particles is zero,

= 0.

5. It is a massless spin 1/2 state that is necessarily present when SUSY is broken

spontaneously.

468

Supersymmetry Demystified

Chapter 15

1. We need two left-chiral superﬁelds of opposite hypercharge to couple with all

the leptons and quarks (which, after SSB of the electroweak gauge symmetry,

produces the mass terms). We could not take the hermitian conjugate of the

ﬁrst Higgs superﬁeld because the superpotential must be holomorphic in the

superﬁelds.

2. The number of free parameters is the same as in the standard model.

3. The lepton and quark superﬁelds have odd R-parities, whereas the Higgs

superﬁeld has even R-parities. The F term of H

◦ H

◦ L

is of dimension

5, gauge invariant, supersymmetric and invariant under R-parity.

4. H

has quantum numbers 1, 2, 1, whereas H

has 1, 2, −1.

5. They are both odd [they transform with a phase exp(±iπ) =−1].

Chapter 16

1. One hundred and ﬁve in general and only ﬁve in mSUGRA.

2. There are four complex Higgs ﬁelds, so eight degrees of freedom. Three

become the longitudinal degrees of freedom of the gauge bosons, leaving ﬁve

observable Higgs ﬁelds.

3. The inverse of the U (1)

coupling constant decreases with energy in both

cases, and the inverse of the strong coupling constant increases. The inverse

of the SU(2)

coupling constant increases in the standard model but decreases

in the MSSM.

4. The mass of the h

has an upper bound. Not only that, but this upper bound is

well within reach of the LHC, which means that it will be possible to conﬁrm

or rule out the MSSM in the near future.

5. Three: δm

, δm

, and b.

APPENDIX D

Solutions to Final Exam

1. [φ] = 1, [χ] = 3/2, [F] = 2, [ζ ] =−1/2.

2. The result is equal to −1 (the minus sign comes from moving θ

to the left of

dθ

3. We have ¯χ ·

λ = ¯χ

˙a

=−

˙a

¯χ

˙a

. We now need to move the index of

λ down

and move up the index of ¯χ using the  symbol. There are several ways to

proceed. One possibility is to write −

˙a

¯χ

˙a

as −

˙a

¯χ

˙a

and then to go through

the following steps:

−

˙a

¯χ

˙a

= 

b ˙a

¯χ

˙a

¯χ

λ · ¯χ

4. The supercharges Q

and Q

†

raise the helicity by one half (wihout affecting

the four-momentum) while Q

and Q

†

lower the helicity by the same amount.

5. It is given by χ

†

i ¯σ

∂

χ. To prove that it is real, we use the fact that when we

apply a hermitian conjugation to the product of two spinors, we switch their

order without introducing a minus sign. We therefore obtain

(χ

†

i ¯σ

∂

χ)

†

=−(∂

†

)i ¯σ

470

Supersymmetry Demystified

where we have used the fact that the σ

are hermitian and the minus sign

comes from the complex conjugation of i . Performing an integration by parts,

we recover the original expression which shows that it is indeed real.

6. The auxiliary ﬁeld allows the supersymmetric algebra to close off-shell for

the spinor χ. In this way, the supersymmetric algebra closes for all the ﬁelds

without the use of any equation of motion.

7. Because the hypercharges of all the particles add up to zero. In such a case,

the supertrace vanishes, and this would imply the existence of some scalar

superpartners lighter than the known fermions, which is ruled out experimen-

tally.

8. Yes, SUSY is broken spontaneously because the minimum of the potential is

not zero.

9. Since ζ is of dimension −1/2 and D is of dimension 2, the transformation must

necessarily contain the photino λ which is the only ﬁeld with a half integer

dimension. For the dimensions to come out right, there must be a derivative

acting on the photino. So we try as a ﬁrst guess δD  ζ∂

λ. But the Lorentz

indices don’t match. We must introduce a set of Pauli matrices to act on the

photino. On a left-chiral spinor, it is the ¯σ

that we must apply. Since ¯σ

transforms like a right-chiral spinor, we must combine with it the hermitian

conjugate of ζ . Or ﬁnal answer is therefore

δ D = Cζ

†

¯σ

∂

λ + h.c.

where we needed to add the complex conjugate to make the transformation

real since D is a real scalar ﬁeld.

10. If invariance under SUSY is not imposed, there are ten free parameters: three

masses and seven coupling constants. When supersymmetry is imposed, there

are only two parameters: one mass and one coupling constant.

11. The constraint is

˙a

 = 0.

12. A Dirac mass term is of the form −m(χ

· χ

+ ¯χ

· ¯χ

), whereas a Majorana

mass term is −m(χ

· χ

+ ¯χ

· ¯χ

). This makes sense because a Majorana

fermion is its own antiparticle, and therfore, χ

= χ

13. This would be inconsistent with SUSY transformations. If we start with a

superﬁeld depending only on x

and θ and apply a supersymmetry transfor-

mation, the resulting superﬁeld will now depend on

θ as well.

14. First, we use the fact that the matrix iσ

is used to raise indices so that

¯χ

= ¯χ

and ¯χ

=−¯χ

. Then we use the fact that barred and unbarred spinor

components are simply related by hermitian conjugation, so we ﬁnally have

¯χ

= (χ

)

†

and ¯χ

=−(χ

)

†

APPENDIX D Solutions to Final Exam

471

15. Gaugino is the generic term for the spinor superpartners of gauge ﬁelds.

Examples of gauginos are the photino, the gluinos, the bino, and the wino.

16. The D term is the coefﬁcient of θ · θ

θ ·

θ/4. Some other references use the

convention that the factor of one-fourth is not included in the deﬁnition of the

D term.

17. Helicity is the projection of the spin along the direction of motion; in other

words, it is the eigenvalue of the operator



S ·

P. Chirality is the eigenvalue

of γ

18. The lagrangian is

∂

†

∂

φ +i ¯χ ¯σ

∂

χ + F

†

19. The LSP is the lightest supersymmetric particle.

20. If R-parity is a valid symmetry, the LSP is stable; i.e., it cannot decay into

any other particle.

21. No, because none of the superﬁelds of the MSSM is invariant under the gauge

group SU(3)

× SU(2)

×U (1)

22. The ﬁeld must have a transformation that contains both ζ (which has a

dimension equal to −1/2) and one of the other ﬁelds, which all have a lower

dimension. Schematically, δ H  ζ L, where we used H to denote the ﬁeld

of highest dimension and L to represent one of the other component ﬁelds,

which has a lower dimension than H. To make the dimensions match, we

need absolutely to include a derivative ∂

(as well as some Pauli matrices to

take care of the Lorentz indices). Therefore, the component ﬁeld of highest

dimension transforms with a total derivative.

23. To eliminate interactions in the superpotential that violate lepton and baryon

number conservation.

24. At the uniﬁcation scale, we necessarily have sin

= 3/8, so θ

sin

−1

(

√

3/8).

25. Because we need to extract the F term of the superpotential, and extracting

the F term increases the dimension by 1 (because the Grassmann variables θ

have dimension −1/2 and extracting the F term results in dropping a factor

θ ·θ/2).

26. Because the tree-level relation implies that the mass of the h

is smaller than

the Z mass, which is ruled out experimentally.

27. Because they have an electric charge or a color charge (except for the sneu-

trino), and we don’t want the color and charge invariances to be broken spon-

taneously because that would conﬂict with observations. On the other hand, a

472

Supersymmetry Demystified

vev for the sneutrino would break R-parity spontaneously, which would cause

problems with lepton and baryon number-violating effects.

28. It is



. Note the factor of one-half!

29. The tree-level relation between the two masses is exactly the same as in the

standard model, namely, m

= m

cos θ

30. Masses for the gauginos (the spin 1/2 superpartners of the gauge bosons),

mass terms for all the scalar ﬁelds (which include the Higgs as well as the

sleptons and the squarks), and holomorphic superrenormalizable interactions

between the scalar ﬁelds.

31. Off-shell, a left-chiral superﬁeld contains two complex scalar ﬁelds and a

left-chiral Weyl fermion which has two complex components. So there are

four bosonic and four fermionic degrees of freedom. On-shell, the auxiliary

complex scalar ﬁeld is eliminated and we are left with only one complex

scalar ﬁeld. On the other hand, the spinor must obey the Weyl equation which

reduces in half the number of fermionic degrees of freedom. The end result

is that there are two bosonic and two fermionic degrees of freedom on-shell.

32. The general superpotential is



ijk



+ c



although the last term is generally not included.

33.

L = W(

)



+ h.c. +





†



34. It transforms as a left-chiral Weyl spinor.

35.

¯χ

˙a

¯χ

˙a

( ¯σ

)

˙ab

36. The particle content is different. Even though the gauge bosons are the same

in both theories, the beta functions are different because different particles

circulate in the loops.

37. The F term is 2A(x) and the D term is 4C(x).

38. The potential is given by

V =|μ|











+ g

2





−







APPENDIX D Solutions to Final Exam

473

This potential is clearly minimized when both ﬁelds are equal to zero, which

implies that the gauge symmetry is not broken. In addition, at the minimum

the potential is equal to zero which means that SUSY is not broken either.

39. The general formula is

R = (−1)

3B+L+2s

For the quarks, s = 1/2, B = 1/3, and L = 0soR =+1. For the leptons,

B = 0, L = 1, and s = 1/2 so we again get R =+1. For the Higgs, B =

L = s = 0sotheyhaveR =+1 as well. The superpartners simply have a

spin s differing by 1/2 in so they all have R =−1. On the other hand, the

gauge supermultiplets are taken to be invariant under R-parity.

40. The only gauge invariant combination out of two of these superﬁelds is H

◦

(plus the hermitian conjugate, as always). We must extract the F term,

which increases the dimension by 1, so the lagrangian is of the form μH

◦ H

with μ having the dimension of a mass. This is of course the μ term of the

MSSM. With three ﬁelds, our only gauge invariant combination is H

◦ H

but this is identically zero since for any column vector A we have A

iτ

A =

0. As for gauge invariant terms containing four superﬁelds, we have two

possibilities, H

◦ H

and H

◦ H

but the second term is

identically zero. After extracting the F term, this will be of dimension 5. So

our lagrangian is



μH

◦ H





+ h.c.

41. A Fayet-Iliopoulos term is a term linear in a left-chiral superﬁeld. Its contri-

bution to a supersymmetric lagrangian is simply m

|

+ h.c. Such a term

is only allowed if the superﬁeld is a gauge invariant. Such a term is absent in

the MSSM because no superﬁeld is invariant under the full gauge group.

42. Moving the θ

through the ﬁrst derivative, we get

−

∂

∂θ



∂

∂θ



=−

∂

∂θ

Now we must recall that θ

= θ

so the ﬁnal result is −1.

43. The chirality operator is simply γ

whereas the helicity operator is



S ·

44. Since the supercharges change a boson into a fermion or vice versa, the

supercharges anticommutes with (−1)

. Therefore, we may write

(−1)

†

+ (−1)

†

=−Q

(−1)

†

+ (−1)

†

474

Supersymmetry Demystified

Taking the trace of this and using the invariance of a trace under cyclic per-

mutation, we immediately get that the two terms cancel out.

45. Because the loop correction to the square of the Higgs mass are power-law

divergent and an extreme ﬁne-tuning of the bare mass must be invoked to

keep the physical mass small. In contrast, the masses of the other fundamental

particles receive logarithmic corrections and therefore do not require delicate

cancellations.

46. With Dirac spinors, we need only consider the contraction of

 with . With

Majorana spinors, we need to consider in addition the contractions of 



and of



47. Using the fact that the hermitian conjugate of a product of spinors components

switches their order without introducing any minus sign, we have

(χ

)

†

= (λ

)

†

(χ

)

†

Recall that taking a hermitian conjugate changes undotted indices into dotted

indices without changing their position and adds a bar, so that we get

˙a

¯χ

˙a

which is by deﬁnition

λ · ¯χ. This is also equal to ¯χ ·

λ but that requires a bit

more work to prove.

48. The gauginos are the spin 1/2 superpartners of the gauge bosons. Like the

gauge bosons, they transform in the adjoint representation of the gauge group.

49. It is simply



scalars

− 2



fermions

where the sum is over the physical (on-shell) states. When SUSY is not

broken, each Weyl spinor is paired with a complex scalar ﬁeld with the same

mass. There are thus two real scalar ﬁelds degenerate in mass with each Weyl

fermion, which leads to a vanishing supertrace.

50. The superpotential must be a holomorphic function of  (it cannot contain

the hermitian conjugate of the superﬁeld). Since  is charged, there is no

possible gauge invariant superpotential.

INDEX

A ﬁeld, one-point function of, in Wess-Zumino

model, 185–189

abelian ﬁeld-strength superﬁeld, in terms of

component ﬁelds, 308–313

abelian gauge invariance, in superﬁeld

formalism, 297–303

abelian gauge multiplet, explicit interactions

between left-chiral multiplet and, 303–306

abelian gauge theory, free supersymmetric,

206–209, 306–308

abelian vector multiplet, combination of, with

chiral multiplet, 222–232

algebra (see SUSY algebra)

anomaly-mediated SUSY breaking, 350

antiparticles, for Dirac spinors, 68–70

antisymmetry, 106–107

auxiliary ﬁelds, 128–130

elimination of, 232–233

introduction of, 209–211

B ﬁeld, propagator of, to one loop,

189–201

Baker-Campbell-Hausdorff formula, 243

Berezin integration, 252

bosons, 2, 3

Burgess, Cliff, 7

Casimir operators, 134, 135

Centre Europ´een de Recherche Nucl´eaire

(CERN), 3

charge conjugation, 68–70

charges, 102–107 (See also supersymmetric

charges)

conserved, 153

explicit representations of, 107–111

ﬁnding algebra without explicit,

111–115

obtaining, from symmetry currents,

151–153

chiral multiplets, 146

combination of abelian vector multiplet

with, 222–232

combination of nonabelian gauge

multiplet with, 233–236

chiral representation, 15

chirality, 18–19

Coleman-Mandula no-go theorem,

156–157

component ﬁelds:

abelian ﬁeld-strength superﬁeld in

terms of, 308–313

SUSY transformations of, 273–277

component ﬁelds (of superﬁeld), 250

condensates, 350

conserved charges, 153

constraints, and superﬁelds, 261–268