Introduction to Supersymmetry

3.4. MASSIVE SUPERMULTIPLETS

15

For N = 2 we already have 16 states with helicities ranging at least from 1 to 1. Such a massive N = 2 multiplet can be viewed as the union of a massless N = 2 vector and hypermultiplet. A generic massive N = 4 multiplet contains 28 = 256 states including at least a helicity 2. Thus such a theory must include a massive spin two particle which is not believed to be possible in quantum eld theory.

If k < N2 of the qn are equal to 2m then we only have 2N 2k oscillators, and the supermultiplets will only contain 22(N k) states. They are called short multiplets or BPS multiplets. If all qn equal 2m, i.e. k = N2 we get the shortest multiplets with only 2N states, exactly as in the massless case. These BPS multiplets are also called ultrashort, and are completely analogous to the massless multiplets.

15

16 CHAPTER 3. THE SUSY ALGEBRA AND ITS REPRESENTATIONS

16

Chapter 4

Superspace and super elds

Since we want to construct supersymmetric quantum eld theories, we have tond representations of the susy algebra on elds. A convenient and compact way to do this is to introduce superspace and super elds, i.e. elds de ned on superspace. This is particularly simple for unextended susy, so we will restrict here to N = 1 superspace and super elds. Then we have two plus two susy generators Q and Q , as well as four generators P of space-time translations. The idea then is to enlarge space-time labelled by the coordinates x by adding two plus two anticommuting Grassmannian coordinates and . Thus coordinates on superspace are (x ; ; ). Rather than elaborating on the meaning of such a

space we will simply use it as a very e cient recepee to perform calculations in supersymmetric theories.

4.1Superspace

As already said, we restrict here to N = 1. The \odd" superspace coordinatesand just behave as constant (x independent) spinors. Recall that as all spinors they anticommute among themselves, i.e. 1 2 = 2 1, and idem for

the . Spinor indices in bilinears are contracted acording to the usual rule, i.e.= = 2 1 2 = +2 2 1 = 2 1 2, and = = 2 1 2 = : : :. One can then easily prove the following useful identities:

_

= 21

21

_

;

=

;

(4.1)

1

=

1

;

_

_ ;

= 2

2

= 21

= 21 :

g

;

17

18			CHAPTER 4. SUPERSPACE AND SUPERFIELDS
				@							_
	Derivatives in and are de ned in an obvious way as				=	and	@			=

_

. Since the 's anticommute, any product involving more than two 's or more

than two 's vanishes. Hence an arbitrary (scalar) function on superspace, i.e. a super eld, can always be expanded as

F (x; ; ) =

f(x) + (x) + (x) + m(x) + n(x)

(4.2)

+

v (x) + (x) + (x) + d(x) :

If F carries extra vector indices then so do the fomponent elds f; ; : : :.

Integration on superspace is de ned for a single Grassmannian variable, say1 as R d 1(a + 1b) = b so that R d 1d 2 2 1 = 1. Then since = 2 2 1 and

= 2 1 2 we de ne d2 = 12 d 1d 2 and d2 = 12 d 2d 1 = [d2 ]y so that

Z

Z

d2 =

d2

= 1 :

(4.3)

It is easy to check that

1

@

@

1

@ @

Z d2 =

;

Z d2 =

_

:

(4.4)

4

@ @

4

_

@

@

Clearly one also has

Z

d2 d2

= 1 :

(4.5)

With these de nitions it is easy to see that one has the hermiticity property

@	y		@
	!	= +			(4.6)

with . Note the plus sign rather than a minus sign as one would expect from (@ )y = @ .

We now want to realise the susy generators Q and their hermitian conjugates Q = (Q )y as di erential operators on superspace. We want that i Q generates a translation in by a constant in nitesimal spinor plus some translation in x . The latter space-time translation is determined by the susy algebra since the commutator of two such susy transformations is a translation in space-time. Thus we want

(1 + i Q)F (x; ; ) = F (x + x; + ; ) :			(4.7)

Hence iQ = @@ + : : : where + : : : must be of the form c( ) P = ic( ) @ with some constant c to be determined. We arrive at the ansatz

Q = i	@	ic( ) @ ! :	(4.8)
	@

18

4.1. SUPERSPACE
Then the hermitian conjugate is
			@			ic ( ) @ ! ;
	Q = i
			@

and they satisfy the susy algebra, in particular

fQ ; Q _ g = 2 _ P = 2i _ @

if Re c = 1. We choose c = 1 so that

@

_

Q

=

i

_ @

@

@

Q

=

i

+

@ :

@

19

(4.9)

(4.10)

(4.11)

We can now give the action on the super eld F and determine x:

_

_

_

(1 + i Q + i Q)F (x ; ;

) = F (x i + i ; + ;

+ ) (4.12)

and the susy variation of a super eld is of course de ned as

;

F = (i Q + i Q)F :

(4.13)

Since a general super eld contains too many component elds to correspond to an irreducible representation of N = 1 susy, it will be very useful to impose susy invariant condition to lower the number of components. To do this, we rstnd covariant derivatives D and D that anticommute with the susy generators

Q and Q. Then ;(D F ) = D ( ;F ) and idem for D . It follows that D F = 0

or D F = 0 are susy invariant constraints one may impose to reduce the number

of components in a super eld. One nds

@

_

D

=

+ i

_ @

@

@

D

= =

+ i

@

@

where

= (D )y and

D

fD ;

_ g = 2i _ @

D

; fD ; D g = fD ; D _ g = 0

fD ; Q g = fD ; Q g = fD ; Q _ g = fD ; Q _ g = 0 :

(4.14)

(4.15)

19

20	CHAPTER 4. SUPERSPACE AND SUPERFIELDS

4.2Chiral super elds

A chiral super eld is de ned by the condition
			D = 0				(4.16)
and an anti-chiral one by
		D = 0 :					(4.17)
This is easily solved by observing that

D = D = D y = D y = 0 ;

(4.18)

y = x + i ; y = x i :

Hence depends only on and y (i.e. all dependence is through y ) and

only on and y . Concentrating on we have the component expansion p

(y; ) = z(y) + 2					(y) f(y)			(4.19)
or Taylor expanding in terms of x; and :
p
						@ z(x) f(x)
(y; ) = z(x) + 2 (x) + i

(4.20)

pi2 @ (x) 14 @2z(x) :

Physically, such a chiral super eld describes one complex scalar z and one Weyl

fermion . The eld f will turn out to be an auxiliary eld. For we similarly

have

p

(y; ) = z(y) + 2 (y) f(y)

p

=

@ z(x) f (x)

(4.21)

z(x) + 2

(x) i

pi

@2z(x) :

+

@

(x) 41

2

Finally, let us nd the explicit susy variations of the component elds as it results from (4.13): First, for chiral super elds it is useful to change variables from x ; ; to y ; ; . Then

@

@

@

Q = i

;

Q = i

+ 2

(4.22)

@

@

@y

so that

(y; ) =

(y; ) i Q + i

@

+ 2i

@

(y; )

Q

@

@y

p

2 f + 2i (@ z + p

@ )

=

2

2

(4.23)

=

2

+

2

2 f + 2

2

@

p

p

p

p

i @

z

ip

:

20

4.3. SUSY INVARIANT ACTIONS

21

Thus we read the susy transformations of the component elds:

p

z

=

2

p

p

=

2i@ z

2f

(4.24)

p

f

=

2i@

:

p

The factors of 2 do appear because of our normalisations of the elds and the de nition of . If desired, they could be absorbed by a rescaling of and .

4.3Susy invariant actions

To construct susy invariant actions we now only need to make a few observations. First, products of super elds are of course super elds. Also, products of (anti) chiral super elds are still (anti) chiral super elds. Typically, one will have a superpotential W ( ) which is again chiral. This W may depend on several di erenti. Using the y and variables one easily Taylor expands

p

W ( ) = W (z(y)) + 2@W i(y)

@zi

	@zi	fi(y) + 2 @zi@zj	i(y) j (y)	(4.25)
	@W	1 @2W

where it is understood that @W=@z and @2W=@z@z are evaluated at z(y). The second and important observation is that any Lagrangian of the form

Z Z Z

d2 d2 F (x; ; ) + d2 W ( ) + d2 [W ( )]y (4.26)

is automatically susy invariant, i.e. it transforms at most by a total derivative in space-time. The proof is very simple. The susy variation of any super eld is given by (4.13) and, since the and are constant spinors and the Q and Q are di erential operators in superspace, it is again a total derivative in all of superspace:

	@		@				@
F =		( F ) +				( F ) +		[ i( )F ] :			(4.27)
	@		@				@x

Integration R d2 d2 only leaves the last term which is a total space-time derivative as claimed. If now F is a chiral super eld like or W ( ) one changes variables to and y and one has

		@		@
	=		( (y; )) +		[ i( ) (y; )] :			(4.28)
		@		@y

21

22					CHAPTER 4. SUPERSPACE AND SUPERFIELDS
Integrating					d2 again only leaves the last term which becomes					@	[: : :] and is
										@x
a total	derivative in space-time. The analogous result holds for an anti chiral su-
				R
per eld		W	(		) = [W ( )]y and integration		d2		. This proves the supersymmetry
of the action resulting from the space-						time integral of the Lagrangian (4.26).
							R

The terms R d2 W ( ) + h:c: in the Lagrangian have the form of a potential. The kinetic terms must be provided by the term R d2 d2 F . The simplest choice is F = y . This is neither chiral nor anti chiral but real. To compute y one must rst expand the y in terms of x . We only need the terms , called the D-term:

y

=

41 zy@2z 41 @2zyz + 21 @ zy@ z + fyf +

i

@

i

@

2

2

i

=

@ zy@

z +

2

(@

@ ) + fyf + total derivative :

(4.29)

Then

S = Z

iy i + Z

d4xd2 d2

d4xd2 W ( i) + h:c:

(4.30)

yields

S = Z

d4xhj@ zij2 i

@W

1

@2W

j +h:c:i : (4.31)

i @ i+fiyfi

fi+h:c:

i

@zi

2

@zi@zj

More generally, one can replace yi i by a (real) K•ahler potential K( yi ; j ). This leads to the non-linear -model discussed later. In any case, the fi have no kinetic term and hence are auxiliary elds. They should be eliminated by substituting their algebraic equations of motion

@W

fiy =

!

(4.32)

@zi

into the action, leading to

i

2 @zi

i j 2

!

S = Z

d4x j@ zij2 i i @

@zi

@zj

@zi@zj

i j :

h

@W

2

1 @2W

1

@2W

y

i

(4.33)

We see that the scalar potential V is determined in terms of the superpotential W as

i		@W			2
			i
X
V =		@z			:	(4.34)

To illustrate this model, consider the simplest case of a single chiral super eldand a cubic superpotential W ( ) = m2 2 + g3 3. Then @W@z = m + g 2 and the

22

4.4. VECTOR SUPERFIELDS											23
action becomes
SWZ = R d4x	h	j@ zj2 i	@		m2jzj2 m2 (
						+ )					(4.35)
		mg(zyz2 + (zy)2z) g2jzj4 + g(z				+ zy				)i	:

Note that the Yukawa interactions appear with a coupling constant g that is related by susy to the bosonic coupling constants mg and g2.

4.4Vector super elds

The N = 1 supermultiplet of next higher spin is the vector multiplet. The corresponding super eld V (x; ; ) is real and has the expansion

V (x; ; ) = C + i i + v

+2i (M + iN) 2i (M iN)

i

+

i

@ i

i

@

(4.36)

+

2

2

D 21 @2C

+

21

where all component elds only depend on x . There are 8 bosonic components (C; D; M; N; v ) and 8 fermionic components ( ; ). These are too many components to describe a single supermultiplet. We want to reduce their number by making use of the supersymmetric generalisation of a gauge transformation. Note that the transformation

V ! V + + y ;

(4.37)

with a chiral super eld, implies the component transformation

v ! v + @ (2Imz)

(4.38)

which is an abelian gauge transformation. We conclude that (4.37) is its desired supersymmetric generalisation. If this transformation (4.37) is a symmetry (actually a gauge symmetry, as we just saw) of the theory then, by an appropriate choice of , one can transform away the components ; C; M; N and one component of v . This choice is called the Wess-Zumino gauge, and it reduces the vector super eld to

								1
	VWZ = v (x) + i (x) i (x) +								D(x) :			(4.39)
								2

23

24 CHAPTER 4. SUPERSPACE AND SUPERFIELDS

Since each term contains at least one , the only non-vanishing power of VWZ is

	VWZ2 =				v v =	1			v v
										(4.40)
						2

and VWZn = 0, n 3.

To construct kinetic terms for the vector eld v one must act on V with the

covariant derivatives D and D. De ne

		1						1
	W =		DDD V		; W =				DDD V :			(4.41)
		4						4

(This is appropriate for abelian gauge theories and will be slightly generalized in the non-abelian case.) Since D3 = D3 = 0, W is chiral and W antichiral.

Furthermore it is clear that they behave as anticommuting Lorentz spinors. Note that they are invariant under the transformation (4.37) since

_

W

!

W

1 DDD

( + y) = W

+

1 D D _ D

4

4

(4.42)

_

_

i

= W + 41 D fD _ ; D g = W +

_ @ D = W

2

since D = D y = 0. It is then easiest to use the WZ-gauge to compute W . To facilitate things further, change variables to y ; ; so that

@

+ 2i _

_

@

;

=

@

D =

D

(4.43)

@

@y

@

and write

VWZ = v (y) + i (y) i (y) + 12 (D(y) i@ v (y)) : (4.44)

Then, using g = 2 , it is straightforward to nd (all arguments are y )

D VWZ =

(

) v + 2i

i

+

D

2i( )

( @

)

+

@ v +

and then, using DD = 4,

W = i (y) + D(y) + i( ) f (y) + ( @ (y))

with

(4.45)

(4.46)

f = @ v @ v

(4.47)

being the abelian eld strength associated with v .

24