CC BY
23
UDC 530.1; 539.1
A DYNAMICAL THEORY FOR FREE MASSIVE SUPERSPIN 3/2
S. J. Gates, Jr., K. Koutrolikos
Center for String and Particle Theory Department of Physics, University of Maryland College Park, MD 20742-4111 USA.'
E-mail: [email protected], [email protected]
We present a new theory for free massive superspin Y = 3/2 irreducible representation of the 4D, N =1 Super-Poincaré group, which has linearized non-minimal supergravity (superhelicity Y = 3/2) as its massless limit. The theory is being described in terms of a real bosonic superfield Haa and two fermionic ones Xa , ua. The new results will illuminate the underlying structure of auxiliary superfields required for the description of higher massive superspin systems.
Keywords: superspin, superspace.
1 Introduction
After four decades of exploring the topic of supersymmetry (SUSY), the problem of writing a manifestly susy-invariant action that describes a free, off-shell massive arbitrary superspin irreducible representation of the Super-Poincaré group still possesses puzzles. Although the non-supersymmetric case of massive higher spin theory has been developed [1,2] and is well understood, the off-shell supersymmetric case has yet to be understood with a comparable level of clarity There has been progress for on-shell supersymmetry [3], but these results do not capture the rich off-shell structure of supersymmetric theories. There is a need for a manifestly susy invariant theory of massive integer and half-integer superspins which includes all the auxiliary superfields a theory of this nature is expected to possess.
Progress in this direction was made with the works presented in [4-6] where free massive irreducible representations of superspin 1 mid 3/2 were constructed. These results provided a proof of concept that constructions like these are possible, but they do not shed light to the heart of the problem which is to determine the set of auxiliary superfields required to describe an arbitrary superspin system with a proper massless limit. Specifically in [4] the focus was on massive extension of linearized old-minimal supergravity and new-minimal supergravity. These theories do not generalize to the arbitrary spin case, therefore the results obtained do not provide clues about the underlying structure of the auxiliary superfields for the general case.
This is not the case with the work presented in [6] where a free massive extension of linearized non-minimal supergravity is derived. Linearized nonminimal supergravity supermultiplet is a member of a tower of irreducible representations that can be
extended to the arbitrary super-helicity and that makes it a good starting point. However, their construction uses a lagrange multiplier technique in order to impose constraints that were not derived in a dynamical way.
We will show that there is an alternative formulation of the theory where all the constraints required, for the description of a free massive irreducible representation of Y=3/2, are dynamically generated from the equations of motion of a set of superfields {Haa, ua}. Superfields Haa, Xa in the massless limit form the free linearized non-minimal theory (superhelicity Y = 3/2) and ua is an auxiliary superfield that decouples when m ^ 0.
Finally, the theory presented here is a free theory without interactions. The full interactions, non-linear problem is still an open and very hard problem and for sure it is one of the motivations for this kind of investigations. In a realistic approach we can not talk about interactions if we have not established the free theories first. The results presented here extend our understanding for the free massive theory of the 3/2
the first non-trivial. Furthermore, we provide clues for some of the degrees of freedom that must be present in the non-linear, interacting theory. These are the superfields that have auxiliary status in the free linearized theory.
Our presentation is organized as follows: In section 2, we quickly review the representation theory of the 4D, N =1 Super-Poincare group for a free massive arbitrary superspin system. In section 3, we present the constraints imposed in the theory in order to have a proper massless limit. In the last section 4 we present the new massive theory for Y = 3/2.
S. J. Gates, Jr., K. Koutrolikos. A dynamical theory for free massive superspin 3/2
2 Arbitrary superspin representation theory Ha(s)a(s) =ffa(s)a(s)- On the other hand the fermionic
The irreducible representations of the Super-Poincare group are labeled by its two Casimir operators. The first one is the mass and the other one is a supersymmetric extension of the Poincare Spin operator. For the massive case the Super-spin Casimir operator takes the form
W2 (3
C = m + 4 + A)P(o)'
A2 + A _ W2 .
P(o)$.
(o) ^a(n)â(m) ^a(n)â(m) , j
2
and describes the highest possible representation (highest superspin)
A
n + m 2
C2$a(n)à(m) _ Y(Y + 1)^a(n)ct(m), Y =
has to satisfy the following constraints:
n + m +1 2
d2 ^ _ 0
D ^a(n)a(m) 0 ,
1)a(m) 0
(5)
d 2 _0
D ^a(n)a(m) 0 , ^YY $^a(n- 1)Ya(m- 1) 0
a(n)a(m)
a(n)a(m),
superfields have odd number of indices and describe integer superpsin systems, Y=s+1. For that case we can pick n=s+ 1, m=s (^a(s+1)<i(s)) and the reality-condition is the Dirac equation idas+1 as+1 ' a(s)a(s
m'
a(s + 1)d(s)~
0
(1)
where W2 is the ordinary spin operator (the square of the Pauli-Lubanski vector), P(o) is the projection operator P(o) = — m DYD2 DY and the parameter A satisfies the equation
(2)
In order to diagonalize C2 we want to diagonalize both W2, P(o). The superfield $a(n)a(m) that does this
W2$a(n)<i(m) _ j (j + 1}m2$a(n)à(m) , (3)
n+m
(4)
where all dotted and undotted indices are fully symmetrized and the spin content of this supermultiplet is j = Y + 1/2, Y, Y, Y — 1/2.
A superfield that describes a superspin Y system has index structure such that n + m = 2Y — 1, where n, m are integers. This Diophantine equation has a finite number of different solutions for (n, m) pairs but the corresponding superfields are all equivalent because we can use the operator to convert one kind of index to another. Therefore we can pick one of them to represent the entire class.
One last comment has to be made about the reality of the representation. The reality condition imposed on the superfield differs with the character of the superfield. The bosonic superfields, have even number of indices therefore describe half-integer superspin systems, Y=s+1/2. In this case we can pick to have n=m=s (Ha(s)ii(s)) and the reality condition is
3 The massless limit
Representation theory tells us the type of superfields and constraints we need to consider in order to describe a specific irreducible representation. We would like to have a dynamical way to derive these constraints, through an action. Very quickly we realize that we need a set of auxiliary superfields to help us generate these constraints, as in the case of non-supersymmetric free massive arbitrary spins. The heart of the problem is to find the minimum number and type of these auxiliary superfields needed. A helpful clue in the process of constructing these massive representations is their massless limit. We demand the massless limit of our theory to give the corresponding massless irreducible representation.
The list of available massless highest superhelicity irreducible representations was presented in [7-10]. There is one infinite tower for theories of integer superhelicity and two different infinite towers for theories of half integer super-helicities. However there are a few theories that do not fit into this pattern, like the old minimal, new minimal and new-new minimal supermultiplets. These are special cases that can not be generalized to the arbitrary superhelicity. If our goal is towards the construction of an arbitrary massive superspin supermultiplet, then it is obvious that we should start with massless theories that are memebers of an infinite tower and not a special case.
The conclusion is that the construction of massive theories must start with the corresponding massless
action and then add to it deformations proportional
m m2
massless limit, along with extra auxiliary superfields if necessary.
4 New massive Y=3/2 theory
We will follow this suggested strategy to build a theory of superspin |. The starting point is the theory of superhelicity and specifically the one that is the linear limit of non-minimal supergravity (s = 1 in [9]). Linear Non-minimal supergravity is formulated in terms of a real bosonic superfield Haa and a fermionic superfield Xa- We will add mass corrections to that action and check if 1) we can make xa vanish on-shell (auxiliary status) and 2) generate the constraints DaHaa=0 UHaa=m2Haa required by representation theory. The rest of the constraints can be generated
m
_m
out of this subset due to the reality of Haa and the D-algebra. The starting action is:
S = J j#aaDYID2DyHaa + aim#aa(IDaxa + cc)
—2Haa ID a D2xa + a2mHaa D2Haa + c.c. -2xaD2Xa + a3mxaXa + c.c.
+ 2XaDaD a Xa + «4 m2Haa HQ
(6)
£(X) = —4D2Xa + 2DaD a X« — 2D2D a H +aimDa Haa + 2a3mxa.
(7b)
A + B = 0, Aa4 — 1=0, a1 =0
and for that choice we get,
Ia = —4m2D2xa + 2Ba3mD2D 2xa +2m2DaDDaXa + 2a3m3Xa .
The equations of motion are:
4HP = 2Dy D 2D7 Hact + 2(DaD 2Xa — D a D2xa)
+aim(DD ai Xa — DaXà ) + 2a2m(D2Haai + D 2#aà ) +2a4m2Haà, (7a)
Now we can use these equations and attempt to remove any Haa-dependence to derive one equation that depends solely on xa- That will tell us if we can pick coefficients in a way that xa vanishes on-shell. Consider the following combination:
Ia = AD2Da EH + BD2D 24x) + m2 4x)
—2 (A + B) □D2DDàHaà + 2 (A + B) D2D2DaDàXà
—Aa1mD2DàDaXà + 2 (Aa4 — 1) m2D2DàHaà
—4 (A + B) ^D2Xa — 4m2D2Xa
+a1m3DàFaà + 2 (Aa1 + Ba3) mD2DD2xa
+2m2DaDà Xai + 2a3m3Xa. (8)
From that it is obvious that there IS cl choice of coefficients that will remove any ffaà dependence:
We must update the action with the addition of the interaction term m uaxa, the kinetic energy terms for ua (the most general quadratic action) and the mass term of ua. The new action is
S = J d8z j HaaiDYD2DYHad
-2Haai (DdD2Xa - DaD2Xd) -2xaD2Xa + a2mHai D2Had + c.c. +2xaDaD * Xai + a4m2Haa Haa +a3mXaXa + YmuaXa + c.c. +61 uaD2ua + 62uaD 2ua + c.c. +63 uaDDaud + 64uaDaDu&
+65muaua + c.c. | . (11)
and the updated equations of motion are
= 2D7D2D7Haa + 2(DaD2Xai - DaD2Xa) +2a2m(D2Ha<i + D 2Haa) + 2a4m2Haa, (12a) 4x) = -4D2Xa + 2DaDaXa - 2D2DaHaa +2a3mXa + Ymua. (12b)
4U = 261D2ua + 262D 2ua + 63Da Daua +64DaDa u^ + 265mua + YmXa. (12c)
We repeat the process of eliminating Ha(i, but since ua does not couple to Haai nothing will be changed regarding the Ha(i-dependent terms. The same choice of coefficients as in (9) must be made to remove Ha(i. The updated expression for /a is
4 = 2Ba3mD2D 2Xa — 4m2D2 Xa + 2m2DaDa Xà +BYmD2D 2 ua + Ym3ua + 2a3m3Xa .
(13)
(9)
(10)
It is clear that there is no freedom left in order to make Xa vanish on-shell. Therefore we must introduce an auxiliary superfield. Its purpose will be to impose a constraint on Xa when it vanishes. That constraint will be used to simplify the above expression for /a and set Xa to zero. But a more careful examination of /a will convince us that there is no single differential constraint on Xa that will make all dynamical terms vanish. The inescapable conclusion is that we have to treat Xa=0 as the desired constraint. This suggests that we must introduce a spinorial superfield ua that couples only with Xa through a mass term muaXa-Hence when ua=0 then immediately we see Xa=0.
Now we want to use the equation of motion of ua to remove any dependences on Xa in order to derive an equation just for ua. For that, consider the combination
Ja = /4 + mK D24u) + mADaDS™ = [2Ba3]D2D 2Xa + [By + 2K62 + A63]mD2Dd ua -[4 - K7]m2D2Xa + [K63 + 2A62]mD2DdDaud + [2 + A7]m2DaDdXd + [A(264 - &3)]DaD2D^u^ + [K65]m2D2u a + [A65]m2DaD d ud + [2a3]m3Xa + Ym3u a. (14)
If we choose
a3 = 0 , -4 + K7 = 0 , 2 + A7 = 0 , (15)
X
Ja = [By + 2K62 + A63]mD2Dd ua + [K65]m2D2u a
+ [Kb3 + 2Ab2]mD2D a D a U à + [Ab5]m2D aD a U, + [A(264 — b3)]D aD 2D^ m^ + Ym3u a.
3/2
As a result, we have the freedom to further choose coefficients in order to cancel all the dynamical terms of ua and force it to vanishe on-shell
BY + 2K62 + A63 2b4 — 63 = 0 ,
Y = 0
0
K63 + 2A62
65 = 0,
0
(17)
□Ha
m2Haa.
1
—2Haa D a D2xa + - uaD 2ua + c.c. 6
— 2(xaD2Xa + c.c.) + 1 UaDa DaUai
+2xaDaD a Xaa + ^ UaDaD a U<i +m2Haa Haa + m(uaxa + c.c.)
This is the superspace action that describes a superspin Y= 2 system with the minimum number of auxiliary superfields and has a massless limit that gives the free linearized non-minimal supergravity. This action is a representative of a family of actions that are all equivalent and connected through superfields redefinitions of the form
ua 0
ua
Xa 0 Haa •
4x)k=x*=o = —2D2DaHaa ^ D2DaHaa = 0,(18a)
4HP k=x*=o = 2DY D 2D7 Haa + 2a4m2Ha<i +2a2m(D2Ha<i + D 2 Haa).
Xa ^ Xa + Ziua + WiDaH( ua ^ ua + Z2Xa + W2DaH(
aai
aa,
(18b)
therefore due to (18a) we get that
D^ = 2a2mDaDD 2Haa + 2a4m2DaHaa. (19)
For a2 =0, a4 = 0 this gives the desired condition DaHaa = 0 and the equation of motion for Haa becomes the Klein-Gordon equation with a4 = 1
(20)
To complete the analysis we look for the consistency and non-trivial solution of the systems of equations (9,15,17) plus a2= 0 Mid a4=1. A solution exists
ai =0 , 6i = free, can be set to zero , y =1 , (21)
a2 =0 ,62 = 1 , A =1 , 6
a3 = 0 ,63 = - , B = —1 , 6
a4 = 1 ,64 = — , K = 4 ,
4 , 4 12 , ,
65 =0 , A = —2
and the final action takes the form
S = i HaaDYD2DYHaa
(22)
where z^d wj € C.
5 Summary and conclusions
We started with the | superhelicity theory of free linearized non-minimal supergravity, formulated in terms of a real vector superfield Haa and a fermionic
Xa
attempt to discover a theory for massive superspin | system, only to find that it is not possible and we need
ua
Xa
Finally using the equations of motion we manage to show that on-shell ua = 0 ^ xa = 0 ^ DaHaa = 0 ^ □Haa = m2Haa.
We have managed to derive yet another formulation of free massive supergravity supermultiplet and most importantly probe into the set of auxiliary superfields required for the construction of higher superspin
ua
trivial auxiliary superfield needed beyond the massless theory. As we go to even higher superspin values we should discover more and more of these objects. The hope is that after the study of some non-trivial low superspin examples, such as the one demonstrated here, we will have a deeper understanding on the number, type and role of these auxiliary objects. When that happens we might be in a position to construct the arbitrary massive superspin irreducible representation in an inductive manner.
Acknowledgments
This research has been supported in part by NSF Grant PHY-09-68854, the J. S. Toll Professorship endowment and the UMCP Center for String & Particle Theory.
References
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Received, 09.11.2014
С. Д. Гейтс мл., К. Кутроликос ДИНАМИЧЕСКАЯ ТЕОРИЯ СВОБОДНОГО МАССИВНОГО СУПЕРСПИНА 3/2
Мы предлагаем новую теорию свободного массивного суперспина Y = 3/2 неприводимого представления 4D, N = 1 супер-Пуанкаре группы, которое линеаризует неминимальную супергравитацию (суперспиральность Y = 3/2) в безмассовом пределе. Теория представлена в терминах реального бозонного суперполя Наа и двух фермионных суперполей Ха , ua. Новые результаты исключают неоднозначность структуры явных суперполей, необходимой для описания систем массивных высших суперспинов.
Ключевые слова: суперспин, суперпространство
Гейтс С. Д. мл., доктор, профессор. Мэрилендекий университет.
College Park, MD 20742-4111 США. E-mail: [email protected]
Кутроликос К., доктор. Мэрилендекий университет.
College Park, MD 20742-4111 США. E-mail: [email protected]
