Ind i8n Journal or Pure & Applied Ph ys ics Vol. 38. June 2000. pp. 359-364

Supersymlnetry breaking

B S Rajput

V ice-Chancellor. Kumaun Uni versi ly. Nainital 263002

1 Introduction Perpl exed and perturbed by the apparent di versiti es

and complexiti es of nature, man at an early stage of awakeni ng co ncei ved the noti on of ulti mate harmon y or sy mmetry of this universe . Symmetry is one idea by wh ich man , through ages, has tried to co mprehend and create order, beauty and perfect ion. Some of these sym­metries are exact sy mmetri es and othe rs are onl y ap­p rox im ate. Exa mpl es o f e xact sy mm e tri es are invariances of a physical sys tem under rotati ons and tran slations in space and time. An example of approxi­mate symmetry is the sy mmetry under space refl ecti on. All these symmetries quoted here are space-time sy m­metri es or ex ternal sy mmetries . There are also the sy m­metries in volv ing in variances of Hamiltonian of the phys ical system under those transformati ons which do not involve space and time coordinates. These sy mme­tries are known as internal ones. Such sy mmetries are much more mysterious than the ex ternal sy mmet ries . Addi tional understanding of these sy mmetri es comes from the stud y of local gauge transformati ons wh ich re late internal quantum numbers to space time depend -

t. . J

ent trans ormatl ons. In some cases the internal sy mmetri es are onl y ap­

prox imate and they are so badl y broken th at these are hardly recognised. To underst and why some symmetries are exact and so me are approx imate, we must look to the dynam ics . It can be done e ithe r on a fundamental leve l or on a phenomenologica l level. More we learn about the properties of fundamental interactions at pheno­menologica ll eve l, bette r we can hope to appreciate how ~ymmetries are broken. But we can be more ambiti ous and try to understand brOKe n sy mmetry at a fundamenta l leve l. O ne of the wavs to do it is within the framework - - . or ri e ld theory by e ithe r constructing a Lagrangian con­t~linin g terms whi ch arc not in var iant under re levant transfornlario ns or by break i ng the sy mmetry spontane­ously where we construct the Lagrang i ~1Il vy h.ich has sy mmetry in questi on. but arrange such that the physica l states of the theory do not obey this sy mmetry.

The understandin g of symmetries of nature could lead to many attempts of unifi cat ion of fundamental forces . The resulting grand unified theories (G UT' s) had remark able success in unifying QeD of strong interac­tions with QFf of e lectro-weak theory with a single couplin g constants. But they too leave some unsettled problems i.e .

(i) There ex ists a desert between weak interact ion!'. mass scale M". and unification mass scal e Mx; (ii ) 0

ex plan a ti o n ex ist s for incre dibl y tin y ra ti o of

M ~ . /M ~ « 1O- 2~) (Gauge hi erarchy problem); (iii )

GUT's do not incorporate gra vity in an y fundamenta l way; (iv) Even in ve ry successful GUT mode l SUe) ), space-ti me and matte r st i II remai n disconnected and non- interactin g entiti es.

To resolve these difficulties the idea of supergrand unifi cati on has been put forward and consequentl y the notion of ultimate symmetry has been visuali zed in terms of supersy mmetry, supergrav ity. and higher d i­mensional space-t i me, i ncorporati ng the interacti on of space-time and matte r as natural framework of super­strings .

2 Supersymmetry Supersymmetry consis ts or transforilla tions of fer­

mi ons to bosons and vice versa. Thi s beautiful sy mmetry between fermions( i.e. mathemati ca l objects which on quanti zati on are assoc iated with an anti-co mmu tator al gebra) and bosons (i.e. the object where qu antization is associated with commutator algebra) in its sirnp\e:t form is an extension of usua l Po incare alge bra. It is such a graded lie al gebra wh ich in vo lves both commu tati on and anti -comlllutati on re lati ons, plays a unique role in particl e phys ics and pro vides a fusion between space­time and internal sY lllmetries overco ming no-go theo­rem l about the possibl e sy mmetries of the S-m<l tr ix. Of al l the graded Lie a l gebra ~, onl y the supersy mmetr ic al gebras (together with their ex tens ions 10 include cen­tra 1 charge ) generate sy mmetries 0 I' S-mat ri x co ns istent with relat ivistic quantum fie ld theory.

360 INDIAN J PURE & APPL PHYS, VOL 38, JUNE 2000

Starting with the pioneer work of Witten", it has been

recognised that supersymmetry could be applied to

quantum mechanics as a limiting case (N = I) of field theory and the subsequent development of supersym­metric quantum mechanics led to an interesting theory . We have developed supersymmetric quantum mechan­ics in complex space-time

1 and also carried out

4 quanti­

zation in supersymmetric complex space-time by

constructing the supersymmetric Hamiltonian H in terms of non-Hermitian supercharge operator Q and Q+ in the following form :

H = ~ IQ , Q + i = ~ fQ Q + , Q + Q 1 ... (2.1)

such that

lQ, Q +] = 0, Q 2 = Q +" = 0

and

fH, Q 1= fH , Q + 1 = () ; l y, Q]=-iQ ; [ y ,Q + J= iQ +

where I -

y =-;:;-I\jf,\I'] L.

with \jf and \1' as fermionic variables describing spin degrees of freedom .

The total hamiltonian H of the supersymmetric sys­

tem may be decomposed into a bosonic part H/3 and a fermionic part HI';

H= H/I + HI'

with I 1 I , I ,2

Hu = "2 p- + "2 i W (q) ;

and

H r = l W /I ( q ) ] \if ' 'I' J = i W" ( (f) Y 2

.. . (2 .2)

where W (q) is superpotential. From Eqs (2 .1) and (2.2) we have:

. Q T = I p - i IV I ( q) ]\V

and

Q = I p + i W F ( q) ]\if Any state I B> satisfying the conditions :

QIR > =()

and

Q ' IB > i-O

is hosonic state for which we have:

1 ) + HIB > '2 QQ IB >

The fermionic state IF> satisfies the conditions:

Q+IF>=O

and

QIF>i-O

which give

I HIF>'2 Q + QIF>

Using these rL'lations, it may readily be shown that all

the eigen values of the operator H are non-negative and

hence energy in supersymmetric theories is always posi­

tive. This eigen value is zero only when:

QIIl>=Q+ln>=O

which is a necessary condition for supersymmetric ground state. As such the ground state of a supersym­

metric state is the true vacuum. The supersymmetry is spontaneously broken when the ground state energy is non-zero.

Using above relations, it may readily be demonstrated

that the operator Q transforms states IF> into states I B > of the eigen energy £ and the operator Q+ transforms

the state I B> into the states IF> i.e. QIF> £112 1 B >

and Q+IB> £1 12 1 F> ... (2.3)

showing that the supersymmetry pairs the bosonic and fermionic states of all positive energy states of H. On the other hand, the zero energy states are not paired in

this way . With Q" = H, each state annihilated by His also annihilated by Q. These states form trivial one

dimensional supersymllletric Illultiplets . There exists the unpaired state (ground state) if and only if the

supersymmetry is an exact symmetry of the system. In other words, the ground states of zero energy preserve

supersymmetry, while those of positive energy breaks it

spontaneously. This situation has been described in Fig. 1.

A supersymllletric invariant Lagrangian may be con­structed out of chiral as well as with vector superfields. Such a Lagrangian contains a kinetic term, the mass term and the interaction term;

.. . (2.4)

whereLkill is D-type term while Lilla"" and Lilli areF-type terms.

When one introduces the gauge transformation , this Lagrangian does not remain invariant and one has (0

reconstruct it in a gauge covariant manner by introduc­ing a gauge field . Usually , gauge invariant interactions

RAJPUT: SUPERSYMMETRY BREAKING 361

are introd uced on repl ac i ng ordi nary deri vati ve by gauge covari an t deri vat i ve. But the kinetic term of chiral superfie ld s does not ha ve any ex plicit derivative term visib le in the space of superrie ld s. This difference in the ex pression for the kinetic term , in terms of superfields, prec ludes the method of replacing ordi nary deri vati ves by gauge covariant derivatives for the introduction of local gauge in va ri ance. The si mplest supersy mmetric generali zati on of sp in-I gauge boson is that of a vec tor superfield consist ing of the spin- I along with spin-Ill component fi elds .

3 Breaking of supersymmctry Supersy mmetry re lates partic les hav ing different

spins followin g different stat ist ics. It demands the equi va lence of fermi ons and bosons. Such an equi va­lence is not observed in nature and hence SUSY should be broken spontaneously i I' we have to incorporate all th e ni ce properties of supersy mmetry into a phys ica l theory and yet to make it rea li sti c one. Moreover, the req uirement that supersy mmetric particles can onl y be prod uced in pair impli es th at every bosonic partic le must have a fe rmi onic partner and vice-versa. Such particles are not found with a requ i red pattern of degeneracies and this type of spectrum of particles is not compatibl e with the observat ions. This apparent lack of degeneracy can in principle be attributed to the spontaneous break down of supersymmetry, preserving Illany usefu l aspects of the theory and suggesting the presence of superpartner (with predictable propert ies) of all the fundamental par­ticl es of nature. SupersYlllllletry has to be badly broken, since no supersy mmetri c partners has been observed, so far, with the lower bounds ror masses of such particles varying bctwecn SOGev to I ()O Gev .

Supersymmetric quantum mechanics, whieh has re­cently attracted Illueh attent ion. serves as theoreti cal

Pol. Eng . V-(0)

laboratory for testing various ideas of supersY l11metr ic breaking mechanism in hi gh energy phys ics and it h a~ enhanced the hope to get better insight into the mecha­nism of SUSY breaking. [n unbroken supersy mmetry there are no qu adrat ic di vergences and the finite induced mass splitting are determined through the mec hani sm of SUSY breakdown. In F ig. I we have shown that a necessary and sufficient condition for supersymmetry to be good sy mmetry is that vacuum should have zero energy. Further, we know that the sca lar potenti al of su persy mmetric Hami Itonian is:

V=FF * +~D 2 .. D . I)

Hence the energy of the vacuum will be non-vanish­ing when one of the auxili ary fields has non- vani shing vacuum expectat ion value (vev) so th at supersymmetry is spontaneously broken and the fermionic partner of the auxiliary fi eld that rece ives a vev is the mass less Go ld­stone fermion of broken supersymmetry.

Such sy mmetry breaking theoret ica ll y may be or three categories:

(i) Supersymmetry may be broken with gauge sy m­metry remaining good.

(ii ) Gauge sym metry may be broken with supersY Ill ­metry remaining good.

(iii ) Supersymmetry and gauge sy mmetry both Illay be broken.

The spontaneous breaking of ordin ary gauge symme­try is well understood but the breaking of supersym­metry imposes additional conditi ons. Such an additi onal conditi on has been analysed in the prev ioLis section where it has been show n that the ground state or zero energy preserve supersymmetry, while those of posit i ve energy break it spontaneously. SupersYl11llletry is a !!Iobal sv mmetrv. Its soontaneous breaking wi ll impl y

V-

( b )

A (Aux . Fieldl ------"---- --- A

Fi g.. i - (;1) Ground Slale preserves SUSY . (h) ground Slale break s SUSY sponlancoui sy

362 INDIAN J PURE & APPL PHYS, VOL 38, JUNE 2000

the existence of zero mass Goldstone mode correspond­ing to fermionic increments.

Corresponding to the above mentioned three catego­ries of symmetry breaking, we have foilowing three well known models exhibiting general spontaneous symme­try breaking in supersymmetric theories.

(i) 0' Raijeartaigh Mechanism (Ref 5)

This mechanism incorporates F-type symmetry breaking, where the auxiliary. field F of the chiral super­field attains vev to . break supersymmetry. This model

contains three chiral superfields <Do, <DI and <1>2 and the interaction is given by the superpotential:

V = A. <D 0 + g <1> () <D I. <1> 2 + m <D I <D 2 ... (3 .2)

where chiral fields <1>" are given by:

<l> ( y , 0 ) = A ( y ) + ..J2 Oa \jI1Y. (y) + ( 0 0 ) F ( y )

with A(y) as complex scalar field, \jIa (y) as a spinor field and F(y) as the auxiliary field. This superpotential has an additional U( I) symmetry under:

<1>()---j<1>() ,$ l---j-<l> I ,$ 2 ---j -$ 2

The F-terms of these three Chiral fields are :

. d V '\ F ' =--= 11.+ (1 AI A1 () dAo ... -

' . d V F; = d A I = g A 2 A () +111 A 2

• d V F ? = -d = g A I A () +m A I

- A2

or in general :

F ; = - ( A. k + m ik A i + g i k A i A j )

the scalar potential corresponding to F-type term is:

V = L. i ( Fi )2

which is obviously positive definite . The potential can vanish only when the conditions

A+ g A IA2=g A 2 Ao+IIIA2= g A IAo+II1A 1=0

are simultaneously satisfied . There is no solution to these equations and thus the potential never vanishes, so that, the SUSY is broken spontaneously . As such, this model has a broken SUSY, although \\ ' L' start with a symmetric theory.

(ii ) Forcl-Tlli/)(}II/o .l' MCc/Wlli.l'1II (Ret: 0)

This mechanism involves D-type symmetry break­ing. In this model the SUSY is spontaneously broken in gauge theories with Abelian gauge groups. It is based on

the observation that OOW component of the vector su­perfield is both supersymmetric and gauge invariant. This term is added to supersymmetric Lagrangian and it

spontaneously breaks the SUSY. In this model the po­tential is:

V I D2 F F * F F * ="2 + I 1+ 2 2 ... (3.3)

where D, FI and F2 are solutions of Euler equations:

e . • D+K+"2(A I A I -A2A2)=0

FI +mA ; =O

F 2 +mA ~ =0 There is no solution tothese equations which leaves

V = 0, and hence SUSY is broken spontaneously. Substituting for the auxiliary fields, the potential V

becomes:

V = '2 K - + . In + '2 e K A I A I + . m- - '2 e K A 2 A 2 1 ?(21 J* (?I J '" I ? . ?

+ 8' e -( A I A I - A 2 A :d - ... (3.4)

Now there arise following two possibilities;

? I (a)m->"2 eK :

In this case gauge symmetry is unbroken. Here both A I and A2 have real masses and the model describes two

complex scalar fields with masses m~ = 11'12 + ~ e K and

? ? I K h . .. Id '\ d In; = 111.- -"2 e ; t fee spmor fie s \jI1,'1'211.; an one

vector field Vm . The masses of spinor and vector fields are unchanged by the symmetry breaking. The situation is described by Fig. I (b).

We also note here that: ? 1 ?

1111 + 111; = 2 111- ... (3.5)

The vector field Vm plays the role of gauge field for

the unbroken U( I) symmetry group. A. is the Goldstone fermion arising from spontaneous broken supersym­metry.

? I (b) 111- <"2 e K:

In this case A I = A2 = () no longer minimizes the potential V. To find the minima we solve the following equations;

d V ( ? I 1 e2

.:. " dA;' = m-+"2 eK AI+4(AIAI - A2A2)AI=O

a V ( ? I 1 e2

* .,. a A; = -m- -"2 e K A2 - 4 (A I A I - A2 A2 ) A2 = 0

These equations give the minima at: AI=0,A2=V

where v is given by:

RAJPUT: SUPERSYMMETR Y BREAKING

.J 2 ( e K - 2 /1 / ) V=± = ±II

e ]n this case both gauge sy mmetry and SUSY are

broken simultaneousl y as described by Fig. 2. This model describes two sp inor fields of mass

~ , I , ? 111- + - e- v- ,. . 2 ; one vector field and one scalar field

~ I , 1

each o f mass "2 e v ; one complex sca lar field of

mass ~ , and one massless Go ldstone spi nor. We

also note that sum of masses squared weighted by the number of degrees of freedom is identical for bosonic and fennionic modes;

4 1 e" v=' + 2 2 11/" = 4(. II? + l ('1 \," ) '2' 2

... (3.6)

which is generalizati on of re lati on given in Eq. (3. 5) . Such re lati onships between bosoni c and fermioni c masses are cornma n in supersy mmetri c theories .

( ii i ) (Jil l.'" Gallg e .'T II IIII I' /'-Y is hmk{'// (U,I 7)

This model is sllpcrsymmetri c ex tension of Higg ' s mec hani sm. ]n this mode l. spontaneous gauge symme­try break ing in sllpersymmetric theories gives rise to a massive vector Illu ltip let. It leads to supersy mll1etric ge neralizat io n o f grand unifi cati on theo ri es. Thi s Illechani sm of sy mmetry breaking al so leads to su-

. I . K- I (I " I d d ] persymmetnc t leon es o ' monopo es an yons. n a si mple man ner th is mechan ic!', has been described in Fig . J.

In general any boundary cond iti on or an y environ­ment which would distingui sh between bosons and fer­mions, would break thi s sy mmetry. In parti cular, since bosons and fe rmi ons respond dilTerentl y to tempera ture. a supersy mmetric sys tem immersed in a thermal bath

III . III ' F' . wou ( oose supersymmetry . -. Inlt e temperature su -persymmetry breaking can al so be di scu<,sed by appl y­in g th e prope rti es o f th ermo- rie ld- dynamics(TFD) whi ch is an a lte rn ati ve wayll of l11 ~ k i n g cal cul ations in quantu m mechani cs . One of the order parameter fo r SUSY brea kin g al fin ite tempnature is the thermal ground state energy. It s no n-vani shing val ue is a sure test ofb reakdowll of S USY. O lle of the important crite­ri a for spontaneous bre;l kdow ll of SUS Y at non-zero temperature i<, th at the therma l ave rage of at leas t one auxi liary fi e ld does not v a ni s h l~ . Thi s cri teri a have en­hanced the hope [0 res tore the SUS Y at finite tempera-

1:;, 17 ture .

-ll +u

Fig. 2 - Gaugc ssymmclry and slIpcrsYllllnctry hOlh brokcn spolllancollsly

-u +u

Fig. :I - Onty gaugc symlllciry hrokcn

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