A model for mass splitting inside goldstone supermultiplets

Volume 164B, number 1,2,3 PHYSICS LETTERS 5 December 1985

A M O D E L FOR MASS S P L I T T I N G INSIDE G O L D S T O N E S U P E R M U L T I P L E T S

G. G I R A R D I a, H. HOGAASEN b and P. SORBA a

" LAPP, BP 909, 74019 Annecy-le-Vieux Cedex, France b Institute of Physics, University of Oslo, Blindern, Oslo 3, Norway

Received 27 June 1985

From a model where supermultiplets of particles which we call urons are confined to the inside of a bag, we obtain relations between the uron mass and the mass of the pseudo Goldstone particles due to the breaking of chiral symmetry on the boundary of the bag. By the same token supersymmetry is also broken and it appears that for small bags the Goldstone bosons are much heavier than their fermion partners.

One of the most attractive ideas in the effort to construct quarks and leptons as composite particles is that these may be quasi Goldstone fermions [1-5] associated with a broken symmetry of a supersymmetric theory. There is in these theories an obvious problem: how does one explain that the true Goldstone bosons, which would correspond to s- quarks and s-leptons, are much more massive than their fermionic partners (quarks and leptons) in the Goldstone supermultiplet? In this letter we wish to present a tentative explanation of this phenomenon, based on a bag model, confining supersymmetric particles within a very small region in space. Our model is a generalization of the quark bag model with chiral symmetry constraints [6,7] that give rise to pseudoscalar Goldstone bosons like the pions. We shall first give a short description of a model of quark confinement where the SU(2) X SU(2) chiral symmetry is broken by the Bogoliubov boundary condition, even if quarks are massless. The axial isospin current is not conserved but the continuity of this current can be restored by introducing a compensating field outside the bag, which is just the pion field. We shall stick to the two phase picture of chiral symmetry realization: inside the bag, massless quarks are our dynamical variables; outside the bag, the massless Goldstone pions carry the axial current. Chiral symmetry, inside, is then explicitly broken by mass terms for the quarks, as a result the pion, outside,

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will acquire a mass and become a pseudo-Goldstone particle. It is not a big surprise [8] that the resulting relation between the quark mass and the pion mass is a Dashen like relation [9] : for small masses the quark mass is proportional to the square of the pion mass. We then generalize the bag model to confine supermultiplets of fields, which we call urons. For sim- plicity, we restrict ourselves to two chiral multiplets with urons of spin 0 and 1/2, and in our model we shall, at this stage, disregard properties due to the in- ternal group. In this approach we find the supercurrents carried by the urons and discuss the supersymmetric version of PCAC. Extending the analysis for confined quarks and Goldstone pions mentioned above [8], we obtain relations between uron masses and masses of the Goldstone particles. As previously done these relations will be deduced by imposing the continuity of the supercurrents and their divergences through the bag surface. The boundary condition at the'bag surface for the confined urons is the generalization of the Bogoliubov boundary condition by a supersymmetry transformation. Besides the breaking of chiral symmetry, as in the usual case, supersymmetry is also broken, in relation with the lack of translation invariance introduced by the bag surface. Therefore, masses will be different for the vari- ous members of the Goldstone supermultiplet. Speci- fically, light urons confined to a tiny region of space give rise to light quasi Goldstone fermions and heavier Goldstone bosons.

0370-2693[85[$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


Let us now turn to the description of the quark bag model as introduction to our model.

When fermions as quarks are confined to a spher- ical region of space r < r B the quark energy can be quantized by the Bogoliubov-MIT boundary condition

- i 7 • hp = ~b (1)

for r = r B. If the Dirac spinor is decomposed as

=ffL + f i R , w i t h ~ b L / R = ~ ( l ~ 7 5 ) f f , (2)

the boundary condition has the form

- i y - ~¢L = fiR, - i 7 " ?~R = ~ L , (3 )

and it is therefore completely adiagonal in a chiral basis. As a consequence chiral symmetry is broken even with zero mass for the fermions. This is reflected in the fact that the radial component of the (isovector) axial current ,1

1 Am ° = (4)

is discontinuous at the bag surface where its divergence has a 6-function singularity.

To generate a continuous (isovector) axial current one now introduces the (pseudo-)Goldstone pion field which for r > r B has the ordinary static source form

7r i = (g/STr/kO[(1 + pr)/r 2 ] e - u r ( o • i)~ i . (5)

Here M is the mass of the source,g the coupling constant and/a the pion mass. Outside the bag the axial current is now carried by the gradient of the pion field,

A = A ' = - . f . V n , r > r B . (6)

Demanding continuity of the axial current through the bag surface, the equalities

A(r B - e) = AQ(r B - e ) = A ( r B +e)=A~r(rB + e) (7)

determine the coupling constant g from the quark wave functions on the inner o f the surface. By further demanding continuity o f the divergence o f the axial current through the surface

1 amA2(r B - e) = 2mqi~ 75 ~Tr~ = ~mA~n(r B + e)

= -flr//27r

one obtains also a relation between the quark mass

,1 We use the index m as vector index as in ref. [10].

and the pion mass. If the quark mass is zero so is the pion mass: the pion is a true Goldstone boson.

In the general case when the chiral symmetry is broken through mass terms [8]

mq = ~m~rX(1 + X')/(2 + 2X + X 2 ) , (8)

where X = ~tr B, r B being the confinement radius as before. For masses small so that X is small,

mq = ~R/~ 2 . (9)

This is the well-known proportionality between fermion and boson masses when chiral symmetry is broken. It should be noted that in the non-perturbative deduction of the mass relation we have only invoked continuity properties of currents in coordinate space.

We now turn to the case when we confine supermultiplets whose members we call urons. This uron bag may be thought of as a model for confinement of urcolour in an asymptotically free theory, in analogy with QCD and the quark bag model ,2 . The resulting confinement region for urons, characterized by r B when r B is extremely small, then separates space into one region r < r B where urons move (almost) free, and one region r 3> r B where the dynamical variables are (pseudo-)Goldstone bosons and fermions that couple to the uron bag surface to give us continuous currents and current divergences.

We shall simplify the problem by considering only two chiral supermultiplets for urons, one right- and one left-handed such that these contain urbosons and urfermions of the two chiralities. We associate to supersymmetric uron chiral and antichiral superfields [10]:

~bL = (L, ~k, EL), q ~ = ( R , p , F R ) , (10)

satisfying D~b L = D 0 ~ = 0. Then the uron Dirac spi- nor appears as ffD = (2) in the Weyl basis. Inside the bag, we have the lagrangian

£ = ~bL~b L + ~bR~b R + m(~bL~b R + ~bROL) , (l l)

where OL (resp. ~R) transform as the fundamental representation of the compact Lie group G L (resp. GR) of dimension n. One sees that for massless urons m = 0, £ satisfies the chiral global invariance G L × GR; this symmetry reduces to GL+ R for m :~ 0.

,2 However, here, we only consider a model with a global symmetry, and no gauge interactions.

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The equations of motion read

~ 2 ¢ [ _ 4 m t ~ = 0 , D2~ R - 4 m ¢ L = 0 , and c.c., (12)

that is:

F R = - m L , FL = - m / ~ ,

EIR --- m 2 R , ElL = m 2 L ,

i ' rmom~ = - m ~ . (13)

To each generator t a (a = 1 ..... n) of GL(R) can be associated a vector and an axial vector current superfield, as follows:

V a = ¢~ta¢ L - ¢~ta¢ R, Aa = ¢[ ta¢ L + ¢~taO R , (14)

where ~+ + i ' L,Rtaq~L,R stands for: Ct,R(ta)i i¢~,R. The above supercurrents fulfil the "conservation equations"

O2Va = O, O2Aa : 8m¢~ta¢ L ,

~ 2 V a = 0 ' ~ 2 A a = S m ¢ ~ t a ¢ R . (15)

From now on, we will omit the generators ta, but the position of the fields will be respected to avoid con- fusion in the matrix products which are understood. It is convenient to associate to each current superfield J the superfield ] m defined as [5]:

j m = _ ~o.m& [D% D &]J, (16)

which satisfies

~m~m = _ ~zi[D 2, ~2 ] j . (17)

We note in particular, using (3.6),

Om I~m = O, Om ~ m = 1. - ~lm ( D 2 ¢ ~ L -- D2¢~¢R) , (18)

that is, considering the explicit form

A m = i ( L ~ m L - R ~ m R ) + f T m 7 5 ~ ,

Ar~ = 2 i ( o m n ) ~ n ( L X~ + p ~ )

+ 2mo~¢.¢(~/~R + L--p~) (19)

of the number and 0# components of the axial supercurrent .~m,

= _2 im~75tk , ~mAn~ = 2mo~aham(~R+ffp~), bm A m (20)

where use is made of conventions o f re f . [10], in particular the metric gmn = ( - 1 , 1, 1, 1).

Now, the boundary conditions expressing the flux conservations

ri ~ri = 0 at r = r B , (21)

i f r is a unit space-like vector normal to the surface of the bag, can be obtained by generalizing to bosons the (usual Bogoliubov-MIT) boundary condition for fermions which reads in two-component spinor no- tation:

- i ~ . P~ =X, - i ~ - ~ = p , r = r B . (22a)

Applying the supersymmetry transformation [10] on (22a), one then gets, after use of the equations of mot ion (13),

m L = a L / ~ r , mR=OR~Or , r = r B . (22b)

It then follows that on the boundary of the bag

i " A = - i~75~k, i • Aa = 2im(Pa, R +LXa) , r = r B , (23)

as well as

V. A = - 2 i m ~75 ¢,

V" A a = 4im2(pa, R + L Xa), r = r B . (24)

It is natural to expect supersymmetry to be broken on the boundary of the bag, since transla- tional invariance is not satisfied on the frontier r = r B . This can be checked explicitly after computation of the vector-spinor current associated to the supersymmetry generator

s m =sn~ +sr~,

with

sr~ = x/~[on-ffm ]t~nL + iomXFL] ,

s~ = X/2[on-o m POnR + iomf fFR] • (25)

The physical effect of the discontinuity at the boundary can be made clear if one makes the replace- ment

X(r) -+ X(r) = O(r B - r)X(r) , (26)

with X = X, p, L, R. Then sin(X, p, L , R ) = s m be- comes sm(OX, Op, OL, OR) - s m, and it is straight- forward to obtain restricting the four-component vec-

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tor s m to the ordinary three-dimensional space vector sJ :

• s / = 0 a t r = r B, OjsJ = 0 , bu tOj~ '7=~0. (27)

More precisely one has

+ (m + ~-)5(r B - r)] (XL + o R ) . (28)

So the divergence of the SUSY current does receive a contr ibut ion from the bag surface. As this lat ter is repelled to infinity, such a term vanishes since the fields are sommable functions, and therefore supersymmetry is restored as expected. One may also show that the flux of the SUSY current is bounded:

f d3r 37J <~C/r B . (29)

Note that the same techniques can be used to explicitly see that, as in the usual MIT bag model, the axial supercurrent is not conserved on the boundary, even in the case of massless uron (m = 0).

Demanding continui ty of the axial current A a outside the bag, one will now consider an axial super- currentA~r a associated via super-PCAC to the "p ion" chiral superfield 7r a = (S o + irr a,~a, F~ra), and such that on the boundary r = r B the axial current pro- vided by the uron field on one hand and by the "p ion" field on the other hand are equal, as well as their divergences. We note that analogously to the MIT bag model, the symmetry breaking on the frontier can be considered as spontaneous for mur = 0 inducing then the masslessness o f the Goldstone boson zr a.

The supersymmetric extension of PCAC reads on

,4~ [51:

- ~ D2A = ~m~rf~rr, at r = r B , (30)

or on the associated current superfield ~rn :

Orn.~rn = ~ im~rf~r(D2zr _ ~2~-), at r = r B. (31)

Because of the breaking of supersymmetry the masses

m S, m'~ and mTr corresponding respectively to the (quasi-Goldstone) S scalar, ~" fermion and (Goldstone) 7r pseudoscalar will be in general different, as well as the corresponding decay constants f s , f~ and f~r' Eq. (31) then reads in components:

amAm = m2fTr 7r, Om Am =-- ~im2 f~'~a ,

Om Amn = gmnm2 f s ~ m S , (32)

where A mn is the Oon-ff component o f / i m . I t appears that a relation between the uron mass

mur and Goldstone fermion ~" mass can be directly deduced from the above equations. Indeed a careful inspection of the components associated to ~ , that is A m given in (19) as well as Om Am given in (20) and (32) joined to the equation of mot ion for 7r,

- io ta Om ~ = m~-~", (33)

leads to the relation ,3 :

rn(RX~ + paL) = - ~ m~ f g 7 3 , at r = r B . (34)

Taking into account the boundary conditions (22) implies also

V" A~ = 4im2(p~d~ + L X#), at r = r B , (35)

that is, by (32)

2m2(0~ 3 + / , Xts) = 1 - n m ~ f ~ 7 r a , a t r = r B . ( 3 6 )

Rewriting eq. (34) and its complex conjugate with the help of eq. (22) and using eq. (33) as well as eq. (36) gives finally

rn~. - 2tour . (37)

We, therefore, obtain a linear relation between the Goldstone fermion mass and the uron mass, in ac- cordance with the supersymmetric version of Dashen's formula [5,11 ]. Now, the boundary condit ion eq. (22b) ensures that the contrubit ion of the urbosons to i • .zi as well as to ~z. A vanishes at the boundary (only the ur-fermions are involved) and the mass relation between urons and external pseudoscalar particles is" the same as the one found before, that is

2 2 mur = ~ m~rrn~rrB(m~rr B + 1)/(2 + 2m~rr B + m~rrB) . (38)

We can immediately check that for r B --> o o

mur = ~ m~r , (39)

and therefore rn~ = m~r as expected because of the restoration of supersymmetry invariance all over space, while for "small" values o f r B

,3 Note that eq. (34) can also be obtained by developing eq. (30).

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m,=G&&, (40)

Assuming uron mass to be small and the bag radius

rB @mu,, we get finally the result

m,-=2mu,9m,=~~. (41)

This is our promised result: we have shown that a

confined supersymmetric uron theory where confine-

ment is realized by a SUSY extension of the Bogoliubov-MIT condition, for very small confinement radii, gives rise to pseudo-Goldstone bosons which are much heavier than their fermionic com- panions. Unfortunately we have been unable to obtain an information on the mass of S which would enable us to evaluate Str7?@ for the Goldstone multiplet, a precious indication on the nature of SUSY breaking.

Let us add that our results do not appear incon- sistent with Witten’s constraints on supersymmetry breaking [ 12 3. In our superbag approach the poten- tial reduces to the non-zero bag pressure constant B. Therefore, there is no zero energy state and the vanishing of the Witten’s index does not allow to ex- clude supersymmetry breaking. This is not the case in SQCD in which a non-vanishing Witten’s index pre- vents dynamical supersymmetry breaking, at least in the massive case [ 131.

In our example, if the bag pressure vanishes, zero energy states are allowed; however, then, the bag evaporates or equivalently rB goes to infinity in which case supersymmetry is restored, as already pointed out.

Our methods were non-perturbative, and invoked continuity properties of currents and their divergences in coordinate space.

It is conceivable that a model for realistic quark and leptons might be constructed on these ideas.

We are indebted to W. Buchmiiller and R. Stora for useful discussions. It is also a pleasure to thank 0. Piguet and X. Tata for interesting remarks. Finally we want to express our gratitude to the Franco- Norvegian Foundation for Scientific Research, Technique and Industrial Development for travel grants and G. Rivibre for help and goodwill.

References

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[3] W.A. Bardeen and V. Vi$nji&Nucl. Phys. B194 (1982) 422; W.A. Bardeen, T.R. Taylor and C.K. Zachos, Nucl. Phys. B231 (1984) 235.

[4] R. Barbieri, A. Masiero and G. Veneziano, Phys. Lett. 128B (1983) 179.

[5] TE. Clark and S.T. Love,Nucl. Phys. B232 (1984) 306; see also G. Shore, Nucl. Phys. B231 (1984) 139.

[6] GE. Brownand M. Rho,Phys. Lett. 82B (1979) 177. [7] R.L. Jaffe, Ettore Majorana School, Vol. 17, ed. A.

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A model for mass splitting inside goldstone supermultiplets