N = 1 and N = 2 supersymmetric positronium

Volume 155B, number 5,6 PHYSICS LETTERS 6 June 1985

N = 1 A N D N = 2 S U P E R S Y M M E T R I C P O S I T R O N I U M

P. DI V E C C H I A and V. S C H U C H H A R D T

Physics Department, University of Wuppertal, Gauss-Strasse 20, D- 5600 Wuppertal 1, Fed. Rep. Germany

Received 27 February 1985

The energy of the N = 1 and N = 2 supersymmetric positronium is computed using the Breit-Fermi lagrangian obtained from photon, photino and scalar exchanges. In particular in the case of the supersymmetric N = 2 QED we find complete degeneracy for para- and ortho-superpositronium.

The most appealing interpretation [ 1 ] of the monojet and multijet events at the collider [2] is in terms of a minimal extension o f the standard model plus soft breaking terms, that survive from the super-Higgs effect at the Fermi scale * ~. If this interpretation is confirmed also quantitatively by future experiments one expects also to find hadrons that contain the scalar quarks as constituents [4].

The properties o f bound states o f heavy quarks have been successfully understood in terms of potential models in analogy to what is done in the case o f positronium [5]. If, however, the underlying theory is supersymmetric apart from soft breaking terms, it would be nice to use potentials that are consistent with supersymmetry.

A first important step has been made by BuchmfiUer, Love and Peccei [6] who extracted the low energy Bre i t -Fe rmi potential in the case o f the N = 1 supersymmetric QED from photon and photino exchanges. They have also shown that the 16 states o f the N --- 1 superpositronium belong to two N = 1 supermultiplets with different energies, that are the supersymmetric completion of respectively the para- and ortho-positronium. The restauration of supersymmetry is however not trivial and can only be obtained if one adds the contr ibution of second order perturbat ion theory.

In this letter we consider the N = 2 supersymmetric extension of QED and we extract the low energy " B r e i t - Fermi" potential . A new feature in this case is the appearance of a complex scalar exchange. Using this potential we can then compute the energy o f the N = 2 superpositronium and after the addition o f the second order perturbation theory we find that now the 16 states of the N = 2 superpositronium have all the same energy forming a massive multiplet of the N = 2 supersymmetry without central charges, that consists o f a vector, 5 scalars and 2 Dirac fermions [7 ]. As a consequence we find no splitting between ortho- and para-positronium in the N = 2 supersymmetnc QED.

Finally in the last part of this letter we construct the composite superfields, that contain dynamical fields corresponding to the states ofsuperposi t ronium. Those are the superfields, that must be used in an effective lagrangian for the low energy bound states of the theory.

We start from the following abelian theory [8] ,2 :

L = LGaug e + L Matter , (1)

where LGaug ~ describes an abelian N = 2 supersymmetric gauge theory:

LGauge = ~ W~Wa IF +'~W& W& I F + q~*~ID , (2)

, l For a review of those models see ref. [3]. #2 In ref. [9] however a slightly different definition of the Dirac spinors and the SU(2) spinors is used.

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and LMatter is given by

LMatter -- (Rle* 2elVR 1 + R~e-2el VR2)ID + [R1R2(m + x/~e2¢ ) + R~R~(rn+x/~e2q~*)][ F . (3)

If we take e I -- e 2 = e lagrangian (1) describes a N = 2 supersymmetric extension of QED. In terms of component fields we get the following lagrangians:

LGauge = - 4 " l r ' u v - - ~ u v - ~ K ~ u K * - ~ i ~ i 3 ' u a u r l i ' (4)

and

LMatter = - (DvZ)+(DuZ) - ½e2(Z+r3Z) 2 - ~e2(Z+'cllZ) 2 - (Z+Z) [m 2 + x/~me2(K + K * ) + 2e22KK * ]

1 - i + - i + ~ {--if21 -- m -- x/~e 2 [~ (1 + 3'5 )K + ½( 1 - 3'5)K*])~0 + ib'x/~(ff75r/~Z + Z i 77 3'5 i f ) , (5)

where Z i is a SU(2) doublet made up with the scalar electrons, ff is a SU(2) singlet describing the electron and 7/i is a SU(2) doublet containing the two photinos defined by

, = . ( 6 )

h a [p~] is the photino appearing in the vector [chiral] multiplet V [~]. The coupling constant ~" in the last term of (5) must be chosen equal to e 1 [e2] if it multiplies a term containing the gaugino ~ [p] and ~11 = (7"1' ¢2)- By construction the previous theory has a N = 1 supersymmetry and for e 2 = 0 (5) reduces to the lagrangian for the matter fields in N = 1 SQED. On the other hand, when e I = e 2 = e, it provides a N = 2 supersymmetric extension of QED. In this case it has also a manifest SU(2) invariance.

In the case of positronium one starts from the scattering amplitude for the process e÷e- ~ e÷e - , from which one derives the "Bre i t -Fe rmi" potential. In our case we must also compute the scattering amplitudes involving the scalar partners of e + and e - . The scattering amplitudes relevant for positronium and its scalar partners are

~i ~16E1E2E3E4~ ~1~ 2 Lel 2 [fi33'Uul Du u(,P3 - P l ) v23'Vv4 - u33'tav4Duv(P3 + P4) v23'vUl] T(e+e - e+e -)

(P3 - P l ) 2 (P3 +P4) 2 '

T(Z~ Zi ~ Z~ Zi) = (16EIE2E4E3)-l /2 ( - i e l 2 [(P4 + P2)uDuz,(P3 - Pl ) (P3 +Pl )v

+ 2(P4 - P3)uDuv(.P3 + P4) (Pl - P2)v] + e2 + 2e] - e24m2 [ 1/(P3 - P l ) 2 + 2/(P3 + p4)21, (8)

T(Z+r A Z ~ Z÷rBZ) = (16EIE2E3E4) -1/2 {[- ie2(p2 + p4)uDu~,(p3 - Pl )(Pl + P3)u

- 2 - A 3 ~ B 3 - - 2,oAI~BI 26B2)), - e ~ - - a m 2 e 2 / ( p 3 - P l ) 2 - 2 e 2 ] 6 a B +,~elo o *,~e2to o +6 A (9)

T (e+e - _+ Z+Z) = (16EIE2E3E4) -1/2 [ -2e2m (fi3v4)/(p3 + p4) 2

+e21(fi33"Uv4)Duv(P3 + P4) (P2 - Pl)v - ( e2 + e2) (fi33'Uv4) (P3 - Pl)u/(P3 - Pl )2] ' (10)

while those relevant for its fermionic partners are

T(e+B - _+ e - B +) = T(e+C - ~ e -C+)= (e 2 _ e 2) 2(16EIE2E3E4 ) - 1/2

X ( fia3'U(Pl - P3)uUl 4 ql3'a(Pl - P3)uv4 + fi43'U(P3 +P4)uUl + Vl3'u(P3+P4)uv4 (11)

\ (P--I - P 3 ) ---~ (Pl - P 3 ) 2 (P3 +P4 )2 (P3 +p4)2 ]

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T(e+B- ._+ e + B - ) = T(e+C - -+ e+C - ) = (16EIE2E3E4)- l /2

-e222mfi3Ul ie21(fi37uUl)Duv(p3 - p l ) (p2 +p4) v - ×

\ (/93 - p l ) 2

T ( e + B -+ e - C ÷) = T ( e - B + -+ e+C - )

VlTU(Pl +P2)u75v4 + = - ( e 2 + e~)(16E1E2E3E4) -1/2 (

\ (Pl +p2) 2

(e~+e2)(V27Uv4)(P2 +Pl )u/

I ' (12)

i]4TU(P4 - p2)u'Y5Ul

(P4 - P2) 2 J o (13)

The scalar fields B and C are those used in ref. [6] and they are related to the fields Z i by the following relations:

B _ = - 2 - 1 / 2 [ ( Z 2 ) * - ( z l ) * I , C_ : - 2 - 1 / 2 i [(Z2) * + ( Z I ) * ] . (14)

Notice that the SU(2) invariance is recovered in (9) i f e 1 = e 2. Proceeding as in ref. [6] it is possible in the nonrelativistic limit to extract the Bre i t -Fe rmi potential , that is

now a matrix in the space of positronium and its partners. We get:

(15) : v < 0 ) l + <l) ,

where B [F] stands for positronium and its bosonic partners [the fermionic partners of positronium] and

V (0) = - ( a I + a2)/r - [(a I - a2)/r]p2/m2 + (a l l2m2r) [0 2 - ( p . t ) / r 2 ] - (a2 / r 3) i ( p . r) /m 2 .

The matrices V~ 1) and V~. 1) are given respectively in tables 1 and 2 and ~i = e2/4rr.

(16)

It is interesting to notice that, if e 1 = e2, in the potential (16) the scalar exchange gives a contribution to the

Coulomb potential that is exactly equal to the one of photon exchange and therefore in the N = 2 SQED the Coulomb interaction in e+e - ~ e+e - for instance is two times bigger than in N = 1 SQED or in QED. The same calculation for the process e e -+ e - e - in the case e I = e 2 shows instead a complete cancellation of the Coulomb potential. This is analogous to what happens in the case of the interaction between monopoles in the P ra sad - Sommerfeld and Bogomolny limit, where due to the combined contr ibut ion of photon and Higgs exchange one has no Coulomb interaction in m o n o p o l e - m o n o p o l e scattering [ 10]. This result can be very important for phe- nomenological applications of the N = 2 supersymmetric gauge theory [ 1 1 ].

By performing first order perturbation theory around the Coulomb term we get the following energies for the various states:

E = E 0 + ~m(a I + a 2 ) 3A , (17)

Table 1 Matrix potential for positronium and itsbosonic partners. The composite state structure and the values of J PC are also given.

States States

e+e - 1 - - e+e-0 -+ Z+ZO ++ Z+r 3 ZO-- Z+rbZ0++0 -+

e%- 1- - (1 la l /3+ 20~2/2) ~5 t7 0 -i(al+a2)ri/2rnr 3 0 0 e+e-0-+ x Or/m 2) 6 (r) (-'*1 +4~2) 0 0 0

x (n/m2)6(r) Z+ZO ++ -i(otl +a2)r//2mr 3 0 (al+6a2)(n/m2)5(r) 0 0 Z+r3Z0 - - 0 0 0 Or/rn2)6(r)(3al) 0 Z+r~Z0-+0 ++ 0 0 0 0 Or/rn2)8(r)6 ab

x (--a i +4a2)

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Volume 155B, number 5,6 PHYSICS LETTERS

Table 2 Matrix potential for the fermionic partners of positronium.

6 June 1985

State State

e+B-+e-B + e+C-+e-C + e+B - _ e-B + e+C - _ e-C +

2 7 1 e+B- + e - B + (n/rn)6(r)(~oq + ~ a 2 ) -i[(at+a2)~2m](~'r)/r 3 0 0 e+C- + e - C + -i[(at+a2)/2m](a'r)/r a (/r/m 2)6 (l') (~ ot 1 + ~ or2) 0 0

2 1 9 e + B - - e - B ÷ 0 0 (n/m )6(r)(- ~ oi l+ ~ct2) -i[(al+a2)/2m](ff.r)/r 3 e+C - - e - C + 0 0 _i[(al+~2)/2m](~.r)/r3 1 9 (n/m2)5(r)(-~al+~tx2)

where A is given for the various states by the expression contained in the brackets appearing in the diagonal ele- ments of tables 1 and 2.

However, as observed in ref. [6], to order m~ 4 we get also a contribution from second order perturbation theory, that comes from the non-diagonal terms in tables 1 and 2. When this contribution is added we get the following energies for the various states:

E( 1 - - ) = E(Z+r3Z) = E ( e + B - + e - B + ) = E(e+C - + e - C +) = E 0 + ~ m ( a 1 + o~2)3(3~ 1 ) , (18)

E ( 0 - + ) = E(Z+Z) = E ( Z + % Z ) = E(e+B - - e -B+) = E ( e + C - - e - C +) = E 0 + ] m ( a I +a2)3(4a2 - a l ) . (19)

l f a 2 = 0 these values coincide with the result of ref. [6]. l f r q = a 2 corresponding to the N = 2 supersymmetric limit, we get that all levels are degenerate and in particular there is no splitting between para- and ortho-positronium. They form a massive multiplet o f the N = 2 supersymmetry without central charges, that contains 1 vector, 5 scalars and 2 Dirac fermions. If or I = ot 2 the extended N = 2 supersymmetry is broken, but the theory is still N = 1 supersymmetric and the states in (18) correspond to those of a massive vector multiplet and o f two degenerate chiral muitiplets.

In the last part of this letter we want to identify the composite superfields, that contain dynamical fields corresponding to the 16 states of superpositronium. The most obvious composite superfield is T = R 1R2, that has already been introduced in ref. [12]. In terms of component fields T is given by:

T = A + + X/~0aX~ + 0 2 G . (20)

1 It is easy to see that A + correspond to the state ~Z+(r 1 + i r2 )Z in table 1 ;y,~ is contained in a combination o f the states (e+B - - e - B +) and (e+C - - e - C +) o f table 2 and G corresponds to - m Z + Z - ~ ~ (1 + 3'5)6 after the use o f the equation of motion for the auxiliary fields of R 1 and R 2 in the small coupling limit. Being Z+Z and ~ ' 5 ~ auxiliary fields o f T they will not have any dynamics. This is however in contradiction with the calculation in superpositronium where we get levels corresponding to those states. Therefore we must introduce another chiral superfield R that has the previous fields in the first component. The obvious candidate is:

R = - ¼ I ) 2 ( R [ R ~ ) = A + v r 2 0 a ( - 0 l ) + 02D - , (21)

the appearance of which was already noticed in the effective lagrangian for the two-dimensional supersymmetric CP N-1 model [ 12] and that was also considered in an appendix o f ref. [ 13]. The fact that such a composite field must be present has been also explicitly found in ref. [ 14], where the effective lagrangian for the composite states o f the supersymmetric extension o f the N a m b u - J o n a - L a s i n i o model has been evaluated in one-loop approximation. Although classically R is a function o f T , from the quantum theory one must consider T and R as independent fields, because they have different renormalization properties. On the other hand this super field has dynamical degrees o f freedom that correspond precisely to the superpartners of the para-positronium, that are not contained in T. In fact A corresponds to the states Z+Z and O - + o f table l , while ~ l is contained in the combination o f the

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states (e+B- - e -B ÷) and (e+C- - e -C÷) , that is orthogonal to the one contained in P. In conclusion if we use the chiral superfields T and R and we consider their component fields as independent,

we reproduce all the states of the N = 1 supersymmetric extension of para-positronium. Another reason to have two chiral supermultiplets is the R invariance of the original theory, that requires for the composite states a Dirac and not a Majorana mass term ,3 .

For the states corresponding to the supersymmetric extension of ortho-positronium one must introduce the following vector composite superfieid ,4.

U = R2R ~ - R 1R~ = A 3 + X/~0aX~ + x/r2 0&X~ + 2-1/202N + 2 -1/2~2N* - 0o~t ~v ~

+ 020,:, I - X / ~ + 2-1 /2 i (ou) &~ 0uX~] + 020a [-X/~q~2 + 2-1/2i(ot~)a& 0UX~] + ~ 0202(D 3 + ½17A3). (22)

After the use of the classical equation of motion in the small coupling limit, it is easy to see that the superfield (22) contains all the states of the super ortho-positronium of eq. (18). In conclusion the states of super positronium can be described by two chiral superfields R and T and by a massive vector superfield U.

Notice that the component fields of R, T and U form multiplets of the internal SU(2) group. However, the various components of the multiplets are contained in different superfields. Therefore if the original theory (5) is invariant under SU(2), all levels must have the same energy. This is in fact what happens in the case a I = a 2.

Using the superfields R, T and U it is possible to write a massive free effective lagrangian for the composite fields, that is SU(2) invariant if the mass for the superfields R and T is equal to the mass of U.

It would be iJlteresting to study the implications of our results for the construction of effective lagrangians, containing the anomaly effect, fo rN = 1 and N = 2 SQCD.

We thank W. Buchmiiller, S. Cecotti, S. Ferrara, L. Girardello and G. Veneziano for very useful discussions.

,3 We thank W. Buchmiilier for a discussion on this question. ,4 The importance of the vector composite superfield has recently been emphasized in ref. [ 15].

References

{ 1 ] E. Reya and D.P. Roy, Phys. Let t. 141B (1984) 442; Phys. Rev. Lett. 52 (I 984) 881 ; J. Ellis and H. Kowaiski, Phys. Lett. 142B (1984) 441 ; Nucl. Phys. B246 (1984) 189.

[2] UA1 CoUab., G. Arnison et al., Phys. Lett. 139B (1984) 115. [3] E.g.R. Barbieri and S. Ferrara, CERN preprint TH 3547 (1983), Surv. High Energy Phys., to be published;

H.P. Nilies, Phys. Rep. 110 (1984) 1 ; H.E. Haber and G.L. Kane, University of Michigan preprint UM-HE-TH-83-17 (1984).

[41 C. Nappi, Phys. Rev. D25 (1982) 84. S.H.H. Tye and C. Rosenfeld, Phys. Rev. Lett. 53 (1984) 2215.

[5 ] See e.g., V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics, Ch. IX, p. 343. [6] W. Buchmiiller, S.T. Love and R. Peccei, Nucl. Phys. B204 (1982) 429. [7] A. Salam and J. Strathdee, Nucl. Phys. B80 (1974) 499; B84 (1975) 127;

see also: P. Fayet, Lectures XVIlth Winter School of Theoretical physics (Karpacz, Poland, February 1980), LPTENS 80•7. [81 P. Fayet, Nucl. Phys. Bl13 (1976) 135. [91 See also: P. Di Vecchia, R. Musto, F. Nicodemi and R. Pettorino, CERN-TH 3905 (May 1984).

[10] N.S. Manton, Nucl. Phys. B126 (1977) 525; C. Montonen and D. Olive, Phys. Lett. 72B (1977) 117.

[ 11 ] F. del Aguila, M. Duncan, B. Grinstein, L. Hall, G.G. Ross and P. West, Harvard University preprint-84-A001. [12 ] A. D'Adda, A. Davis, P. Di Vecchia and P. Salomonson, Nucl. Phys. B222 (1983) 45. [13] T.R. Taylor, G. Veneziano and S. Yankielowicz, Nucl. Phys. B218 (1983) 493. [141 W. Buchmiilier and U. Ellwanger, Nucl. Phys. B245 (1984) 237. [ 15 ] E. Guadagnini and K. Konishi, University of Pisa preprint (1984).

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N = 1 and N = 2 supersymmetric positronium