On finite quantum field theories

Volume 147B, number 1,2,3

PHYSICS LETTERS

1 November 1984

ON FINITE QUANTUM FIELD THEORIES S. RAJPOOT and J.G. TAYLOR Department of Mathematics, King's College, London, England Received 7 July 1984

The properties that make massless versions ofN = 4 super Yang-Mills theory and a class ofN = 2 supersymmetric theories finite are: (i) a universal coupling for the gauge and matter interactions, (ii) anomaly-free representations to which the bosonic and fermionic matter belong, and, (iii) no charge renormalisation, i.e./3(g) = 0. It was conjectured that field theories constructed out ofN = 1 matter multiplets are also finite if they too share the above properties. Explicit calculations have verified these theories to be ffmite up to two loops. The implications of the f'miteness conditions forN = 1 finite field theories with SU(M) gauge symmetry are discussed.

The fact that field theories can be made ultraviolet f'mite by the use of extended supersymmetry has once again revived interest in supersymmetry. The N = 4 super Yang-Mills theory with four supersymmetries was the first to be proved ultraviolet ffmite [ 1 ]. It had been known that the N = 2 theory was also fmite bey o n d one loop [2]. It is a trivial matter to make it finite at one loop by adding enough matter multiplets (hypermultiplets) so that the one-loop ~-function of the gauge coupling vanishes [3]. I t is natural to enquire whether these N = 2 and N = 4 theories are the only candidates for f'mite field theories(excluding gravity) or are there others? In particular, can one make Finite field theories out o f N = 1 vector and chiral superfields? The answer lies in the properties that make the N = 4 a n d N = 2 field theories finite. These were discovered by studying the N = 4 super Yang-MiUs theory which is better understood even in the presence of explicit mass terms for the scalars and fermions of the theory. It was found [4] that the properties that make the N = 4 super Yang-Mills theory finite are: (i) It has a single coupling constant for gauge and matter interactions. (ii) The representations of the theory are anomalyfree. (iii) The coupling constant renormalisation vanishes. (iv) I f masses are explicitly introduced into the theory [5], then these are required to satisfy the masssquared super-trace sum rule 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

J:~0,½ ( - 1 ) 2 J + 1 ( 2 J + 1)Mj2= 0 . Finite N = 2 field theories also satisfy the above properties. To demonstrate how these properties operate, consider the N = 4 and N = 2 theories in the absence of any scale. Invariance under extended supersymmetry and the additional symmetry on the supersymmetry charges require these theories to possess a single coupling constant between the gauge and matter fields. The second property i.e. anomaly-free representation is trivially satisfied by the N = 4 theory. This is because all matter belongs to the adjoint representation of the gauge group and the anomaly of adjoint representations is zero. In the case of N = 2 theory, gauge and fermionic matter belong to the vector and matter hypermultiplets (V, Hi). The vector hypermultiplet V (consisting of one vector and one chiral N = 1 supermultiplets) belongs to the adjoint representation and is anomaly-free. The central charge constraint requires the hypermultiplet (H) to consist o f two N = 1 matter multiplets, one in representation R and the other in the conjugate representation R o f the gauge group G. The anomalies of R and R are related;

.4 (Ri) : -.4 (Ri),

(1)

where A (Ri) is the anomaly index o f the representation Ri. Since the anomaly indices of a reducible representation are additive, the hypermultiplet is also anomaly-free. 91

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PHYSICS LETTERS

Note that the N = 2 theory can also admit matter hypermultiplets that are pseudo-real [6]. These representations are pseudo self conjugate and hence anomaly-free by definition. Finally, the vanishing of charge renormalisation at the one-loop level [property (iii)] Fixes the representation content of the theory. In the N = 4 case this condition is saturated by three N = 1 matter multiplets in the adjoint representation of G. In t h e N = 2 case, the number of matter hypermultiplets is fixed by . ~ T(Hi) = 2C2(G),

(2)

l

where T(Hi) and C2(G) are the second indices of the hypermultiplet and the vector multiplet. For the conventional hypermultiplet T(H/) = 2T(Ri) and for the pseudo-real hypermultiplet T(H o = T(Pi) where T is the second index of the representation Ri (Pi). In the N = 4 and N = 2 f'mite theories, the condition that the charge renormalisation vanishes at one loop ensures that it vanishes to all higher orders. Clearly, the missing member of the class of finite field theories to which t h e N = 2 a n d N = 4 theories belong is finite field theories built out o f N = 1 superfields in arbitrary representations of G and having properties (i), (ii) and (iii) in common with the N = 2 a n d N = 4 theories. This member was first proposed in ref. [4]. An explicit calculation has revealed this class of N = 1 field theories to be finite to two loops [7]. In fact one-loop finiteness ensures two-loop finiteness. In the absence of any mass scales in the N = 1 f'mite field theories, apart from having a unique coupling as required by (i), one also requires to satisfy anomaly cancellation . ~ A (Ri) = 0 ,

(3)

i

and no charge renormalisation T(Ri) = 3 C2(G),

(4)

by properties (i) and (ii) for ffmiteness. The anomaly index of a representation is proportional to the thirdorder Casimir invariant of the group. It is known that the third-order Casimir invariant of all classical groups except for the SU(M) (M > 1) groups are zero. All orthogonal groups are anomaly-free except for 0(6). Since 0(6) is locally isomorphic to SU(4), this case is 92

1 November 1984

covered in the discussion on anomalies in SU(M). A general lagrangian in terms o f N = 1 superfields can contain terms of at most cubic. The term linear in superfields is forbidden due to non-abelian gauge invariance. In the absence of non-abelian gauge symmetry, the couplings grow asymptotically and hence there is no finiteness. In the absence of mass terms, the only interaction term is the cubic L(3 ) = (g/3!) haL~MNT r ( R a " q~L)(Rb " ¢ f l ) ( R c " ¢/V). (5) The interaction term enjoys the usual properties of a Yukawa term of being symmetric and gauge invariant. abc • The matrix hLM N is constrained by ~

M,n

~bc

_

, , L M N n L , M N - 2~LL,8

actt

C2(G).

(6)

b,e

For particular choice of the representation matrices R a, L(3 ) of eq. (5) reproduces the interaction term of the known theories. Thus if the representation matrices belong to the adjoint representation of the gau~e group G, then R~ = ~ and only the term ~Tr~Ta [T °, Tc]) = i f abc is allowed to contribute. Hence ha/~N = i e L m N f abc. If ~bL, 4~r are in representation R and R of G [~bL in Rk and q~Min Rk], then hLMN = 6 L k 6 N k 6 0 M and the interaction term of eq. (5) reproduces the interaction term of the N = 2 theory [~0 is in the adjoint representation, the multiplets ~L, q ~ are equal in number and the number is constrained by the fmiteness condition of eq. (2)]. The requirements of eqs. (3) and (4) for finiteness restrict the allowed representations in f'mite N = 1 theories only to low dimensionality ones. In what follows the representations allowed by the finiteness condition [eqs. (3) and (4)] and the number of families contained in these representations of the SU(M) group are examined. The Young tableaux, dimensionality d(Ri) the anomaly indexA (R/) and the second index T(Ri) of the SU(M) representations allowed by the f'miteness condition of eq. (4) are given in table 1. The range of M for the allowed representations is restricted so as not to duplicate the representations. For example for M < 8, the fourth-rank antisymmetric tensor is equivalent to the third-rank conjugate antisymmetric tensor and so on. As the dimension of the representation increases, so does their anomaly index [A (Ri)] and the second index [T(Ri)]. This and the fact that there

Volume 147B, number 1,2,3

PHYSICS LETTERS

1 November 1984

Table 1 The dimensionality, anomaly index A (Ri) and the second index T(Ri) of the SU (M) representations Ri satisfying the one-loop finiteness condition. Ri

d(Ri)

A (Ri)

T(Ri)

Range of M

o

M

1

~]

M(M-1)/2

n-4

112 (M- 2)/2

M> 2 M~ 4

rr~

M(M+ 1)/2

M+ 4

(M+ 2)12

M ;~ 3

~]

M(M-1)(M-2)[3!

(M-3)(M-6)/2

(M-2)(M-3)/4

6
M(M-1)(M-2)(M-3)/4!

(M-3)(M-4)(M-8)/3!

( M - 2 ) ( M - 3 ) ( M - 4 ) / ( 2 . 3!)

8
[~

M(M2-1)13

M2 - 9

(M2 -3)•2

M< 6

[~

M2(M2 - 1)/12

M(M 2 - 16)/3

M(M 2 -4)/6

M< 4

(M2 - 1)

0

M [T(adj) = C2(G)]

M~>2

M-1 ! boxes n

<17

(adjoint representation) should be enough representations to accommodate at least three families of quarks and leptons restricts one to use only the lowest dimensionality representation for SU(M). The representations employed will be the fundamental, second- and third-rank antisymmetric, and the adjoint representation. Also SU(M) groups with rank greater or equal to four will be employed in order to encompass the SU(3) X SU(2) X U(1) strong and electroweak interactions. The groups and the fermion content are discussed in turn. Denote the multiplicity of the fundamental, second-rank, third-rank and adjoint representations by (nl, n2, n3, n). The multiplicity of the conjugate representations is denoted by (nl, n2, ~3)- The anomaly and finiteness constraints of eqs. (3) and (4) imply (nl - nl) + (n2 - K2)(M - 4) + (n 3 - ~3)(M - 3)(M - 6)/2 = 0 ,

(7)

~(nl + n2) + (n2 + n2)(M - 2)/2 + (n 3 + n3)(34- 2)(M - 3)/4 +nM = 3M.

(8)

The quarks and leptons required to describe the particle spectrum transform according to the complex representation of the SU(3) X SU(2) X U(1) subgroup of the unifying gauge group of one's choice. A family of quarks and leptons transforms like (3,2) + (1,2) +

2(3,1) + (1,1) under SU(3) X SU(2) X U(1). These complex representations form left-handed complex and 10 representations of SU(5). In discussing groups larger than SU(5) the number of complex representations as allowed by eqs. (7) and (8) will be decomposed according to their SU(5) subgroup. This will produce a number of { 5,3,10,]-0} representations and singlets. Next the "survival hypothesis" or the hypothesis of "pairing off" is invoked. Every 5 will combine with a to form four-component fermions with mass as large as the natural scale present in the theory. Similarly for the 1O's and 1--0's.In this procedure the 5's and the 10's that do not pair off will be identified as massless fermions that survive at ordinary energies. Out of these every set consisting of a 5 and 10 will constitute a family of massless fermions. Note that if the hypothesis of "pairing o f f ' is not invoked then there can be more fermion families accompanied by their twin-mirror families. In this scheme there is the problem of explaining the non-observation of mirror fermions. Therefore the former scheme giving massless fermions at low energies is preferred. Consider first an SU(M) f'mite field theory consisthag of one adjoint representation and a number of fundamental and second-rank antisymmetric representations that satisfy eqs. (7) and (8). The adjoint representation is required for the descent of SU(5) to SU(3) X SU(2) X U(1) in the intermediate stages of the hier93

Volume 147B, number 1,2,3

PHYSICSLETTERS

archy

SU(M) - ,

1 November 1984 {7-M,3,3,2},

... -, SU(5)

-,

SU(3) x SU(2) x u(1)

{10 - 2M,2,4,2},

-* SU(3) X U(1)em •

(9)

{6, 2 M - 2,2,0},

{2,10 - 2M, 2,4} ;

Out of these, the solutions that occur in all SU(M) groups with rank greater or equal to four are

{2M-1,M+3,0,1},

{ 2 M - 2,6,0,2} ;

{9-M, 2M-3,8,0},

M = 5,6,7,

M=5.

For this case the solutions to eqs. (7) and (8) are given as integer {nl, K1, n2, n2} for SU(M) groups with rank i>4. SU(M): ( M + 3 , 2 M - 1 , l , O } ,

{3,7-M,2,3} ;

{M+ 3, 2 M - 1 , 1 , 0 } ,

{6, 2 M - 6,3,0},

{ 2 M - 1 , M + 3, 0,1},

{ 2 M - 2, 6,0,2},

(3/+2,M+2,1,1},

( 3 M - 3 , 9-M,3,0} ;

{M+1,5,1,2},

M~<9,

{5,M+1,2,1},

{4,4,2,2},

{2M-6,6,0,3},

(6, 2 M - 2, 2,0}.

( 1 2 - 2 M , 2M--4,4,0}, ( 2 M - 4 , 12-2M,0,4}; M~<6, { 3 M - 15, 2 M - 5 , 5 , 0 } , {2M-5, 3M-15,0,5}; M=5, {M+ 2 , M + 2 , 1 , I } ,

{4,4,2,2};

{ 6 - M , 6 - M , 3 , 3 } ; M = 5,6, {5,M+1,2,1},

{8-M,M,3,1},

{3/+ 1,5,1,2};

{M, 8-M,1,3} ; M<~8,

{ l l - 2 M , M-1,4,1}, { M - l , 11-2M,1,4}; M=5,

It can be shown that these contain only two families of massless fermions if the survival "hypothesis" scenario is adapted. The SU(M) groups that contain three or more families of massless ferrnions based on the same scenario are given in table 2. Notable consequences of the restrictive nature of these N = 1 £mite theories are: (a) The remarkable result that three massless families can be accommodated in groups up to and including SU(9). (b) A maximum of five massless families can be accommodated in the N = 1 finite SU(5) model. This constraint is compatible with the one deduced from cosmological considerations, the amount of helium abun-

Table 2 N = 1 f'mite SU(M) models consisting of one adjoint representation and a number of fundamental and second-rank antisymmetric representations (bar denotes conjugate representations). Number of families

SU(M)-group (M>5)

Numberof representations []

R

~

adjoint

7

4

3

0

1

4

1

4

1

1

SU(6) SU(7) SU(8) SU(9)

9 11 13 15

3 2 1 0

3 3 3 3

0 0 0 0

1 1 1 1

SU(5) SU(6)

6 8

2 0

4 4

0 0

1 1

6(5) + 2(5) + 4(10)

SU(5)

5

0

5

0

1

5(~)+ 5(10)

SU(5)

94

SU(5) fermionic content in o, ~, Iq,~

7(5) + 4(5) + 3(10) 4(5) + 1(5) + 4(10) + 1 (1-0) 9(5) + 6(5) + 3(10) + 12(1) 11(5)+ 8(5)+ 3(10)+ 29(1) 13(5) + 10(5) + 3(10) + 51(1) 15(5) + 12(5)+ 3(10)+ 78(1)

8(5)+ 4(5)+ 4(10)+ 8(1)

Volume 147B, number 1,2,3

PHYSICS LETTERS

1 November 1984

Table 3 SU(M) models containing three massless SU(5) fermion families in antisymmetrie representations upto and including rank three. Gauge group

Representations

0

~

SU (5) fermionie content (excluding adjoint)

~

B

~

~

adjoint

SU(6)

0 0

6 6

3 3

0 0

1 0

0 1

1 1

3 (5) + 6 (5) + 4 (10) + 1 (1-0)+ 6 (1)

SU(7) SU(8) su(9)

0 1 3

8 10 12

2 1 0

0 0 0

1 1 1

0 0 0

1 1 1

5(5)+ 8(5)+ 4(10) + 1(1--0)+ 18(1) 7(5)+ 10(5)+ 4(10) + 1(1-0)+ 37(1) 9(5) + 12(5) + 4 (10) + 1 (1-0)+ 64(1)

dance from the time o f nucleosynthesis in the early universe. It is to be noted that there is no restriction on the allowed SU(M) groups if only two massless families of quarks and leptons are required. Now consider the presence of the third-rank antisymmetric representation. One adjoint representation be taken to be present to break SU(5) to SU(3) X SU(2) X U(1) in the hierarchy shown in eq. (9). The number of solutions to eqs. (8) and (9) are quite large. Therefore only the solutions that contain three massless fermion families are presented in table 3. Since the third-rank antisymmetric tensor is equivalent to the 10 f o r M = 5, this case has already been dealt with previously. Therefore the solutions in table 3 begin with M = 6. This still only goes up to M = 9. The most interesting outcome of these finite theories is their restriction on the choice of the group and the number of massless fermion families they admit. All these models contain the necessary scalar representations for breaking the gauge symmetry spontaneously as shown in eq. (9). Which of these models gives the right mass spectrum for the quark and leptons requires further study. In conclusion, it was conjectured by us that finite field theories can be built out o f N = 1 superfields provided they too have the following properties that make massless versions of the N = 2 a n d N = 4 theories finite: (i) A universal coupling.

(ii) Anomaly-free representations. (iii) Representation content that gives 13(g) = 0. Explicit calculations have verified this conjecture to be correct up to two loops. In this letter, N = 1 theories based on SU(M) groups and containing three or more SU(5) families have been listed. One of us (S.R.) would like to thank the SERC for financial support while this work was being completed.

References [1] S. Mandelstam, NucL Phys. B123 (1983) 149; L. Brink, O. Lindgren and B.E.W. Nilsson, Nucl. Phys. B212 (1983) 401. [2] M.T. Gdsaru, W. Siegel and M. Ro~ek, Nucl. Phys. B159 (1979) 429. [3] P. Howe, K. SteRe and P. West, Phys. Lett. 124B (1983) 55; S. Rajpoot and I.G. Koh, Phys. Lett. 135B (1984) 397. [4] S. Rajpoot, J.G. Taylor and M. Zaimi, Some applications of finite field theories, contributed paper Brighton Conf. (July 1983), conference preprint number 0172. [5] M.A. Namazie, A. Salam and J. Strathdee, Phys. Rev. D28 (1983) 1481; S. Rajpoot, J.G. Taylor and M. Zaimi, Phys. LetL 127B (1983) 347; S. Rajpoot and J.G. Taylor, Phys. LetL 128B (1983) 299. [6] J.P. Derendinger, S. Ferrara and A. Masiero, Phys. LetL 143B (1984) 133. [7] S. Rajpoot and J.G. Taylor, Towards finite quantum field theories, King's College prepdnt; S. Hamidi, J. Pafera and J.H. Schwartz, CALT-68-115, and references thereirL

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