N = 1 supersymmetry and the phenomenological pion physics effective action

PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

ELSEVIER

N = 1 Supersymmetry and the Phenomenological Pion Physics Effective Action S. James Gates, Jr. ~ *~ ~Department of Physics, University of Maryland, College Park, MD 20742-4111, USA In this presentation, I will show that it is possible to construct an extension of the phenomenological pion physics effective action by use of chiral and nonminimal 4D, N = 1 scalar multiplets. The result obtained differs drastically from the standard low-energs, N -- 1 SQCD theory first suggested by Veneziano and Yankielowicz.

1. Review of F u n d a m e n t a l a n d Low Energy Effective Theories of H a d r o n Physics The fundanmntal description of strongly interacting particles is believed to be provided by the QCD action (below I write it using the conventions of Superspace [1]) which involves gluons (G~_(x)), right-handed quarks (QR~(x)) and lefthanded quarks (QL h(x))

[ -zTr[QR~V~QR

1] + QL ~_QL J J ,

(1)

V a =-- O~ +igGa_ , Gab =- - i [ V a , Vb] , (2) where G~, QRa and QL'~ are all matrices whose explicit forms are given in the appendix. In writing the action I have neglected M0, a 6 x 6 diagonal matrix that describes quark masses. The usual Dirac spinor quark field (Q) is simply defined by -=

,

(3)

QL'a(X)

so that Qn~ and QL "~refer to its distinct chiral components• However in the laboratory at low energies, free quarks interacting with gluons axe not seen. Instead one observes baryons and mesons whose *[email protected] tNSF grant # PHY-96-43219 and NATO Grant CRG-930789 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(97)00654-3

dynamics are described by an effective action Fel! QCD" The effective Lagrangian £eQIIcD(FeQIICD ef$ f d4 x£QCD) is a very complicated quantity. I

want to split this according to

feff

QCD = c

U

(Barvon)+ £~/c/o (Meson)

+ £~/C/D(Baryon -- Meson Interaction)

(4)

In the subsequent discussion, I will only concentrate on £ ~ D ( M e s o n ) in its simplest approximation but in a imaginary world of unbroken 4D, N = I supersymmetry! For weakly coupled theories described by starting from a Lagrangian £Clo•ssical, it is possible to (approximately) calculate F ~If. For strongly coupled theories, the perturbative approach usually breaks down and there is no "first principles" guidance for finding F e/f. (Apparently, 4D, N=2 SUSY YM is an exception [2]!) Experimental guidance becomes essential since QCD is a strongly coupled theory at low energies. In comparison with the case of QCD where QRa, QL"a and G~ are matrices in color SUc(3) space, the fundamental entities of the pion effective action are matrices in a flavor SUf(3) space, 1 rPt

(5)

Here tl, ..., t8 are proportional to the Gell-Mann SU(3) matrices and f= is the pion decay constant. Exponentiation of the Lie algebraic element in (5) via the definition U ~ exp[ii~TH~ti] then leads to elements in the SU(3) group. An important additional symmetry of the theory is rigid SUL(3) ~) SUR(3) invariance (with constant real

172

S.J GatesJr.~NuclearPhysicsB (Proc. Suppl.) 62A-C(1998) 171-181

parameters ~,i and mation

a i) generated

by the transfor-

(U)' = exp[-iSiti] U exp[iaiti]

(6)

I will restrict my considerations solely to the physics of massless pion octets (neglecting p, etc.) and their possible SUSY generalization in the following discussion. It is convenient for me to begin with an expansion of the pion effective action in powers of f2, F(Pion) = S(2 ) -[-3(4 ) q-8(6)Jr.... The leading term in this expansion takes the form

= -~'~2 f2 / d4x Tr[ (Oa--U-1 ) (Oa_.U) ]

S(2)

.(7)

This is the famous non-linear cr-model [3] describing the flavor SUL(3) (~ SUR(3) invariant physics of the pion octet. It is only the leading term in an infinite expansion. For purposes that will become clear later, let me make some simple observations that follow from calculus via the "chain rule,"

Luetwyler [6] in a way most convenient for our purposes. As was first shown by Witten [4], the celebrated WZNW term is simplest when formulated by using the Vainberg construction. Thus I define U =_exp[iyf~lII] and in terms of the extended group element (U)~ the WZNW term is given by SWZNW -~ --iNc[2"5!] -1

//01 d4x dy

W4 : ~abcd(0a~r--1)(0b6)(0c~-l)

(0d~) . (10)

I may rewrite this by again using the chain rule and noting that our definitions imply that the pullbacks are y-independent,

SWZNW = - i N c

[2.5!]-1

f d4xeala_2~4 4

×

o~u-

ou 017

(o~n')

=

(o~u)(o~n ~)

i

-

j----1 ,

(S)

flijkl(II) ~- foldy Tr[(V-lOy~')(OiV -1)

and from which it follows that I may re-write the action S(2) in the form,

x (OjU) (OkU-1) (OtU)]

OiU-1 = (Oi V -1) (0aYI i ) ,

8(2) =

f

J d~

y~)(n) (o~-n') (o~_n')

~/(j2) ~ ~_2 E2,I~[(0iU_I)(0iV)]

(9)

In the limit I am investigating, S c - a = 31 + 82 + 83 where,

S1---- LI / d4x ( 'l~[ (oau-1) (OaU ) ] ) 2

The factors of (OaIIi ) are often called "the pullback from the space-time manifold to the SUf (3) manifold." In our attempt to find 4D, N = 1 supersymmetrical generalizations of these result, the pullbacks will play an important role. It is convenient to also define an auxiliary variable Zi j via the equation Zij - (OiU-1 ) (OsU), so that Continuing to the next term in the expansion I write, 3(4) = SWZNW + SO-L, where the first of these is the well-known Wess-Zumino-NovikovWitten term [4,5] and the second is an action whose form has been discussed by Gasser and

.(11)

$2

f

L2 J d4x T~[ (O~U-~ ) (ObU)] × ~.[(o~-u -~ ) (oh-u)]

83

,

,

L3 f d~x ~[ (o~w-1 ) (Oh_U) × (0-~V -1 ) (0bv)]

, (12)

Momentarily concentrating only on 32, I write

S2

L2 f d~x ~[ (O~V-1 ) (Ob_V)] ¢--~ × ,~- "m[(o~u-') (odu)]

. (13)

S.J Gates Jr./Nuclear Physics B (Proc. Suppl.) 62A-C (I 998) 171-181 Upon introduction of irreducible Lorentz projection operators p(0)a_~...~, p(1)_%...~ and p(2) a~...~ defined by

p(O) a b c d =

p(2)

abcd_

¼~ab ~d_

__ ~b_ _ ~1 [ no~

is obviously in the same class as the J-tensors above. It is associated with the existence of the e-tensor replacing one of the projection operators with four _a-indices. By applying the same type of reasoning to the entire p/on effective action, I conclude that it can be written as a Laurent series in ]2 oo

had_ ~_b

..[_

173

s(/)]

se.(p o ) = s . + Z /=2

½~_b~_d]

_

,

(14)

Ia_cIbd = p(O) a_b~_I + pO) abcd ,

,

(19)

S(i) = / d4x E Lk(A'B) p(A)a-I""a2' A,B,k

and further using the identity

+ p(2) abed

"4- 3WZNW

(15)

X

p(=)k,il .-.i2i

=, k)(Z )]

'~kl ...k21

2i

it is clear that $2 can be written as the sum of three other actions. Thus SG-L = 82 + 8a where

2

$2 --- / d4x E L2(A) p(A)a-~'"~ A=O 4

=> [

] (g

,(16)

j=l 2

& = ld4x ~_, La(A)p(A)~_~ ....~ A=0 4 ×

,(lr)

j=l

j----1

So if I regard H i as the coordinate of some space, then v.7.(A'B ~1 • "g2p'k) is a 2p-th order tensor (constructed as a polynomial in Zij) defined over that space. T(A,B) axe at present The dimensionless constants ~k only determined by phenomenology. The value of the "L" associated with eabed is determined by topological considerations. A theoretical derivation of all such coefficients would represent a major advance in understanding QCD. In terms of geometry (as I noted already) ~ 2 ) is to be regarded as a metric. All of the rest of the J ' s must be some algebro-geometrical tensors in the space described by ~.(.2) So I conjecture that ~.? the pion effective action may be regarded as the solution to some geometrical problem. At present I do not know how to solve this problem or even state it. •

which utilize the additional definitions .7(A, 2) ~'il i2 i3 i4 .7(A, 3)

~]/1 i2 i3 i4

~-- Z i l i2

Tr [ Zis u

-~ g i l i9 --'~i3 i4

L~o) -

4Ll+L2

]

,

2. Preliminaries of Fundamental and Low Energy Supersymmetric Models of Hadron Physics

, L~1) ---- L2 ,

L~2) -- L2 , L~0 -

L3

(18)

In principle, one can also introduce flavor SU(3) irreducible projection operators p(B)/.~/.2 i? i.~ J1 32 J3 34 that may be inserted between the multiplication of the fl-coefficients and the "pull back" products. In this way of thinking, the WZNW term

In a hypothetical world with unbroken 4D, N = 1 supersymmetry, the fundamental action for a QCD-like theory (SQCD) has long been presumed to have the structure

174

S.J Gates Jr.~Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

V~ -

)

e - ~'Y D ~ e ½v , V.~ =- - (V~ t ,

-

w o _= - [ V a ,

(21)

Thus the octet of gluons are replaced by an octet of gluon superfields V 1 , . . . , V 8. Similarly, the quark matter field are replaced by two eovaNantly chiral superfields (QR and QL with V'~QL = VgQR = 0). The usual quark fields in the nonsupersymmetric model are embedded in the supersymmetric theory with the identifications

-

(

)'

determines the renormalizable interactions of a theory of chiral scalar multiplets. This last requirement cannot be satisfied by any of the existing literature on 4D, N = 1 SQCD effective actions [8,9]. To achieve this I have extended a nmch overlooked class of models [10] that I call "CNM (Chiral-Nonminimal) models" and these take advantage of a little used representation of 4D, N = I SUSY ... the complex linear superfield [11]. The simplest 4D, N = 1 supermultiplet, the chiral scalar multiplet [12] (usually denoted by ~), contains component fields A = ~ [, ~b~ = Da~[ and F ~. D~¢[ which occur as

(22)

It is a matter of direct calculation to show that the superfield action above contains precisely the non-supersymmetric standard QCD terms of (1) together with additional terms describing the dynamics of gluinos and squarks. About a year and a half ago, I [7] started to wonder if the structure of the phenomenological pion effective action (discussed in the last section) could consistently occur in a model with manifest 4D, N = 1 supersymmetry. To my surprise, the answer is yes but to do so most simply requires tools (i.e. 4D, N = 1 superfield representations) that are not widely known. In the pion effective action above, apart from the constants L~A'B), everything is determined by ffi(j2)' ~.(A,B?k) and $1 ~...,~2p ~ij k l (which are functions of IIi(x)) and the pulli~'"i~ - (,1-Iq=l0a.II/i). In the simback factor f~a~..-~, plest supersymmetric taeory, tiae ordinary fields IIi(x) should be replaced by chiral scalar multiplets. As an additional assumption, I will impose the restriction that the supersymmetric versions of .-].(A,B,k) "] il ...i2n and flijkl should be expressed as the real or imaginary parts of strictly chiral superfields in an appropriate 4D, N =1 superspace action. Thus, it is my goal to seek a description where the higher order derivative terms of the pion effective action are embedded in a 4D, N = 1 supersymmetric theory via holomorphic functions much as the holomorphic "super-potential"

+ FF]

,

(23)

if I pick the standard superfield action for a massless free chiral scalar multiplet. The superfield equation of motion that follows from this is D2(I) = 0 or in terms of components O~-O~A = 0 , i O~_~b~ = 0 , F = 0 .

(24)

The 'chirality' of this multiplet is defined by the differential equation Dg(I) = 0. A long time ago [10,11], the question of the number of distinct off-shell 4D, N = 1 supersymmetric (0, 1/2) representations was studied. Among the results found was that there exist one such multiplet (called the "nonminimal multiplet" and denoted by P,) that is curiously related to the free massless scalar multiplet via Poincar~ duality [13]. The fields of the nonminimal multiplet are B - g[, ~ _ = D~E[, p~ - D~E[, H --D~E [, p~ - D~D~E[ and ~ - ½D~-D.~D~E[ which appear as SNM = -- / d 4 x d 2 8 d 2 - ~ E

=

f d%

[ -

-

-

HH

+ 2~-pa__

--0~

Ot

S.J. Gates Jr.~Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-18I

-k_

+3p

+

]

, (25)

Bianchi identity Ot'H -

so that the superfield equation of motion is D hE = 0 or in terms of components O~O~B = 0 , i0~¢ ~ = 0 , (26)

H = Pa_ = Po = 3~ = 0

The duality is seen by writing the defining constraint on this multiplet (i.e. D2E = 0 and E # which identifies E as a complex linear superfield). Summarizing as a table I have Constraint Chiral Nonminimal

D~

= 0

D2E = 0

Equation D2~ = 0 D~E=O

F~_ =

[ ½F

~_~o.

(27)

It is apparent that due to its definition, Fa_ satisfies a differential constraint (that I may call its "Bianchi identity"). I can display both equations in a simple table. Bianchi identity

Equation of motion

OaFb -- ObFa = 0

O~-F~_ = 0

T a b l e II On the other hand, the Kalb-Ramond field is described by the action 3 $2 :

= 0

Equation of motion OaHb -- ObHa = 0

Table I I I The Poincard duality between the two fields (~ and b_ab) is seen by noting that the equation of motion for one of the fields has the same form as the Bianchi identity for the other one and viceversa. However, this is exactly what appears in Table I for the chiral and nonminimal multiplets. Therefore, just as St and 82 provide dual descriptions of a spin-0 state, S c and SNM provide dual descriptions of (0 +, 0-,1/2) states. 3. E m b e d d i n g the P i o n Physics M o d e l in a 4D, N = 1 S u p e r s y m m e t r i c a l Theory

Table I It is useful to review Poincard duality in another context. The usual description of a massless scalar field is provided by

= - f

a

175

By utilizing the existence of nonminimal multiplets, my goal of describing a manifestly 4D, N = 1 superfield theory that contains precisely the structure of the pion effective action has a strikingly simple solution [7]. This begins by defining what I call "the CNM Mapping of Pion Physics." To all higher derivative terms, I apply the following mapping operation (denoted by _GC) described below ~ c : exp[:ki 1 Wt 1 __~ r:

[ ]:t:oltI expL/.cos( s) , (29)

where (I)I corresponds to a set of chiral scalar supermultiplets. In addition, tl are exactly the same matrices as in the non-supersymmetric case. It is also necessary to introduce a mixing angle 7s- Due to this operation, it is clear that the analogs of the higher order fl-polynomials (as well as 3) have a completely unambiguous definition in terms of chiral superfields.

. / d4x [1H~-H~_]

t ~b~cd H~_ = ~_b~dc,-o--

=

1

,~.C.ab

cdHbCd - -

(28)

It is apparent that due to its definition, H a satisfies a differential constraint (that is its "Bianchi identity"). Once again I can display both equations in a simple table. 3Note that even the change of sign between the actions $1 and S2 can also be seen between S c and ~NM.

JI( A,B, k ( ¢~'~ x--.I2. ~, ]

'

~IJKL(~2)

(30)

Thus, I obtain a set of holomorphic tensors that control the higher order interactions in exactly the same manner as the superpotential controls the interactions of a renormalizable theory involving purely scalar multiplets. It can be seen that the nonminimal multiplet does not enter into the above considerations.

S.J Gates Jr./Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

176

However, the pion effective action also required a set of pullback factors. It is at this step I define q : f ] - - [ ~

~I1,.,Iq

Gsc j=l q--1 C"fl

~

11 j=2

It should now be apparent why I re-wrote the pion effective action in such a seemingly unnatural manner by explicitly separating out the ordinary pullback factors (rI~=l Oc_.jH'J). After this is done, the step to supersymmetrize becomes marvelously simple. It is here that the nonminimal multiplets make their appearance. The super~.-~I1...Iq symmetric pullback _c~~ defined by equation (31) also has the feature that it transforms holomorphicallyunder the rigid SU/(3) ~ SUR(3) superfield transformations and it is a chirai superfield (although latter property is not a critical feature of the model). -~(A,B,k), Since the conditions D~ (~IJKL, I1...12n Ii ...lq ~c,...c_q) = 0 are satisfied, for the integration measure I define

:/ d4x

f

f

(32)

It is well known that the coupling constant g that appears in (21) may be regarded as being complex unlike its analog in (1). Thus in going from the non-supersymmetric theory to the supersymmetrical one, the dimensionless coupling constant of the Yang-Mills theory can be analytically continued. This same feature is true of most of the dimensionless c o n s t a n t s Lk (A'B). This suggests that Gc :Lk(A, B) should be interpreted as the complexification of the real parameters Lk (A'B) which appear in the pion effective action. The one exception to this is the coupling constant associated with p(1)a~ ...~4. The reason for this is that if the dimensionless constant associated with this term acquires an imaginary part, it then changes the value of the WZNW term, which is not allowed. In fact applying this mapping operator to the higher derivative non-supersymmetric pion effective action yields a supersymmetric action with

the property that all the transcendental functions that appear in it are holomorphic functions. This supersymmetrical theory may properly be called "holomorphic Schiral Perturbation theory" (Schiral -- SUSY Chiral). Thus, for every term in (19) except the leading term, I can explicitly write a 4D, N -- 1 supersymmetric term. For the leading term I propose GC: $~ = S~(4~, E) where

where U is the superfield group element obtained after the application of the CNM map to the usual group element. My motivation for making this choice is based on by several observations; (a.) SUL(3) ~) SUR(3) invariance, (b.) N = 2 SUSY KVM models [15], (c.) simplicity. Below I discuss how each of these motivates the choice above. The SUL(3)~)SUR(3) invariance of the component theory survives the above embedding into a 4D, N = 1 supersymmetrical one as I will now show. I simply apply ~ff to the group elements 4 in (6) for the chiral multiplets and this gives a transformation law for the chiral superfield group elements. In terms of the chiral superfields ~I this must correspond to some coordinate transformation of the form (¢I)~ __ kI(~). This coordinate transformation is clearly parametrized by a and and is continuously connected to the identity transformation. The last feature means that it must be possible to find a set of vector fields in the space of the tangent planes to the CLmanifold that generate the required coordinate transformation. Starting from the fact that the infinitesimal law can be written as

(~SUL(3)® suR(3)¢ ' -

- i[f, cos(Ts)]

=

a(A)~A)(~)

[SJ(L-I)j' - a J (R-I)j ' ], (34)

4It is critical that the SUb(3)@ SUR(3) transformation parameters remain as real constants, however.

S.J Gates Jr.~Nuclear Physics B (Proc. Suppl.) 62A-C (1998)171-181

in term of the left and right chiral superfield Manrer-Cartan forms LI K and RI K, it follows that k'(@) = exp[a(A)~A)0K ]@t

,

y--lDo£ y : [h COS(:s)]ll( DOlOI )

( D~U)U-1

,

= [f, cos(~s)]-:(D~O I ) × LIK(O) tK

actions I first proposed 6 in 1984 [15]. This action may be written as

(35)

(where OK = 0/0oK). The left and right chiral superfield Maurer-Cartan (LI K and RI g) forms are defined by

X RIK(O) tK

177

,

(36)

and can be calculated as powers series from

:I /

d20 W aI o~ W2o'~] H(~ ) + h.c.~ J . (38)

As is apparent, both $~ and SKVM have the form of integro-differential operators acting on a scalar function. In the former case I have used the Pernici-Riva K/~ler potential because of its obvious SUL(3)~ SUR(3) invariance. In SeibergWitten theory the K~hler potential is fixed by elliptic curves. I note that the transformation law of E I in the former case and that of W ~ I in the latter case are both fixed by exactly the same geometrical property, i.e. the transformation law of

0~. LIM(O)

=

((72) -1

foldy Tr[tM(evZXtI)]

RIM(O) -- LIM(-@)

,

,

(37)

where AtM -- [f~cos(Ts)]-l[OItl,tM], A2tM = AAtM, etc. and the constant C2 is determined so that LMI(0) = RMI(0) = 6MI. All of the higher derivative terms of the super field action will then be SUL(3)~ SUn(3) invariant if the nonminimal superfield transforms The fl and J tena~s (EL) ' = ( O I k i ) ~ I. sors all transform holomorphically with factors of (~ kL) -1 and the pullback fill...let :v..~ with factors of (~ kL). This last statement is true even though this pullback has both dotted and undotted spinor indices. This feature is impossible to achieve in a theory that solely uses chiral scalar multiplets. The SUL(3)~ SUR(3) invariant K~ihler potential Tr[UtU] was first suggested by Pernici and Riva [14] so that the invariance of the leading term is obvious. My second motivation for choosing the suggested form for the leading term is its similarity to the actions for K~ihlerian Vector Multiplet (KVM) models 5 of whose 4D, N = 1 superfield 5By choosing a special choice of H ( ¢ ) one arrives at the familiar N = 1 'prepotential' formulation of SeibergWitten theory.

The final reason that Sa seems reasonable is its simplicity. Even though tl are the usual hermitian matrix generations of SU(3) in,

U(O)

[

OIti

l

exp[ f~cos(Ts) ]

'

(39)

since @I are complex objects, b" # U -1 so that UtU # I. Thus in principle I may replace UtU by higher powers (U?U)p in the K~ihler potential. Even more complicated possibilities such as choosing the K~hler potential as Tr[(UfU) p] Tr[(UtU) q] etc. might be considered. 4. Consequences of the 4D, N -- 1 Supersymmetric C N M P i o n M o d e l The model that I have presented in the previous section (which I call the SUSY CNM pion model) clearly shows that it is mathematicallypossible to supersymmetrize the entire Laurent series expansion that is the pion effective action subject to the assumption that the higher derivative interactions are solelydetermined by a set of holomorphic tensors. These holomorphic tensors are the analytical continuation of similar quantities that occur in the non-supersymmetrical theory. 6The N = 2 formulation as well as component formulations both with and without supergravity were also suggested independently by others [16].

178

S.j Gates Jr.~Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

The SUSY CNM Pion model is not in the same class as the standard low-energy N = 1 SQCD model. The simplest way to see this is to perform a superfield duality transformation (which can formally be carried out) to replace the nonminimal multiplets by chiral multiplets. When this is done the resulting dual SUSY CNM Pion model (or C 2 Pion model) has twice the number of propagating fields as appear in the standard low-energy N = 1 SUSY QCD model. One way to understand this difference is to ask, "Should the fundamental N = 1 SUSY QCD theory upon confinement yield either Majorana-like or alternately Dirac-like pioniniT? For the standard lowenergy N = 1 SQCD model, the answer is the former. For the SUSY CNM Pion model it is the latter. This difference in the spectra can be seen even without performing the duality transformation. The spin-0 fields in the SUSY CNM Pion model occur as • II = £(z)

= A I( z ) +

i[ III(x)cos(Ts) +

eI(x)sin('~s)]

,

ZII __= BI(x) = ~ I ( x ) +

i[-III(x)sin(Ts) + OI(x)cos(Ts) ] ,

(40)

in terms of two real SU! (3) octets of spin-0 + fields A I and B I as well as two real SUf(3) octets of spin-0- fields H I and 01 . This is similar to the structure seen in the MSSM where for each known Dirac fermion there occur two scalar fields and two pseudoscalar fields. Accordingly, the SUSY CNM Pion model contains a Dirac-like set of pionini (denoted by gI(x))

gI(x) -~

~I(x)

D ~ xI ]

(41)

that transform holomorphically under infinitesimal SUL(3) ~ SUn(3) transformations (gl), = ~L(OLKI) . A remarkable property (which provided my initial motivation to consider CNM models) is that 7Pionini are the spin-l/2 superpartners of the pions.

none of the auxiliary fields that appear in the superfield pair (¢I, Et) propagate even though the complete action contains arbitrarily high order derivative interactions for the propagating fields! I call this property "auxiliary freedom." It turns out that the holomorphy of the fl and J tensors provide precisely the conditions needed so that the equations of motion of the auxiliary fields remain algebraic. Interestingly enough the condition of auxiliary freedom is violated in SeibergWitten theory. This result is implicitly contained in the fact that the analog of the $(4) term [17] describes propagation of the F auxiliary fields in the chiral scalar multiplets. The mixing angle 7s plays a curious role in this model. It turns out that it must be non-vanishing in order for the model to contain a set of fields that may be identified as the pion octet. Basically this occurs because for arbitrary values of this parameter, the term with the correct pion-like coupling in the action SWZNW is proportional to sin2(27s). If sin2(27s) ~ 0, I can choose the normalization of the superfield action so that the required single-valued behavior of the generating functional, as first noted by Novikov [5] in the general case and Witten [4] in the case of QCD, is not disturbed. In the usual Standard Model there is also another angle that (on theoretical grounds alone) satisfies the same condition. This is the weak mixing angle 8w. It is highly amusing to note that the structure of G~ : Seff(Pion) is vaguely reminiscent of 2D NSR heterotic string a-models. Here the chiral scalar multiplets play the role of 2D scalar multiplets and the NM-multiplets play the role of the 2D minus spinor multiplets. The former transform like the coordinates of the SUL(3) ~ SUR(3) manifold and the latter as covectors. All right-handed spinors occur in chiral superfields and left-handed spinors are in complex linear superfields. For 2D heterotic models, all right-handed spinors occur in minus spinor superfields and left-handed spinors in scalar or lefton superfields. So I am left with a mathematical curiosity. The class of CNM models seems very suited to solving the problem of beginning with a nonsupersymmetric effective action and embedding

S.J Gates Jr/Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

it within a supersymmetric model. The standard non-perturbative low-energy N = 1 SQCD model [9] has a long and much studied history. It has passed a remarkable number of self-consistency tests. After the stunning revelations of SeibergWitten theory [2], it seems more secure than ever. In the face of all of this overwhelming evidence to support it, the only point which has not been so thoroughly investigated is the structure of any higher derivative terms that are expected to accompany it. On the other hand, the CNM Pion model (and its superfield dual C 2 Pion model) is precisely such that its higher derivative terms may be regarded as the analytic continuation of the ordinary pion effective action in such a way that holomorphic functions play a crucial role. I find the emergence of such a dichotomy extremely puzzling. A possible resolution to this might occur if I consider a different fundamental superfield theory as the starting point. Instead of (21), I can define an alternate s Poincar~-dual fundamental theory by ~CNM-SQCD

+

(42)

This is a heterodexterous model that differs from the standard version of SQCD by assigning righthanded quarks to covariantly chiral multiplets (Qa~ - V~¢[, V~,(I) = 0) and left-handed quarks to covariantly complex linear multiplets (QL h =- ~ , E I, ~ 2 E = 0) so that the Dirac quark field transforms holomorphically under the SUSY SUe(3) gauge group, a property that is not shared by the Dirac quark field in the usual N = 1 SQCD theory. REFERENCES . S.J.Gates, Jr., M.T.Grisaru, M.Ro~ek, and W.Siegel, Superspace, Benjamin-Cummings Publishing Company (1983), Reading, MA. sI anticipated this by introducing the notation ~C2_QCD for the action in (21).

179

2. N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19; ERRATUM-ibid. B430 (1994) 485; Nucl. Phys. B431 (1994) 484; N. Seiberg, Rutger preprint, RU-94-64, May 1994, Talk given at Particles, Strings, and Cosmology (PASCOS 1994), Syracuse, New York, 19-24 May, 1994; idem. Nucl. Phys. B431 (1994) 551. 3. J. Schwinger, Ann. Phys. 2 (1957) 407; M. Gell-Mann and M. L6vy, Nuovo Cim. 16 (1960) 705; S. Weinberg, Phys. Rev. Lett. 18 (1967) 185; J. Schwinger, Phys. Rev. Lett. 18 (1967) 923; idem. Phys. Lett. 24B (1967) 473; J. Wess and B. Zumino, Phys. Rev. 163 (1967) 1727; J. Bell and R. Jackiw, Nuovo Cim. 60A (1967) 47; J. Cronin, Phys. Rev. 161 (1967) 1483; S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C. Callen, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247; R. Dashen, Phys. Rev. 183 (1969) 1245. See also the extensive references on chiral perturbation theory and effective Lagrangians in J. Donoghue, E. Golowich and B. Holstein Dynamics o] the Standard Model, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, (Cambridge Univ. Press, 1992). 4. J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95; A. P. Balachandran et. al. Phys. Rev. D27 (1983) 1369; E. Witten, Nucl. Phys. B223 (1983) 422, 433. 5. S. Novikov, Sov. Math. Dokl. 24 (1981) 222. 6. J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465. 7. S.J.Gates, Jr., Phys. Lett. 365B (1996) 132; idem. Nucl. Phys. B396 (1997) 145; S.J.Gates, Jr., M.T.Grisaru, M. KnuttWehlau, M. Ro~ek and 0. Soloviev, Phys. Lett. 396B (1997) 167. 8. G.Veneziano and S.Yankielowicz, Phys. Lett. l 1 3 B (1982) 321; T.R.Taylor, G.Veneziano and S. Yankielowicz, Nucl. Phys. B219 (1983) 493. 9. See the reviews which follow and references therein. D.Amati, K. Konishi, Y.Meurice, G.C.Rossi and G.Veneziano Nonperturbative

Aspects in Supersymmetric Gauge Theories Phys. Rept. 162 (1988) 169-248; M. Shifman,

180

S.J GatesJr.~Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

Nonperturbative Dynamics in Supersymmetric Gauge Theories, (hep/th9704114), Lectures given at International School of Physics, 'Enrico Fermi', Course 80: Selected Topics in Nonperturbative QCD, Varenna, Italy, 27 Jun - 7 Jul 1995, and at Summer School in High-energy Physics and Cosmology, Trieste, Italy, 10 Jun - 26 Jul 1996. 10. S. J. Gates, Jr., M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace Benjamin Cummings, (1983) Reading, MA., pp. 148-157, 199-200; B. B. Deo and S. J. Gates, Jr., Nucl. Phys. B254 (1985) 187. 11. S. J. Gates, Jr. and W. Siegel, Nucl. Phys. B187 (1981) 389. 12. Y.Golfand and E.Likhtman, Psi'ma ZhETF, 13 (1971) 452, idem. JEPT Lett. 13 (1971) 323; J. Wess and B. Zumino, Nucl. Phys. BT'0 (1974) 39. 13. U.LindstrSm and M. Ro~ek, Nucl. Phys. B222 (1983) 285. 14. M.Pernici and F.Riva, Nucl. Phys. B267 (1986) 61. 15. S. J. Gates, Jr., Nucl. Phys. B238 (1984) 349. 16. G. Sierra and P.K. Townsend, "An Introduclion to N = 2 Rigid Supersymmetry," in Supersymmetry and Supergravity 1983, ed. B. Milewski (World Scientific, Singapore, 1983), p. 396; B. de Wit, P.G. Lauwers, R. Philippe, Su S.-Q. and A. Van Proeyen, Phys. Lett. 134B (1984) 37; B. de Wit and A. Van Proeyen, Nucl. Phys. B245 (1984) 89. 17. B. de Wit, M.T.Grisaru and M.Rotek, Phys. Lett. 374B (1996) 297; A. De Giovanni, M.T.Grisaru, M.Ro~ek, R. yon Unge and D. Zanon, "The N=2 Super Yang-Mills LowEnergy Effective Action at Two Loops," hepth/9706013; Sergei V. Ketov, "On the Nextto-Leading Order Corrections to the Effective Action in N = 2 Gauge Theories," (Preprint No. DESY-97-103), (hep-th/9706079).

181

S.J. Gates Jr./Nuclear Physics B (Proc. Suppl.) 62A-C (1998) 171-181

Appendix: Explicit Matrices Throughout the text a number of matrices were used. Their explicit forms are given below. Ga 3 + G~_=

QRa

~

-

Q L~ -

V -

_

½g

Ga 1 -

_

Ulc~ u2a

i Ga 2

_

G a 4 + iGa_ 5

_

Clc~

Sla

$1c~

?-t3 a

c2a c3 a

s2a s3 a

t2(~ t3 a

ul. u2 a" u3 "~

dl"~ d2 ~" d3 "~

Cl"~ c2 ~ c3 "~

Sl"~ s2 ~" s3 "a

-~

v3 +

V1 + {V2 V 4 4- i V 5

Ga 4 Gs

Ga 6 + i G a 7

dla d2a d3 a

7r-

II -

~7~Ga s

+ 7~

t l "~ t2 ~" t3 "~

i Ga 5 / -~

a V ~

bla b2a 53 a

)

QRb --- - ( Q t ~ a ) t ,

bl"~) b2 ~" b3 "~

QLa

K°

v1_ 8

- V3 ÷

Vs + i V 7

v4_ivo )

V6 -

)

{V7

- V 8V~ ~

-

-(QL'a) t

,