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The one-loop effective potential in superspace
Nuclear Physics B214 (1983) 465-480 ~'3North-Holland Publishing Company
THE ONE-LOOP EFFECTIVE POTENTIAL IN SUPERSPACE M.T. GRISARUt California Institute of Technology, Pasadena, CA 91125
F. RIVA and D. ZANON lstituto di Fisica dell'Universita di Milano and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy
Received 26 August 1982
Using superfields and supergraphswe evaluate the superspace one-loop effectivepotential for several supersymmetricmodels: self-interactingchiral superfields, supersymmetricYang-Mills, and supersymmetric QED.
I. Introduction
In conventional field theory, the calculation of (one-loop) effective potentials has been reduced to a simple algorithm [1] which gives the result as a sum of separate contributions from spin-0, spin-½, and spin-1 fields. The algorithm usually requires that vector field contributions be evaluated in Landau gauge. In general, the effective potential is gauge dependent and some care is required to extract from it gauge independent, physical results. The algorithm can equally well be used in supersymmetric theories by working with component fields (including auxiliary fields) of scalar multiplets (A, A~ ~b, F, if) or vector multiplets in Wess-Zumino gauge (A~, k, D). If only scalar multiplets are present this is a perfectly satisfactory procedure [2]. However, if vector multiplets are present the result is not guaranteed to be supersymmetric. Working in Wess-Zumino gauge and also fixing the gauge of the component vector field breaks supersymmetry. Physical quantities will be gauge independent and supersymmetric, but such a result is not manifest [3] at the effective potential level, and can be obtained only by imposing the supersymmetry Ward identities. On the other hand, a superfield formalism is guaranteed to preserve supersymmetry and in general seems preferable for handling supersymmetric systems. * Fairchild Scholar. On leave of absence from Brandeis University, Waltham, MA 02254. Supported in part by NSF grant no. PHY 79-20801. 465
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M.T. Grisaru et al. / One-loop effective potential in superspace
In this paper we address ourselves to the problem of obtaining the one-loop effective potential in superspace for a variety of models: self-interacting scalar multiplets, Yang-Mills multiplets, and supersymmetric QED. We also consider some situations with broken supersymmetry, for those cases where the breaking can be expressed in a supersymmetric fashion, by coupling to external spurion superfields [4]. Our primary purpose is not to rederive component results, but rather obtain expressions which can be manipulated directly in terms of superfields. Unfortunately, we have not been able to obtain a general algorithm valid for arbitrary models of interacting chiral (scalar multiplet) and real gauge (vector multiplet) superfields in arbitrary representations of internal symmetry groups. This may be due to our low-brow (diagrammatic) methods for evaluating one-loop contributions. Undoubtedly a more efficient way is based on direct functional integration and operatorial techniques in superspace, but such techniques are not yet sufficiently developed. Our paper is organised as follows: in sect. 2 we consider contributions to the effective potential from chiral multiplets and rederive the results of Huq [2]. We also discuss some cases with broken supersymmetry. In sect. 3 we discuss supersymmetric QED, while in sect. 4 we treat a Yang-Mills multiplet. In the appendices we present our notation and conventions, some calculational details, and an alternative derivation of the results of sect. 4.
2. Chirai loop contributions We compute the effective potential by evaluating the one-loop effective action and setting to zero external momenta, as well as the spinor and vector components of the external superfields. These will therefore depend only on the spin-zero physical and auxiliary fields. At zero momentum this is a supersymmetric procedure. We begin with a simple case of a chiral superfield 99 interacting with a classical (external) real gauge superfield V = V~T~where the T ' s are generators of some gauge group and the chiral superfield carries a representation of the group. As we will see, the computation of the effective potential for other situations can be reduced to this one. The action is
S =fd4xd40~eV~.
(2.1)
It is invariant under gauge transformation of the external field
e v __, ei.7,eVe -iA,
(2.2)
where A is chiral. Such transformations can be absorbed by a field redefinition with unit jacobian qo' = e - iaqo, ~p' = Upei•. (2.3) Therefore the effective potential will be gauge invariant, and in particular can be evaluated in Wess-Zumino gauge. Setting external momenta, spinor and vector fields
M.T. Grisaruet al. / One-loopeffectivepotential in superspace tO zero, we have in this gauge V = F = In A, where
a =
0202D, and
467
the one-loop effective potential is
DUpe/d'xd'° ~(1 +02g2D)w.
(2.4)
G r a p h s contributing to A are shown in fig. 1. The D-algebra in the loop integral is trivial. Since 0 3 = 0 3 = 0, all the D 2,/if2 operators must go on the external lines except for one D2/~ 2 factor needed for the P-loop integral, and we get (D202 =/~2/~2 = 1 at zero external m o m e n t u m )
Here, and in what follows, the trace includes integration over the loop m o m e n t u m with d 4 p / ( 2 ~ ' ) 4. We can rewrite F in a manifestly gauge-invariant form in terms of the connection F,~ = ie - VD~eV and the field strength W,~ = / ) 2 ( e - VD~eV), using the definition of the D - c o m p o n e n t D = D~W~lo=o:
F = - tr f d4O OZUe l n ( 1 - D ~ W J p 2 )
= _itr f d40
F°Wo 2 ,m /{1 1
O~ ) . p2
(2.6)
W e note that in the first form of this equation, with 0 2 0 2 = 34(8) one simply has to evaluate the rest of the integrand at O = 0 = 0. We consider n o w the most general renormalisable action for self-interacting chiral superfields:
S [~, ¢p]= fd4x
d40 1
~
D
Fig. 1. Chiral superfield loop with gauge superfield background.
M.T. Grisaru et al. / One-loop effective potential in superspace
468
where a = 1,2 .... n, and m~b, ~kabc, a r e symmetric in their indices. In order to perform one-loop calculations, we make the standard splitting q~ = f p Q - t - f p e x t and look at the terms quadratic in the quantum fields (dropping the Q on ~o):
s2=fd4xd4Off~-fd4xd20(½mab~aepb +½X~b<~]xt~b~<)+h.c..
(2.8)
~ext is a chiral superfield with spinor field and external momentum set to zero:
(2.9)
q9ext = a + 6 0 2 .
Defining Xab ~
m~b + X~b<(a<+ bc02)
-Aa~ + FabO2,
(2.10)
we obtain
s2: f d4xd40~p~-½ f d4xd20~Xeg + h.c..
(2.11)
(For notational simplicity we do not write explicit indices, but they are to be understood.) Using the usual Feynman rules for chiral superfields (see (A.8) and (A.9) in the appendix), we obtain a general contribution to the one-loop effective potential with n X and n ~ external lines. The integration by parts rules allow us to move the D 2,/if2 factors as shown in fig. 2a. Clearly we can perform the 0 integration on the internal lines 1-2, 3-4, etc., and the graphical result is shown in fig. 2b. Therefore, we can equivalently consider the following quadratic action (p2 _= _ rq)
s~=Sd4xd4O~7~(l.-I-~)¢P=Sd4xd40~evB~,
(2.12)
2
~x
X
(2a)
(2b)
Fig. 2. Chiral superfield loop with chiral superfield background.
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M.T. Grisaru et al. / One-loop effective potential in superspace
where VB=ln 1 + 7~ )
(2.13)
'
is an effective external real superfield with spinors, vector field and momenta set to zero. We can use now (2.6) to evaluate the effective potential. To make contact with component calculations we consider first the Wess-Zumino action (one chiral multiplet); then X is a single superfield. In this case (note that acting on external fields at zero momentum (D,/if)-- 0) we have D~W~ D2/ff2VB. We must evaluate at 0 = t~ = 0 the expression =
F=-trln
1
p2
In 1+
,
(2.14)
with X = A + 02F. Observing that at 0 = 0 (or equivalently at zero external momentum) D 2 = _ ¼@2 and that ~ 0 2f(0 2) I0= o = f'(0) we obtain immediately
F= -trln(1
ffF ) (p2 + ZA)2 '
(2.15)
Returning to the general case we have
D a W~=D a--2 D (e -- V"D~eV")
= 1+ 7
71+77
D2x
(2.16)
Thus
I'= -trfd4OO2021n[1 __(p2 =
-trfd4OO202{ln[(P 2+ X:~)(P2+ ~'X)-- (P 2 q- x x ) D 2 x ( p - l n ( p z + XX)(P 2 + XX)),
2q- XX.) ID2x] (2.17)
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M.T. Grisaruet al. / One-loopeffectivepotentialin superspace
i.e., evaluating at 0 = 0, F = - t r { l n [ ( p 2 + AA-)(p 2 + A - A ) - (p2 +
A.~)ff(p2 + A~)-lF]
-ln(p2 + A.,~)(p2 +.,4A)) = _ tr(ln(p2 +
xz)_ln(p2 + y2)),
(2.18)
where we have defined xE_(AA-F
-F) XA
y2_(A( '
0 ) XA
(2.19) "
This is Huq's result [2]; the X and Y contributions are due to bosons and fermions, respectively. The above result can readily be applied to the computation of the one-loop effective potential of a theory with explicit soft breaking of supersymmetry [4]. A superfield calculation is still possible since the breaking term can be written, in a supersymmetric fashion, as a coupling of the quantum fields to an external superfield with a suitable, fixed value. There are essentially two possibilities of interest: Coupling of the form f d'x d40 ~Ucpto an external real superfield U = ~20202, which gives (additional) equal mass terms to the scalars and pseudoscalars but not to the fermion of a chiral multiplet, and coupling of the form fd4xd20 rlep2 to an external chiral superfield ,/=/~202 which gives mass terms opposite in sign to the scalars and pseudoscalars. Obviously this second case is contained already in our general discussion, as an additional contribution to X, so we need only consider the first case. Thus, we discuss the action
s2 =f d'xd"O( w + Cvw)-½ f d'xd2Owxw +
h.c.
(2.20)
It is convenient to introduce two sets of chiral multiplets fPl and fP2, define _ ~2 02, and consider instead the following action:
S~ = f d4xd4O( ~p,~&+ ~2qh)+ f d4xd2O( ~l~'q~2- ½q~lXcP,)+ h.c..
(2.21)
At one loop the action in (2.21) is completely equivalent to (2.20), as can be seen by examining, for example, the corresponding Feynman graphs. With the redefinitions
(q~'q~2) --> q~' (x_XT °
~ X,
(2.22)
M.T. Grisaru et al. / One-loop effective potential in superspace
471
we are again back to (2.11). The effective potential is thus given by (2.18), where X has to be understood as in (2.22). The explicit expression in components is F = - t r ( l n [ ( p 2 + AA--/~2)(p 2 +A-A - #2)
-Fff]
-ln(p2 + AA-)(p2 + A-A)).
(2.23)
It is interesting to observe what effect the addition of the supersymmetry breaking terms has on the divergence structure of the effective potential. In the absence of either U or 77breaking terms, inserting (2.19) into (2.18) and expanding the integrand in powers of p - 2 we have F= -tr
(X
2
--
y 2 ) + p4 (
--
(2.24)
[ p4
The cancellation of the quadratic divergences is due to tr X 2 = tr y2. The logarithmic divergence proportional to Fff reflects the need for wave function renormalization in this model. Adding the ~ symmetry breaking term amounts to shifting F ~ F + #2. Therefore, in the above formula we will obtain now not only the Fff divergence, but also a divergence proportional to F + ff (as well as a field independent divergence). Adding the U term however gives tr X 2 - tr y2 = 2 , u 2 which leads to a field-independent quadratic divergence as well as a new logarithmic divergence. All these results are consistent with the general superfield analysis of soft breaking [4].
3. SupersymmetrieQED The supersymmetric extension of QED [5] is described by the following action
s= fd'xd2OW"W + fd4xd40[ffeVS + Te-VTl,
(3.1)
where S and T are two chiral superfields and W ~ = D2D~V, V being an abelian vector multiplet superfield. We can write (3.1) in a more compact form, by redefining •
Thus
S= fd4xd2OW"W. + fd4xd40~pe%p.
(3.3)
M.T. Grisaruet al. / One-loopeffectivepotentialin superspace
472
Adding to this action the supersymmetric gauge fixing performing the quantum-classical splitting cp ~ c p + X,
term
fdaxd40VD2D2V and
V ~ V+ U,
(3.4)
we obtain the action quadratic in the quantum fields cp and V
S 2 =fd4xd40[V[]V+ ~pe%p+ ~Ve%p + ~VeVx + ½~eVV2x]
= fdax d40 [V[]V + C~e%+ ½~eVx V2 + ~o3e%pV+ Upo3eVxV].
(3.5)
Now, completing the square in V, and shifting V by a real superfield 1
V' = V+ 2
1
~ + ½~eUx
[ x ° 3 e ~ + ~°3e~x]'
(3.6)
we obtain
s2= fd"xd40{V'[•+ eUx]V' +~p[eV - 1 (03eVx)(xeV03)
1 (xo3eVcP)2
l(CPo3eUx) 2 )
4 []+l~eV X
4 []+½~eVx
" (3.7)
The one-loop contribution from the real superfield V' is clearly zero, since no D's are present in the loop. Thus, in order to evaluate the one-loop effective potential, we need only look at the following action: (3.8) where VB and ~ are 2 by 2 matrices defined by VB ln[e U _ 1 ( 03eVx)(2e%3 ) 2 [] + ½~e~x
(3.9)
;~= ½/~2 (°3eUx)(xe%3) [] + ½£eVx
(3.10)
t
and
M.T. Grisaru et al. / One-loop effective potential in superspace
473
Essentially we have now the problem of computing the effective potential for a chiral loop with external vector multiplet and scalar multiplet lines. We can replace [] ~ _ p 2 and perform a field redefinition ~ --* eAcp,
Cp--* cpeA ,
(3.11)
such that eA-eVBea = e °292°.
(3.12)
~ e a x ea --=X"
(3.13)
This redefines
At this point, our quadratic action is exactly of the same form as the action in (2.20) and therefore the one-loop effective potential can be calculated. However, we have not been able to obtain a simple closed form.
4. Supersymmetric Yang-Mills theory We will consider now supersymmetric Yang-Mills theory, in the absence of chiral superfields. In a component calculation, in Wess-Zumino gauge, the effective potential can only depend on the auxiliary field D. On the other hand, D does not couple to A a or ~, nor to the Faddeev-Popov ghosts of the vector field and therefore one would expect the effective potential to be zero. However, this method of calculation is clearly not supersymmetric, with the vector gauge-fixing term breaking supersymmetry. As we will see, a superfield calculation gives a non-zero result for the effective potential. While one does not expect this result to have any direct physical significance, it is a warning that one cannot trust Wess-Zumino gauge calculations. We can obtain significant simplifications by calculating the effective potential in the background field formalism. The D-algebra is much simpler than when using conventional methods. Furthermore, since we can choose for the quantum gauge fields a background covariant gauge fixing term, the effective potential will be gauge invariant. This allows us to do the actual computation with the background field in a fixed, suitable gauge. The new feature of the superfield background-quantum splitting is that it involves the exponentials of the fields [6], that is: e v ~ ee,eVeffB,
eVB = ee,effB.
(4.1)
Choosing a suitable background covariantized gauge fixing term, we obtain the Yang-Mills action in a manifestly background covariant form. Up to quadratic
474
M.T. Grisaru et al. / One-loop effective potential in superspace
terms in the quantum fields we have [6] $2= tr f d 4 x d 4 0 [ V ( - ' X7~V a - W~V~ + W ~ V a ) V + beV, b + g'eV, c + c'eVBg] (4.2) where V describes the quantum Yang-Mills multiplet, c, g, c,, g, are the FaddeevPopov ghosts, b and 6 are the Nielsen-Kallosh ghosts. VB is the background Yang-Mills field which also appears in the field strength W~ and the covariant derivatives V~, XT~,Va (see (A.6)). This action is manifestly invariant under gauge transformations of the background field. The one-loop effective potential will be given by the sum of contributions from chiral ghost loops, and the V-loop. The contribution from the chiral fields has already been discussed. Therefore, we will concentrate here on the vector loop contribution. We use the fact that, as mentioned above, the effective potential is gauge invariant. We then perform the calculation with VB in Wess-Zumino gauge, and covariantize the result at the end. In this gauge, the background field is simply VB = 8 2 / ~ 2 D . After some algebra, we obtain for the quadratic part of the vector action the following expression:
S~=trfd4xd40(½V[-fq+
D(OaOa-O
-&0&)]V)
(4.3)
and the contribution to the effective potential is
I]
[ D
F v = - ½ t r l n 1 + ~-~(O~a~- OaO~
= - ½ t r E1 ( - ) 'n+ '
D
" ( 0 ~0~ - 0 - ~ -Oa) n .
(4.4)
" ~D. - 0 a-Da); one could do the D-algebra on the (Note that 0 ~ 0 ~ - 0a0 a --- -2~(0 graphs but there is little advantage in doing so.) We now have n
k
=
11
a
n-k
(0 Oa) ,
(4.5 t
k=0
(0o o)
= 0o0o
2 ~- l _ 1 0202 .
(4.6)
The trace operation on the 8-variables in (4.4) requires at least two factors of 8~ and
475
M.T. Grisaru et al. / One-loop effective potential in superspace
two factors of Oa; thus, inserting the 0 2 coefficient of (4.5) (and the corresponding one for the (OaOa) ~ quantity) into (4.4), we obtain:
FV=_½tr~ (-)m+l(D] 4+m m=O 4 + m
=-
+(7) 1 - D2
\~]
m+4
(k+2)(--)k(2k+l--1)(2m-k+l--1)
k=O~
-¼1n(1-
D2
(4.7)
The detailed evaluation of the series in (4.7) is given in the appendix. The contribution from the chiral ghosts is just three times that for a single chiral field (with a minus sign because of statistics). We can therefore write the complete one-loop effective potential, in a manifestly supersymmetric and background gauge invariant manner, as
r=-itrfd4Or~w~( [ ( (•aWa)2
o2
1 In 1
(v p4wo)
-31n(1
p2
(
- Iln 1 - 4
2]
(v p4wo)
(4.8)
Since the ghosts are in a real representation of the gauge group, the last term can be rewritten as
( °2)
~ln 1
(V W~)
p4
(4.9)
This last result can also be derived using improved Feynman rules for covariantly chiral superfields (see appendix C).
5. Conclusions
We have obtained superfield expressions for the one-loop effective potential for several supersymmetric systems. The results are manifestly supersymmetric but, for example in the case of QED, seem more complicated than the corresponding component results. We suspect that a method of calculating which uses functional methods in superspace, rather than diagrammatic techniques, would give simpler results. In particular it might allow obtaining a general algorithm for the one-loop potential, similar to that for component calculations, which would apply directly to any supersymmetric model.
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M.T. Grisaru et al. / One-loop effective potential in superspace
We draw attention to the results for supersymmetric Yang-Mills. In a component calculation, in Wess-Zumino gauge, we would obtain a zero result. In a supersymmetric gauge, we find a non-zero result, depending on the auxiliary field D. This of course is a reflection of the gauge dependence of the effective potential, but it should also serve as a warning that Wess-Zumino gauge results for the effective potential in supersymmetric theories should not be trusted.
Appendix A We present here a brief description of our notation. We use two component spinors and work with euclidean space-time coordinates. The spinor covariant derivatives are defined as .-2 D,~= ½i( O,, + tO
a,~a),
/~=-½i(ea+iO'~a,~a),
(A.1)
where 0~ is the spinorial derivative and we use the abbreviation 0~a--o~,~0a; o~a = (1, o ) ~ with o the usual Pauli matrices. The fundamental anticommutation relations of the D's are
(D,~,Da}=iO. a ,
( D,~, DO) = ( / ) a , / ) ~ ) = 0 •
(A.2)
The conventions for 0-integration are
fd2002
=fd2002=
1.
(A.3)
Therefore, when also integrated over the space-time variable x the integral over 0 is equivalent to a covariant derivative:
f d20 ~ D 2 ' fdz# /ffz.
(A.4)
For a Yang-Mills superfield, the covariant derivatives are defined in chiral representation, as
VA=DA--iFA,
A-(a,&,a),
(a.5) (A.6)
The field strength is given by W = [ ~-d, (V-a,
XT,~)]=D2(e-VD,~eV),
(A.7)
M.T. Grisaruet al. / One-loopeffectivepotential in superspace
477
The massless rp~ propagator and ~3 vertex are given by [6] 3ab~22 34(01 -- 02),
(A.8)
f d404 D234( O, - 04 ) D234( 02 - 04) 34(03 - 04),
(a.9)
(/if2 __, D 2 for a ~p3 vertex). An additional factor D 2 (/if2) must be removed from the vertex if one of the lines is external. The Yang-Mills propagator is - 1/p234(01 - 02). All vertices are integrated with d40, and the D's are integrated by parts until only one line has any D's acting on it. We then use [6] D2DZ3"(01 - 02)1a,=02 = 1, (A.10) while loops with less than two Ds and two/~s give zero.
Appendix B We give here some details of the calculation of the series in (4.7). We have to evaluate I=
~
(--)mxm ~
( m + 3)!(--) k
k=0 (k + 2 ~ ( ( m Z ; +
m=0
= ~
(__)mxm+k
re,k=0
(2 k + l - 1)(2 ' ~ - k + l - l) 2) !
(re+k+3)!
(2k+l
1)(2m+l
1) "
(B.1)
( k + 2)!(m + 2)!
We can express (m + k + 3)! using the integral representation for the factorial n!= With the definition ~
G(z)=
/o d u e Zm
Y'~ ( m + 2 ) !
m=O
Uu~.
(B.2)
1
z z (ez-l-z)'
(B.3)
we have oo
I = f o ° ~ d u e - " u 3 y~ ( - ) " re,k=0 =fo°~due
(2 k+m+2 -- 2 k+l -- 2 ''+l + 1) (ux) m+k (k + 2)](m + 2)!
Uu3 [ 4 G ( 2 x u ) G ( - 2xu) - 2 G ( 2 x u ) G ( - xu) - 2G( - 2 x u ) G ( x u ) + G ( x u ) G ( - x u ) ]
_
1_._~[ ~ d u e - " ( 6 + e R x U + e _ 2 : , U _ 4 e X , , _ 4 e _ X , , ) . 4x 4 Jo u
(B.4)
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M.T. Grisaru et al. / One-loop effective potential in superspace
We can now use the following relations:
f0 °
due,,_ le_p, - F p,~ (a)
p
- a
= 1-alnp+O(a
(B.5)
2)
and write 1
o~
I=lim~..4f duu"-le-U(6+e2X"+e-2XU-4eX~-4e-X") ~-~o 4x ao = lim
r(a---~)[6 + (1 - 2 x ) - "
+(1 + 2 x ) - ~ - 4 ( 1 -
= lira
r(a----~)[-aln(1
+4aln(1 - x:)]
a--,0
a~0
4x 4
4X 4
-4x:)
x)-~-4(1
+x)-"]
= ~4 [In(1 - x 2) - ¼1n(1 - 4xZ)].
(B.6)
Appendix C
In a recent paper [7] new covariant Feynman rules for chiral fields in a Yang-Mills background were obtained, for those cases where the chiral fields are in a real representation of the group. The complete one-loop contribution of a covariantly chiral superfield is given by the following Feynman rules: propagators:
~2 ~ 4 ( 0 -- 0 ' ) .
one vertex:
/52( 72 - D 2) --/52A
other vertices:
O + - [] = ~:ra V a -- [] "J¢- W a ~Ta -- 1 ( ~TaWCt ) ~ j~
The vertices can be written explicitly in the form A - (A~D~ + A ) ,
A " - - 2 i F ~,
B = B~D~ + B,
A = - r " r . - iD~F~,
(C.1)
B ~ =- 2 W ~ ,
B - D '~W,~ - 2iW'~F,, + 2D"F'~D~F,~ - 4 D a F " O , ~ - 20"a--DaF,"
(C.2)
We observe that ( A D 2 )" = A n " -,/~2.
(C.3)
M.T. Grisaru et al. / One-loop effective potential in superspace
479
r+a
Fig. 3. Yang-Millssuperfieldwith Yang-Millsbackground. A one-loop diagram is shown in fig. 3. There is only one/~2 in the loop and this means that all the D's but two must be integrated by parts outside the loop. The one-loop contribution to the effective potential can be written as
oc
F = tr~l
1 B n - lA]ff2
2n
(C.4)
p2.
We need to evaluate the coefficient of the D 2 term in B" - IA, and this can be done easily by induction. Setting
(C.5)
B " - tA = a . D e + b~.D,~ + c,,,
we obtain
a.+, = (B'~D,~ + B ) a . - ~ B' ~'b..,
a l=O,
b
b =A °,
c.+,=(B'~n.+B)c.,
c,=A.
(C.6)
We obtain a recursion relation for the a. coefficients:
a.+ 1 = ( B ' ~ D , ~ + B ) a . - I B ' ~ B , ~ B " - 2 A ,
a, = 0 .
(C.7)
Using (C.1) and (C.2) with external V, we can equivalently write
a.+, : , . W " [ a . -
2i(D~W~)"-VWVFv],
(C.8)
a.+ 1 - - i ( D'~W,~)n-IW#F# = D,~W'~[a.- i ( D # W ~ ) ~-2WVFv] = (D,,W'~)" ' ( 2 i W ~ F ~ - iW~F~),
(C.9)
with the final result
a.+ 1 = " [ t ( V a W,~) n - - I + ,(. D,~W a ) n-- ']
Wfli~fl,
lq>O.
(C.lO)
M.T. Grisaru et al. / One-loop effective potential in superspace
480
The effective potential is then F=
f tr
oo 1
d40EI 2n
an p2n
DYW - 2
(DaW~)
k +--~j+ln
=-½itrfd4OW~F, I(DoW~)2 In ( ( D ~ W 'P') 2 ) 1 which is equivalent to (2.6) for real representations. References [1] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887; R. Jackiw, Phys. Rev. D9 (1974) 1686 [2] M. Huq, Phys. Rev. DI6 (1977) 1733 [3] W. Lang, Nucl. Phys. BII4 (1976) 123 [4] D. Capper, J. Phys. G Nucl. Phys. 3 (1977) 731; L. Girardello and M.T. Grisaru, Nucl. Phys. B194 (1982) 65 [5] P. Fayet and S. Ferrara, Phys. Rep. 32c (1977) 1 [6] M.T. Grisaru, M. Ro~ek, and W. Siegel, Nucl. Phys. B159 (1979) 149 [7] M. T. Grisaru and W. Siegel, Nucl. Phys. B201 (1982) 292
1 ,
(C.I1)
THE ONE-LOOP EFFECTIVE POTENTIAL IN SUPERSPACE M.T. GRISARUt California Institute of Technology, Pasadena, CA 91125
F. RIVA and D. ZANON lstituto di Fisica dell'Universita di Milano and Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy
Received 26 August 1982
Using superfields and supergraphswe evaluate the superspace one-loop effectivepotential for several supersymmetricmodels: self-interactingchiral superfields, supersymmetricYang-Mills, and supersymmetric QED.
I. Introduction
In conventional field theory, the calculation of (one-loop) effective potentials has been reduced to a simple algorithm [1] which gives the result as a sum of separate contributions from spin-0, spin-½, and spin-1 fields. The algorithm usually requires that vector field contributions be evaluated in Landau gauge. In general, the effective potential is gauge dependent and some care is required to extract from it gauge independent, physical results. The algorithm can equally well be used in supersymmetric theories by working with component fields (including auxiliary fields) of scalar multiplets (A, A~ ~b, F, if) or vector multiplets in Wess-Zumino gauge (A~, k, D). If only scalar multiplets are present this is a perfectly satisfactory procedure [2]. However, if vector multiplets are present the result is not guaranteed to be supersymmetric. Working in Wess-Zumino gauge and also fixing the gauge of the component vector field breaks supersymmetry. Physical quantities will be gauge independent and supersymmetric, but such a result is not manifest [3] at the effective potential level, and can be obtained only by imposing the supersymmetry Ward identities. On the other hand, a superfield formalism is guaranteed to preserve supersymmetry and in general seems preferable for handling supersymmetric systems. * Fairchild Scholar. On leave of absence from Brandeis University, Waltham, MA 02254. Supported in part by NSF grant no. PHY 79-20801. 465
466
M.T. Grisaru et al. / One-loop effective potential in superspace
In this paper we address ourselves to the problem of obtaining the one-loop effective potential in superspace for a variety of models: self-interacting scalar multiplets, Yang-Mills multiplets, and supersymmetric QED. We also consider some situations with broken supersymmetry, for those cases where the breaking can be expressed in a supersymmetric fashion, by coupling to external spurion superfields [4]. Our primary purpose is not to rederive component results, but rather obtain expressions which can be manipulated directly in terms of superfields. Unfortunately, we have not been able to obtain a general algorithm valid for arbitrary models of interacting chiral (scalar multiplet) and real gauge (vector multiplet) superfields in arbitrary representations of internal symmetry groups. This may be due to our low-brow (diagrammatic) methods for evaluating one-loop contributions. Undoubtedly a more efficient way is based on direct functional integration and operatorial techniques in superspace, but such techniques are not yet sufficiently developed. Our paper is organised as follows: in sect. 2 we consider contributions to the effective potential from chiral multiplets and rederive the results of Huq [2]. We also discuss some cases with broken supersymmetry. In sect. 3 we discuss supersymmetric QED, while in sect. 4 we treat a Yang-Mills multiplet. In the appendices we present our notation and conventions, some calculational details, and an alternative derivation of the results of sect. 4.
2. Chirai loop contributions We compute the effective potential by evaluating the one-loop effective action and setting to zero external momenta, as well as the spinor and vector components of the external superfields. These will therefore depend only on the spin-zero physical and auxiliary fields. At zero momentum this is a supersymmetric procedure. We begin with a simple case of a chiral superfield 99 interacting with a classical (external) real gauge superfield V = V~T~where the T ' s are generators of some gauge group and the chiral superfield carries a representation of the group. As we will see, the computation of the effective potential for other situations can be reduced to this one. The action is
S =fd4xd40~eV~.
(2.1)
It is invariant under gauge transformation of the external field
e v __, ei.7,eVe -iA,
(2.2)
where A is chiral. Such transformations can be absorbed by a field redefinition with unit jacobian qo' = e - iaqo, ~p' = Upei•. (2.3) Therefore the effective potential will be gauge invariant, and in particular can be evaluated in Wess-Zumino gauge. Setting external momenta, spinor and vector fields
M.T. Grisaruet al. / One-loopeffectivepotential in superspace tO zero, we have in this gauge V = F = In A, where
a =
0202D, and
467
the one-loop effective potential is
DUpe/d'xd'° ~(1 +02g2D)w.
(2.4)
G r a p h s contributing to A are shown in fig. 1. The D-algebra in the loop integral is trivial. Since 0 3 = 0 3 = 0, all the D 2,/if2 operators must go on the external lines except for one D2/~ 2 factor needed for the P-loop integral, and we get (D202 =/~2/~2 = 1 at zero external m o m e n t u m )
Here, and in what follows, the trace includes integration over the loop m o m e n t u m with d 4 p / ( 2 ~ ' ) 4. We can rewrite F in a manifestly gauge-invariant form in terms of the connection F,~ = ie - VD~eV and the field strength W,~ = / ) 2 ( e - VD~eV), using the definition of the D - c o m p o n e n t D = D~W~lo=o:
F = - tr f d4O OZUe l n ( 1 - D ~ W J p 2 )
= _itr f d40
F°Wo 2 ,m /{1 1
O~ ) . p2
(2.6)
W e note that in the first form of this equation, with 0 2 0 2 = 34(8) one simply has to evaluate the rest of the integrand at O = 0 = 0. We consider n o w the most general renormalisable action for self-interacting chiral superfields:
S [~, ¢p]= fd4x
d40 1
~
D
Fig. 1. Chiral superfield loop with gauge superfield background.
M.T. Grisaru et al. / One-loop effective potential in superspace
468
where a = 1,2 .... n, and m~b, ~kabc, a r e symmetric in their indices. In order to perform one-loop calculations, we make the standard splitting q~ = f p Q - t - f p e x t and look at the terms quadratic in the quantum fields (dropping the Q on ~o):
s2=fd4xd4Off~-fd4xd20(½mab~aepb +½X~b<~]xt~b~<)+h.c..
(2.8)
~ext is a chiral superfield with spinor field and external momentum set to zero:
(2.9)
q9ext = a + 6 0 2 .
Defining Xab ~
m~b + X~b<(a<+ bc02)
-Aa~ + FabO2,
(2.10)
we obtain
s2: f d4xd40~p~-½ f d4xd20~Xeg + h.c..
(2.11)
(For notational simplicity we do not write explicit indices, but they are to be understood.) Using the usual Feynman rules for chiral superfields (see (A.8) and (A.9) in the appendix), we obtain a general contribution to the one-loop effective potential with n X and n ~ external lines. The integration by parts rules allow us to move the D 2,/if2 factors as shown in fig. 2a. Clearly we can perform the 0 integration on the internal lines 1-2, 3-4, etc., and the graphical result is shown in fig. 2b. Therefore, we can equivalently consider the following quadratic action (p2 _= _ rq)
s~=Sd4xd4O~7~(l.-I-~)¢P=Sd4xd40~evB~,
(2.12)
2
~x
X
(2a)
(2b)
Fig. 2. Chiral superfield loop with chiral superfield background.
469
M.T. Grisaru et al. / One-loop effective potential in superspace
where VB=ln 1 + 7~ )
(2.13)
'
is an effective external real superfield with spinors, vector field and momenta set to zero. We can use now (2.6) to evaluate the effective potential. To make contact with component calculations we consider first the Wess-Zumino action (one chiral multiplet); then X is a single superfield. In this case (note that acting on external fields at zero momentum (D,/if)-- 0) we have D~W~ D2/ff2VB. We must evaluate at 0 = t~ = 0 the expression =
F=-trln
1
p2
In 1+
,
(2.14)
with X = A + 02F. Observing that at 0 = 0 (or equivalently at zero external momentum) D 2 = _ ¼@2 and that ~ 0 2f(0 2) I0= o = f'(0) we obtain immediately
F= -trln(1
ffF ) (p2 + ZA)2 '
(2.15)
Returning to the general case we have
D a W~=D a--2 D (e -- V"D~eV")
= 1+ 7
71+77
D2x
(2.16)
Thus
I'= -trfd4OO2021n[1 __(p2 =
-trfd4OO202{ln[(P 2+ X:~)(P2+ ~'X)-- (P 2 q- x x ) D 2 x ( p - l n ( p z + XX)(P 2 + XX)),
2q- XX.) ID2x] (2.17)
470
M.T. Grisaruet al. / One-loopeffectivepotentialin superspace
i.e., evaluating at 0 = 0, F = - t r { l n [ ( p 2 + AA-)(p 2 + A - A ) - (p2 +
A.~)ff(p2 + A~)-lF]
-ln(p2 + A.,~)(p2 +.,4A)) = _ tr(ln(p2 +
xz)_ln(p2 + y2)),
(2.18)
where we have defined xE_(AA-F
-F) XA
y2_(A( '
0 ) XA
(2.19) "
This is Huq's result [2]; the X and Y contributions are due to bosons and fermions, respectively. The above result can readily be applied to the computation of the one-loop effective potential of a theory with explicit soft breaking of supersymmetry [4]. A superfield calculation is still possible since the breaking term can be written, in a supersymmetric fashion, as a coupling of the quantum fields to an external superfield with a suitable, fixed value. There are essentially two possibilities of interest: Coupling of the form f d'x d40 ~Ucpto an external real superfield U = ~20202, which gives (additional) equal mass terms to the scalars and pseudoscalars but not to the fermion of a chiral multiplet, and coupling of the form fd4xd20 rlep2 to an external chiral superfield ,/=/~202 which gives mass terms opposite in sign to the scalars and pseudoscalars. Obviously this second case is contained already in our general discussion, as an additional contribution to X, so we need only consider the first case. Thus, we discuss the action
s2 =f d'xd"O( w + Cvw)-½ f d'xd2Owxw +
h.c.
(2.20)
It is convenient to introduce two sets of chiral multiplets fPl and fP2, define _ ~2 02, and consider instead the following action:
S~ = f d4xd4O( ~p,~&+ ~2qh)+ f d4xd2O( ~l~'q~2- ½q~lXcP,)+ h.c..
(2.21)
At one loop the action in (2.21) is completely equivalent to (2.20), as can be seen by examining, for example, the corresponding Feynman graphs. With the redefinitions
(q~'q~2) --> q~' (x_XT °
~ X,
(2.22)
M.T. Grisaru et al. / One-loop effective potential in superspace
471
we are again back to (2.11). The effective potential is thus given by (2.18), where X has to be understood as in (2.22). The explicit expression in components is F = - t r ( l n [ ( p 2 + AA--/~2)(p 2 +A-A - #2)
-Fff]
-ln(p2 + AA-)(p2 + A-A)).
(2.23)
It is interesting to observe what effect the addition of the supersymmetry breaking terms has on the divergence structure of the effective potential. In the absence of either U or 77breaking terms, inserting (2.19) into (2.18) and expanding the integrand in powers of p - 2 we have F= -tr
(X
2
--
y 2 ) + p4 (
--
(2.24)
[ p4
The cancellation of the quadratic divergences is due to tr X 2 = tr y2. The logarithmic divergence proportional to Fff reflects the need for wave function renormalization in this model. Adding the ~ symmetry breaking term amounts to shifting F ~ F + #2. Therefore, in the above formula we will obtain now not only the Fff divergence, but also a divergence proportional to F + ff (as well as a field independent divergence). Adding the U term however gives tr X 2 - tr y2 = 2 , u 2 which leads to a field-independent quadratic divergence as well as a new logarithmic divergence. All these results are consistent with the general superfield analysis of soft breaking [4].
3. SupersymmetrieQED The supersymmetric extension of QED [5] is described by the following action
s= fd'xd2OW"W + fd4xd40[ffeVS + Te-VTl,
(3.1)
where S and T are two chiral superfields and W ~ = D2D~V, V being an abelian vector multiplet superfield. We can write (3.1) in a more compact form, by redefining •
Thus
S= fd4xd2OW"W. + fd4xd40~pe%p.
(3.3)
M.T. Grisaruet al. / One-loopeffectivepotentialin superspace
472
Adding to this action the supersymmetric gauge fixing performing the quantum-classical splitting cp ~ c p + X,
term
fdaxd40VD2D2V and
V ~ V+ U,
(3.4)
we obtain the action quadratic in the quantum fields cp and V
S 2 =fd4xd40[V[]V+ ~pe%p+ ~Ve%p + ~VeVx + ½~eVV2x]
= fdax d40 [V[]V + C~e%+ ½~eVx V2 + ~o3e%pV+ Upo3eVxV].
(3.5)
Now, completing the square in V, and shifting V by a real superfield 1
V' = V+ 2
1
~ + ½~eUx
[ x ° 3 e ~ + ~°3e~x]'
(3.6)
we obtain
s2= fd"xd40{V'[•+ eUx]V' +~p[eV - 1 (03eVx)(xeV03)
1 (xo3eVcP)2
l(CPo3eUx) 2 )
4 []+l~eV X
4 []+½~eVx
" (3.7)
The one-loop contribution from the real superfield V' is clearly zero, since no D's are present in the loop. Thus, in order to evaluate the one-loop effective potential, we need only look at the following action: (3.8) where VB and ~ are 2 by 2 matrices defined by VB ln[e U _ 1 ( 03eVx)(2e%3 ) 2 [] + ½~e~x
(3.9)
;~= ½/~2 (°3eUx)(xe%3) [] + ½£eVx
(3.10)
t
and
M.T. Grisaru et al. / One-loop effective potential in superspace
473
Essentially we have now the problem of computing the effective potential for a chiral loop with external vector multiplet and scalar multiplet lines. We can replace [] ~ _ p 2 and perform a field redefinition ~ --* eAcp,
Cp--* cpeA ,
(3.11)
such that eA-eVBea = e °292°.
(3.12)
~ e a x ea --=X"
(3.13)
This redefines
At this point, our quadratic action is exactly of the same form as the action in (2.20) and therefore the one-loop effective potential can be calculated. However, we have not been able to obtain a simple closed form.
4. Supersymmetric Yang-Mills theory We will consider now supersymmetric Yang-Mills theory, in the absence of chiral superfields. In a component calculation, in Wess-Zumino gauge, the effective potential can only depend on the auxiliary field D. On the other hand, D does not couple to A a or ~, nor to the Faddeev-Popov ghosts of the vector field and therefore one would expect the effective potential to be zero. However, this method of calculation is clearly not supersymmetric, with the vector gauge-fixing term breaking supersymmetry. As we will see, a superfield calculation gives a non-zero result for the effective potential. While one does not expect this result to have any direct physical significance, it is a warning that one cannot trust Wess-Zumino gauge calculations. We can obtain significant simplifications by calculating the effective potential in the background field formalism. The D-algebra is much simpler than when using conventional methods. Furthermore, since we can choose for the quantum gauge fields a background covariant gauge fixing term, the effective potential will be gauge invariant. This allows us to do the actual computation with the background field in a fixed, suitable gauge. The new feature of the superfield background-quantum splitting is that it involves the exponentials of the fields [6], that is: e v ~ ee,eVeffB,
eVB = ee,effB.
(4.1)
Choosing a suitable background covariantized gauge fixing term, we obtain the Yang-Mills action in a manifestly background covariant form. Up to quadratic
474
M.T. Grisaru et al. / One-loop effective potential in superspace
terms in the quantum fields we have [6] $2= tr f d 4 x d 4 0 [ V ( - ' X7~V a - W~V~ + W ~ V a ) V + beV, b + g'eV, c + c'eVBg] (4.2) where V describes the quantum Yang-Mills multiplet, c, g, c,, g, are the FaddeevPopov ghosts, b and 6 are the Nielsen-Kallosh ghosts. VB is the background Yang-Mills field which also appears in the field strength W~ and the covariant derivatives V~, XT~,Va (see (A.6)). This action is manifestly invariant under gauge transformations of the background field. The one-loop effective potential will be given by the sum of contributions from chiral ghost loops, and the V-loop. The contribution from the chiral fields has already been discussed. Therefore, we will concentrate here on the vector loop contribution. We use the fact that, as mentioned above, the effective potential is gauge invariant. We then perform the calculation with VB in Wess-Zumino gauge, and covariantize the result at the end. In this gauge, the background field is simply VB = 8 2 / ~ 2 D . After some algebra, we obtain for the quadratic part of the vector action the following expression:
S~=trfd4xd40(½V[-fq+
D(OaOa-O
-&0&)]V)
(4.3)
and the contribution to the effective potential is
I]
[ D
F v = - ½ t r l n 1 + ~-~(O~a~- OaO~
= - ½ t r E1 ( - ) 'n+ '
D
" ( 0 ~0~ - 0 - ~ -Oa) n .
(4.4)
" ~D. - 0 a-Da); one could do the D-algebra on the (Note that 0 ~ 0 ~ - 0a0 a --- -2~(0 graphs but there is little advantage in doing so.) We now have n
k
=
11
a
n-k
(0 Oa) ,
(4.5 t
k=0
(0o o)
= 0o0o
2 ~- l _ 1 0202 .
(4.6)
The trace operation on the 8-variables in (4.4) requires at least two factors of 8~ and
475
M.T. Grisaru et al. / One-loop effective potential in superspace
two factors of Oa; thus, inserting the 0 2 coefficient of (4.5) (and the corresponding one for the (OaOa) ~ quantity) into (4.4), we obtain:
FV=_½tr~ (-)m+l(D] 4+m m=O 4 + m
=-
+(7) 1 - D2
\~]
m+4
(k+2)(--)k(2k+l--1)(2m-k+l--1)
k=O~
-¼1n(1-
D2
(4.7)
The detailed evaluation of the series in (4.7) is given in the appendix. The contribution from the chiral ghosts is just three times that for a single chiral field (with a minus sign because of statistics). We can therefore write the complete one-loop effective potential, in a manifestly supersymmetric and background gauge invariant manner, as
r=-itrfd4Or~w~( [ ( (•aWa)2
o2
1 In 1
(v p4wo)
-31n(1
p2
(
- Iln 1 - 4
2]
(v p4wo)
(4.8)
Since the ghosts are in a real representation of the gauge group, the last term can be rewritten as
( °2)
~ln 1
(V W~)
p4
(4.9)
This last result can also be derived using improved Feynman rules for covariantly chiral superfields (see appendix C).
5. Conclusions
We have obtained superfield expressions for the one-loop effective potential for several supersymmetric systems. The results are manifestly supersymmetric but, for example in the case of QED, seem more complicated than the corresponding component results. We suspect that a method of calculating which uses functional methods in superspace, rather than diagrammatic techniques, would give simpler results. In particular it might allow obtaining a general algorithm for the one-loop potential, similar to that for component calculations, which would apply directly to any supersymmetric model.
476
M.T. Grisaru et al. / One-loop effective potential in superspace
We draw attention to the results for supersymmetric Yang-Mills. In a component calculation, in Wess-Zumino gauge, we would obtain a zero result. In a supersymmetric gauge, we find a non-zero result, depending on the auxiliary field D. This of course is a reflection of the gauge dependence of the effective potential, but it should also serve as a warning that Wess-Zumino gauge results for the effective potential in supersymmetric theories should not be trusted.
Appendix A We present here a brief description of our notation. We use two component spinors and work with euclidean space-time coordinates. The spinor covariant derivatives are defined as .-2 D,~= ½i( O,, + tO
a,~a),
/~=-½i(ea+iO'~a,~a),
(A.1)
where 0~ is the spinorial derivative and we use the abbreviation 0~a--o~,~0a; o~a = (1, o ) ~ with o the usual Pauli matrices. The fundamental anticommutation relations of the D's are
(D,~,Da}=iO. a ,
( D,~, DO) = ( / ) a , / ) ~ ) = 0 •
(A.2)
The conventions for 0-integration are
fd2002
=fd2002=
1.
(A.3)
Therefore, when also integrated over the space-time variable x the integral over 0 is equivalent to a covariant derivative:
f d20 ~ D 2 ' fdz# /ffz.
(A.4)
For a Yang-Mills superfield, the covariant derivatives are defined in chiral representation, as
VA=DA--iFA,
A-(a,&,a),
(a.5) (A.6)
The field strength is given by W = [ ~-d, (V-a,
XT,~)]=D2(e-VD,~eV),
(A.7)
M.T. Grisaruet al. / One-loopeffectivepotential in superspace
477
The massless rp~ propagator and ~3 vertex are given by [6] 3ab~22 34(01 -- 02),
(A.8)
f d404 D234( O, - 04 ) D234( 02 - 04) 34(03 - 04),
(a.9)
(/if2 __, D 2 for a ~p3 vertex). An additional factor D 2 (/if2) must be removed from the vertex if one of the lines is external. The Yang-Mills propagator is - 1/p234(01 - 02). All vertices are integrated with d40, and the D's are integrated by parts until only one line has any D's acting on it. We then use [6] D2DZ3"(01 - 02)1a,=02 = 1, (A.10) while loops with less than two Ds and two/~s give zero.
Appendix B We give here some details of the calculation of the series in (4.7). We have to evaluate I=
~
(--)mxm ~
( m + 3)!(--) k
k=0 (k + 2 ~ ( ( m Z ; +
m=0
= ~
(__)mxm+k
re,k=0
(2 k + l - 1)(2 ' ~ - k + l - l) 2) !
(re+k+3)!
(2k+l
1)(2m+l
1) "
(B.1)
( k + 2)!(m + 2)!
We can express (m + k + 3)! using the integral representation for the factorial n!= With the definition ~
G(z)=
/o d u e Zm
Y'~ ( m + 2 ) !
m=O
Uu~.
(B.2)
1
z z (ez-l-z)'
(B.3)
we have oo
I = f o ° ~ d u e - " u 3 y~ ( - ) " re,k=0 =fo°~due
(2 k+m+2 -- 2 k+l -- 2 ''+l + 1) (ux) m+k (k + 2)](m + 2)!
Uu3 [ 4 G ( 2 x u ) G ( - 2xu) - 2 G ( 2 x u ) G ( - xu) - 2G( - 2 x u ) G ( x u ) + G ( x u ) G ( - x u ) ]
_
1_._~[ ~ d u e - " ( 6 + e R x U + e _ 2 : , U _ 4 e X , , _ 4 e _ X , , ) . 4x 4 Jo u
(B.4)
478
M.T. Grisaru et al. / One-loop effective potential in superspace
We can now use the following relations:
f0 °
due,,_ le_p, - F p,~ (a)
p
- a
= 1-alnp+O(a
(B.5)
2)
and write 1
o~
I=lim~..4f duu"-le-U(6+e2X"+e-2XU-4eX~-4e-X") ~-~o 4x ao = lim
r(a---~)[6 + (1 - 2 x ) - "
+(1 + 2 x ) - ~ - 4 ( 1 -
= lira
r(a----~)[-aln(1
+4aln(1 - x:)]
a--,0
a~0
4x 4
4X 4
-4x:)
x)-~-4(1
+x)-"]
= ~4 [In(1 - x 2) - ¼1n(1 - 4xZ)].
(B.6)
Appendix C
In a recent paper [7] new covariant Feynman rules for chiral fields in a Yang-Mills background were obtained, for those cases where the chiral fields are in a real representation of the group. The complete one-loop contribution of a covariantly chiral superfield is given by the following Feynman rules: propagators:
~2 ~ 4 ( 0 -- 0 ' ) .
one vertex:
/52( 72 - D 2) --/52A
other vertices:
O + - [] = ~:ra V a -- [] "J¢- W a ~Ta -- 1 ( ~TaWCt ) ~ j~
The vertices can be written explicitly in the form A - (A~D~ + A ) ,
A " - - 2 i F ~,
B = B~D~ + B,
A = - r " r . - iD~F~,
(C.1)
B ~ =- 2 W ~ ,
B - D '~W,~ - 2iW'~F,, + 2D"F'~D~F,~ - 4 D a F " O , ~ - 20"a--DaF,"
(C.2)
We observe that ( A D 2 )" = A n " -,/~2.
(C.3)
M.T. Grisaru et al. / One-loop effective potential in superspace
479
r+a
Fig. 3. Yang-Millssuperfieldwith Yang-Millsbackground. A one-loop diagram is shown in fig. 3. There is only one/~2 in the loop and this means that all the D's but two must be integrated by parts outside the loop. The one-loop contribution to the effective potential can be written as
oc
F = tr~l
1 B n - lA]ff2
2n
(C.4)
p2.
We need to evaluate the coefficient of the D 2 term in B" - IA, and this can be done easily by induction. Setting
(C.5)
B " - tA = a . D e + b~.D,~ + c,,,
we obtain
a.+, = (B'~D,~ + B ) a . - ~ B' ~'b..,
a l=O,
b
b =A °,
c.+,=(B'~n.+B)c.,
c,=A.
(C.6)
We obtain a recursion relation for the a. coefficients:
a.+ 1 = ( B ' ~ D , ~ + B ) a . - I B ' ~ B , ~ B " - 2 A ,
a, = 0 .
(C.7)
Using (C.1) and (C.2) with external V, we can equivalently write
a.+, : , . W " [ a . -
2i(D~W~)"-VWVFv],
(C.8)
a.+ 1 - - i ( D'~W,~)n-IW#F# = D,~W'~[a.- i ( D # W ~ ) ~-2WVFv] = (D,,W'~)" ' ( 2 i W ~ F ~ - iW~F~),
(C.9)
with the final result
a.+ 1 = " [ t ( V a W,~) n - - I + ,(. D,~W a ) n-- ']
Wfli~fl,
lq>O.
(C.lO)
M.T. Grisaru et al. / One-loop effective potential in superspace
480
The effective potential is then F=
f tr
oo 1
d40EI 2n
an p2n
DYW - 2
(DaW~)
k +--~j+ln
=-½itrfd4OW~F, I(DoW~)2 In ( ( D ~ W 'P') 2 ) 1 which is equivalent to (2.6) for real representations. References [1] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; S. Weinberg, Phys. Rev. D7 (1973) 2887; R. Jackiw, Phys. Rev. D9 (1974) 1686 [2] M. Huq, Phys. Rev. DI6 (1977) 1733 [3] W. Lang, Nucl. Phys. BII4 (1976) 123 [4] D. Capper, J. Phys. G Nucl. Phys. 3 (1977) 731; L. Girardello and M.T. Grisaru, Nucl. Phys. B194 (1982) 65 [5] P. Fayet and S. Ferrara, Phys. Rep. 32c (1977) 1 [6] M.T. Grisaru, M. Ro~ek, and W. Siegel, Nucl. Phys. B159 (1979) 149 [7] M. T. Grisaru and W. Siegel, Nucl. Phys. B201 (1982) 292
1 ,
(C.I1)