Volume 176, number 3,4
PHYSICS LETTERS B
28 August 1986
THE EFFECTIVE LAGRANGIAN OF S U P E R S Y M M E T R I C Y A N G  M I L L S THEORY
Theophil O H R N D O R F CERN, CH1211 Geneva 23, Switzerland
Received 5 June 1986 The oneloop effective action of the supersymmetric YangMills theory in the presence of a covariantly constant YangMills superfield is calculated. This is done with the help of Schwinger's proper time method. The propagator of the YangMills superfield in the presence of a nonvanishing background superfield is derived in a closed form. The effect of an unstable mode is isolated. 1. Introduction and results. The last few years have seen an enormous growth in the interest in supersymmetry. Almost all of the efforts to reconcile supersymmetry with the observable world indicate that supersymmetry will at best be phenomenologically relevant in the form of an effective, nonrenormalizable lowenergy field theory. However, in the most preferable case, such a lowenergy theory should be calculable from first principles, i.e., from a fundamental supersymmetric theory, which has all the good features of a wellbehaved quantum field theory. In order to implement these ideas, a clear understanding of the relations between the fundamental level and the effective level of supersymmetric field theories is required. There is, in particular, the need to develop calculational techniques which allow to derive supersymmetric effective actions in an economical fashion. As far as superfield formulations are available, superfield techniques have proved to be in most respects superior to manipulations on the component level. Not only do they provide a convenient framework to construct supersymmetric lagrangians, superfields are indispensable in doing perturbation theory. However, the calculation of an effective theory is a problem which, in general, goes beyond standard manipulations in perturbation theory. There are some attempts in the literature [13] to transfer various techniques like the heat kernel method or the method originally used by Coleman
and Weinberg to calculate effective potentials into the framework of superspace. Unfortunately, such a transfer is only possible at the price of breaking supersymmetry superficially by the approximation which has to be employed to perform such a transfer successfully. In nonsupersymmetric theories, a gauge covariant method to calculate effective actions nonperturbatively is Schwinger's operator technique [4]. Starting from the supersymmetric proper time method [5], this technique has recently been generalized to supersymmetric QED (SQED) [6]. This method supersedes previous attempts to calculate effective actions, in that it is fully supersymmetric. In the present paper, we shall apply Schwinger's method to supersymmetric YangMills theory (SYM). The purpose of this article is twofold. At first, we would like to demonstrate the flexibility of Schwinger's method when combined with superspace techniques. Secondly, by calculating the effective action of SYM we hope to gain some understanding necessary to apply the method to theories which exhibit even larger gauge invariances, notably supergravity theories. It will turn out that the calculation is less involved in the present case, compared with previous work [6]. This is due to the fact that SYM is formulated in terms of a vector superfield, whereas SQED requires chiral matter superfields. Accordingly the superspace heat kernel of SYM is unconstrained. This fact allows us to write it down 421
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explicitly. As a byproduct the propagator of SYM in the presence of a constant background field can be obtained in a closed form. Due to the UV properties of SYM, the effective action does not require any UV subtractions. However, there are potential IR singularities. We shall show how these singularities can be avoided. It is well known that in nonsupersymmetric YangMills theory these IR singularities can be attributed to an unstable mode. It leads to an imaginary part of the effective action [7]. We shall isolate the effect of the corresponding unstable mode of SYM and calculate the imaginary part which contributes to the effective action. In the present paper, we shall employ the conventions of ref. [6].
2. Backgroundquantum splitting. Let us begin by reviewing the derivation of the general form of the oneloop effective action which will be the starting point of our considerations in the following section. The procedure is well known [8], we shall therefore sketch its derivation only briefly. The action of SYM is given by S = ¼trfd4x d20 W 2 + h.c.,
(2.1)
28 August 1986
  i U Q, w e get
~=eV~
e v',
~a=~,
(2.6)
where e v' =
e UQ e vo,
and the covariant derivatives ~ , and ~ contain only the background gauge connections. In order to quantize the theory, the action (2.1) has to be augmented by gauge fixing and ghost terms. To maintain invariance under gauge transformations of the background field, we add a gauge fixing term which is covariantized with respect to the background field
× {~ff, [ ~ ,
vQ]}.
(2.7)
In the quadratic approximation the ghosts contribute a piece to the effective action which is identical to the one already computed in ref. [6] for SQED. Therefore, we shall simply drop all the ghost contributions in the present work. Inserting the decomposition (2.6) into (2.1) and keeping only contributions quadratic in the quantum field V' we obtain
where W~:=  ~ [ ~ ,
{~,
~}],
(2.2)
where
and
1
~ : = e  V D ~ e +v,
~ : = e  ~  D ~ e +U
(2.3)
The prepotentials U and V take values in the fundamental representation of the gauge group. A gauge transformation acts on V like
eV~eV'=eiYieVeiK,
D~A = 0,
(2.4)
and similarly for U. The prepotentials U and V are now split into a background piece UB(V B) and a quantum piece u Q ( v Q) according to e V = e v " e v~,
e U = e U"e UQ.
(2.5)
Performing a gauge transformation with K = 422
(2.8)
s + SGF =
2
1W~_½W~,
(2.9)
and 9,:= 2i[~,
~].
(2.10)
All the covariant derivatives and fields appearing in (2.9) have to be tzken in the adjoint representation of the gauge group. As V' is a vector superfield, it is readily integrated out to yield the effective action F =  ½i Yr log [].
(2.11)
The trace in the foregoing expression has to be taken over the full superspace.
3. The supersymmetric heat kernel The effective action F is related to the superspace heat kernel
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PHYSICS LETTERSB
h(z 1, z2; r), taken at coinciding arguments z := z 1 = z z, via the proper time integral Tr log [] = Tr fe~ drr h (
z; r ) .
(3.1)
The heat kernel h(zl, z2; r) is defined by ( i d / d r  ~)h(z,, z2; r) = 0,
(3.2a)
h(z,, z2; 0) = 88(Zl  z2).
(3.2b)
The Feynman iE prescription [] , [] + ie is understood in (3.1) and (3.2). The information encoded in h(z~, z2; r) does not only allow to calculate F, h(za, z2; r) is closely related to the propagator associated with [] c drh(z~, z2; ~). 0o
(nie)~=if_
(3.3)
Note that in contrast to the SQED heat kernel of ref. [6], the heat kernel h(za, z2; r) is defined on the full superspace. This fact implies that here we have much more liberty in formulating a supersymmetric constant field approximation. We shall require that ~ J := { N,, W ~ } * 0,
(3.4a)
~a ~ := {~a, W~} ~ 0,
(3.48)
by an ordinary function, we have to make use of the Heisenberg representation of the distance (z~ z2). It turns out to be convenient to introduce the supersymmetric invariant distances ~ := (Xl  x2)~ + 2i0,10a2  2i020l,
(3.8)
A .'=0~02,
(3.9)
~a:=0202.
Defining the Heisenberg representation of ~ , A and ~a in the same manner as for ~ we get ~a( T ) ~a =2[R
'(exp(irR)l)]ab~bxa(r),
(3 .lO) A"(r) = (Wg~l(exp(½irN)  1))" + a ",
(3.11)
Aa(r)=((exp(½irC~ )  I ) ~ W )
(3.12)
~ B ..= ~ a , xo(
)
(3.13)
= fo dr' X'a(r'),
(3.14)
=
(3.15)
In (3.15) the Heisenberg representations of W and W appear ,
= (W e x p ( ~ i r ~ ) )
3W~ = 0~WB= 0,
(3.5)
as well as the derivatives of ff~¢ and ~ j vanishing. Using the Jacobi identity in the same way as in ref. [9], one may show that in this approximation the superfields W and W anticommute among each other, i.e., are effectively abelian. We shall next calculate the heat kernel h(z 1, z2; r) almost along the same lines as in ref. [6]. In contrast to ref. [6], here we shall only need the Heisenberg representation ~ , ( ¢ ) of the covariant derivative N, = [[], .9~],
~(0) =~,
+~a,
where
whereas
i(d/dr)~,(~)
28 August 1986
,
Wa(r) = ( e x p (  ½ir~')W) a.
(3.16)
It is obvious that
(~(r)h(z 1, Zz; r ) = 0.
(3.17)
We can now invert (3.10) to express ~a in terms of ~a and x~(r). Making use of this representation of the action of Na(0) on h(z~, z2; r) one gets
[]h(z,,
z2;
,), (3.18)
(3.6) with
whereas ~ , and ~a are not transformed into the Heisenberg picture. Eq. (3.6) is solved by ~, (r) = (exp(irR)) ,~b.
(3.7)
In order to substitute the ~ 2 part of [] in (3.2a)
C3:= ¼tr[R cth(½irR)]
~z(~x(r))~[R 2 sh z(½irR)]~h × ( ~  x(r)) b.
(3.19) 423
Volume 176, number 3,4
PHYSICS LETTERSB
28 August 1986
With the help of the relation (3.18) the heat equation (3.2a) is integrated formally
Inserting this expression into (3.20) we obtain the heat kernel
h(zl, 22; +) = U(r)exp(if'dr' U(r')rS(r')U(r'))
h(=,, z2; ~)
X
(3.20)
G(Z1, Z2) ,
U(rr')~(r')U(r'r))
× exp(if'dr'
in terms of the differential operator
U(r) := e x p [  ½ir(W~ W ~ ) ] .
(3.21)
Analyzing the behaviour of (3.20) for small values of r, one recognizes that the initial condition (3.2b) is satisfied, provided
G(z,, z2) [ ;,= o =  [i/(4"rr) 2] a2~ 2.
(3.22)
Though (3.20) is the general solution of the heat equation after substituting (3.18), it is not a solution of (3.2a) as long as the function C(z 1, z2) remains unspecified. This peculiar situation arises because the substitution (3.18) is valid only for solutions of the heat equation (3.2). All that we have to do to obtain a solution of (3.2) is to find a suitable function C(z 1, z2) such that (3.17) is valid. Using the same method as in ref. [6], it is straightforward to show that (3.17) is satisfied, provided the function C(z 1, z2) respects the relation
~,C(z,, z 2 ) = ¼~'RhoC(z ,,
z2).
(3.23)
To solve (3.23) one first factorizes a path dependent phase factor
(,24, from C(z 1, z2) where the path cd is a straight line in superspace running from z: to z 1. This procedure transforms the covariant derivatives N, appearing on the lefthand side of (3.23) into the SchwingerFock gauge, where a representation of the gauge connection in terms of the field strength can be given. This has been worked out in ref. [10]. The resulting differential equation is solved easily, giving
C(Zl ' z 2 ) = e x p ( _ f d~A @a(~)] i A2~2 (4~r)2
(3.26)
where the covariant derivatives appearing in U(r) have to be taken in the SchwingerFock gauge. The derivatives embodied in U(r) appearing outside the d r ' integral are easily isolated by rewriting U(r) as
u(~) = e x p (  ½ i f o ' d ~ ' U(r')(WO WO)U(r')) xexp{  ½ir(WD  WD)}.
(3.27)
The second exponential does not contribute in (3.26). Explicit expressions for ,# and 0 have been given in ref. [10]. Eq. (3.26) together with (3.27) provides an explicit representation of the Green's function associated with [] via (3.3).
4. The effective action. The effective action F is determined by the coincidence limit of the heat kernel h(z l, z2; r). As the coincidence limit of ,~2(r) (~2(r)) is proportional to w Z ( w 2 ) , W and W can be set to zero everywhere else in (3.26). In this way the effective action becomes F
1 1 fd40 d4x/0 dr 2 (4qr) 2
a_ ~~
xexp(if'dr'~(r')) X [ W~¢ 1(exp(½irfq)  1)] 2 × [ ( e x p (  ½ir~)  1) ~~W] 2
"
(3.25) 424
XA2( r )~2( r )U( r ),
This expression may be rewritten as
(4.1)
/"
1
f d40
(~2) B = _ 4 a ~ B 2.
d4 X W 2 ~ 2
dT{  [Tr' G 2
×
28 August 1986
PHYSICS LETTERSB
Volume 176, number 3,4

In the presence of a "pure magnetic" background superfield the effective action becomes
Tr' ~2]
oo T
× [Tr'(ch½iT
)  Tr'(ch½i
C)]'
× [0

E. B,
 ch(iTB)]2}.
(4.6) We can now analyze the behaviour of (4.6) for large values of s. Subtracting and adding the expression ½ei'B [ e x p (  ¼iT I D l)  e x p (  iTB)] 2,
(4.7)
where I Ol refers to the lowest component of D, inside the curly bracket of (4.6) we obtain /" = /'stable [' /'unstable"
(4.8)
In E~tab~ewe can rotate the contour of integration to obtain 2
Fstable 
(16~r) 2
fd4Od4 X W 2 ~ 2 B

B
~'~ ds
× Jo 7{shl(Bs)[ch(~Ds) s ch(Bs)]2   ~ e  m [exp(~s I DI)  exp(Bs)]2}. (4.9) /'unstable is given by 'unstable 
__
{ + ( 2 i / T ) B ( ~ D 2  B 2) 2
(4.3)
and ah
dr
d4x W2~2
Xshl(iBT)[ch(liTo)
(4.2)
where G, G and D have been defined in ref. [6]. In the proper time formalism UV singularities show up as singularities in the proper time integral for small values of ~'. As is readily seen from (4.2), F is UV finite. However, this does not imply that the Bfunction of SYM vanishes to one loop, as we have dropped the ghost contributions which are UV singular [11]. To show that (4.2) represents a welldefined, real action, one would like to rotate the tiTcontour into T = is, 0 4 s ~ + oo. At this stage, one encounters a difficulty typical of massless YangMills theories [7]. The contour cannot be deformed immediately as the integrand does not vanish at infinity. In ref. [7] it was shown that in the nonsupersymmetric case an unstable mode of the analogue of the operator [] is responsible for this behaviour. This mode must be treated separately. In the nonsupersymmetric case, it leads to an imaginary contribution to F, even in the presence of a pure magnetic background field. Let us analyze the effect of the unstable mode in the present case. For simplicity, we shall restrict our considerations to the case of a "pure magnetic" background superfield. We have in general
RabR
(16~)2
j _ oo ~
× T r ' [ ( G '2 ~D2) l(ch(¼iTD)
R.hR a b ½( E 2  B 2 ) ,
1 fd40
F
×Tr'[(G21D2)l(ch(li'rD)ch(½i~G))]
ch(½iTG))] },
(4.5b)
(4.4)
1
1
fd40
d4xW2~
2
4 (16~.) 2
X fo °° dT __ 71_ B ( 1 D 2
B2 ) 2 ei~B
T IT
where ,~,b is the dual of Rob. E and B are the supersymmetric generalizations of the electric and the magnetic field. Setting E to zero we get (G 2 ) ~ = _ 48,#B 2,
(4.5a)
×{exp[iT(~lDIB)]l}
z.
(4.10)
Due to the Feynman i~ prescription, this expression is well defined. It can be evaluated to give 425
Volume 176, number 3,4
Funstable
1
1
4 (16~r) 2
PHYSICS LETTERS B
fd40
d4xW2~ 2
References
×n(~o 2Bz)2(B log IDI2B2B +½IDI l o g Z i DILDI _4
B
)•
(4.11)
O n e recognizes that in the present case, the u n stable m o d e does n o t lead to an i m a g i n a r y contrib u t i o n to the effective action if I D [  2B > 0 for the lowest c o m p o n e n t s of these superfields.
Note added After the c o m p l e t i o n of this work, the a u t h o r b e c a m e aware of ref. [12] where the effective l a g r a n g i a n of S Y M has b e e n c o m p u t e d for a pure m a g n e t i c b a c k g r o u n d field. T h e results of the present p a p e r indicate that the effective l a g r a n g i a n changes substantially once a full superfield is a d o p t e d as b a c k g r o u n d .
426
28 August 1986
[1] M. Huq, Phys. Rev. D 16 (1977) 1733. [2] M.T. Grisaru, F. Riva and D. Zanon, Nucl. Phys. B 214 (1983) 465. [3] K. Shizuya and Y. Yasui, Phys. Rev. D 29 (1983) 1160. [4] J. Schwinger, Phys. Rev. D 82 (1960) 664. [5] J. Honerkamp, M. Schlindwein, F. Krause and M. Scheunert, Nucl. Phys. B 95 (1975) 397. [6] Th. Ohrndorf, Nucl. Phys. B 273 (1986) 165. [7] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376. [8] M.T. Grisaru, W. Siegel, and M. Ro6ek, Nucl. Phys. B 159 (1979) 429. [9] M.R. Brown and M.J. Duff, Phys. Rev. D 11 (1975) 2124. [10] Th. Ohrndorf, Nucl. Phys. B 268 (1986) 654. [111 I.M. McArthur, Phys. Lett. B 128 (1983) 194. [12] P. Roy, Z. Phys. C., Particles and Fields 30 (1986) 79.