OHSTPY-HEP-T-01-22

Two-loop Super Yang Mills effective action

and interaction between D3-branes

I.L. Buchbinder,
A.Yu. Petrov
and
A.A. Tseytlin^{1}^{1}1Also at Imperial College,
London and
Lebedev Physics Institute, Moscow

Department of Theoretical Physics

Tomsk State Pedagogical University

Tomsk 634041, Russia

Instituto de Física, Universidade de São Paulo

P.O. Box 66318, 05315-970, São Paulo, Brasil

Department of Physics

The Ohio State University

Columbus, OH 43210-1106, USA

We compute the leading low-energy term in the planar part of the 2-loop contribution to the effective action of SYM theory in 4 dimensions, assuming that the gauge group is broken to by a constant scalar background . While the leading 1-loop correction is the familiar term, the 2-loop expression starts with . The 1-loop constant is known to be equal to the coefficient of the term in the Born-Infeld action for a probe D3-brane separated by distance from a large number of coincident D3-branes. We show that the same is true also for the 2-loop constant : it matches the coefficient of the term in the D3-brane probe action. In the context of the AdS/CFT correspondence, this agreement suggests a non-renormalization of the coefficient of the term beyond two loops. Thus the result of hep-th/9706072 about the agreement between the term in the D0-brane supergravity interaction potential and the corresponding 2-loop term in the 1+0 dimensional reduction of SYM theory has indeed a direct generalization to 1+3 dimensions, as conjectured earlier in hep-th/9709087. We also discuss the issue of gauge theory – supergravity correspondence for higher order (, etc.) terms.

## 1 Introduction

The remarkable relation between supersymmetric gauge theories and supergravity implied by existence of D-branes in string theory [2, 3, 4, 5] motives detailed study of quantum corrections in super Yang-Mills theory, and, in particular, their non-renormalization properties. One aspect of this relation which will be of interest to us here is a correspondence between the super Yang-Mills theory and type IIB supergravity descriptions of subleading terms in the interaction potential between parallel D3-branes (see, e.g., [6, 7, 8, 9, 10, 11, 12, 13] for related discussions of interactions between D-branes).

Consider the supergravity-implied action for a D3-brane probe in curved background produced by a large number of coincident D3-branes. Ignoring higher-derivative (“acceleration”) terms, it is given by the Born-Infeld action in the corresponding curved metric and the “electric” part of the R-R 4-form potential,

(1.1) |

Here , ,
and , .^{1}^{1}1We set string tension
to be 1. In general,
appears in front of and
in the relation between
the scale in the supergravity expressions and the scalar
expectation
value in the SYM expressions.
In addition, the action contains also the
“magnetic” interaction part given
by the Chern-Simons term,
.
In what follows we shall consider the case when
, i.e. ignore all scalar derivative terms.
We shall assume that
, i.e. that one can drop 1 in
the harmonic function , so that (1.1)
becomes the same as the action for a D3-brane probe in
the
space (oriented parallel to the boundary
of )

(1.2) |

Expanding in powers of , we get

(1.3) |

where is the trace of the matrix product in Lorentz
indices,
i.e.
^{2}^{2}2Note that the sum of
the and can be represented as follows:

(1.4) |

From the weakly-coupled flat-space string theory point of view, the leading-order interactions between separated D-branes are described by the “disc with holes” diagrams (i.e. annulus, etc). The limit of small separation should be represented by loop corrections in SYM theory, while the limit of large separation – by classical supergravity exchanges. If the coefficient of a particular term in the string interaction potential (like term in [7]) happens not to depend on the distance (i.e. on dimensionless ratio of separation and ) then its coefficient should be the same in the quantum SYM and the classical supergravity expressions for the interaction.

In the SYM theory language, computing the interaction potential between a stack of D3-branes and a parallel D3-brane probe carrying constant background field corresponds to computing the quantum effective action in constant scalar background which breaks to and in constant gauge field . For the interactions between D3-branes, i.e. in the case of finite SYM theory, the expansion of in powers of the dimensionless ratio has the following general form

(1.5) |

In the planar (large , fixed ) approximation the functions should depend only on .

In more detail, the planar -loop diagrams we are interested in are the ones where the background field legs are attached to the “outer” boundary only (in double-line notation). In D-brane interaction picture, this corresponds to one boundary of the -loop graph attached to one D-brane (which carries background) and all other boundaries attached to coincident “empty” D-branes. This produces the factor of .

The comparison of (1.5) with the supergravity expression (1.4) is done for and . Naively, it would work term-by-term if , i.e. if the -th term in (1.5) would receive contribution only from the -th loop order.

This is indeed what is known to happen for the leading
term^{3}^{3}3Since we
set (see footnote 1)
in what follows we shall not distinguish between and .
which appears
only at the first loop order, and not at higher orders
due to the existence of a non-renormalization
theorem [15].^{4}^{4}4The absence of the 2-loop
correction was proved
in 1+3 dimensional SYM theory in [16],
and similar result was found in 1+0 dimensional theory
[17]. For indications of existence
of more general
non-renormalization theorems see [18].
Moreover, the one-loop
coefficient of the term in the
SYM effective action
[19, 20] is in
precise agreement with the supergravity
expression (see, e.g., [6, 13, 10]).

In the D3-brane case, there is also another viewpoint suggesting a correspondence between the classical supergravity D3-brane probe action (1.2) and the quantum SYM effective action (1.5) – the AdS/CFT conjecture [5, 21]. In the present context it implies that the supergravity action (1.2) should agree with the strong ‘t Hooft coupling limit of the planar (large ) part of the SYM action (1.5). The AdS/CFT conjecture thus imposes a weaker restriction that , with being directly related to in (1.2). The simplest possibility to satisfy this condition would be realized if the functions of coupling in front of some of the terms in (1.5) would receive contributions only from the particular orders in perturbation theory, and with the right coefficients to match (1.2), i.e. if they would not be renormalized by all higher-loop corrections. As we shall see, while this is likely to be the case at the order, the situation for , etc., terms is bound to be more complicated.

### 1.1 The term

In this paper we shall study this correspondence for the term by explicitly computing its 2-loop coefficient in the SYM theory. The Lorentz structure of this term in the SYM effective action is the same as in the BI action (1.3) (the form of the abelian term is, in fact, fixed uniquely by the supersymmetry), and the planar () part of its coefficient will turn out to be exactly the same as appearing in (1.4).

This precise agreement between the supergravity and the SYM
actions at the
level we shall establish below
was conjectured earlier in [13],^{5}^{5}5It was
further conjectured in [13]
that all terms in the BI action may be reproduced
from the SYM effective action (see also [10, 5, 22]
for similar conjectures). We shall present a more
precise version of this conjecture in section 1.2 and
section 5.
being
motivated by the agreement
[12] between the term in the
interaction
potential between D0-branes in the supergravity
description and the corresponding
2-loop term in the effective action
of maximally supersymmetric 1+0 dimensional SYM
theory.^{6}^{6}6This
conjecture was tested in [13]
(in the general non-abelian case)
by demonstrating that
there is a universal expression
on the SYM side which
reproduces subleading terms in the supergravity potentials
between various bound-state configurations of branes.
Since
brane systems with different amounts of supersymmetry
are described by very different SYM backgrounds,
the assumption that
all of the corresponding interaction potentials originate
from a single universal SYM expression
provided highly non-trivial constraints on the
structure of the latter.

Combined with the known fact that the abelian term does not appear at the one-loop SYM effective action [20, 11] (see eq.(1.8) below), this suggests that this 2-loop coefficient should be exact, i.e. the abelian term should not receive contributions from all higher () loop orders. Indeed, from the point of view of the AdS/CFT correspondence, higher order corrections to in (1.5) would dominate over the two-loop one for , spoiling the interpretation of (1.4) as the strong-coupling limit of (1.5).

One should thus expect the existence of a new non-renormalization theorem for the abelian term in (computed in the planar approximation), analogous to the well-known theorem of [15]. The reasoning used in [15] (based on scale invariance and supersymmetry) does not, however, seem to be enough to prove the non-renormalization of the term. It is most likely that one needs to use the full power of 16 supersymmetries of the theory (which are realized in a “deformed” way). One may then expect to show that the supersymmetry demands that the coefficient of the term should be rigidly fixed in terms of the -coefficient (proportional to its square); then the fact the term appears only at the 1-loop order would imply that the term should be present only at the 2-loop order.

One possible way to demonstrate this would be
to apply the component approach,
by generalizing to 1+3 dimensions
what was done for the term
in 1+0 dimensions [23] (see also [24]),
i.e. by deforming the
supersymmetry transformation rules
order by order in and trying to show that
the coefficient of the term in
is completely fixed by the
supersymmetry in terms of the
coefficient of the term.
In effect, this is what was already done in [25]
in SYM theory in a background.
It was shown there that the structure of the abelian
and terms in the
effective action which starts with the super Maxwell term
is completely fixed by the (deformed)
supersymmetry
to be the same as in the BI action,^{7}^{7}7In
the role of or a fundamental scale
is played by an UV cutoff (or ),
powers of which multiply .
with the coefficient
of the term being related to that of the
term in
precisely the same way as it comes out of
the expansion of the BI action.^{8}^{8}8Let us mention also
that the coefficient of the term is fixed uniquely
in terms of the coefficient of that of term
(to be exactly as in the BI action) by the condition of
self-duality of the SYM effective action written in terms of
superfields [26, 27].
We thus expect that
the arguments of [25, 23] may indeed
have a direct counterpart in theory,
relating the and coefficients and
and thus
providing a proof of
the non-renormalization of the
term beyond the 2-loop order.

### 1.2 Comments on higher-order terms

The obvious question then is what happens at the next – order: should one expect that the coefficients of these terms in the SYM effective action are again receiving contributions only from the corresponding – 3-loop – graphs and that they are in agreement with the supergravity expression (1.2)? That the story for the term should be different from the one for the (and ) term is indicated by the fact that the 1-loop SYM effective action [20, 11] is already containing a non-trivial term (see (1.8)). One could still hope that the term will not receive corrections beyond the 3-loop order, so that the 3-loop contribution will dominate over the 1-loop and 2-loop ones in the supergravity limit ().

However, the situation is more complicated since, in contrast to what was the case for the and terms, the supersymmetry alone does not constrain the structure of the invariant in a unique way – the terms in the BI action and in the 1-loop SYM effective action have, in fact, different Lorentz index structures!

Indeed, considering four Euclidean dimensions and choosing the gauge field background to have canonical block-diagonal form with non-zero entries and one can explicitly compare the expansions of the BI action and the 1-loop SYM effective action. For the BI action we get (we use a constant instead of in (1.2) to facilitate comparison with the SYM expression)

(1.6) |

The 1-loop Euclidean Schwinger-type effective action
for the SYM theory (with gauge group broken
to by a scalar field background )
depending on a constant
gauge field strength
parametrized by
has the following form [20, 11]
^{9}^{9}9We shall
consider the background representing
a single D3-brane with gauge field
separated (in one of the 6 transverse directions)
by a distance
from coincident D3-branes.
The background matrix in the fundamental
representation is then
and the corresponding
matrix is ,
where is the trace in the fundamental representation.
Only this traceless part of the background couples to quantum fields
and thus enters the quantum effective action
(the planar part of which will be proportional to a single trace
of in the fundamental representaion).
The (Minkowski space) YM
Lagrangian will be
where and the
generators in the fundamental representation
are normalized so that .
This normalization is convenient for comparison with the kinetic
term in the
BI action for a collection of D3-branes
( should be evaluated
on the diagonal background
matrix in the fundamental representation).
In this case
Comparing to the supergravity expression (1.2) note that
for one has
;
note also that going from Euclidean to Minkowski signature
notation one should change
the overall sign of the action.

(1.7) |

(1.8) |

The term in (1.8) cancels out, but term is present and has the structure different from that of the term in (1.6).

Let and denote the bosonic parts of the two abelian super-invariants appearing at the order in the expansions of the BI action (1.6) and the SYM 1-loop action (1.8) respectively. We propose the following conjecture about the SYM effective action (which replaces the earlier conjecture in [13]): (i) the coefficient of receives contribution only from the 3-loop order with coefficient which is in precise agreement with the one in the supergravity BI action (1.2); (ii) the coefficient of receives contributions from all loop orders, but the planar part of the resulting non-trivial function in (1.5) goes to zero in the limit , so that the requirement of the AdS/CFT correspondence is satisfied.

The possible existence of the two independent invariants – one of which has “protected” coefficient and another does not is reminiscent of the situation for the invariants in type IIA string theory (see [28] and refs. there). While a “universality” or “BPS saturation” of coefficients of terms with higher than 6 powers of may seem less plausible, there are, in fact, string-theory examples of specific higher-order terms that receive contributions only from one particular loop order, to all orders in loop expansion [29].

The existence of several independent super-invariants appearing at the same -order of low-energy expansion of SYM effective action with only one invariant having “protected” coefficient and matching onto the term appearing in the expansion of the BI action may then be a general pattern. Moreover, it should probably apply also in the case of non-abelian backgrounds, here starting already at the order.

Indeed, the non-abelian (or multi-)
term is likely to be a combination
of the two superinvariants differing by
‘‘commutator’’ terms absent in the abelian limit.
It is natural to expect that there is, in fact,
such commutator term in the 1-loop SYM effective
action.^{10}^{10}10The
precise structure and coefficient
of this term can be determined using the general methods
of [30, 31, 32],
or by computing the non-abelian
1-loop term in the 1+0
dimensional theory
[33]
and lifting it to 1+9 or 1+3 dimensions.
Thus
the full non-abelian term in the SYM
effective action will no longer be protected. This
may be related to an observation in
[34]
that supersymmetry does not
completely
determine the coefficient of the
term in 1+0 dimensions in the case of
more general (“-particle”)
backgrounds.^{11}^{11}11A possibility of existence,
in , case,
of “unprotected”
non-abelian
tensor structures already at “” order
(and that the proof of the non-renormalization theorem of
[15] applies in the
or “two brane” case only)
was suggested in [35].
Particular diagonal
(“three D0-brane”) backgrounds
in 1+0 dimensional 2-loop SYM effective action were
considered in
[36, 37, 38] and the agreement with supergravity
was demonstrated.

### 1.3 Superspace form of the term

Before describing the content of the technical part of the paper let us discuss the superspace form of the term in which it will be computed below.

Using superfield notation, the expansion of the BI action containing the sum of the and terms in (1.3) can be written as (cf. [39])

(1.9) |

where is the abelian superfield strength. We assume Minkowski signature choice as in (1.3) and the same superspace conventions as in [40, 27] (in particular, ).

To reproduce this expression by a 2-loop computation on the SYM side we shall use the superfield formulation (with the harmonic superspace description for the quantum fields in the context of background field method). We shall assume that only one chiral superfield has a non-zero constant expectation value, namely, the one contained in the vector superfield, i.e. only the latter and not the hypermultiplet will have a non-vanishing background value. Thus we will be interested in the case when the gauge group (or , which is the same as we will consider only the leading large approximation) is spontaneously broken down to by the abelian constant (in -space) superfield background with non-zero components being (same as in [40])

(1.10) |

It is understood that the background function should be multiplied by the diagonal matrix (generating the relevant abelian subgroup of )

(1.11) |

which represents the configuration of coincident D3-branes separated by a distance from the single D3-brane carrying the background field.

Assuming the background (1.10), the combination
of the superconformal invariants
representing the sum of the two unique abelian
and corrections in (1.3),(1.9)
may be written in the manifestly
supersymmetric form as follows
^{12}^{12}12We use that
for the given background

(1.12) | |||||

where (for the relation between the (1.9) and (1.12) forms of the term in Appendix A)

(1.13) |

is a spurious scale which drops out
after one integrates over ’s –
it gets replaced by
in going from the
(1.12)
to (1.9) form.^{13}^{13}13For comparison, the superfield form of
the
two abelian invariants discussed above is [40, 27]:

The low-energy expansion of the quantum SYM effective action turns out to have the same structure (1.12). The coefficient of the 1-loop term

Our aim will be to show that the two-loop correction to the SYM effective action in background has the superspace form as the

The rest of the paper is organized as follows. In section 2 we shall consider the super Yang-Mills theory formulated in terms of unconstrained superfields in harmonic superspace [45], i.e. represented as the SYM theory coupled to hypermultiplet. Then we shall briefly describe the superfield background field method allowing one to carry out the calculation of the effective action in a way preserving manifest supersymmetry and gauge symmetry. Section 3 will be devoted to the evaluation of the hypermultiplet and ghost corrections. We shall find that their contributions to the leading part of the 2-loop low-energy effective action vanish. In section 4 we shall compute the pure SYM contribution to the 2-loop SYM effective action. Section 5 will contain a summary and some concluding remarks on possible generalizations. Appendix A will present a relation between and forms of the . Appendix B will describe some details of calculations of integrals over the harmonic superspace.

## 2 background superfield expansion

The aim of this work is to calculate the leading two-loop correction to the low-energy SYM effective action in the sector where only the vector multiplet has a non-trivial background. Our starting point will be the formulation of SYM theory in harmonic superspace [45]. In terms of superfields SYM is simply SYM theory interacting with one adjoint hypermultiplet with the action [46, 16]

(2.1) |

is superfield strength expressed in terms of vector superfield , and is the hypermultiplet field taking values in the adjoint representation of gauge algebra, with . Here and below we use the notation introduced in [45, 46, 16].

The most natural way to calculate loop corrections in this model is to use the background field method which guarantees manifest supersymmetry and gauge covariance at each step of the calculation. We shall do background-quantum splitting of the gauge superfield in the form [46, 16] with in the right-hand side being a background and being a quantum superfield. The hypermultiplet will have no background, i.e. will be treated as a quantum superfield.

As explained above,
we are interested in the large part of the 2-loop
effective action in the case of the background
(1.10),(1.11) corresponding
to coincident
D3-branes separated by a distance
from one D3-brane carrying
field. The background matrix is the traceless part of the
matrix .^{15}^{15}15
Writtent in adjoint representation it is just a combination
of differences of its diagonal elements,
,
i.e. it contains zeros and pairs .
Since we are after the planar contribution only
we may just consider the theory and ignore the subtraction of
traces in the background field
expressions and propagators.
The relevant planar 2-loop graphs (represented
in double-line notation) will have the following structure
(see Fig.1):
the background fields will be
attached only to the “outer” cycle of the diagram
(representing, in string theory language,
the loop lying on the single separated D3-brane),
with two internal
cycles (lying on coincident D3-branes) each producing a
trace factor of .^{16}^{16}16In more general case when the probe is
a cluster of D3-branes the planar contribution
will be proportional to (products of background matrices in
fundamental representation).

To develop perturbation theory one needs background dependent superfield propagators. They can be found by analogy with [16]

(2.2) |

The first line here defines the SYM propagator, the second – hypermultiplet propagator and third – the ghost propagator. The index means that the corresponding superfields are taken in the so called -frame [45].

We have suppressed the indices of the fundamental representation, (), i.e. the propagators carry the two pairs of indices . In the absence of the background each propagator is proportional to (dropping trace terms subleading at large ). In the presence of our background the propagator will remain to be diagonal, with + background-dependent terms, background-dependent terms, , where are indices of the fundamental representation of the unbroken group and is the free propagator. Thus all one is to do is to separately take into account the two type of contractions – with the index “1” (which involve the background-dependent propagator) and with the indices taking valies (which involve the free propagator). The part will not contribute to the leading large part of the diagram. It is obvious also that the only part of that will be contributing to the relevant diagrams (with all background dependence at outer line only) will be background-dependent terms. It is effectively this non-trivial background-dependent block of the propagator that will be discussed below. The role of the remaining free index contractions is simply to produce the factor of .

Since the background superfields will be on-shell [44], i.e. will be subject to the equations of motion (at the end, satisfying (1.10))

(2.3) |

and are also abelian, one is able to show that , . Then the operator [46] in (2.2) takes the form

(2.4) |

Within the background field method, the 2-loop corrections to the effective action are given by the “vacuum” diagrams containing only cubic and quartic vertices of quantum superfields. The supergraphs with cubic vertices have the form