Reggeization of N=8 Supergravity and N=4 Yang--Mills Theory

0.14 MB·English

by  Howard J. Schnitzer

BRX TH-583
Reggeization of =8 Supergravity and
=4 YanNg–Mills Theory
N
Howard J. Schnitzer1
Theoretical Physics Group
Martin Fisher School of Physics, Brandeis University
Waltham, MA 02454
7 Abstract
0
0 We show that the gluon of =4 Yang–Mills theory lies on a Regge trajectory, which
2 N
then implies that the graviton of =8 supergravity also lies on a Regge trajectory.
N
n This is consistent with the conjecture that =8 supergravity is ultraviolet finite in
a perturbation theory. N
J

3
2
1 Introduction

1
v
7
There has been a great deal of interest in the possibility that =8 supergravity (sugra) has the
1 N
2 same degree of divergence as =4 super Yang–Mills theory (YM), and thus is ultraviolet finite
N
1 in four-dimensions [1]. In particular the =8 sugra perturbation expansion is closely related to
0 N
=4 YM amplitudes [1, 2]. It has been argued by Green, Russo and Vanhove [3] that the dualities
7 N
0 of M-theory imply that the four-graviton amplitude of =8 sugra is ultraviolet finite in four
N
/ dimensions. Further they argue [4] that even without the duality conjectures, four-dimensional
h
t =8 sugra might be ultraviolet finite at least up to eight loops. Clearly the relationship of these
- N
p two theories is providing new insights. In this paper we explore the Reggeization program in both
e
theories.
h
:
v In our work in collaboration with Grisaru and Tsao [5], and with Grisaru [6, 7], we showed
i
X that the elementary fields of renormalizable non-Abelian gauge theories in four-dimensions lie on
r Regge trajectories. In addition, our old preliminary study [7] of =8 sugra suggested that this
a N
might be true there as well. In this paper we extend Ref. [7] to show that the Reggeization of the
gluon in =4 YM implies the Reggeization of the graviton in =8 sugra.
N N
The calculations leading to this conclusion involves arcane technology (to the present gener-
ation of theorists). Therefore we provide an overview of these methods in Sec. 2, which involves
(perhapsunfamiliar)conceptsofMandelstamcounting,nonsensehelicitystates,etc. However, since
we cannot provide a complete review of these techniques, the reader should refer to the original
papers, especially ref. [7] for additional technical details.
In Sec. 3 we apply the formulism of Sec. 2 to the Reggeization of the gluon in N=4 YM and
the graviton in =8 sugra. Concluding remarks are in Sec. 4, while several Appendices collect
N
useful formulae needed in the calculations. Some of the information in the appendices is repeated
in the text for clarity of presentation.
1email: [email protected]
Supported in part bythe DOEundergrant DE-FG02-92ER40706
1
2 Regge Behavior: An Overview
Thereexistdifferentways offindingRegge poles inLagrangian fieldtheory. Onemethodconsists of
summing Feynman diagrams for large momentum transfer at fixed s in leading logarithm approxi-
mation and recognizing that a Regge trajectory α(s) corresponds to asymptotic behavior tα(s).
∼
Another involves solving analyticity-unitarity integral equations for the analytic continuation of
the scattering amplitude f(s,J) and looking for Regge poles directly. In =8 sugra the second
N
methodis applicable, usingas inputonly knowledge of the Bornapproximation (as we explain later
in this section.) [Given the close relationship of =4 YM scattering amplitudes to those of =8
N N
sugra, summation of leading logarithms in the latter theory may also be possible.] We have used
the second method extensively [5]–[7] in the past to find Regge poles in Yang–Mills theories, which
gave the same result as diagram summation in all cases where a comparison can be made. We
emphasize that the existence and number of Regge trajectories in a neighborhood of small integral
or half-integral J is independent of the size of the coupling. As a consequence we have control of
local properties in J of Regge trajectories, but not global ones.
One assumes that the (kinematical singularity free) scattering amplitude F(s,J) can be
continued to large Re J without encountering singularities in the angular momentum plane, and
that it can be continued to the left of Re J = N. Here one may encounter singularities such as
poles and cuts. In particular if one continues F(s,J) to an integer J = j N, one may ask if
≤
?
F(s,J) = f (s) (2.1)
j
where the scattering amplitude f(s,z) computed from diagrams has the form
1 1
f (s) = dzP (z)f(s,z)
j j
2
Z−1
N
= b (s)δ +analytic in j, (2.2)
n jn
n=0
X
where z = cosθ and P (z) is a Legendre polynomial. [In renormalizable field theories N 1, and if
j
≤
=8 sugra is finite, or at most log. divergent, presumablyN 2.] Thepresence of kronecker delta
N ≤
terms seems to make the equality (2.1) unlikely, yet one can prove that in certain case that F(s,j)
and f (s) must coincide. If they do coincide at some value of j, one says that f (s) “Reggeizes” at
j j
J = j, and that F(s,J) is analytic in the neighborhood of such j.
Mandelstam [8] has given certain criteria for establishing whether or not kronecker delta
singularities are present at a given j, based on the following counting argument. Both f (s) and
j
F(s,J) have s-plane analyticity and unitarity properties which require that they satisfy certain
s-channel dispersion relations, in which the inhomogeneous terms are the same in both cases, as
is the unitarity condition. Where the solutions for f (s) and F(s,J) may differ is in the value
j
of possible subtraction constants in the dispersion relations, and in the positions and residues of
CDD poles, which correspond to singularities not resolved by analyticity and unitarity. However
both f (s) and F(s,J) are subject to identical kinematical constraints. If the number of these
j
constraints equals or exceeds the number of free parameters, the amplitudes must coincide. In this
case one understands the kronecker delta as the boundary value of an analytic function, e.g.,
α(s) J
− −−→ δJj . (2.3)
α(s) j α(s)→J
−
2
[Infieldtheorythefreeparameters aremassesandcouplingconstants.] SinceMandelstam counting
[8]iskinematical,itistruetoallordersofperturbationtheory. However,theMandelstamprocedure
must be carried out for each j separately.
The proof of Mandelstam is delicate as it involves the unitarity of the theory, which appears
to eliminate non-renormalizable theories, such as massive YM theories without a Higgs mechanism
[9]. Nevertheless in this paper we show that the graviton in =8 sugra Reggeizes as a consequence
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of the Reggeization of the =4 YM massless gluon.
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We consider the scattering of massless particles with spin (p +p p +p ) as functions of
1 2 3 4
→
the Mandelstam variables
s = (p +p )2 = 4q2
1 2
s
t = (p p )2 = (1 z) (2.4)
1 3
− −2 −
s
u = (p p )2 = (1+z)
1 4
− −2
where q is the center-of-mass momentum, z = cosθ, with θ the scattering angle. One considers the
integral equation satisfied by the two-body, kinematical singularity free, helicity amplitudes
F˜ (s,J) = V (s,J)
λ3λ4,λ1λ2 λ3λ4,λ1λ2
ds′
+ ρ(s′)F˜ (s′,J)F˜ (s′,J) (2.5)
s′ s λ3λ4,λ5λ6 λ5λ6,λ1λ
λX5λ6Z −
or as matrices
ds′
F˜µλ = Vµλ+ s′ sF˜µλ′ ρλ′µ′F˜µ′λ (2.6)
Z −
where λ = λ λ , µ = λ λ , and ρ is a phase-space factor. The unitarity condition couples
1 2 3 4
− −
particles of different helicity. In the continuation to small j one reaches a point where j < λ , µ
| | | |
for some value of λ and/or µ. The corresponding amplitudes are unphysical, i.e., states λ ,λ >
1 2
|
or λ ,λ > with λ or µ < j are “nonsense” states.
3 4
| | | | |
V (s,J) can be obtained to any given order of perturbation theory from diagrams. In
µλ
particular if we keep only the lowest order diagrams,
V (s,J) Q (z (s))v (s) (2.7)
µλ ∼ J−λm 0 µλ
where λ = max (λ , µ ) is an integer or half-integer, and Q is a Legendre function of the 2nd
m
| | | |
kind, noting that Q functions have poles at negative integers. Imagine writing (2.6) for sufficiently
large J so that all helicity states are sense, and continuing J to the neighborhood of some small
physical value of j where some of the helicity states are nonsense. At such values of j some of the
matrix elements of V are singular since Q develops poles at negative values. Denote F˜ by
µλ J−λ µλ
F˜ λ , µ j
ss
| | | | ≤
F˜ λ j, µ > j
ns   | | ≤ | |
F˜sn  if  λ > j, µ j (2.8)
| | | | ≤
F˜ λ > j, µ > j .
nn   | | | |
3
In the neighborhood of j, the integral equation is of the form
F˜ F˜ v δ v (J j)−1/2
ss sn = − ss Jj sn −
F˜ F˜ v (J j)−1/2 v (J j)−1
ns nn ! ns nn !
− −
ds′
+ (F˜)ρ (F˜) . (2.9)
(s′ s)
Z −
The Born approximation quantities v , v , v , and v are essentially polynomials in s. The
ss sn ns nn
solution to (2.9) is
K(s)
F˜ss(s,J) = vsn′ J j v(s)K(s)+ (g4) vns
(cid:20) − − O (cid:21)n′n
+ regular in J (2.10)
where K(s) is a known integral common to all channels. Here F˜ consists of those helicity ampli-
ss
tudes physical at J = j.
We find Regge poles with trajectories
α(s) = j +(eigenvalues of v ) K(s)+ (g4). (2.11)
nn
× O
The number of Regge trajectories at j is equal to the rank of the nonsense-nonsense matrix v ,
nn
with trajectories given by (2.11). Eqn. (2.11) has the same accuracy as the leading logarithm
approximation in summing diagrams. Therefore complete information about the trajectories (but
not the residue) in the neighborhood of j can be obtained by studying v . If we keep only
nn
contributions of the t and u channel poles in Born approximation, this gives the location of the
− −
Regge poles for values such that α(s )= j, with α′(s ) correct to (g4).
o 0
O
Consider (2.10) for J j. Consistency for Reggeization to occur at j requires the matrix
→
factorization
vss = vsn′(v−1)n′nvns (2.12)
at J = j. [This should notbeconfused with the tree factorization of the Born approximation.] One
has the following statement [10]. A necessary and sufficient condition for (2.12) to hold is that the
rank of the matrix
v v
ss sn = v (2.13)
v v
ns nn !
equals that of the nonsense-nonsense matrix v . [Recall that the rank v is equal to the number
nn nn
of Regge trajectories at j.]
In the next section we apply the above formalism to gluon-gluon scattering in =4 YM, and
N
graviton-graviton scattering in =8 sugra. We show that the gluon pole at j = 1 Reggeizes, with
N
rank v = 1 and rank v = 1, and thus (2.12) is satisfied. As a consequence, we show that these
nn
relations in =4 YM also imply that the graviton Reggeizes in =8 sugra. That is (2.12) being
N N
satisfied for =4 YM also implies that the analogous factorization condition holds for =8 sugra.
N N
In both theories the rank v = 1 and rank v = 1, though dim v differs.
nn
4
3 Helicity Amplitudes
A. Generalities
In this section we present the relevant information to justify our claim of the Reggeization
of the graviton and gluon in =8 sugra and =4 YM respectively. Since both theories are
N N
supersymmetric, pseudohelicity conservation applies. For the scattering amplitude
F(λ ,λ ; λ ,λ )
3 4 1 2
with helicities λ , , λ one defines the pseudohelicity
1 4
···
P(λ ,λ )= λ +λ (3.1)
1 2 1 2
for the initial state, and similarly for the final state. Supersymmetry requires
P(λ ,λ )= P(λ ,λ ) . (3.2)
1 2 3 4
That is
F(λ ,λ ;λ ,λ ) = 0 if λ +λ = λ +λ . (3.3)
3 4 1 2 1 2 3 4
6
As a result, we only need consider P=0 states in our discussion of the Reggeization of the graviton
and of the gluon.
InordertoextractReggebehavioronemustdealwithkinematicalsingularityfreeamplitudes,
as these are the ones that satisfy unitarity and analyticity. Given
F(λ ,λ ;λ ,λ )= F (s,t,u) (3.4)
3 4 1 2 µλ
whereµ = λ λ , λ = λ λ ,ands,t,uaretheMandelstamvariables,thekinematicalsingularity
3 4 1 2
− −
free amplitudes F˜ are obtained from
µλ
θ |λ+µ| θ |λ−µ|
F (s,t,u) = √2 cos √2 sin F˜ (s,t,u)
µλ µλ
2 2
(cid:20) (cid:21) (cid:20) (cid:21)
|λ+µ| |λ−µ|
u t
= (√2)|λ+µ|+|λ−µ| − − F˜ (s,t,u) . (3.5)
µλ
"r s # "r s #
A typical singularity free amplitude has the form in Born approximation
a b c
F˜ = + + . (3.6)
µλ
t u s
Given the kinematical singularity free amplitudes, one obtains the angular momentum projection
1 1
FJ = dzCJ (z)F˜ (z) (3.7)
λµ 2 λµ λµ
Z−1
where the matrices CJ we need are tabulated in Appendix C, with t = s (1 z);
λµ −2 −
u= s (1+z); z = cosθ. Given the CJ , one computes the projection (3.7) using
−2 λµ
1 1 dzP (z)
ℓ
= Q (a) . (3.8)
ℓ
2 a z
Z−1 −
5
For the continuation to small j use
Q (z) = π(cotπℓ)P (z)+Q (z) (3.9)
−ℓ ℓ−1 ℓ−1
−
where
1
πcotπℓ (3.10)
− ℓ→∼ℓ0 (ℓ ℓ0)
−
with ℓ an integer.
The P = 0, kinematic singularity free Born amplitudes for =8 sugra and =4 YM are
N N
listed in Appendices A and B respectively. Throughout, the explicit factors of κ or g2 are omitted,
though it is obvious how to restore these if needed. Theamplitudes relevant for the Reggeization of
the gluon or graviton are the P=0, flavor singlet amplitudes, tabulated in (D.1)–(D.3) and (E.1)–
(E.6) respectively. The angular momentum projections near j = 1 and j = 2 are to be found in
(D.4)–(D.6) and (E.7)–(E.12) respectively.
We nowturntoprovidingadditional informationfor therelevant P=0, two-body amplitudes.
B. =4 YM
N
The P=0 states that contribute are 1, 1 > and 1/2a, 1/2 >; a = 1 to 4, i.e., λ=2
b
| − | −
and 1, with the former nonsense at j=1 and the latter sense, where the helicity 1/2 fermion has
“flavor” a. The kinematical singularity free amplitudes F˜(1, 1;1, 1); F˜(1/2a, 1/2 ;1, 1) and
b
− − − −
F˜(1/2a, 1/2 ;1/2 , 1/2d) are to be found in (B.1)–(B.3), which leads to a 2 2 scattering matrix
b c
− − ×
for P=0.
C. =8 sugra
N
As a result of the KLT relations [11], the tree amplitudes for the 4-point functions of =8
N
sugra can be expressed in terms of the square of the tree amplitudes of =4 YM. The relevant
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P=0 =8 states are 2, 2 >, 3/2A, 3/2 > and 1[AB], 1 >, A=1 to 8. Schematically the
B [CD]
N | − | − | −
flavor singlet, P=0 Born amplitudes we need are
M˜(2, 2;2, 2) F˜(1, 1;1, 1)F˜(1, 1;1, 1)
− − ∼ − − − −
M˜(2, 2;3/2, 3/2) F˜(1, 1;1, 1)F˜(1, 1;1/2, 1/2)
− − ∼ − − − −
M˜(3/2, 3/2;3/2, 3/2) F˜(1, 1;1, 1)F˜(1/2, 1/2;1/2, 1/2)
− − ∼ − − − −
M˜(2, 2;1, 1) F˜(1, 1;1/2, 1/2)F˜(1, 1;1/2, 1/2)
− − ∼ − − − −
M˜(3/2, 3/2;1, 1) F˜(1, 1;1/2, 1/2)F˜(1/2, 1/2;1/2, 1/2)
− − ∼ − − − −
M˜(1, 1;1, 1) F˜(1/2, 1/2;1/2, 1/2) F˜(1/2, 1/2;1/2, 1/2) (3.11)
− − ∼ − − − −
where the left-hand side are =8 sugra amplitudes, and the right-side are =4 YM amplitudes.
N N
Eqn. (3.11) implies a 3 3 P=0 scattering matrix. The structure (3.11) manifests itself in the
×
projections near J=2 for the sugra amplitudes, and its relationship to the projections near J=1 for
the YM amplitudes. This is evident in comparing (D.4)–(D.6) with (E.7)–(E.11).
Another way of presenting [1] the KLT relations [11] is
C C C
F˜ = s + t + u (3.12)
− s t u
(cid:20) (cid:21)
6
C2 C2 C2
M˜ = s + t + u (3.13)
−" s t u #
which can be confirmed using Appendices A and B.
D. The helicity matrices
Generically, for each j the Born approximation is of the form
v δ v (J j)−1/2
FJ = ss Jj sn − (3.14)
λµ v (J j)−1/2 v
ns nn !
−
which is the first term on the right-hand side of (2.8) given in terms of the submatrices v , v =
ss ns
v ; v .
sn nn
For =4 YM, we obtain the helicity matrix
N
v v
v = nn sn (3.15)
v v
ns ss !
fromtheP=0statesnearj = 1. Explicitvaluesarein(D.7)–(D.9), whichwerepeatforconvenience
v = 4
nn
16
v = v =
ns sn
√3
64
v = . (3.16)
ss
3
From (3.15) and (3.16) we have
rankv = 1
nn
rankv = 1
v = v (v )−1v . (3.17)
ss sn nn ns
Eqn. (3.17) combined with Mandelstam counting implies that the gluon must Reggeize.
For =8 sugra, the analogous helicity matrix for P=0 is [using V to avoid confusion with
N
(3.15)]
V V
V = nn sn (3.18)
V V
ns ss !
near j = 2. Explicit values are in (F.1)–(F.4), with the KLT relations [11] evident in (F.1)–(F.4).
Using (F.1)–(F.4), (3.12) and (3.15)–(3.17) we find that
rankV = 1
nn
rankV = 1 (3.19)
We conclude that the graviton must Reggeize, as a consequence of the Reggeization of the gluon!
7
4 Concluding Remarks
In this paper we have shown that the gluon of =4 YM theory lies on a Regge trajectory. The
N
factorization condition for this to hold, (3.17), also implies that the graviton of =8 sugra lies on
N
a Regge trajectory, c.f. (F.5)–(F.7). In Ref. [7] we only verified that the rank V = 1, but did not
nn
checkthefactorizationconditionforV,sincewepresumedthat =8sugrawasnon-renormalizable.
N
Here we verify that the factorization condition for =4 YM, (3.17) then also implies detV = 1
nn
N
and rank V = 1, leading to the conclusion that the graviton must Reggeize, which is consistent
with the speculation that =8 sugra is ultraviolet finite. It should be emphasized that this is not
N
a holographic result, as both theories are considered in perturbation theory.
The renormalizability vs. non-renormalizability of a field theory is not a trivial issue for the
Reggeization program, since at face value there are an infinite numberof free parameters for a non-
renormalizable theory, and thus Mandelstam counting would not apply. An example is massive
YM theory without the Higgs mechanism. It is known that the factorization condition (2.12) fails
in this case [9], and thus there the gluon does not Reggeize. However, =8 sugra appears to evade
N
the difficulties exampled by that massive YM example.
Thecomputationpresentedinthispaperisequivalenttotheleadinglogarithmapproximation
in the summation of an infinite set of diagrams. It is therefore reasonable to expect that a leading
logarithm summation of diagrams in =8 graviton-graviton scattering will reproduce our results.
N
The Regge trajectories, computed in perturbation theory, will not lead to recurrences in weak-
coupling, but rather are analogous to the Regge trajectories of potential scattering. All orders in
perturbation theory continued to strong-coupling, may well produce Regge recurrences.
TheotherfundamentalfieldsoftheLagrangiansof =8sugraand =4YMshouldReggeize
N N
as well, as a consequence of the unbroken SU(8) and SU(4) flavor symmetries (respectively) of the
theories.
Further exploration of the possible implications of an ultraviolet finite =8 sugra promises
N
to be a fruitful enterprize. It is an important issue of principle to know whether or not =8 sugra
N
is a finite quantum theory of gravity distinct from string theory. Given that =8 sugra contains
N
Regge poles, we speculate that is is not distinct.
Acknowledgements
We thank Albion Lawrence for stimulating conversations. We are also appreciative of our old
collaboration with Marc Grisaru on this subject.
8
Appendix A
=8 sugra: Kinematical Free Amplitudes: P = 0
N
s2 1 1
M˜(2, 2;2, 2) = − + , (A.1)
− − 16 t u
(cid:20) (cid:21)
3A 3 3 3A
M˜ − , − ;2, 2 = M˜ 2, 2; , −
2 2 B − ! − − 2B 2 !
δAs2 1 1
= − B + , (A.2)
16 u t
(cid:20) (cid:21)
3A 3 3 3D 1 s3 δAδD δAδD
M˜ , − ; , − = B C + C B , (A.3)
2 2 B 2C 2 ! 8 u !" s t #
M˜(1AD, 1 ;2, 2) = M˜(2, 2;1 , 1BC)∗
BC AD
− − − −
s2 1 1
= − + δAD , (A.4)
16 t u BC
(cid:18) (cid:19)
∗
3A 3 3 3B
M˜ , − ;1 , 1DE = M˜ 1CF, 1 ; ,−
CF DE
2 2 B − ! − − 2A 2 !
1 s3 1 1
= [δAδDE δAδDE] δAδDE , (A.5)
8 u !(cid:26)t F BC − C BF − s B CF(cid:27)
where throughout (*) is an SU(8) conjugation which raises and lowers indices only:
s2 1 1
M˜(1AH, 1 ;1 , 1FG) = − εAHFGMNOPε
BC DE BCDEMNOP
− − 4 u4!
(cid:26)
1 1
+ δAHδFG + δAHδFG (A.6)
s BC DE t DE BC
(cid:27)
with δAB = δAδB δAδB , (A.7)
CD C D − D C
δABC = δAδBC +δBδCA +δCδAB
DEF D EF D EF D EF
1
= εABCGHKLMε , (A.8)
DEFGHKLM
5!
1
δABDDE = εABCDEMNOε (A.9)
FGHKL 3! FGHKLMNO
where the flavor indices are A= 1 to 8.
9
Appendix B
=4 YM: Singularity-Free Amplitudes
N
Pseudohelicity (0)
α β
F˜(1, 1;1, 1) = s + (B.1)
− − t u
(cid:20) (cid:21)
1a 1 α β
F˜ , ;1, 1 = δa s + (B.2)
2 −2b − b t u
(cid:18) (cid:19) (cid:20) (cid:21)
1a 1 1 1d α β
F˜ , ; , = 2 sδaδd+tδaδd + (B.3)
2 −2b 2c 2 ! h c b b ci(cid:20)t u(cid:21)
where
α = f f
ikn njℓ
β = f f (B.4)
iℓn njk
are products of the structure constants of the gauge group, and the flavor indices are a= 1 to 4.
10