Structure Constants in N 4SYM and Separation of Variables Carlos Bercini Instituto de F sica Te orica UNESP ICTP South American Institute for Fundamental Research

2025-05-02 0 0 842.61KB 14 页 10玖币
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Structure Constants in N= 4 SYM and Separation of Variables
Carlos Bercini
Instituto de F´ısica Te´orica, UNESP, ICTP South American Institute for Fundamental Research,
Rua Dr Bento Teobaldo Ferraz 271, 01140-070, S˜ao Paulo, Brazil
Alexandre Homrich
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Perimeter Institute for Theoretical Physics, 31 Caroline St N Waterloo, Ontario N2L 2Y5, Canada
Pedro Vieira
Perimeter Institute for Theoretical Physics, 31 Caroline St N Waterloo, Ontario N2L 2Y5, Canada
Instituto de F´ısica Te´orica, UNESP, ICTP South American Institute for Fundamental Research,
Rua Dr Bento Teobaldo Ferraz 271, 01140-070, S˜ao Paulo, Brazil
We propose a new framework for computing three-point functions in planar N= 4 super Yang-
Mills where these correlators take the form of multiple integrals of Separation of Variables type. We
test this formalism at weak coupling at leading and next-to-leading orders in a non-compact SL(2)
sector of the theory and all the way to next-to-next-to-leading orders for a compact SU(2) sector.
We find evidence that wrapping effects can also be incorporated.
I. INTRODUCTION
Solving planar N= 4 SYM in a satisfactory way would
mean efficiently computing both the spectrum as well as
higher point correlation functions – starting with three
points – at any value of the ’t Hooft coupling.
A formalism for computing three point functions by
means of integrability exists. It is the so called hexagon
approach [1]. It is conjectured to hold for any coupling
indeed but it is not easy to use, at least not by the re-
markable standards of the spectrum quantum spectral
curve approach [2]. With hexagons one needs to go over
infinitely many sums and integrals to produce such corre-
lators. At weak coupling perturbation theory these sums
and integrals truncate [3–5] and we can use hexagons to
produce a wealth of data to test any putative new frame-
work. We will do it all the time below.
Here we suggest a new approach for correlation func-
tions in N= 4 SYM based on the Baxter Q-functions.
The final representations are of so-called separation of
variables (SoV) type where these Baxter functions are
integrated against simple universal measures to produce
the structure constants.
Q-functions play the central role in the quantum spec-
tral curve, the top of the line tech for computing the
dimension of any single trace operator in this conformal
gauge theory so it is only natural to look for a similar cen-
tral role for these objects in the context of other physical
observables such as the OPE structure constants.
In the most conventional integrable spin chains, Q-
functions are polynomials whose roots are the so called
Bethe roots vk. In SYM these polynomials are present
at leading order at weak coupling but at higher cou-
pling they get dressed by quantum non-polynomial fac-
tors. This is expected; as we crank up the coupling we
are no longer dealing with a spin chain or with a classi-
cal string but something in between and so these Baxter
FIG. 1. Three point functions of operators with sizes L1,
L2and L3. At tree level there are `ij = (Li+Lj−Lk)/2
propagators between operators iand j; these integers `ij are
called the bridge lengths. We will usually use L≡L1for the
length of the non-BPS operator and `≡`12 for its so-called
left adjacent bridge (the right adjacent bridge will have length
L−`); the bottom bridge we will often denote as `B≡`23.
polynomials get naturally deformed. For the so-called
SL(2) sector of the theory which includes all operators of
the schematic form Tr(DJZL) + permutations we have
Q(u)≡
J
Y
k=1
u−vk
qx+
kx−
k
eg2
2Q+
1H+
1(u)+ g2
2Q−
1H−
1(u)+O(g4)
(1)
where the charges Q±
1are simple functions of the
Zhukowsky variables x±
k=x±(vk) and H±
kare harmonic
functions, see appendix A 1.
An operator is given by a Q-function (1). What we
are after is thus a functional eating these functions and
spitting out a number, the structure constant.
arXiv:2210.04923v2 [hep-th] 23 Nov 2022
2
For the most part we will consider a single non-BPS
operator and two BPS operators. The geometry of the
three point function depends on the size of all these op-
erators or – equivalently – on the so-called bridges that
connect them as reviewed in figure 1.
The proposal is that (the square of) the structure con-
stant is given by a ratio of SoV like scalar products
C2
•◦◦ =(J!)2
(2J)! hQ,1i2
`
hQ,QiL
(2)
Here J=J+g2Q+
1and the scalar product
hQ1,Q2i`≡J1+J2+`−1
`−1Zdµ`−1
`−1
Y
i=1 Q1(ui)Q2(ui)
(3)
with a nice factorized measure
dµ`=
`
Y
i=1
duiµ1(ui)
`−1
Y
i=1
`
Y
j=i+1
µ2(ui, uj) (4)
which is constructed out of the building blocks (using,
su, cu, tufor sinh(πu),cosh(πu),tanh(πu))
µ1(u)= π
2c2
u1 + g2π23t2
u−1+...(5)
µ2(u, v)= π(u−v)su−v
cucv1+g2π2(tu+tv)2−4
3+...(6)
valid to leading order (LO) and next-to-leading order
(NLO); the dots in the above expressions are NNLO cor-
rections which start at O(g4). The structure of the result
might be corrected at subleading orders as explained be-
low. The contours in (2) are the real axis for all ui.
We can prove (2) by exhaustion by comparing it
with hexagon produced data for numerous L’s and `’s
and for different operators corresponding to different Q-
functions. We did it; (2) is correct. We can also establish
it more honestly as discussed in the next section.
Representations like (2) are the main results of this
letter. In section III we present an SU(2) counterpart
of this representation in (19); we managed to fully test
it to LO, NLO and NNLO. We compare these two rank
one sectors in section IV. In the discussion section V we
discuss further generalizations such as multiple spinning
operators and speculate on all loop structures we expect
to find. Many appendices complement the main text.
II. SL(2)
At leading order, that is at tree level, correlation func-
tions are given by Wick contractions. Each operator can
be through of as a spin chain and these Wick contrac-
tions are thus given by spin chain scalar products [6–8].
For the SL(2) sector such scalar products can be cast as
SoV integrals [9] – see also [10] in the N= 4 context.
Once we properly normalize all these scalar products as
in [7] to extract the structure constant we precisely end
up with (2) for g= 0.
What we find remarkable is that this classical g= 0 ex-
pression seems to have a nice quantum lift as anticipated
in the introduction.
Let us first discuss the simplest possible case where a
twist two operator (L= 2) splits evenly into two BPS
operators (`= 1) so that the proposal (2) simply reads
C2
•◦◦ =J!2
(2J+ 1)! Zdu µ1(u)Q(u)2−1
.(7)
We derived the single-particle measure µ1as well as the
quantum deformed Baxter functions Qin two ways. Both
are based on looking for a measure such that, for two
different twist-two states (i.e. with different spins Jand
J0), we have an orthogonality relation
Zµ1(u)QJ(u)QJ0(u)∝δJJ0,(8)
a powerful relation which has been extensively exploited
by [11–13] in numerous SoV studies, most of which for
rational spin chains or for the fishnet reduction [14] of
SYM, see most notably [15].
The first uses the fact that the Bethe roots vkof
twist two operators are given by the zeros of Hahn
polynomials, a fact that persists at NLO. We have
that Qk(u−vk) is proportional to the Hahn polyno-
mial [16, 17] pJ(u|a, b, b, a) where a−b= 2i√2gand
a+b= 1 + 4g2H1(J). Hahn polynomials are orthogonal
with a simple measure as reviewed in appendix B 1 which
allows us to derive (5) up to some simple tuning due to
the mild Jdependence in the polynomial parameters a, b,
see appendix for details. It is the Jdependence that ren-
ders the derivation non-trivial and which is responsible
for the needed modification of the Baxter functions in (1).
The second derivation follows [12] closely (the novelty
being the extension to g2corrections) and makes use of
the Baxter equation [18]
B ◦ Q=T(u)Q(u) (9)
where Q(u) are the Baxter polynomials (i.e. just the
parentheses in (1)) and the Baxter operator
B ≡ (x+)L1−g2
x−(u)(Q+
1−iQ−
1)ei∂u+c.c. (10)
Note that at g= 0 we have x+=u+i/2 and most impor-
tantly, Bbecomes a simple linear operator but as we turn
on gcorrections this is no longer true since the charges
Q±
1depend on Q. Consider first g= 0 and L= 2 so that
the transfer matrix is T(u)=2u2+cJ. Multiplying (9)
by the sought after measure µ1and by another Baxter
polynomial with a different spin, subtracting that to the
same thing with the spin swapped and integrating yields
(cJ−cJ0)Zµ1QJQJ0=Zµ1QJ0(B◦QJ)−(B◦QJ0)QJ.
(11)
3
If we manage to make Bself-adjoint we will thus have
the required orthogonality. An i-periodic µ1would do
the job since under shifts of contour the two terms
of Baxter would swap and cancel in the right hand
side. In detail, QJ0(u)(u±i
2)2QJ(u±i)µ1(u) becomes
QJ0(u∓i)(u∓i
2)2QJ(u)µ1(u∓i) under a shift of contour
by ∓ileading to an interchange (and thus cancellation)
of the two terms in (11) once we use that the measure
is periodic. To make this manipulation kosher we need
to make sure no singularities are picked when deforming
the contour and to make the measure acceptable we need
to make sure it decays fast enough at infinity so that it
can be integrated against polynomials of arbitrary de-
gree. Both this problems are solved at once with
µ1(u) = π/2
cosh2(πu)+O(g2).(12)
The function decays faster than any polynomial and the
double poles at ±i/2 precisely cancel the double zeroes
in the potential terms (u∓i/2)Lwhen L= 2 so that
they lead to no extra contribution when deforming the
contours. A periodic function without these double poles
would not decay fast enough and a function with more
than double poles would lead to extra contributions when
deforming the contours; (12) is the sweet spot.
Turning on gcorrections is not complicated. The re-
definition (1) brings the Baxter operator to a linear op-
erator again but introduces some further poles at ±i/2
in the (no longer polynomial) Baxter Qfunctions so that
the measure now needs some extra poles to cancel the
contribution of these when deforming the contour. This
explains (1) as well as (5); more details in appendix B 2.
Having derived the measure µ1and the Baxter poly-
nomial dressing it remains to fix the overall normaliza-
tion of the structure constant to be sure everything is in
order. Evaluating the SoV integral using the loop cor-
rected Bethe roots for various spins immediately leads to
the cute result
Zdu µ1(u)Q(u)2−1
=1 + g24H2(J) + 8H1(J)
2J+1 +. . .
2J+ 1
(13)
which is indeed almost the correct loop level structure
constant computed in [19]. The prefactor in (7) with J
deformed into Jin a reciprocity reminiscent fashion [20]
neatly combines with this expression to give the full NLO
structure constants for twist-two operators.
At this point it is straightforward to guess the general
structure constant for any SL(2) operators of any twists
by simply taking the tree level g= 0 result and deform-
ing the new ingredient for higher twists – the two particle
measure µ2– by a bunch of hyperbolic tangents follow-
ing what was derived for twist two. The coefficients of
these hyperbolic tangents are then fixed by requiring or-
thogonality between any two different Baxter solutions.
The last line deformation of the Baxter functions in (1) –
which was invisible for twist two operators for which odd
charges vanish – is also fixed by imposing orthogonality.
In the end, we just need to check that with a minimal
reciprocity friendly prefactor as in (2) we precisely agree
with perturbative data produced by hexagons. We do.
Even without matching with data, there is a nice self-
consistency check of the full construction including the
deformed pre-factor: The structure should be invariant
under swapping left and right bridges `↔L−`; we
checked that this is indeed realized by our expressions
once the prefactors are deformed as in (2).
We made some progress at higher loops, in particular
at NNLO (two loops) and for the smallest possible sizes
and bridge lengths. For twist L= 2 operators for ex-
ample we found a Baxter function dressing as well as an
orthogonal one-particle measure realizing (8) as
Q(u)≡
J
Y
k=1
u−vk
qx+
kx−
k×e1
2g2Q+
1H+
1+1
8g4Q+
1Q+
2H+
1−1
2g4Q+
1H+
3
and
µ1(u) = π
2c2
u1 + π2g23t2
u−1+π4g45
6−7t2
u+11
2t4
u
−g4
8H+
1Q+
1(v1)Q+
2(v2) + Q+
1(v2)Q+
2(v1),(14)
where we use the fact that Q−
k= 0 for twist-2 operators.
The last line is exotic as it depends now on the charges
of the two operators in (8) with Bethe roots v1and v2
respectively. Note, however, that when considering the
pairings hQ,1iand hQ,Qi a single Baxter function shows
up and thus this mixing term can be absorbed as new fac-
tor dressing Q. One would then have different dressings
in each pairing, a phenomenon we observe in the next
section as well (in the SU(2) sector).
The first line in (14) is also not any random combi-
nation of trigonometric functions. Take the tree level
measure 1/cosh2(πu) – which is periodic with period i
and has poles at all the imaginary half integers – and
promote it to a periodic function where all these poles
are opened up into small cuts following [15], see figure 2,
ˆµ1≡Idv
2πi
π/2
cosh2(π(u−v))
1
x(v).(15)
The integration contour encircles the Zhukowsky cut v∈
[−2g, 2g]. Evaluating this integral in perturbation the-
ory precisely reproduces the first line in (14) up to an
overall normalization constant! It is tempting to conjec-
ture that the finite coupling expression (15) might play
an important role in an all loop SoV formulation. We
make further comments on NNLO structure constants in
the discussion.
Other sectors such as the SU(2) sector might also hint
at other important structures in a putative all loop for-
mulation. This is what we turn to now.
摘要：

StructureConstantsinN=4SYMandSeparationofVariablesCarlosBerciniInstitutodeFsicaTeorica,UNESP,ICTPSouthAmericanInstituteforFundamentalResearch,RuaDrBentoTeobaldoFerraz271,01140-070,S~aoPaulo,BrazilAlexandreHomrichDepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,Ontario,N2L3G1,CanadaPe...

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