Asymptotic Freedom in Yang-Mills
from Open String Loops
(arXiv:0808.0458)
Charles Thorn
IAS, University of Florida
BNL, 8 May 2009
Introduction: Field/String Duality
Much to learn from close relationship of string and field theory.
AdS/CFT asserts equivalence, with QFT the simpler description
for N g 2 ≪ 1, and string description simpler for N g 2 ≫ 1
My work with Bardakci attacks N g 2 = O(1) by mapping planar
diagrams of Yang-Mills to a worldsheet template.
Summing planar open string multi-loop diagrms no harder in
this framework, but conceptually more promising
Today’s talk devoted to exploring the field theory limit of the
open string planar 1-loop diagrams.
1
Brief Review of Field/String Duality
’t Hooft’s N → ∞ (SU(N) Chan-Paton):
X
X
(Planar Open String Loops)D3 ≡
(Closed String Trees)D3source
Left Side =⇒
N = ∞ Gauge Theory in 4d
′
α →0
Right Side
=⇒
α′ →0
g 2 N →∞
Classical gravity
If g 2 N = O(1), right side stays stringy as α′ → 0.
I.e. must solve classical closed string field eqs.
My suggestion:
N → ∞ QCD by direct planar graph summation at α′ > 0.
2
Simplest Open String for Pure 4D Yang-Mills
Even G-parity sector of Neveu-Schwarz open string in D = 4
No fermion R sector, no extra dimensions
Gauge group: SU(N) Chan-Paton factors
Rest of talk: One-loop Yang-Mills from NS+ Model
Method of Metsaev&Tseytlin applied to bosonic string:
Nucl Phys B298 (1988) 109.
The following 23 slides include a complete calculation
of asymptotic freedom: Don’t try to follow every detail!
I will make them available.
3
Some Notation
±
=
µ
Klm
≡
k
k0 ± k3
k 1 + (−)ik 2
∧(∨)
√
√
,
k
=
2
2
µ
+ µ
kl+ km
− km
kl
for each pair (l, m)
Let ǫµm be a gluon polarization with ǫ+
m = 0.
∧ ∨
∨ ∧
+
Then ǫ−
m = (km ǫ + km ǫ )/km
∨ ∧
∧ ∨
ǫm
ǫm + Kml
Kml
k l · ǫm =
+
km
P3
Consider 3 lightlike momenta with i=1 kiµ = 0
∧,∨
∧,∨
∧,∨
for µ = ∧, ∨, +. Then K12
= K23
= K31
, and
3
X
i=1
ki−
∧
∨
K12
K12
= + + +
k1 k2 (k1 + k2+ )
∧
Conservation ⇔ K12
=0
4
Gluon Vertex Operator in NS+ Model
r
√
2
′ k · Hǫ · H)eik·x
V (k, ǫ) = g
(ǫ
·
P
+
2α
α′
= h0, k1 |ǫ1 · b1/2 V (k2 , ǫ2 )ǫ3 · b−1/2 |0, k3 i
1
2
= 2g(ǫ1 · ǫ3 k3 · ǫ2 + k2 · ǫ1 ǫ2 · ǫ3 − k2 · ǫ3 ǫ1 · ǫ2 )
5
Choose Lightcone gauge ǫ+
i = 0; then
ǫ∧
1
ǫ1 · ǫ2
∨
∨
∨
∧
= ǫ∧
=
ǫ
=
0,
ǫ
=
ǫ
=
ǫ
2
3
1
2
3 =1
= 0, ǫ1 · ǫ3 = ǫ2 · ǫ3 = 1
k1+ + k2+ ∧
= 2g + + K12
k1 k2
1
2
Taking this vertex in planar graphs (Nc → ∞) shows that
gs2 Nc
g2
αs Nc ≡
=
4π
2π
gs is conventional QCD coupling,
Nc is number of colors.
6
NS+ at 1 loop, D < 10 (Goddard and Waltz, 1971)
One loop planar M gluon NS+ amplitude for D < 10:
A1
loop
= (g
√
2α′ )M
M+ − M−
2
where, in cylinder variables, ln q = 2π 2 / ln w,
7
M
+
= 2
r
1
8π 2 α′
D/2 Z Y
M
k=2
dθk
Z
1
0
dq
q
−π −(D−1)/8 (D−9)/16
q
w
(1 − w1/2 )
ln q
Q
(1 + q 2r )D−1 Y
2α′ kl ·km
r
Q
[ψ(θm − θl , q)]
2n
D−1
n (1 − q )
l<m
hP̂1 P̂2 · · · P̂M i+
q −(D−1)/8 reflects closed string tachyon: α′ p2 = (D − 1)/4
Graviton is massive: α′ p2 = (D − 1)/4 − 2 = (D − 9)/4
8
M
−
= 2
r
1
8π 2 α′
D/2 Z Y
M
k=2
dθk
Z
1
0
dq
q
−π (D−1)/2 (D−9)/16
2
w
(1 + w1/2 )
ln q
Q
2n D−1 Y
(1
+
q
)
2α′ kl ·km
Qn
[ψ(θm − θl , q)]
2n )D−1
(1
−
q
n
l<m
hP̂1 P̂2 · · · P̂M i−
where
θ Y (1 − q 2n eiθ )(1 − q 2n e−iθ )
ψ(θ, q) = sin
2 n
(1 − q 2n )2
√
P̂ = ǫ · P + 2α′ k · Hǫ · H
Range of integration: 0 = θ1 < θ2 < · · · < θN < 2π
9
h· · ·i evaluated via Wick expansion with contractions:
"
#
∞
2n
X
X
√
1
2q
θil
ki
2α′
hPl i =
cot
+
sin nθil
2n
2
2
1−q
n=1
i
∞
X
2q 2n
1
2 θil
−
n
cos nθil
hPi Pl i = hPi ihPl i + csc
2n
4
2
1
−
q
n=1
hHi Hj i
hHi Hj i
+
−
=
=
X q 2r sin rθji
1
−2
2r
2 sin(θji /2)
1
+
q
r
X q 2n sin nθji
cos(θji /2)
−2
2n
2 sin(θji /2)
1
+
q
n
Space-time indices are suppressed
In these formulas r ranges over positive half odd integers, n over
positive integers, and l, m ∈ [1, · · · , M ].
10
The UV Cutoff
Metsaev and Tseytlin: − ln w is like a Schwinger parameter T :
Z ∞
α′
−(L0 −1/2)T /α′
=
dT e
L0 − 1/2
0
UV cutoff:
T >
1
1
2 ′ 2
⇔
−
ln
w
>
⇔
−
ln
q
<
2π
αΛ
2
′
2
Λ
αΛ
Field Theory limit is α′ → 0 at fixed Λ.
11
1 Loop 2 Gluon Amplitude
hP̂1 P̂2 i = hǫ1 · P1 ǫ2 · P2 i + 2α′ hk1 · H1 ǫ1 · H1 k2 · H2 ǫ2 · H2 i
"
∞
X
2nq 2n cos nθ
1
2 θ
csc
−
hǫ1 · P1 ǫ2 · P2 i = ǫ1 · ǫ2
4
2 n=1 1 − q 2n
"
#2
∞
1
θ X 2q 2n sin nθ
′
−2α k2 · ǫ1 k1 · ǫ2
cot +
2
2 n=1 1 − q 2n
#
hk1 · H1 ǫ1 · H1 k2 · H2 ǫ2 · H2 i = (k2 · ǫ1 k1 · ǫ2 − k1 · k2 ǫ1 · ǫ2 )C ±
"
#2
2r
X q sin rθ
1
+
C
=
−2
2r
2 sin(θ/2)
1
+
q
r
"
#2
2n
X q sin nθ
cos(θ/2)
−
C
=
−2
2n
2 sin(θ/2)
1
+
q
n
12
Divergences in θ Integral
Momentum conservation: k2 = −k1 , k1 · k2 = ki · ǫj = 0
θ integral reduces to:
#
Z 2π "
∞
2n
X 2nq cos nθ
1
2 θ
ǫ1 · ǫ2
csc
−
=∞
dθ
2n
4
2
1
−
q
0
n=1
Goddard and Neveu/Scherk suggest calculating with
k1 + k2 = p 6= 0, and p → 0 at the end. (GNS regularization)
Z 2π
Z π
′
dθ
dθ
2 θ
2α k1 ·k2
α′ p2 −2
csc ψ(q, θ)
=
(sin θ)
+ O(p2 )
4
2
2
0
0
Γ(1/2)Γ(−1/2 + α′ p2 /2)
2
+
O(p
)
=
2Γ(α′ p2 /2)
∼ −πα′ p2 /2 + O(p2 )
As first noted by Neveu and Scherk for the bosonic string (1972)
13
Poles in p2
Γ(−1/2 + α′ p2 /2) has poles: α′ p2 = 1 − 2n, n = 0, 1, 2, . . .
p
k1
k2
NS Model provides a factor 1 − α′ p2 , killing tachyon pole.
Vanishing of this diagram at p = 0 ⇒ gluon massless.
All statements hold at finite α′ > 0.
14
M+
2 ∼
Z
[dq]+ πα′ (ǫ1 · ǫ2 p2 − 2p · ǫ1 p · ǫ2 )


∞
∞
2n
2r
X
X
1
4q
4q

× − +
+
2n
2
2r
2
2 n=1 (1 − q )
(1 + q )
r=1/2
M−
2 ∼
Z
[dq]− πα′ (ǫ1 · ǫ2 p2 − 2p · ǫ1 p · ǫ2 )
#
"∞
∞
2n
2n
X
X
4q
4q
+
×
2n )2
2n )2
(1
−
q
(1
+
q
n=1
n=1
Here [dq]± is shorthand for dq times
the θ independent factors of the integrands.
15
1-loop 3 Gluon Amplitude
Important point: this amplitude includes both 1-particle reducible
and irreducible contributions:
=
=
=
Will normalization of reducible contributions be that of S-matrix
or that of Green function?
(Recall S-matrix wave function factor in QFT is Z n/2 , whereas
Green function supplies Z n .)
Shall see GNS reg gives precisely S-matrix normalizaton!
16
√
Coefficient of ǫ1 · ǫ2 2α′ k1 · ǫ3 due to bosonic variables:
Z
Z 2π
Z θ3
3
CBose
=
[dq]
dθ3
dθ2
0
0
"
#
∞
2n
X
2q
1
2 θ2
csc
−
n
cos nθ2
2n
4
2
1
−
q
n=1
#
"
∞
2n
X 2q
sin(θ2 /2)
+
(sin nθ32 − sin nθ3 )
2n
2 sin(θ3 /2) sin(θ32 /2) n=1 1 − q
ψ(q, θ2 )
2α′ k1 ·k2
ψ(q, θ3 )
2α′ k1 ·k3
ψ(q, θ32 )
2α′ k2 ·k3
The whole answer for bosonic string. Need fermi part for NS+
Metsaev and Tseytlin postulate: “1PIR amp” obtained
by setting exponents to zero and replacing singular terms
by their formal expansions
17
1
2 θ
csc
4
2
→
−
∞
X
n cos nθ,
n=1
∞
X
θ
1
cot
→
sin nθ
2
2
n=1
Then the θ integrals are elementary with the result
#
"
∞
∞
2
X 1 + q 2n
X
q 2n
1
2π
→ 2π − + 4
2n
2n )2
1
−
q
2
(1
−
q
n=1
n=1
P
where the formal sum n 1 has been interpreted as ζ(0) = −1/2.
Prescriptions give the correct result in gauges where Z1 = Z3 .
Unsatisfactory points:
• 1PIR not a Gauge Invariant concept: Need calculate
complete contribution to S-matrix.
• More care needed for θ near 0, 2π.
18
Handling θ divergences
Most singular part: Put
α1
ki2
= 2α′ k1 · k2 ,
α2 = 2α′ k1 · k3 ,
= 0,
k1 + k2 + k3 = p
α1 = 2α′ k2 · k3
Then (Neveu and Scherk)
π
θ3
1
dθ3
dθ2 [sin θ2 ]α1 −1 [sin θ3 ]α2 −1 [sin θ32 ]α3 −1 =
2
0
√ 0
π Γ(−(1 − α′ p2 )/2)Γ(−α1 /2)Γ(−α2 /2)Γ(−α3 /2)
4 Γ(−(α1 + α2 )/2)Γ(−(α2 + α3 )/2)Γ(−(α3 + α1 )/2)
π
3
→ + = −π 1 −
,
for p → 0
2
2
Z
Z
Compared to −π for corresponding term in “1PIR”
19
3
1PR contributions to CBose
Contributions discarded in 1PIR formally O(p2 ), but enhanced
by pole singularities from integration near θ2 , θ32 , 2π − θ3 ∼ 0
E.g. integrating near θ2 ∼ 0 gives
#
"
Z 2π
∞
2n
X
1
2q
1
2 θ3
csc
−
cos nθ3
n
dθ3
2n
2α′ k1 · k2 0
4
2
1
−
q
n=1
#2α′ (k1 +k2 )·k3
"
∞
θ3 Y (1 − q 2n eiθ3 )(1 − q 2n e−iθ3 )
sin
2 n=1
(1 − q 2n )2
"
#
∞
2n
X
p · k3
q
1
∼ −π
− +4
2n )2
p · k3 + p2 /2
2
(1
−
q
n=1
#
"
∞
2n
X
q
1
→ −π − + 4
2n )2
2
(1
−
q
n=1
The other two singular regions give the same.
20
3
CBose
"
∞
X
#
q 2n
1
→ (2 − 3) π [dq] − + 4
2n )2
2
(1
−
q
n=1
"
#
Z
∞
X
1
q 2n
→ −π [dq] − + 4
2n )2
2
(1
−
q
n=1
Z
Note that agreement of the π/2 term with it’s exact
evaluation
(explained earlier) confirms the interpretation of
P
“ n 1” as ζ(−1/2) by Metsaev and Tseytlin.
Contributions of the H correlators all notionally of order p2 ,
and require enhancement by pole singularities.
21
We simply quote the final result for fermi contributions:


Z
∞
X
q 2r
3,+
+

CFermi → −π [dq] 4
(1 + q 2r )2
r=1/2
#
"
Z
∞
2n
X
q
3,−
− 1
+4
CFermi → −π [dq]
2n )2
2
(1
+
q
n=1
and the total contributions:


Z
∞
∞
2n
X
X
q 2r
1
q
3,+
+

C
→ −π [dq] − + 4
+4
2n
2
2r
2
2
(1 − q )
(1 + q )
n=1
r=1/2
" ∞
#
Z
∞
X
X
q 2n
q 2n
3,−
−
C
→ −π [dq] 4
+4
2n
2
2n )2
(1
−
q
)
(1
+
q
n=1
n=1
Quantities in [· · ·] identical to those in 2 gluon function!
22
Field Theory Limit
Need q integration near q ∼ 1 or w < exp(−1/α′ Λ2 )
[dq]±
=
∼
√
1+D/2
√ Q
r D−1
1 ∓ w (1 ± w )
2(g
dw
2π
√
−
(1 − wn )D−1
ln w
2w
w
(8π 2 α′ )D/2
√
1+D/2
√ 3
′
2π
2(g 2α ) 1 ± (D − 2) w
dw
√
−
ln w
2w
w
(8π 2 α′ )D/2
2α′ )3
Last line is valid for w ≪ 1. Compare bosonic string measure:
√
1+D/2
1 + (D − 2)w
dw
2(g
2π
−
[dq]B ∼
w
ln w
2w
(8π 2 α′ )D/2
2α′ )3
in the same limit.
23
Also need:
X
q 2n
4
(1 − q 2n )2
n
4
X
r
4
X
n
∼
1 ln w ln2 w
ln2 w
2
+
+
−
w
+
O(w
)
6
2π 2
24π 2
π2
q 2r
(1 + q 2r )2
2
ln
w
ln w
∼ − 2 − w1/2 2 + O(w)
2π
π
q 2n
(1 + q 2n )2
2
ln
w
1 ln w
1/2
∼ − −
+w
+ O(w)
2
2π 2
π2
Which enter in the combinations:
√ 1
q
w
1
1
q
2
+
− +4
+
4
∼
−
−
ln
w
2n
2
2r
2
2
2
2
(1 − q )
(1 + q )
3
24π
π
n=1
r=1/2
√ ∞
∞
2n
2n
X
X
1
1
q
w
q
2
+
ln
w
+
4
∼
−
+
4
2n
2
2n
2
2
2
(1 − q )
(1 + q )
3
24π
π
n=1
n=1
∞
X
2n
∞
X
2r
24
C3+
−
2
C3−
√
′ 2
1+D/2
3 Z exp(−1/α Λ )
′
dw
(g 2α )
2π
∼ −π
−
2w
ln w
(8π 2 α′ )D/2 ǫ
√ √ 1
w
1
1 + (D − 2) w
2
√
−
ln
w
− +
2
2
3
24π
π
w
√ √ 1
1
1 − (D − 2) w
w
2
√
− +
−
+
ln
w
3
24π 2
π2
w
√
′ 2
1+D/2
3 Z exp(−1/α Λ )
′
(g 2α )
dw
2π
∼ π
−
w
ln w
(8π 2 α′ )D/2 ǫ
D − 2 D − 26 2
−
ln w
3
24π 2
1PIR part (a la Metsaev/Tseytlin) of (D − 26) is −2(D − 2)
1PR = D − 26 + 2(D − 2) = 3(D − 10)
25
Put ln w = −1/α′ λ, dw/w = dλ/α′ λ2 , ln ǫ = −1/α′ λ0 :
C3+
−
2
C3−
3
∼
∼
for D = 4. The tree
gR
=
Nc αs (λ0 )
π
=
2−D/2
Λ2
g (4π)
dλ
√
λ
2 2α′
λ0
D − 2 ′2 D/2 D − 26 (D−4)/2
α λ
−
λ
2
3
24π
3
2
g
1 ′2 4
22
Λ
√
α Λ +
ln
24π 2
λ0
2 2α′ 3
√
contribution to this quantity is 2g/ 2α′
2
2
Λ
11g
4
+
O(g
)
ln
g 1+
2
48π
λ0
2
1
gR
12
=
+O
2π 2
11 ln λ0 /M 2
ln2 λ0 /M 2
26
Z
Summary
• Studied field theory limit of 1-loop NS+ model
• UV cutoff on q: − ln q < 2π 2 α′ Λ
• GNS regularization, used throughout, consistently handles all the formally IR divergent θ integrations.
• Gluon self-energy vanishes and coupling renomalization
correctly given
• This study is preliminary to setting up a systematic
formalism to sum planar open string diagrams by,
for example, employing the lightcone worldsheet.
27