Monopoles, bions, and other oddballsin confinement or conformality
In collaboration with E. Poppitz
Mithat Unsal, SLAC, Stanford University
arXiv:0906.5156, arXiv:0812.2085, Index theorem for topological excitations on R**3 x S**1
arXiv:0910.1245, Conformality or confinement (II): One-flavor CFTs and mixed-representation QCD
Earlier collaborators on related topics: M. Shifman, L. Yaffe
1Monday, February 1, 2010
This is a talk about nonperturbative gauge dynamicsThings that one would like to understand in any gauge theory:
- does it confine? how? why? - does it break its (super) symmetries?- is it conformal? Why?- what are the spectrum, interactions...?
tough to address, in almost all theories but relevant:
2Monday, February 1, 2010
• Hadron physics, dynamics of YM, QCD-like and chiral theories
• Problem of electro-weak symmetry breaking (EWSB), scenarios in TeV scale physics. (where Higgs is a composite) to be tested at LHC.
• e.g. technicolor: naive scaled-up QCD fails with EW precision data, fails to produce acceptable spectrum. (walking, conformal...)
• Theories with interesting long distance behavior, scale invariance. (frustrated spin systems?)
Motivations
3Monday, February 1, 2010
what I’ll talk about applies to any of the above theories
SUSY - very “friendly” beautiful - exact results
pure YM - formal
QCD-like - hard, leave it to lattice folks chiral limit $$$
non-SUSY chiral gauge theories - even lattice not practical ...nobody talks about them anymore
conventional wisdom:
4Monday, February 1, 2010
• •confined
Nf!!IR!freeIR!CFT "
NAFfN!
f
Conformality or confinement problem
Mechanisms of confinement and conformality: What distinguishes two theories, one just below the conformal boundary and confines, and the other slightly above the conformal window boundary? In other words, why does a confining gauge theory confine and why does an IR-CFT, with an almost identical microscopic matter content, flows to a CFT?
Lower boundary of conformal window: What is the physics determining the boundary of conformal window?
Conceptually, two problems of out-standing importance in gauge theories:
Phases of non-abelian gauge theories
5Monday, February 1, 2010
• Progress in understanding of confinement mechanism(s) in vector-like and chiral theories
• Generalized QCD (different physical IR behavior, scale invariance
• Simplest chiral gauge theory example
• Conformality or confinement?
Outline
6Monday, February 1, 2010
First, the key players: (and outline)
3d Polyakov model & “monopole-instanton”-inducedconfinement Polyakov, 1977
“monopole-instantons” on R3 x S1 K. Lee, P. Yi, 1997P. van Baal, 1998
the relevant index theorem Nye, Singer, 2000MU, Poppitz, 2008
center-symmetry on R3 x S1 - adjoint fermions ordouble-trace deformations Shifman, MU, 2008MU, Yaffe, 2008
“bions”, “triplets”, “quintets”... - new non-self-dualtopological excitations and confinement MU, 2007MU, Poppitz, 2009
(1)
(2)
(3)
(5)
(4)
7Monday, February 1, 2010
A broad-brush overview of some recent progress.
R4 Locally 4d. R3 ! S1
Take advantage of circle (as control parameter) AF.
Traditional: thermal compactification
R4
R3 ! S1
R3Phase transition: Bad for our goal, but inevitable.
However: There are useful ways to go around this thermal impass.
8Monday, February 1, 2010
1) Twisted Partition Function Circle compactification--pbc for fermions
2) Deformation theory
a small step in the desired direction: One of the two always guarantees that small and large circle physics are connected in the sense of center symmetry and confinement.
“NEW” METHODS
9Monday, February 1, 2010
why bother?
various “deformations” of 4d field theories have been useful to studyaspects of nonperturbative dynamics.
especially true in supersymmetry, where consistency with allcalculable deformations play an important role, e.g.:- circle compactification of N=2 4d SYM(Seiberg, Witten, 96)- circle compactification of N=1 4d SYM(Aharony, Intriligator, Hanany, Seiberg, Strassler, 97; Dorey, Hollowood, Khoze, Mattis, 99)in the supersymmetric case, using holomorphy, one argues thatwith supersymmetric b.c. there is a smooth 4d limit
for nonsupersymmetric theories, its utility is understood recently.
CIRCLE COMPACTIFICATION
10Monday, February 1, 2010
Raison d'être of deformation theory at finite N
One can show the mass gap and linear confinement (similar to Seiberg-Witten and Polyakov solutions). Although the region of validity does not extend to large circle, it is continuously connected to it with no gauge invariant order parameter distinguishing the two regimes.
*
0
β=1/Τ
1/Λ
R4
Deformation
confined
B−path
deconfined
A−path
YM Smooth connection to the target theory. A new method to avoid singularities.
11Monday, February 1, 2010
SYM!= SYM +
!
R3!S1P [U(x)]
P [U ] = A2
!2"4
!N/2"!
n=1
1n4
|tr (Un)|2
Deformed YM theory at finite N Shifman-MU, Yaffe-MU
Lattice studies by Ogilvie, Myers, Meisinger backs-up the smoothness conjecture.
*
0
β=1/Τ
1/Λ
R4
Deformation
confined
B−path
deconfined
A−path
YM
Ogilvie, Myers also independently proposed the above deformations to study phases of partially broken center.
Theorist-path
Theory-path
12Monday, February 1, 2010
deformation equivalence
ordinary Yang−Mills deformed Yang−Mills
orbifoldequivalence
combineddeformation−orbifold
∞
c
∞
0
L
0
L
equivalence
Homage to source of inspiration: At large N, volume independence is an exact property, a theorem. Solution of small volume theory implies the solution of the theory on R4. First working example of EK-reduction (25 years after the birth of idea) QCD(Adj) with pbc: The most insightful/friendly QCD-like theory.
80’s: EK, QEK, TEK...Eguchi-Kawai, EK,Gonzalez-Arroyo, Okawa, TEK,Bhanot, Heller, Neuberger, QEK,Gross-KitazawaParisi et.al.Tan et.al
Last few years: QCD(adj), deformations:
Pavel Kovtun, MU, Yaffe, Shifman,MU.Barak Bringoltz, Sharpe,D’elia et.al,Bedaque et.al.Hanada et.al. Narayanan et.al.
Raison d'être of deformation theory at infinite N
QCD(adj)QCD(S/AS)
13Monday, February 1, 2010
SU(N) QCD(adj)
S =!
R3!S1
1g2
tr"14F 2
MN + i!I "MDM!I
#short distance
Center (SU(nf )! Z2Ncnf )/ZnfChiral
ZNc
Solve it by using twisted partition function.
techicolor: minimal walking for 4 flavors? AF-boundary: 5.5 flavors
14Monday, February 1, 2010
With deformation or pbc for adjoint fermions, eigenvalues repel. Minimum at
At weak coupling, the fluctuations are small, a “Higgs regime”
Spatial Wilson line/non-thermal Polyakov loop
U = Diag(1, ei2!/N , . . . , ei2!(N!1)/N )
!trU" = 0
Georgi-Glashow model with compact adjoint Higgs field.
Compactness implies N types of monopoles, rather than N-1.
SU(N)! [U(1)]N!1
15Monday, February 1, 2010
Small volume theory becomes solvable in the same sense as Polyakov model or Seiberg-Witten theory by abelian duality.
Abelian Dualities*
0
β=1/Τ
1/Λ
R4
Deformation
confined
B−path
deconfined
A−path
YM
16Monday, February 1, 2010
1
!µ
1/L
g2(µ)
G
H
Perturbation theory
IR in perturbation theory is a free theory of “photons”. Is this perturbative fixed point destabilized non-perturbatively?
G = SU(2), H = U(1)
17Monday, February 1, 2010
Reminder: Abelian duality and Polyakov model
Free Maxwell theory is dual to the free scalar theory.
F = !d!
U(1)flux : ! ! ! " "
The masslessness of the dual scalar is protected by a continuous shift symmetry
Noether current of dual theory:
Jµ = !µ" = 12#µ!"F!" = Fµ
!µJµ = !µFµ = 0
Its conservation implies the absence of magnetic monopoles in original theory
Topological current vanishes by Bianchi identity.
18Monday, February 1, 2010
!µJµ = !µFµ = "m(x)
The presence of the monopoles in the original theory implies reduction of the continuous shift symmetry into a discrete one. Polyakov mechanism.
L = 12 (!")2 ! e!S0(ei! + e!i!)
The dual theory
Discrete shift symmetry: ! ! ! + 2"
U(1)flux if present, forbids (magnetic) flux carrying operators.
Physics of Debye mechanism
Proliferation of monopoles
19Monday, February 1, 2010
In theories with massless fermions, monopoles operators has fermionic zero modes.
e!S0ei! ! . . .!! "# $fermion zero modes
What is the role of monopoles? Is a new mechanism at work?
Hence, cannot generate confinement and mass gap.
We needed an index theorem for any topological excitation on S1 * R3. After some literature search, we found
20Monday, February 1, 2010
Last formula in the paper, but not your most “friendly” formula
21Monday, February 1, 2010
• Thankfully, in collaboration with Erich Poppitz, we were able to rederive the Nye-Singer formula by using quantum field theory methods and in full generality such that it will be useful for concrete gauge theory applications.
• Some relevant index theorems...
• Atiyah-M.I. Singer 75
• Callias 78 E. Weinberg 80 (physics derivation)
• Nye-A.M.Singer 00, Poppitz-MU 08
• Our strategy was similar to E. Weinberg. Results interpolates nicely.
R3
R4 :
R3 ! S1
22Monday, February 1, 2010
BPS KK
BPS KK(2,0) (!2, 0)
(1, 1/2) (!1, 1/2)
(!1, !1/2) (1, !1/2)
Magnetic Bions
Magnetic Monopoles
e!S0ei! detI,J !I!J ,
e!S0ei! detI,J !I !J
e!2S0(e2i! + e!2i!)
!"S2 F,
"R3!S1 FF
#
Discrete shift symmetry : ! ! ! + "
fermionic zero modes
!I ! ei 2!8 !I
(Z2)!
Topological excitations in QCD(adj)
relevant index theoremsCallias 78
Nye-A.M.Singer, 00E. Weinberg 80Poppitz, MU 08
Atiyah-M.I.Singer 75
Crucial earlier work: van Baal et.al. and Lu, Yi, 9723Monday, February 1, 2010
LdQCD =12(!")2 ! b e!2S0 cos 2" + i#I$µ!µ#I + c e!S0 cos "(det
I,J#I#J + c.c.)
magnetic bions magnetic monopoles
Same mechanism in N=1 SYM.
Dual Formulation of QCD(adj)
Also see Hollowood, Khoze, ... 99
Earlier in N=1 SYM, the bosonic potential was derived using supersymmetry and SW-curves, F, M theories, field theory methods. However, the physical origin of it remained elusive till this work.
Proliferation of magnetic bions
Self-dualNon-selfdual
m! !1L
e!S0(L) =1L
e! 8!2
g2(L)N = !(!L)(8!2NWf )/3 ,
Increasing for Nf<4Decreasing for 4<Nf<5.5
24Monday, February 1, 2010
L!"1
calculable
QCD(adj)
0
<trU>
<tr
<det tr
##>
## >
B) Discrete chiral symmetry is always broken.
A) Mass gap in gauge sector due to magnetic bion mechanism, so is linear confinement, and stable flux tubes.
C) Continuous chiral symmetry is unbroken at small radius, hence massless fermions in the spectrum.
Red line N=1SYM. Confinement without chiral
symmetry breaking
/N
25Monday, February 1, 2010
α2α1
a) Magnetic monopoles
c) Magnetic triplets
b)Magnetic bion
−α2α3
α2α1
−α3 −α3 −α3 −α3
α1
[I1, I2, I3] = [4, 4, 2], Iinst =3!
i=1
Ii ,
QCD(S) topological excitations/confinement
The mechanism of confinement in sextet QCDTestable towards the chiral limit of the lattice theory
26Monday, February 1, 2010
Chiral SU(2) with J=3/2
Shifman, M.U. 08 for new techniques applied to chiral quiver gauge theories, (not included in this talk) Poppitz, MU, relatively simpler applications.
Well-defined, gauge and global (Witten) anomaly free.No framework to address its dynamics until recently. SUSY version: Simplest dynamical SUSY? (Controversial: Intriligator, Seiberg, Shenker, 94, EP, MU )
Instantons:
Symmetry: Z10
e!Sinst!10
27Monday, February 1, 2010
Chiral SU(2) with J=3/2
M1 = e!S0ei!!4, M1 = e!S0e!i!!4,
M2 = e!S0e!i!!6, M2 = e!S0ei!!6,
relevant index theorem, Nye-Singer, 00Poppitz, MU
Monopole operators
e!5S0 cos 5!Mass gap magnetic quintet op.
Z5 : !4 ! ei 2!5 !4, " ! " " 2#
5
Topological symmetry
28Monday, February 1, 2010
KK
BPS
!"
S2!
B,
"F #F
$=
%±5, ±1
2
&
[M1]3[M2]2 ! [BPS]3[KK]2
In the absence of fermion zero modes, the constituents of the magnetic quintet interact repulsively.
An oddball composite. Compare with Cooper pairs!
ISS 94: Does not confine, does not break susy.29Monday, February 1, 2010
Theory Confinementmechanismon R3 ! S1
Index for monopoles[I1, I2, . . . , IN ]
Index for instanton Iinst. (Mass Gap)2
YM monopoles [0, . . . , 0] 0 e!S0
QCD(F) monopoles [2, 0, . . . , 0] 2 e!S0
SYM/QCD(Adj) magneticbions
[2, 2, . . . , 2] 2N e!2S0
QCD(BF) magneticbions
[2, 2, . . . , 2] 2N e!2S0
QCD(AS) bions andmonopoles
[2, 2, . . . , 2, 0, 0] 2N " 4 e!2S0 , e!S0
QCD(S) bions andtriplets
[2, 2, . . . , 2, 4, 4] 2N + 4 e!2S0 , e!3S0
SU(2) YM I = 32 magnetic
quintets[4, 6] 10 e!5S0
chiral [SU(N)]K magneticbions
[2, 2, , . . . , 2] 2N e!2S0
AS + (N " 4)F bions and amonopole
[1, 1, , . . . , 1, 0, 0] +[0, 0, . . . , 0, N " 4, 0]
(N " 2)AS + (N " 4)F e!2S0 , e!S0 ,
S + (N + 4)F bions andtriplets
[1, 1, , . . . , 1, 2, 2] +[0, 0, . . . , 0, N + 4, 0]
(N + 2)AS + (N + 4)F e!2S0 , e!3S0 ,
Table 1: Topological excitations which determine the mass gap for gauge fluc-tuations and chiral symmetry realization in vectorlike and chiral gauge theorieson R3 ! S1.
To the surprise of the past
Nye-Singer, E. Poppitz, MU Atiyah-Singer
More refined data
30Monday, February 1, 2010
Theory Confinementmechanismon R3 ! S1
Index for monopoles[I1, I2, . . . , IN ]
Index for instanton Iinst. (Mass Gap)2
YM monopoles [0, . . . , 0] 0 e!S0
QCD(F) monopoles [2, 0, . . . , 0] 2 e!S0
SYM/QCD(Adj) magneticbions
[2, 2, . . . , 2] 2N e!2S0
QCD(BF) magneticbions
[2, 2, . . . , 2] 2N e!2S0
QCD(AS) bions andmonopoles
[2, 2, . . . , 2, 0, 0] 2N " 4 e!2S0 , e!S0
QCD(S) bions andtriplets
[2, 2, . . . , 2, 4, 4] 2N + 4 e!2S0 , e!3S0
SU(2) YM I = 32 magnetic
quintets[4, 6] 10 e!5S0
chiral [SU(N)]K magneticbions
[2, 2, , . . . , 2] 2N e!2S0
AS + (N " 4)F bions and amonopole
[1, 1, , . . . , 1, 0, 0] +[0, 0, . . . , 0, N " 4, 0]
(N " 2)AS + (N " 4)F e!2S0 , e!S0 ,
S + (N + 4)F bions andtriplets
[1, 1, , . . . , 1, 2, 2] +[0, 0, . . . , 0, N + 4, 0]
(N + 2)AS + (N + 4)F e!2S0 , e!3S0 ,
Table 1: Topological excitations which determine the mass gap for gauge fluc-tuations and chiral symmetry realization in vectorlike and chiral gauge theorieson R3 ! S1.
To the surprise of the past
Nye-Singer, E. Poppitz, MU Atiyah-Singer
More refined data
LGY, MU, 08
MU, 07
MS, MU, 08
MS, MU, 08
MS, MU, 08
EP, MU, 09
EP, MU, 09
EP, MU, 09
MS, MU, 08
MS, MU, 08
31Monday, February 1, 2010
Conformality or confinement:
Mechanisms of confinement and conformality: What distinguishes two theories, one just below the conformal boundary and confines, and the other slightly above the conformal window boundary? In other words, why does a confining gauge theory confine and why does an IR-CFT, with an almost identical microscopic matter content, flows to a CFT?
Lower boundary of conformal window: What is the physics determining the boundary of conformal window?
Conceptually, two problem of out-standing importance in gauge theories:
m!1gauge fluc.(R
4) =!
finite Nf < N"f confined
! N"f < Nf < NAF
f IR" CFT
Map the problem to the mass gap for gauge fluctuations:
A priori, not a smart strategy.32Monday, February 1, 2010
Mass gap for gauge fluctuations
d)
0 LNΛ
1
0 LNΛ
Abelian confinement
1
Semi−classical
Non−abelian confinement
Mas
s gap
for
gau
ge fl
uctu
atio
ns
Lightest glueball
a) b)
0 LNΛ
1
0 LNΛ
1
c)
33Monday, February 1, 2010
of bions (or monopoles)
* QCD (R)
*
Nf
Nf
1/(ΝΛ)
CFTConfined
Nf=2
relevance irrelevance
b)a)
*
Nf
Nf
1/(ΝΛ)
CFTConfined
Nf=2
R4 Nf AF Nf
AFL L
4R
χ
confined with SB
confined without SB
χ
Main idea of our proposal
Crucial data: Index theorem on R3*S1, the knowledge of mechanism of confinement, and one-loop beta function.
34Monday, February 1, 2010
QCD(F/S/AS/Adj):Estimates and comparisons Below, I will present the estimates based on this idea and compare it various other approaches. In particular:
1) Truncated SD (ladder, rainbow) approximation.
QCD(F): Appelquist, Lane, Mahanta, and MiranskyTwo-index cases: Sannino, Dietrich.
2) NSVZ-inspired conjecture: Sannino, Ryttov.
Crucial data for 1) and 2): Two-loop (or conjectured all orders) beta function, anomalous dimension of fermion bilinear.
Caveat: chiral gauge theories.
M.E.Peskin, Chiral Symmetry And Chiral Symmetry Breaking, Les Houches, 1982 (Up-to date review)
• •!!!"#=0
Nf!!IR$free!!!"=0 %
NAFfN!
f
3)World-line formalism: Armoni.
35Monday, February 1, 2010
QCD(S/AS/Adj):Estimates and comparisons N Deformation theory (bions) Ladder (SD)-approx. NSVZ-inspired: ! = 2/! = 1 NAF
f
3 2.40 2.50 1.65/2.2 3.304 2.66 2.78 1.83/2.44 3.665 2.85 2.97 1.96/2.62 3.9210 3.33 3.47 2.29/3.05 4.58! 4 4.15 2.75/3.66 5.5
Table 1: Estimates for lower boundary of conformal window in QCD(S), N"f <
NDf < 5.5
!1! 2
N+2
".
N Deformation theory (bions) Ladder (SD)-approx. NSVZ-inspired: ! = 2/! = 1 NAFf
4 8 8.10 5.50/7.33 115 6.66 6.80 4.58/6.00 9.166 6 6.15 4.12/5.5 8.2510 5 5.15 3.43/4.58 6.87! 4 4.15 2.75/3.66 5.50
Table 2: Estimates for lower boundary of conformal window in QCD(AS), N"f <
NDf < 5.5
!1 + 2
N#2
".
N Deformation theory (bions) Ladder (SD)-approx. NSVZ-inspired: ! = 2/ ! = 1 NAFf
any N 4 4.15 2.75/3.66 5.5
Table 3: Estimates for lower boundary of conformal window in QCD(adj), N"f <
NWf < 5.5. In QCD(adj), we count the number of Weyl fermions as opposed to
Dirac, since the adjoint representation is real.
36Monday, February 1, 2010
QCD(F)
The above estimates from DT are for class a and a+c, respectively.
N D.T. 1a/1c Ladder (SD)-approx. Functional RG NSVZ-inspired: ! = 2/! = 1 NAFf
2 5/8 7.85 8.25 5.5/7.33 113 7.5/12 11.91 10 8.25/11 16.54 10/16 15.93 13.5 11/14.66 225 12.5/20 19.95 16.25 13.75/18.33 27.510 25/40 39.97 n/a 27.5/36.66 55! 2.5N/4N 4N " (2.75# 3.25)N 2.75N/3.66N 5.5N
Table 1: Estimates for lower boundary of conformal window for QCD(F), N$f <
NDf < 5.5N
37Monday, February 1, 2010
QCD(F)
QCD(adj)QCD(S)
Dashed line: Truncated SD (ladder, rainbow) approximation.
QCD(F): Appelquist, Lane, Mahanta, Miransky.....Two-index: Sannino et.al.
Estimates of the conformal window from deformation theory
BUT WHY?
2 3 4 5 6 7 80
2
4
6
8
10
12
14
16
18
N
Nf
QCD(AS)
38Monday, February 1, 2010
!(g2) =32
(g2N)8"2
[1 + O(g2N)]. Anomalous dimension of fermion bilinear
If !(L) < 1, no "SB
Monopole action: S0(L) =8!2
g2(L)N
!(L)! 1" S0(L)# 1" e!S0 ! 1, dilute gas of monopoles and bions!(L) $ 1 " S0(L) $ 1 " e!S0 $ 1, non% dilute
!("")S0 = !(g2(L))S0(g2(L)) ! 1Needs refinement, but it seems to be on the right path.
means non-abelian confinement cannot set-in.
How can we relate perturbation theory to non-pert. physics?
39Monday, February 1, 2010
• There is now a window through which we can look into non-abelian gauge theories and understand their internal goings-on. Whether the theory is chiral, pure glue, or supersymmetric is immaterial. We always gain a semi-classical window (in some theories smoothly connected to R4 physics.)
• Deformation theory is complementary to lattice gauge theory. Sometimes lattice is more powerful, and sometime otherwise. Currently, DT is the only dynamical framework for chiral gauge theories. It may also be more useful in the strict chiral limit of vector-like theories.
• Most important: We learned the existence of a large class of new non-self dual topological excitations through this program in the last two years.
• Shed light into the mechanisms of conformality and confinement in gauge theories. This is tied with the (IR)relevance of topological excitations.
Conclusions
40Monday, February 1, 2010