Dual of QCD with One Adjoint Fermion

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by  Matin Mojaza

CP3 - Origins
Particle Physics & Origin of Mass
Dual of QCD with One Adjoint Fermion
ojaza(cid:114) ∗ ardecchia(cid:114) † ica(cid:114) ‡ annino(cid:114)§
Matin M , Marco N , Claudio P , and Francesco S
(cid:114)Centre for Particle Physics Phenomenology, CP3 -Origins,
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark.
Abstract
WeconstructthemagneticdualofQCDwithoneadjointWeylfermion. Thedualisaconsistent
solutionofthe’tHooftanomalymatchingconditions,allowsforflavordecouplingandremarkably
1 constitutes the first nonsupersymmetric dual valid for any number of colors. The dual allows to
1
0 bound the anomalous dimension of the Dirac fermion mass operator to be less than one in the
2
conformalwindow.
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Electronicaddress: [email protected]
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Electronicaddress: [email protected]
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Electronicaddress: [email protected]
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Electronicaddress: [email protected]
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I. INTRODUCTION
One of the most fascinating possibilities is that generic asymptotically free gauge
theories have magnetic duals. In fact, in the late nineties, in a series of ground breaking
papers Seiberg [1, 2] provided strong support for the existence of a consistent picture of
such a duality within a supersymmetric framework. Supersymmetry is, however, quite
special and the existence of such a duality does not automatically imply the existence
of nonsupersymmetric duals. One of the most relevant results put forward by Seiberg
hasbeentheidentificationoftheboundaryoftheconformalwindowforsupersymmetric
QCD as function of the number of flavors and colors. The dual theories proposed by
Seiberg pass a set of mathematical consistency relations known as ’t Hooft anomaly
matchingconditions[3]. Anotherimportanttoolhasbeentheknowledgeoftheall-orders
supersymmetricbetafunction[4–6]. Recentlyweprovidedseveralanalyticpredictionsfor
ff
theconformalwindowofnonsupersymmetricgaugetheoriesusingdi erentapproaches
[7–16].
Weinitiatedin[17]theexplorationofthepossibleexistenceofaQCDnonsupersymmet-
ricgaugedualprovidingaconsistentpictureofthephasediagramasfunctionofnumber
of colors and flavors. Arguably the existence of a possible dual of a generic nonsuper-
symmetricasymptoticallyfreegaugetheoryabletoreproduceitsinfrareddynamicsmust
match the ’t Hooft anomaly conditions [3]. We have exhibited several solutions of these
conditions for QCD in [17]. An earlier exploration already appeared in [18]. In [19] we
havealsoanalyzedtheorieswithfermionstransformingaccordingtohigherdimensional
representations. Some of these theories have been used to construct sensible extensions
of the standard model of particle interactions of technicolor type passing precision data
and known as Minimal Walking Technicolor models [7, 8]. Other interesting studies
of technicolor dynamics making use of higher dimensional representations appeared in
[20]. ThesearetheknownextensionoftechnicolortypepossessingthesmallestintrinsicS
parameter [21–24] while being able to display simultaneously (near) conformal behavior
before adding the backreaction due to traditional type extended technicolor interactions
(ETC) [25]. For recent analysis of the relevant properties of the S parameter, using also
ff
duality arguments we refer to [26–28]. As for the issue of the e ects of the ETC interac-
tionsonthetechnicolordynamicsandassociatedconformalwindow,ithasrecentlybeen
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arguedin[29]thatbeforeaddingtheETCinteractionsthetechnicolortheory,inisolation,
shouldalreadybeconformalforthecombinedmodeltobephenomenologicallyviable. A
deeperunderstandingofthegaugedynamicsof(near)conformalgaugetheoriesisthere-
fore needed making the study of gauge duals very relevant. For example, the magnetic
dualallowstopredict,inprinciple,thecriticalnumberofflavorsbelowwhichtheelectric
theory looses large distance conformality. It has also been show in [27] that it is possible
to use gauge dualities to compute the S parameter in the nonperturbative regime of the
electrictheory.
Here we put the idea of nonsupersymmetric gauge duality on a much firmer ground
by showing that for certain scalarless gauge theories with a spectrum similar to the one
ofQCDthegaugedualpassesalargenumberofconsistencychecks.
The theory we choose is QCD with N Dirac flavors and one adjoint Weyl fermion. A
f
relevantfeatureofthistheoryisthatitpossessesthesameglobalsymmetryofsuperQCD
despite the fact that squarks are absent. This means that there are four extra anomaly
constraintsnotpresentinthecaseofordinaryQCD,moreoverwewillshowthatthedual
canbeconstructedforanynumberofcolorsgreaterthantwo.
The magnetic dual is a new gauge theory featuring magnetic quarks and a Weyl ad-
joint fermion, new gauge singlet fermions which can be identified as states composite
of the electric variables, as well as scalar states needed to mediate the interactions be-
tween the magnetic quarks and the gauge singlet fermions. The new scalars allow for
a consistent flavor decoupling which was an important consistency check in the case of
supersymmetry.
We will show that the dual allows to bound the anomalous dimension of the Dirac
fermionmassoperatortobelessthanoneintheconformalwindow,andalsoestimatethe
critical number of flavors below which large distance conformality is lost in the electric
variables.
II. THEELECTRICTHEORY:QCDWITHONEADJOINTWEYLFERMION
The electric theory is constituted by a scalarless SU(N) gauge theory with N Dirac
f
fermionsandNlargerthantwo,asinQCD,butwithanextraWeylfermiontransforming
accordingtotheadjointrepresentationofthegaugegroup. Thequantumglobalsymmetry
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is:
× × × .
SU (N ) SU (N ) U (1) U (1) (1)
L f R f V AF
At the classical level there is one more U (1) symmetry destroyed by quantum correc-
A
tions due to the Adler-Bell-Jackiw anomaly. Therefore of the three independent U(1)
symmetriesonlytwosurvive,avectorlikeU (1)andanaxial-likeanomalyfree(AF)one
V
indicated with U (1). The spectrum of the theory and the global transformations are
AF
summarizedintableI.
Fields [SU(N)] SU (N ) SU (N ) U (1) U (1)
L f R f V AF
λ
Adj 1 1 0 1
Q 1 1 − N
N
f
Q(cid:101) 1 −1 − N
N
f
Gµ Adj 1 1 0 0
TABLEI:Fieldcontentoftheelectrictheoryandfieldtranformationproperties. The
squaredbracketsrefertothegaugegroup.
Theglobalanomaliesare:
N N2
SUL/R(Nf)3 = ±N , UV(1)SUL/R(Nf)2 = ± , UAF(1)SUL/R(Nf)2 = − , (2)
2 2N
f
N4
U (1)3 = N2 −1−2 , U (1)U (1)2 = −2N2 , [Gravity]2U (1) = −(N2 +1) (3)
AF AF V AF
N2
f
The first two anomalies are the same as in QCD and they are associated to the triangle
diagrams featuring three SU(N ) generators (either all right or all left) at the vertices , or
f
twoSU(N )generators(allrightorallleft)andoneU (1)charge.
f V
III. SOLUTIONS OF THE ’T HOOFT ANOMALY CONDITIONS FOR ANY NUMBER OF
COLORS
Weseeksolutionsoftheanomalymatchingconditionsintheconformalwindowofthe
ffi
electric theory. This means that we consider a su ciently large number of Dirac flavors
sothattheelectriccouplingconstantfreezesatlargedistances.
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Following Seiberg we assume the dual to be a new SU(X) gauge theory with global
× × ×
symmetry group SU (N ) SU (N ) U (1) U (1) featuring magnetic quarks q and
L f R f V AF
(cid:101)q transforming in the fundamental representation of SU(X) together with a magnetic
λ
Weyl fermion in the adjoint representation. We also introduce a minimal set of gauge
m
singletfermionicparticleswhichcanbeviewedascompositestatesintermsoftheelectric
variablesbutareconsideredelementaryinthedualdescription. Welimittogaugesinglet
fermionicstateswhichcanbeinterpretedasmadebythreeelectricfieldsforanynumber
ofcolors1. Theideabehindthischoiceisthatcompositestatesmadebymorefieldscould
beconstructedalsointhedualtheoryfromthenewelementaryfields.
ff
WesummarizethedualspectrumintableII.Thedualgaugegroupcanbedi erentfrom
Fields Compositeeq. [SU(X)] SU (N ) SU (N ) U (1) U (1) #ofcopies
L f R f V AF
λ − Adj 1 1 0 z(cid:48) 1
m
q − 1 y − X z 1
N
f
(cid:101)q − 1 −y − X z 1
N
f
M QλQ(cid:101) 1 0 1−2 N (cid:96)
N M
f
M(cid:101) QQ(cid:101)λ 1 0 1+2 N (cid:96)
Nf M(cid:101)
ΛS QλQ,Q(cid:101)λQ(cid:101),λλλ 1 1 1 0 −1 (cid:96)ΛS
ΛL QλQ 1 Adj 1 0 −1 (cid:96)ΛL
ΛR Q(cid:101)λQ(cid:101) 1 1 Adj 0 −1 (cid:96)ΛR
ΛG λGG 1 1 1 0 1 (cid:96)ΛG
Λ λλλ (cid:96)
1 1 1 0 3 Λ
TABLEII:Masslessspectrumofthecandidatemagneticfermionsandtheir
transformationpropertiesunderthedualgaugeandglobalsymmetrygroup. Thelast
columnrepresentsthemultiplicityofeachstateandeachstateisaWeylfermioninthe
/ ,
(1 2 0)representationoftheLorentzgroup.
1NotethatforN =3ordinarybaryonsarealsomadebythreestates,howeversincewewillfindconsistent
solutions of the ’t Hooft anomaly matching conditions for any N we have not included them in the
spectrum.
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the electric one since the only physical quantities are gauge singlets. On the other hand
theglobalsymmetriesmustbetheoriginalones2 giventhatcanbephysicallyprobed.
We wish to find solutions to the ’t Hooft anomaly conditions valid for any number of
(cid:48)
colors. z ,zandyarearbitraryAbelianchargesforthemagneticquarksandadjontfermion
while the ones of the SU(X) magnetic singlets are derived from the electric constituents.
From table II we write below all the relevant anomalies for the dual theory which we
requiretomatchtheelectriconesgivenineqs. (2)and(3)
(cid:32) (cid:33)
1 X
[SU(X)]2U (1) =Xz(cid:48) + − z N ×2 = X(z(cid:48) −z) := 0 (4)
AF f
2 N
f
SUL/R(Nf)3 =±[Nf((cid:96)M −(cid:96)M(cid:101))−X] := ±N (5)
1 N
SUL/R(Nf)2UV(1) =± yX := ± (6)
2 2
(cid:32) (cid:33)
1 X 1 N 1 N
SUL/R(Nf)2UAF(1) =2 −N z X+ 2(cid:96)MNf(1−2N )+ 2(cid:96)M(cid:101)Nf(1+2N )−(cid:96)ΛL/RNf
f f f
N2
= −
: (7)
2N
f
(cid:32) (cid:33)
X
U (1)2U (1) =2× y2 − z XN := −2N2 (8)
V AF f
N
f
(cid:32) (cid:33)
3
X N N
U (1)3 =(X2 −1)z(cid:48)3 +XN − z ×2+(cid:96) N2(1−2 )3 +(cid:96) N2(1+2 )3
AF f N M f N M(cid:101) f N
f f f
+(cid:96)Λ (N2 −1)(−1)3 +(cid:96)Λ (N2 −1)(−1)3 +(cid:96)Λ +(−1)3(cid:96)Λ +33(cid:96)Λ
L f R f G S
N4
:= N2 −1−2 (9)
N2
f
(cid:32) (cid:33)
X N N
[Gravity]2U (1) =(X2 −1)z(cid:48) +XN − z ×2+(cid:96) N2(1−2 )+(cid:96) N2(1+2 )
AF f N M f N M(cid:101) f N
f f f
−(cid:96)Λ (N2 −1)−(cid:96)Λ (N2 −1)+(cid:96)Λ −(cid:96)Λ +3(cid:96)Λ := −N2 −1. (10)
L f R f G S
The electric U (1), anomaly free condition, i.e. [SU(X)]2U (1) = 0 provides the first
AF AF
(cid:48) =
nontrivialconstraint,i.e. z z.
Consider now the ’t Hooft anomaly matching conditions from SU (N )2U (1) and the
L f V
U (1)2U (1):
V AF
Xy N
= , X2y2z = N2 . (11)
2 2
2Theremightbeexceptionssuchaspossibleenhancedglobalsymmetriesbutitisnotthecasehere.
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whichcanbesimultaneouslysolvedfor
= = / .
z 1 and y N X (12)
The difference between the SU (N )2U (1) and SU (N )2U (1) anomalies forces
L f AF R f AF
(cid:96) = (cid:96)
Λ Λ .
R L
(cid:16) (cid:17)
,(cid:96) ,(cid:96) ,(cid:96) ,(cid:96) ,(cid:96) ,(cid:96)
At this point we remain with 7 unknowns X Λ Λ Λ Λ Λ Λ and 4 indepen-
M M(cid:101) L S G
(cid:96) ,(cid:96) ,(cid:96)
dent anomaly matching conditions. We can use the 4 equations to solve for Λ Λ Λ
M M(cid:101) L
(cid:96)
and Λ. Notice that the system is linear in these 4 unknowns. Obviously we expect
that the solution is not unique and we can parametrize the set of all the solutions using
,(cid:96) ,(cid:96)
combinations of the remaining three unknowns: X Λ Λ . Before showing the result
S G
we already know that the solution of the system can depend only on X and linearly on
≡ (cid:96) −(cid:96) Λ Λ
zD ΛG ΛS. ThesolutiondependsonlyonzD because S and G formvectorlikepairs.
(cid:96) +(cid:96)
Thesum Λ Λ isnotconstrainedbythe’tHooftanomalymatchingconditions.
G S
Hereisthesolution:
(cid:16) (cid:17)(cid:16) (cid:17)
(N+X)(3N−X) N+N +X X+N−N −6z N2N2
(cid:96) = f f D f
Λ (13)
6N 2(3N2 −2)
f
(cid:16) (cid:17)(cid:16) (cid:17)
(N+X)(3N−X) N+N +X X+N−N −4z N2
(cid:96) = − f f D f
Λ (14)
L 4N 2(3N2 −2)
f
(cid:16) (cid:17)(cid:16) (cid:17)
(N+X) 3N2 +3NN −2NX−4+X2 −N X X+N+N +4z N2
(cid:96) = f f f D f
(15)
M 4N 2(3N2 −2)
f
(cid:16) (cid:17)(cid:16) (cid:17)
(N+X) 3N2 −3NN −2NX−4+X2 +N X X+N−N +4z N2
(cid:96) = f f f D f
(16)
M(cid:101) 4N 2(3N2 −2)
f
The four equations are further constraint by being nonnegative integers. The problem is
thenreducedtofindingcombinationsofX andz fulfillingthisrequirement.
D
In this way we have completely characterized all the possible solutions of the ’t Hooft
anomaly matching in a dual theory with matter content presented in Table II. We notice
(cid:96) =
thatonecanreproduceaSeiberg-likesolutionfor 1andtheremainingindicessetto
M
zero.
As we shall show, the above equations can be rewritten in a more transparent way.
Thefollowinglinearcombinationofthefirsttwoequations(13)and(14)gives:
= − (cid:96) − (cid:96) .
zD 3 Λ 2 ΛL (17)
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Therefore,imposingpositivityofthemultiplicities,wecanonlyhavesolutionsforwhich
≤
z 0. Furthermore,wecanalsoformanotherlinearcombinationindependentonz :
D D
(cid:16) (cid:17)(cid:16) (cid:17)
+ − + + + −
(N X)(X 3N) X N N X N N
f f
(cid:96)Λ +N2(cid:96)Λ = . (18)
L 12N 2
f
ff (cid:96) (cid:96)
Itismoreconvenienttoworkwiththesumanddi erenceof and :
M M(cid:101)
+
N X
(cid:96) −(cid:96) = =
d (19)
M M(cid:101) N
f
(cid:96)M +(cid:96)M(cid:101) = d2 +2(cid:96)ΛL, (20)
wherewehavedefineddinthefirstequation. dmustbothbeapositiveintegerimplying
≥ − ≥
X N N. The positivity condition of eq. (18) further requires the inequality X 3N
f
> −
whenX N N.
f
In summary, the original set of ’t Hooft anomaly matching conditions can be neatly
rewrittenas:
(cid:96) −(cid:96) =
d (21)
M M(cid:101)
(cid:96)M +(cid:96)M(cid:101) = d2 +2(cid:96)ΛL (22)
− + −
(d 1)d(d 1)(dN 4N)N
(cid:96)Λ = f f −N2(cid:96)Λ (23)
12 L
= − (cid:96) − (cid:96) .
zD 3 Λ 2 ΛL (24)
(cid:96)
In this form, the constraint on the ’s being positive integers is always fulfilled and
(cid:96)
thus all solutions are given in terms of the generic parameters d and Λ . One immediate
L
−
consequenceisthatthedualgaugegroupSU(X)isSU(dN N)withdapositiveinteger.
f
AsweshallshowintheAppendixAthisstructureofthegaugegroupadmitsaninvolution
of the duality operation, meaning that if one dualizes again it is possible to recover the
SU(N) group. This, however, is only a necessary condition for the complete involution
ofthedualityoperationtoholdexactly,meaningthatonehasstilltodemonstratethat,at
thefixedpointthedualofthedualisactuallytheelectrictheory.
=
The first, and most relevant solution, is obtained for d 1 and it mimics Seiberg’s
solutionforthedualofsuperQCD.Wehave,infact,that:
= ⇒ = − , (cid:96) = , (cid:96) = (cid:96) = (cid:96) = = .
d 1 X Nf N M 1 Λ ΛL M(cid:101) zD 0 (25)
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= (cid:96)
No other solutions exist for d 1, i.e. Λ vanishes. The dual gauge group is therefore
L
−
SU(N N).
f
− = (cid:96) − (cid:96) = (cid:96) + (cid:96) ff
We now show that the minimal value of zD ΛS ΛG 2 ΛL 3 Λ di erent from
− = , , (cid:96) (cid:96)
zero is 5. Solutions for which zD 2 3 4 require either Λ or ΛL to be zero, but one
findsfrom(23),that:
(cid:96) = ⇒ (cid:96) =
for Λ 0 min( Λ ) 3 (26)
L
(cid:96) = ⇒ (cid:96) =
for Λ 0 min( Λ) 5 (27)
L
− = (cid:96) = (cid:96) = .
thus min( zD) 5 with Λ ΛL 1 (28)
(cid:96) (cid:96)
For any d greater than one for which Λ and Λ assume values lower than the ones
L
provided above there is no solution to the ’t Hooft anomaly matching conditions. This
(cid:96)
can be shown by first noting that by setting one of the two s to zero in (23) the minimal
(cid:96) = (cid:96)
value of the other is always obtained by setting d 2. The second step is to set Λ to
zeroandthereforeobtain:
(cid:32) (cid:33)
N N
f − f = (cid:96) .
2 Λ (29)
N N L
(cid:96)
It is clear that the minimum Λ must be three for this equation to hold. When setting
L
(cid:96) =
instead Λ 0weobtain:
L
(cid:16) (cid:17)
(cid:96) = − .
Λ Nf 2N Nf (30)
(cid:96) >
This equation, given that the s are positive integers, requires N 2N and therefore the
f
minimumN mustbe:
f
= + ,
N 2N 1 (31)
f
whichmeans:
(cid:96) = + ,
Λ 2N 1 (32)
(cid:96) > (cid:96) (cid:96) =
and therefore Λ 5. If both Λ and Λ are equal to zero one either recovers the d 1 or
L
≥
onecansaturatetheinequalityX 3N andfindthefollowingsolutions:
N d2 +d d2 −d
> (cid:96) = (cid:96) = = ⇒ = , = , (cid:96) = , (cid:96) =
for d 1 and Λ ΛL zD 0 X 3N d 4N M 2 M(cid:101) 2
f
(33)
These solutions are, however, unnatural since X is N independent and furthermore d
f
is by definition restricted to be an integer. A solution of this type implies that in the
conformal window it is impossible to change the number of flavors in the electric theory
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and remain with an integer d for the same number of colors assuming one had started
with a number of flavors for which d was an integer. Therefore, we do not consider
≤ −
these solutions to represent viable magnetic duals. Solutions with z 5 are certainly
D
=
non-minimalstronglyindicatingthatthed 1solutionistherelevantone.
Requiring the electric theory to be asymptotically free imposes further general con-
ffi
straints. We start with recalling the first coe cient of the beta function for the electric
theorywhichis:
2
βe = 3N− N . (34)
0 3 f
For the theory to be asymptotically free, one imposes βe ≥ 0, leading to the constraint on
0
N :
f
9
≤ .
N N (35)
f
2
(cid:96) ≥
For Λ 0fromeq. (23)wefindthecondition:
− + −
(d 1)d(d 1)(dN 4N)N
f f ≥ N2(cid:96)Λ ,
12 L
leadingtotheconstraint,alreadynotedearlier:
N
(cid:44) , ≥ .
ford 1 d 4
N
f
α = /
Wecansetanotherconstrainton N N rewritingtheinequalityaboveas:
f
(cid:96)
4 12 Λ
α2 − α− L ≥ 0,
d (d2 −1)d2
whichleadsto
(cid:112)
α = Nf ≥ 2d2 −1+ (d2 −1)(d2 −1+3(cid:96)ΛL) .
N (d2 −1)d
/
CombiningthetwoconstraintsonN N weobtain:
f
(cid:112)
2d2 −1+ (d2 −1)(d2 −1+3(cid:96)ΛL) ≤ Nf ≤ 9 with d ≥ 4 N . (36)
(d2 −1)d N 2 N
f
(cid:96)
Expressing instead the bound on Λ , we see that it is bounded from above, hence
L
constrainingthenumberofpossiblesolutionstobefiniteforafixedvalueofd:
(cid:96)Λ ≤ 1 d(d2 −1)(dNf −4)Nf.
L 12 N N
Themostnaturalsolutionistheonewiththelowestvalueassumedbyallindiceswhich
= −
corresponds to d 1 and therefore has associated gauge group SU(N N). We will,
f
therefore, concentrate on this theory below and summarize here the fermionic spectrum
whichresemblesthesupersymmetricversionofthetheory.
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