6D Flux Compactification: Chirality and Symmetry Breaking

1
6D Flux CompactificationChirality and Symmetry
Breaking

• S. Rigolin
• Universidad Autonoma de Madrid and IFT

• J. Alfaro et al., JHEP 0701 (2007) 005 -
  arXivhep-ph/0606070
• M. Salvatori, arXivhep-ph/0611309, accepted by
  JHEP
• A. Faedo, D. Hernandez, R.S., M. Salvatori, work
  in progress

Thanks to B. Gavela and M. Salvatori for useful
discussions
2
Contents

• Framework Gauge-Higgs Unifications
• The Hierarchy problem and Gauge-Higgs
  Unification
• Brief Introduction on Compactification
• 5D vs 6D Compactification SS Boundary
  Conditions
• 6D Compactification t Hooft Consistency
  condition
• 6D Compactification with (magnetic) Flux
• The problem of chirality Magnetic Flux and
  Chirality
• Gauge Symmetry Breaking and t Hooft flux
• Phenomenological analysis of the SU(2) gauge
  sector
• Conclusions Outlook

3
Introduction The Hierarchy Problem

• The Hierarchy problem in the SM
• Contrary to Gauge Bosons and Fermions masses the
  SM Scalar masses are not PROTECTED by any
  symmetry
• If the scalar sector is coupled to some (New)
  Physics at a scale L, higher order contributions
  shift the Higgs mass
• The Higgs mass term has a QUADRATICALLY DIVERGENT
  contribution from any high energy scale
• At least GRAVITY should be included and so Mh
  MPl, unless some LARGE unwanted FINE TUNING is
  present

4
Status on SM Higgs Searches

• Direct Limit (LEPII)
• mH gt 114.4 GeV
• Radiative Corrections (LEP, SLD, Tevatron)
• log(mH) 1.93 0.17
• Complete Fit (Direct Searches Rad. Corr.s)
• mH lt 199 GeV (95 CL)

Low Higgs mass is unnatural HIERACHY PROBLEM
5

• Mechanism to stabilize the Higgs sector
• SUPERSYMMETRY Quadratic contributions exactly
  cancel (in the SUSY limit) between Fermions and
  Bosons diagrams
• CUSTODIAL SYMMETRY The Higgs is a Goldstone
  Boson of a spontaneously (softly) broken global
  symmetry. A shift symmetry preserves the
  lightness of the Higgs
• GAUGE-HIGGS UNIFICATION spin-1 and spin-0 bosons
  are partners of the same Higher Dimensional Gauge
  field. Gauge Symmetry preserves the lightness of
  the Higgs sector

U(1)
6
Gauge-Higgs Unification Framework

• A (4d) dimensional SU(N) gauge field is
  equivalent to

• 4D vector boson degree of freedom ? 1
• 4D scalars degree of freedom ? d

• The Scalar Components can play the role of the
  Higgs
• Gauge-Higgs Unification
• (4d) Gauge Symmetry protects the Higgs from
  quadratic
• divergences

Fairly-Manton 79
7
Hierarchy Problem

• Finite (non-local) contributions to scalar masses
  arise during the compactification procedure (6D
  Lorentz breaking)
• Split gauge boson (massless) from scalar boson
  masses (finite)
• DO NOT introduce local counterterms (quadratic
  divergences)
• The compactification scale MC 1/R natural
  scale for the Higgs

Extra-dimensional Solution
?
R
X
y1
y2
y
?
8
Solution or Sobstitution ?

• At the end we have only traded MPl for MC 1/R
• Compactification scale cut-off the Higgs mass MC
  MH
• Why should MC 1 TeV and not MC MPl ? What
  does it stabilize the Compactification Scale to
  the TeV scale ?
• The real fundamental scale is
  while
• is only an artifacft of our limited
  understanding of Nature
• However similar considerations can be applied to
  SUSY
• When SUSY SBT are introduced
• Why should MSBT 1 TeV ? What does it stabilize
  MSBT ?
• By the way Exp. Higgs data already shows some
  need of Fine Tuning -LITTLE
  HIERARCHY PROBLEM

9

• 3 problems have to be solved
• Mechanism for HIDING the Extra Dimensions
• No experimental evidence of Extra Dimensions at
  energies presently available MC 1/R 1 TeV
• Mechanism for BREAKING Gauge Symmetry
• No scalar potential to drive Electro-Weak
  symmetry breaking is introduced
• For model building reasons one has to start from
  larger gauge group (right Higgs representation,
  Unification, ...)
• Mechanism for OBTAINING chiral fermions
• SM interactions are chiral while higher
  dimensional fermions always reduce to 4D
  vector-like fermions
• Higher dimensional CPT symmetry breaking (flux or
  orbifold compactification)

Luscher, Hosotani83
Randjbar-Daemi, Salam, Strathdee 83
Dixon, Harvey, Vafa, Witten 85
10
(1) Basics on 5D Compactification
y
y
y2pR

• Periodic Boundary Conditions

• One 4D massless state tower of massive
  KK-modes
• The 4D theory has an unbroken symmetry group G

Kaluza 19, Klein 26
11

• General (Scherk-Schwarz) Boundary Conditions
• In non-simple connected space only the Lagrangian
  has to be single-valued not the fields
  themselves individually!

Fundamental representation

• SS Boundary Conditions as Symmetry Breaking
  Mechanism
• If a ? 0 (T ? 1) one break the Global Symmetries
  (Flavour, Supersymmetry) through Boundary
  Bonditions
• SS Boundary Conditions can break the Gauge
  Symmetry
• The (non-integrable) phase a can be associated to
  the vev of the scalar components
  (Continuous Wilson Line)
• a is fixed minimizing the one-loop effective
  potential Dynamical (Spontaneous) Symmetry
  Breaking - Hosotani mechanism

Scherk , Schwarz 79
Luscher, Hosotani83
12
(1) Basics on 6D Compactification
R2
(y1,y2)

• Periodic Boundary Conditions
• (along both the coordinates)

13

• General (Scherk-Schwarz) Boundary Conditions

• Can we choose arbitrary B.C. T1, T2 along (y1,
  y2) ?

t Hooft Consistency Condition
t Hooft 79, t Hooft 81
(2pR1,2pR2)
(0,2pR2)
t1
t2
t2
t1
(0,0)
(2pR1,0)
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t Hooft Consistency Condition

• Fields in the Adjoint representation
• The general BCs (internal automorphism) are given
  by
• and satisfy the following consistency condition
• The general solution is given by
• with in or
  in

U(1) Abelian Flux
SU(N) t Hooft Flux
15

• Fields in the Fundamental representation
• The general BCs (internal automorphism) are given
  by
• and satisfy the following consistency condition
• with the only possible solution

The presence of fields in the fundamental
imposes trivial U(N) or SU(N) t Hooft
Consistency Condition
Is it possible to have simoultaneously
non-trivial t Hooft flux and fields in the
adjoint fundamental representation ?
16
t Hooft Flux with U(N) Gauge Group

• Lets consider the U(N) trivial case
• We can always split U(N) U(1) x SU(N) as so
  that the consistency condition reads
• with

• We can choose the SU(N) twists so that the
  non-trivial
• t Hooft Flux m is compensated by the Abelian
  Magnetic Flux

17
t Hooft (magnetic) Flux

• Embeddings of translations (TWISTS) have to
  commute modulo an element of the center (i.e.
  identity) of SU(N)

Twist Algebra

• Boundary Conditions are generally referred
  (lattice) as
• Untwisted B.C. if m0 Trivial t Hooft Flux
• Twisted B.C. if m?0 Non Trivial t Hooft Flux
• The t Hooft (magnetic) flux m
• Is an integer number keeping values (0,,N-1) mod
  N
• Is a topological quantity that identifies
  equivalence classes of possible vacuum solutions
• Symmetry Breaking and Chirality depend on m

18
Boundary Conditions vs Magnetic Flux

• Consider a scalar field coupled to an U(1)
  background magnetic field Bi (living on the
  torus) with constant Field Strenght B12

• Such background field Bi is not periodic under yi
  ? yi Li

• These transformations can be interpreted as an
  U(1) gauge shift

19

• The Scalar Lagrangian has to be Single-Valued on
  the torus
• and this imposes that also the scalar field
  has to transform under a fundamental translation
  in a U(1)-like manner
• The Twist operators can be written in terms of Bi
  as
• The t Hooft Abelian consistency condition reads
• The Abelian Magnetic Flux is quantized

Abelian magnetic Flux
t Hooft (magnetic) Flux
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(2) Chirality and Extra-Dimensions

• NO-GO THEOREM It is not possible to obtain
  trivially 4D
• chiral theories starting from (4D) (chiral)
  fermions
• 4D theories can be CHIRAL (the SM indeed is
  chiral)
• 4D spinorial representation is a (4-dim)
  REDUCIBLE rep.
• in eigenstates of the chiral operator
• 4D Dirac fermion can be decomposed in 2 Weyl
  chiral components (Left and Right)

21

• Starting from a 5D (Hermitian) Lagrangian is not
  possible to obtain a CHIRAL 4D theory
• 5D spinorial representation is a (4-dim)
  IRREDUCIBLE representation (Mm,5)
• and no other gamma matrix can be introduced
  (that anticommutes with gM and commutes with
  sMN)
• CHIRALITY cannot be DEFINED in 5D and so one can
  only start from vector-like 5D theories
• When compactifying to 4D both the Left and Right
  states acquire a 0-mode

?
Only 4D vector-like theories can be obtained (QED
but not SM)
22

• Starting from a 6D (Hermitian) Lagrangian is not
  possible to obtain a 4D CHIRAL theory
• 6D spinorial rep. is a (8-dim) REDUCIBLE rep.
  (Mm,5,6)
• in eigenstate of the CHIRAL operator
• 6D Dirac fermion can be decomposed in 4 Weyl
  components, but a 6D chiral fermion contains 2
  Weyl fermions (L and R)

Even starting from a 6D CHIRAL theory, after
compactification only 4D vector-like theories
can be obtained (QED but not SM)
23

• This no-go theorem can be circumvented relaxing
• invariance under CPT (and/or Lorentz Invariance)
• ORBIFOLD COMPACTIFICATION Explicitly breaks the
  Extra-Dimensional Lorentz invariance. Only fields
  of a defined parity survive the orbifold
  projection. For example in 5D one can impose the
  following orbifold condition (parity)
• Once compactified to 4D only the LEFT (selecting
  -1) mode has a massles mode, resulting in a 4D
  chiral theory
• This mechanism can be used with any number of ED

Witten 83
Dixon, Harvey, Vafa, Witten 85
24

• FLUX COMPACTIFICATION in 6D the presence of an
  external (background) magnetic field breaks the
  ED CPT invariance. The presence of a fixed
  direction permit to distinguish L vs R modes
• we can write the following Dirac equation

B
L
R
Background Flux splits Left and Right modes
in such a way that both chiralities cannot have
simoultaneously a 4D 0-mode
Randjbar-Daemi, Salam, Strathdee 83
25
Magnetic Field Adjoint vs Fundamental

• Chirality is related to the commutator of the
  Covariant Derivatives D5, D6 in the specific
  representation of G
• Fundamental Representation of U(N)
• Adjoint Representation of U(N)

Only fields in the fundamental representation are
sensitive to the abelian part of the flux
26
(3) SU(N) Gauge Theory on a Torus

• Lets study the case of SU(N) gauge theory on a
  Torus

• The SU(N) Gauge Field SS Boudary Conditions

leave the 6D Yang-Mills Lagrangian single-valued

• The SU(N) t Hooft Consistency Condition reads

Describe which are the the possible vacuum
configurations and their (residual) symmetries
27

• In the presence of an external background Bi(y)
  living on a torus

Do not contribute to the vacuum energy
the 4D Lagrangian reads

• In absence of 4D instantons and assuming 4D
  Lorentz invariance the energy of the vacuum
  solutions is vanishing (TrFij0)

• The fluctuation fields have to develop (infinite)
  VEVs to compensate the presence of the external
  background magnetic field

Olesen-Nielsen Instability
No SU(N) Magnetic Flux
28

• If you are still skeptical work out the
  effective 4D theory (include all KK
• and Landau modes you can) calculate and miminize
  the 4D potential

The system responds to the instability with an
infinite set of vevs so to cancel the original
external background
29
Constant vs Non constant BC (SU(N))

• We proved that in the stable vacua BTOT0,
  consequently
• with all gauge transformations U(y)
  compatible with BC Vi(y)
• One can show that for SU(N) on a 2-dim torus such
  U(y) exist
• To classify all the possible SU(N) vacuum
  configurations and their symmetries is useful to
  go to the symmetric gauge

Ambjorn 80, Salvatori 06
30
Symmetry Breaking Pattern m0

• The translations V1 and V2 commute and they can
  be chosen in the commuting sub-algebra of SU(N)

• The parameters a1 and a2 are free at tree-level
  and are
• fixed once the one-loop effective potential
  is minimized
• (Hosotani Mechanism). If ai?0 the symmetry is
  broken

• The Symmetry Breaking is Rank Preserving
  (Hosotani)

1
2
31
Symmetry Breaking pattern m?0

• The translations V1 and V2 DO NOT commute and
  they cannot be chosen in the commuting
  sub-algebra of SU(N)

• For a given m the possible Ti have been
  classified in terms of
• 2 constant matrices P,Q and 4 integer
  coefficients (si, ti)

• The parameters ai are no longer arbitrary (also
  at tree level)
• but fixed by previous conditions (Discrete
  Wilson Lines)

32

• This induces a Rank Reducing Symmetry Breaking
  pattern

• If K gt1 there is a residual gauge invariance
• The wi are in general non trivial elements of
  SU(K ) and we can apply to them the discussion
  done for m0 (wi commute)
• A second dynamical (spontaneous) symmetry
  breaking a la
• Hosotani is possible for SU(K )

• The complete Symmetry Breaking pattern now reads

• The bosons spectrum it is given by

33
SU(2) Phenomenological Analysis

• Previuos discussion mainly based on Theoretical
  Arguments
• One can study the 4D Effective Field Theory in
  the easiest case (i.e. starting with an SU(2) 6D
  YM Lagrangian)
• Expand the 6D fields in KK (in the following with
  index n,m) and Landau (in the following with
  index j) modes
• Integrate over the extra-dimensions
• Minimize the 4D scalar potential and study vector
  and scalar spectrum in order to determine the
  residual symmetries
• The m0 case
• Correspond to the UNBROKEN case (tree-level)
• One expects to recover the full symmetry case and
  corresponding spectrum

34
Minimum of the Potential (m0)
Confirms that the stable vacuum has NO MAGNETIC
FLUX
35
Mass of the Lightest state (m0)
Confirms that there is at least 1 massless mode
36
Vector and Scalar Spectrum (m0)

• Agreement between theoretical and numerical
  spectra
• All tree-level vacua are degenerate at
  tree-level the PC select a specific value
  (gauge) for the SS phase breaking SU(2) ? U(1)

37
Phenomenological Analysis m?0

• From the previous theoretical analysis
• If m?0 then SU(2) -gt 0
• The minimum of the symmetry is the one expected
  for restaurating a 0 energy vacua level
• The symmetry is (explicitely) broken even at the
  classical level. No residual symmetry is present
  (K1)
• The expected spectrum has no 0-modes in agreement
  with the general theoretical calculation
• In the SU(2) case 1-loop effects cannot produce
  further symmetry breaking (no Hosotani
  mechanism)
• Larger groups are being analyzed

38
Minimum of the Potential (m ? 0)
Confirms that the stable vacuum has NO MAGNETIC
FLUX
39
Mass of the Lightest state (m ? 0)
Confirms that there are no massless modes
40
Vector and Scalar Spectrum (m?0)

• Agreement between theoretical and numerical
  spectra
• The symmetry group si completely broken SU(2) ?
  Ø

41
Conclusions

• Gauge-Higgs unification framework
• Possible solution of the Hierarchy Problem
• Discussed Scherk-Schwarz compactifications in 5D
  and 6D
• Novelty of 6D by t Hooft Consistency Conditions
• Interpretation of Consistency Conditions as
  Magnetic Flux
• Chirality Problem in Extra-Dimensions
• 6D Chirality through (Magnetic) Flux
  Compactication
• SU(N) Gauge Theory Vacua and Symmetry
• Trivial t Hooft flux m0 Spontaneous (Hosotani)
  Mechanism
• Non trivial t Hooft flux m?0 Rank Lowering
  Symmetry Breaking Mechanism (Explicit
  Spontaneous)
• Phenomenological analysis trivial for SU(N) but
  could be used fore more general groups
  (non-vanishing p1(G))

42
Outlook

• Still quite far from a semi-realistic framework
  
• Presence of degenerate Vector and Scalar Bosons
• After all symmetry breaking massless partner of
  photon exists
• One-loop effects break this degeneracy (enough
  ?)
• Needed U(N) gauge group for chirality
• At most we can break to U(1) x SU(N) ? U(1) x
  U(1)
• Degenerate Photon and Z
• Introduce 3 Families and Flavour Structure