Bound state spectrum of theories with a Brout-Englert-Higgs ...funqcd2018/Talks/...Bound state...

Bound state spectrum of theories with a Brout-Englert-Higgs effect

René SondenheimerFSU Jena

666. WE-Heraeus-SeminarFrom correlation functions to QCD phenomenology

Bad Honnef, 5th of April 2018

& L. Egger, A. Maas arXiv:1701.02881

& A. Maas, P. Törek arXiv:1709.07477, arXiv:1710.01941

2

Gauge invariance

• Physical observable is gauge invariant!

2

Gauge invariance

• Physical observable is gauge invariant!

• QED: Dressing by Dirac phase factor

2

Gauge invariance

• Physical observable is gauge invariant!

• QCD: Confinement

• QED: Dressing by Dirac phase factor

2

Gauge invariance

• Physical observable is gauge invariant!

• Weak interaction?

• QCD: Confinement

• QED: Dressing by Dirac phase factor

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

• Mass for the W’s from the Higgs kinetic term

(Dµ�)†Dµ� =

1

2

g2v2

4W i

µWi µ + ...

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

• Mass for the W’s from the Higgs kinetic term

(Dµ�)†Dµ� =

1

2

g2v2

4W i

µWi µ + ...

BUT!

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

• Mass for the W’s from the Higgs kinetic term

(Dµ�)†Dµ� =

1

2

g2v2

4W i

µWi µ + ...

BUT!• Spectrum qualitatively the same in QCD and Higgs-like region

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

• Mass for the W’s from the Higgs kinetic term

(Dµ�)†Dµ� =

1

2

g2v2

4W i

µWi µ + ...

BUT!• Spectrum qualitatively the same in QCD and Higgs-like region

Osterwalder,Seiler ’77; Fradkin,Shenker ’78

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

• Mass for the W’s from the Higgs kinetic term

(Dµ�)†Dµ� =

1

2

g2v2

4W i

µWi µ + ...

BUT!• Spectrum qualitatively the same in QCD and Higgs-like region

Osterwalder,Seiler ’77; Fradkin,Shenker ’78

• “Local symmetry breaking” vs Elitzur’s theorem

Ambiguities beyond perturbation theory

�(x) =1p2

✓'+(x)

v + h(x) + '0(x)

◆

f

UL

• Mass for the W’s from the Higgs kinetic term

(Dµ�)†Dµ� =

1

2

g2v2

4W i

µWi µ + ...

BUT!• Spectrum qualitatively the same in QCD and Higgs-like region

Osterwalder,Seiler ’77; Fradkin,Shenker ’78

• “Local symmetry breaking” vs Elitzur’s theorem

• Gribov-Singer ambiguity

4

Fröhlich-Morchio-Strocchi mechanism Fröhlich et al ’80, ‘81

4

Fröhlich-Morchio-Strocchi mechanism

O(x) = (�†�)(x)

1. Construct a gauge-invariante operator

Fröhlich et al ’80, ‘81

4

Fröhlich-Morchio-Strocchi mechanism

O(x) = (�†�)(x)

� = v + '

hO(x)O(y)i = const.+ v2hh(x)h(y)i+ v3h'i+ vh'3i+ h'4

i

1. Construct a gauge-invariante operator

2. Expand Higgs field in correlator in fluctuations around the vev

Fröhlich et al ’80, ‘81

4

Fröhlich-Morchio-Strocchi mechanism

O(x) = (�†�)(x)

� = v + '

hO(x)O(y)i = const.+ v2hh(x)h(y)i+ v3h'i+ vh'3i+ h'4

i

hO(x)O(y)i = v2hh(x)h(y)itl + hh(x)h(y)i2tl +O(g2,�)

1. Construct a gauge-invariante operator

2. Expand Higgs field in correlator in fluctuations around the vev

3. Perform standard perturbation theory on the right-hand side

Fröhlich et al ’80, ‘81

4

Fröhlich-Morchio-Strocchi mechanism

O(x) = (�†�)(x)

� = v + '

hO(x)O(y)i = const.+ v2hh(x)h(y)i+ v3h'i+ vh'3i+ h'4

i

hO(x)O(y)i = v2hh(x)h(y)itl + hh(x)h(y)i2tl +O(g2,�)

1. Construct a gauge-invariante operator

2. Expand Higgs field in correlator in fluctuations around the vev

3. Perform standard perturbation theory on the right-hand side

4. Compare poles on both sides

Fröhlich et al ’80, ‘81

4

Fröhlich-Morchio-Strocchi mechanism

• Confirmed on the lattice for SU(2)-Higgs theory

O(x) = (�†�)(x)

� = v + '

hO(x)O(y)i = const.+ v2hh(x)h(y)i+ v3h'i+ vh'3i+ h'4

i

hO(x)O(y)i = v2hh(x)h(y)itl + hh(x)h(y)i2tl +O(g2,�)

1. Construct a gauge-invariante operator

2. Expand Higgs field in correlator in fluctuations around the vev

3. Perform standard perturbation theory on the right-hand side

4. Compare poles on both sides

Fröhlich et al ’80, ‘81

Maas ’12

5

Fröhlich-Morchio-Strocchi mechanism

• W-Higgs sector of the standard model

• Local SU(2)L Symmetry

L = �1

4W i

µ⌫Wiµ⌫ + (Dµ�)

†Dµ�� U(�†�)

Wµ ! UWµU�1 � i

g(@µU)U�1, � ! U�

5

Fröhlich-Morchio-Strocchi mechanism

• W-Higgs sector of the standard model

• Local SU(2)L Symmetry

L = �1

4W i

µ⌫Wiµ⌫ + (Dµ�)

†Dµ�� U(�†�)

Wµ ! UWµU�1 � i

g(@µU)U�1, � ! U�

• Additional global SU(2)cust. symmetry

Wµ ! Wµ, � ! ↵�+ ��⇤

5

Fröhlich-Morchio-Strocchi mechanism

• W-Higgs sector of the standard model

• Local SU(2)L Symmetry

L = �1

4W i

µ⌫Wiµ⌫ + (Dµ�)

†Dµ�� U(�†�)

Wµ ! UWµU�1 � i

g(@µU)U�1, � ! U�

• Additional global SU(2)cust. symmetry

• Convenient to rewrite Higgs-Lagrangian

Wµ ! Wµ, � ! ↵�+ ��⇤

5

L = Tr[@µX†@µX]� U(TrX†X)

Fröhlich-Morchio-Strocchi mechanism

• W-Higgs sector of the standard model

• Local SU(2)L Symmetry

L = �1

4W i

µ⌫Wiµ⌫ + (Dµ�)

†Dµ�� U(�†�)

Wµ ! UWµU�1 � i

g(@µU)U�1, � ! U�

• Additional global SU(2)cust. symmetry

• Convenient to rewrite Higgs-Lagrangian

where X =�� �

�=

✓�⇤2 �1

��⇤1 �2

◆=

✓v 00 v

◆+O(')

Wµ ! Wµ, � ! ↵�+ ��⇤

6

Fröhlich-Morchio-Strocchi mechanism

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

• Possible to construct a custodial triplet which expands to the elementary gauge triplet

hOiµ(x)O

j⌫(y)i = g

2v4hW

iµ(x)W

j⌫ (y)i�

ii�jj +O('2)

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

• Flavor = weak gauge charge in a given generation

• Possible to construct a custodial triplet which expands to the elementary gauge triplet

hOiµ(x)O

j⌫(y)i = g

2v4hW

iµ(x)W

j⌫ (y)i�

ii�jj +O('2)

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

• Flavor = weak gauge charge in a given generation• Necessary to construct suitable gauge-invariant state which

emulate elementary fermions

• Possible to construct a custodial triplet which expands to the elementary gauge triplet

hOiµ(x)O

j⌫(y)i = g

2v4hW

iµ(x)W

j⌫ (y)i�

ii�jj +O('2)

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

• Flavor = weak gauge charge in a given generation• Necessary to construct suitable gauge-invariant state which

emulate elementary fermions

• Possible to construct a custodial triplet which expands to the elementary gauge triplet

hOiµ(x)O

j⌫(y)i = g

2v4hW

iµ(x)W

j⌫ (y)i�

ii�jj +O('2)

• E.g., (left-handed) electron and neutrino

O⌫e = X

†✓⌫

e

◆=

✓�2⌫ � �1e

�⇤1⌫ + �

⇤2e

◆= v

✓⌫

e

◆+O(')

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

• Pole of bound state the same as for the elementary fields

• Flavor = weak gauge charge in a given generation• Necessary to construct suitable gauge-invariant state which

emulate elementary fermions

• Possible to construct a custodial triplet which expands to the elementary gauge triplet

hOiµ(x)O

j⌫(y)i = g

2v4hW

iµ(x)W

j⌫ (y)i�

ii�jj +O('2)

• E.g., (left-handed) electron and neutrino

O⌫e = X

†✓⌫

e

◆=

✓�2⌫ � �1e

�⇤1⌫ + �

⇤2e

◆= v

✓⌫

e

◆+O(')

6

Fröhlich-Morchio-Strocchi mechanism• Full symmetry acting on the Higgs field: SU(2)L x SU(2)cust.

X ! UL X Ucust X =

✓�⇤2 �1

��⇤1 �2

◆

• Pole of bound state the same as for the elementary fields

• Mapping of local to global multiplets

• Flavor = weak gauge charge in a given generation• Necessary to construct suitable gauge-invariant state which

emulate elementary fermions

• Possible to construct a custodial triplet which expands to the elementary gauge triplet

hOiµ(x)O

j⌫(y)i = g

2v4hW

iµ(x)W

j⌫ (y)i�

ii�jj +O('2)

• E.g., (left-handed) electron and neutrino

O⌫e = X

†✓⌫

e

◆=

✓�2⌫ � �1e

�⇤1⌫ + �

⇤2e

◆= v

✓⌫

e

◆+O(')

7

Fermions - Quarks Egger,Maas,RS’17

7

Fermions - Quarks

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

• Obvious for baryons (strong indices suppressed)cijklqiqjqkX

†il

or qiqjqkX†iiX†

jjX†

kk

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

• Obvious for baryons (strong indices suppressed)cijklqiqjqkX

†il

or qiqjqkX†iiX†

jjX†

kk

cijkl = a1 ✏ij�kl + a2 ✏ik�jl + a3 ✏jk�il and i = 1

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

• Obvious for baryons (strong indices suppressed)cijklqiqjqkX

†il

or qiqjqkX†iiX†

jjX†

kk

• Proton: cijkl = a1 ✏ij�kl + a2 ✏ik�jl + a3 ✏jk�il and i = 1

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

• Obvious for baryons (strong indices suppressed)cijklqiqjqkX

†il

or qiqjqkX†iiX†

jjX†

kk

• Proton: cijkl = a1 ✏ij�kl + a2 ✏ik�jl + a3 ✏jk�il and i = 1

• Some mesons are weak-gauge singlets, e.g.,

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

• Obvious for baryons (strong indices suppressed)cijklqiqjqkX

†il

or qiqjqkX†iiX†

jjX†

kk

• Proton: cijkl = a1 ✏ij�kl + a2 ✏ik�jl + a3 ✏jk�il and i = 1

• Some mesons are weak-gauge singlets, e.g., !�meson (uu+ dd)

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

7

Fermions - Quarks

• Quark sector different from lepton sector, bound in hadrons

• Hadrons singlets of strong interaction, not necessarily of weak interaction

• Obvious for baryons (strong indices suppressed)cijklqiqjqkX

†il

or qiqjqkX†iiX†

jjX†

kk

• Proton: cijkl = a1 ✏ij�kl + a2 ✏ik�jl + a3 ✏jk�il and i = 1

• Some mesons are weak-gauge singlets, e.g.,

• Not true for all mesons, e.g., pions

!�meson (uu+ dd)

⇡+ : Oud2 O

ud1 (⇠ du)

Oud = X

†✓u

d

◆=

✓�2u� �1d

�⇤1u+ �

⇤2d

◆= v

✓u

d

◆+O(')

Egger,Maas,RS’17

8

Beyond the standard model Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

Aµi

mA 2(N � 1)

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

Aµi

mA 2(N � 1)

Aµi

r2(N � 1)

NmA 1

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

Aµi

mA 2(N � 1)

Aµi

r2(N � 1)

NmA 1

1i�†Dµ�r

2(N � 1)

NmA

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

Aµi

mA 2(N � 1)

Aµi

r2(N � 1)

NmA 1

1i�†Dµ�r

2(N � 1)

NmA

1/1(N � 1)mAO±1

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

Aµi

mA 2(N � 1)

Aµi

r2(N � 1)

NmA 1

1i�†Dµ�r

2(N � 1)

NmA

1/1(N � 1)mAO±1

1/1O±1 (N � 1)mA

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

Maas, Törek ’16, Maas, RS, Törek ’17

8

Beyond the standard model

• SU(N) gauge theory + Higgs in fundamental representation

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

1�

mh 1�†�

0 (N � 1)2 � 1Aµi

Aµi

mA 2(N � 1)

Aµi

r2(N � 1)

NmA 1

1i�†Dµ�r

2(N � 1)

NmA

1/1(N � 1)mAO±1

1/1O±1 (N � 1)mA

• Local and global symmetry group do not match for N > 2: SU(N) vs U(1)

• Nonperturbative check for N=3

Maas, Törek ’16, Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUT Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUTh�i⇠vh⌃i⇠w

w � vSU(5) SU(3)xSU(2)xU(1) SU(3)xU(1)

Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUTh�i⇠vh⌃i⇠w

w � vSU(5) SU(3)xSU(2)xU(1) SU(3)xU(1)

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

⇠w

⇠w⇠w

'a 6�i

�i

�i ⇠w

831

1�

Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUTh�i⇠vh⌃i⇠w

w � vSU(5) SU(3)xSU(2)xU(1) SU(3)xU(1)

Aµ

ZµW±µ

Xµ

Y µ⇠w⇠w

0mW

mZ

11/1166

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

⇠w

⇠w⇠w

'a 6�i

�i

�i ⇠w

831

1�

Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUTh�i⇠vh⌃i⇠w

w � vSU(5) SU(3)xSU(2)xU(1) SU(3)xU(1)

Aµ

ZµW±µ

Xµ

Y µ⇠w⇠w

0mW

mZ

11/1166

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

⇠w

⇠w⇠w

'a 6�i

�i

�i ⇠w

831

1�

• Global symmetry: U(1)xZ2

Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUTh�i⇠vh⌃i⇠w

w � vSU(5) SU(3)xSU(2)xU(1) SU(3)xU(1)

O0+ mh

O0� mh

O±1,+

O±1,�

1

1

1/11/1⇠w

⇠w

Aµ

ZµW±µ

Xµ

Y µ⇠w⇠w

0mW

mZ

11/1166

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

⇠w

⇠w⇠w

'a 6�i

�i

�i ⇠w

831

1�

• Global symmetry: U(1)xZ2

Maas, RS, Törek ’17

9

Beyond the standard model - SU(5) GUTh�i⇠vh⌃i⇠w

w � vSU(5) SU(3)xSU(2)xU(1) SU(3)xU(1)

O0+ mh

O0� mh

O±1,+

O±1,�

1

1

1/11/1⇠w

⇠w

Aµ

ZµW±µ

Xµ

Y µ⇠w⇠w

0mW

mZ

11/1166

Mass MassField OperatorDegeneracyJP

h

Degeneracy

0+ mh 1

⇠w

⇠w⇠w

'a 6�i

�i

�i ⇠w

831

1�

⇠w⇠w

O0+

O0�

00

11

1/11/1O±1,+

O±1,+

• Global symmetry: U(1)xZ2

Maas, RS, Törek ’17

10

Summary

• Observable spectrum must be gauge invariant

• Non-Abelian gauge theory: composite operator

• FMS mechanism provides a mapping of the local to the global multiplets

• Same results in leading order for the standard model

• BSM model building may be affected

• Verification requieres non-perturbative methods