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,+
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Mass MassField OperatorDegeneracyJP
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• 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