Introduction to Supersymmetry in Elementary Particle Physics Simon Albino
II. Institute for Theoretical Physics at DESY, University of Hamburg
[email protected] Every Thursday 8:30-10:00am, starting 17.04.2008, in seminar room 5, DESY Lecture notes at www.desy.de/~simon/susy.html
Abstract
These lectures aim towards supersymmetry relevant for near-future high energy experiments, but some technical footing in supersymmetry and in symmetries in general is given first. We discuss various motivations for and consequences of a fermion-boson symmetry. The two most physically relevant types of supermultiplet are discussed, followed by a redetermination of their content and properties from the simpler superfield formalism in superspace, in which supersymmetry is naturally manifest. The construction of supersymmetric Lagrangians is determined, from which the minimally supersymmetric extension of the Standard Model and its consequences for grand unification are derived. The physically required soft supersymmetry breaking is applied to the MSSM to obtain constraints on the mass eigenstates and spectrum. We will begin with a self contained development of continuous internal and external symmetries of particles in general, followed by a determination of the external symmetry properties of fermions and bosons permitted in a relativistic quantum (field) theory, and we highlight the importance of the Lagrangian formalism in the implementation of symmetries and its applicability to the Standard Model and the MSSM.
Contents
1
2
Quantum mechanics of particles
1
1.1
Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Fermionic and bosonic particles
3
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Symmetries in QM
5
2.1
Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
(Matrix) Representations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
External symmetries
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15
2.3.1
Rotation group representations
2.3.2
Poincar´e and Lorentz groups
2.3.3
Relativistic quantum mechanical particles
2.3.4
Quantum field theory
2.3.5
Causal field theory .
2.3.6
Antiparticles
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15
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19
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21
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24
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26
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27
2.4
2.3.7
Spin in relativistic quantum mechanics .
2.3.8
Irreducible representation for fields .
2.3.9
Massive particles
2.3.10
Massless particles
2.3.11
Spin-statistics connection .
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28
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30
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31
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32
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External symmetries: fermions
34
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36
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36
2.4.1
Spin
1 2
fields
2.4.2
Spin
1 2
in general representations
2.4.3
The Dirac field
2.4.4
The Dirac equation
2.4.5
Dirac field equal time anticommutation relations
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43
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47
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49
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50
2.5
External symmetries: bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.6
The Lagrangian Formalism
54
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2.6.1
Generic quantum mechanics .
2.6.2
Relativistic quantum mechanics
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54 55
2.7
Path-Integral Methods
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57
2.8
Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.9
3
2.8.1
Abelian gauge invariance
2.8.2
Non-Abelian gauge invariance
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60
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62
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64
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66
The Standard Model 2.9.1
Higgs mechanism .
2.9.2
Some remaining features
2.9.3
Grand unification
2.9.4
Anomalies
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70
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72
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74
Supersymmetry: development
75
3.1
Why SUSY?
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75
3.2
Haag-Lopuszanski-Sohnius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.3
Supermultiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.3.1 3.4
Field supermultiplets (the left-chiral supermultiplet) .
Superfields and Superspace
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90
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94
3.5
3.4.1
Chiral superfield
3.4.2
Supersymmetric Actions
4
5
98
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101
Spontaneous supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
3.5.1 3.6
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O’Raifeartaigh Models
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106
Supersymmetric gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
3.6.1
Gauge-invariant Lagrangians
3.6.2
Spontaneous supersymmetry breaking in gauge theories
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The Minimally Supersymmetric Standard Model
112 118
120
4.1
Left-chiral superfields
4.2
Supersymmetry and strong-electroweak unification
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124
4.3
Supersymmetry breaking in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
4.4
Electroweak symmetry breaking in the MSSM
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134
4.5
Sparticle mass eigenstates
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139
Supergravity
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120
141
1 1.1
Quantum mechanics of particles Basic principles
Physical states represented by directions of vectors (rays) |ii in Hilbert space of universe. Write conjugate transpose |ii† as hi|, scalar product |ji† · |ii as hj|ii. Physical observable represented by Hermitian operator A = A† such that hAii = hi|A|ii. Functions of observables represented by same functions of their operators, f (A). Errors hi|A2|ii − hAi2i etc. vanish when |ii = |ai, where A|ai = a|ai , i.e. |ai is A eigenstate, real eigenvalue a. |ai form complete basis. If 2 observables A, B do not commute, [A, B] 6= 0, eigenstates of A do not coincide with those of B. If basis |X(a, b)i are A, B eigenstates, any |ii =
If A, B commute, their eigenstates coincide. AB|ai = BA|ai = aB|ai, so B|ai ∝ |ai.
P
X
CiX |Xi obeys [A, B]|ii = 0 =⇒ [A, B] = 0.
1
Completeness relation: Expand |ii =
P
a Wia |ai
P
a |aiha|
2 = 1.
then act from left with ha′ | −→ Wia′ = ha′ |ii, so |ii =
P
a |aiha|ii
Probability to observe system in eigenstate |ai of A to be in eigenstate |bi of B: Pa→b = |hb|ai|2. Pa→b are the only physically meaningful quantities, thus |ii and eIα |ii for any α represent same state. P
a Pb→a = 1 for some state |bi as expected.
Principle of reversibility: Pa→b = Pb→a.
|bi =
P
a ha|bi|ai. Act from left with hb| gives 1 =
hb|ai = ha|bi∗
P
a ha|bihb|ai.
Time dependence: Time evolution of states: |i, ti = e−IHt|ii, H is Hamiltonian with energy eigenstates. Probability system in state |ii observed in state |ji time t later = |Mi→j |2, transition amplitude Mi→j = hj|e−IHt|ii . Average value of observable Q evolves in time as hQii(t) = hi|eIHtQe−IHt|ii. Q is conserved ⇐⇒ [Q, H] = 0 (hQi(t) independent of t).
1.2
Fermionic and bosonic particles
3
Particle’s eigenvalues = σ. Particle states |σ, σ ′, ...i completely span Hilbert space. Vacuum is |0i = |i. |σ, σ ′, ...i = ±|σ ′, σ, ...i for bosons/fermions. Pauli exclusion principle: |σ, σ, σ ′...i = 0 if σ fermionic. |σ, σ ′ , ...i and |σ ′ , σ, ...i are same state, |σ, σ ′ , ...i = eIα |σ ′ , σ, ...i = e2Iα |σ, σ ′ , ...i =⇒ eIα = ±1.
Creation/annihilation operators: a†σ |σ ′, σ ′′, ...i = |σ, σ ′, σ ′′, ...i, so |σ, σ ′, ...i = a†σ a†σ′ . . . |0i . [a†σ , a†σ′ ]∓ = [aσ , aσ′ ]∓ = 0 (bosons/fermions). |σ, σ ′ , ...i = a†σ a†σ′ | . . .i = ±|σ ′ , σ, ...i = ±a†σ′ a†σ | . . .i
aσ removes σ particle =⇒ aσ |0i = 0 . E.g. (aσ |σ ′ , σ ′′ i)† · |σ ′′′ i = hσ ′ , σ ′′ |(a†σ |σ ′′′ i) = 0 unless σ = σ ′ , σ ′′′ = σ ′′ or σ = σ ′′ , σ ′′′ = σ ′ . i.e. aσ |σ ′ , σ ′′ i = δσσ′′ |σ ′ i ± δσσ′ |σ ′′ i.
[aσ , a†σ′ ]∓ = δσσ′ . e.g. 2 fermions a†σ′′ a†σ′′′ |0i: Operator a†σ aσ′ replaces any σ ′ with σ, must still vanish when σ ′′ = σ ′′′ . Check: (a†σ aσ′ )a†σ′′ a†σ′′′ |0i = −a†σ a†σ′′ aσ′ a†σ′′′ |0i + δσ′ σ′′ a†σ a†σ′′′ |0i = −δσ′ σ′′′ a†σ a†σ′′ |0i + δσ′ σ′′ a†σ a†σ′′′ |0i.
Expansion of observables: Q =
P∞ P∞ N =0
† ′ ;σ ...σ a ′ M =0 CN M ;σ1′ ...σN 1 M σ1
. . . a†σ′ aσM N
4 . . . aσ1 .
Can always tune the CN M to give any values for h0|aσ1′ . . . aσL′ Qa†σ1 . . . a†σK |0i.
Commutations with additive observables: [Q, a†σ ] = q(σ)a†σ (no sum) , where Q is an observable such that for |σ, σ ′, . . .i, total Q = q(σ) + q(σ ′) + . . . and Q|0i = 0 (e.g. energy). Check for each particle state: Qa†σ |0i = q(σ)a†σ |0i + a†σ Q|0i = q(σ)a†σ |0i, Qa†σ a†σ′ |0i = a†σ Qa†σ′ |0i + q(σ)a†σ a†σ′ |0i = a†σ a†σ′ Q|0i + q(σ ′ )a†σ a†σ′ |0i + q(σ)a†σ a†σ′ |0i = (q(σ) + q(σ ′ ))a†σ a†σ′ |0i etc.
Note: Conjugate transpose is [Q, aσ ] = −q(σ)aσ . Number operator for particles with eigenvalues σ is a†σ aσ (no sum).
Additive observable: Q =
P
σ
Qσ a†σ aσ .
E.g. (a†σ aσ )a†σ a†σ a†σ′ |0i = 2a†σ a†σ a†σ′ |0i.
E.g. Q = H, (free) Hamiltonian, Qσ = Eσ , energy eigenvalues.
2 2.1
Symmetries in QM Unitary operators
Symmetry is powerful tool: e.g. relates different processes. Symmetry transformation is change in our point of view (e.g. spatial rotation / translation), does not change experimental results. i.e. all |ii −→ U |ii does not change any |hj|ii|2. Continuous symmetry groups G require U unitary: hj|ii → hj|U †U |ii = hj|ii, so U †U = U U † = 1 , includes U = 1. Wigner + Weinberg: General physical symmetry groups require U unitary, or antiunitary: hj|ii → hj|U † U |ii = hj|ii∗ = hi|ji, e.g. (discrete) time reversal.
hAi unaffected (and hf (A)i in general), so must have A → U AU † . Transition amplitude Mi→j unaffected by symmetry transformation, i.e. hj|U †e−IHtU |ii = hj|e−IHt|ii, which requires [U, H] = 0 if time translation and symmetry transformation commute .
5
6 Parameterize unitary operators as U = U (α), αi real, i = 1, ..., d(G). d(G) is dimension of G, minimum no. of paramenters required to distinguish elements. Choose group identity at α = 0, i.e. U (0) = 1. So for small αi, can write U (α) ≈ 1 + Itiαi . ti are the linearly independent generators of G. Since U †U = 1, ti = t†i , i.e. ti are Hermitian . [U, H] = 0 =⇒ [ti, H] = 0, so conserved observables are generators. Can replace all ti → t′i = Mij tj if M invertible and real, then αi′ = αj Mji−1 real. In general, U (α)U (β) = U (γ(α, β)) (up to possible phase eIρ, removable by enlarging group).
7 U (α) = exp [Itiαi] if Abelian limit is obeyed: for βi ∝ αi, U (α)U (β) = U (α + β) (usually true for physical symmetries, e.g. rotation about same line / translation in same direction). Can write U (α) = [U (α/N )]N , then for N → ∞ is [1 + Iti αi /N + O(1/N 2 )]N = exp [Iti αi ] + O
Lie algebra: [ti, tj ] = ICijk tk , where Cijk are the structure constants appearing in U (α)U (β) = U (α + β + 12 ICαβ + cubic and higher) (where (Cαβ)k = Cijk αiβj , and no O(α2) ensures U (α)U (0) = U (α), likewise no O(β 2)). LHS: eIαt eIβt ≈ 1 + Iαt − 21 (αt)2 × 1 + Iβt − 21 (βt)2 ≈ 1 + I(αt + βt) −
1 2
h
i (αt) + (βt) + 2(αt)(βt) 2
2
1
RHS: eI(α+β+ 2 ICαβ)t ≈ 1 + I(α + β + 21 ICαβ)t − ≈ 1 + I(αt + βt +
1 2 ICαβt)
1 2
−
2 (α + β + 21 ICαβ)t 1 2
h
i (αt) + (βt) + (αt)(βt) + (βt)(αt) , 2
2
i.e. 21 ICijk αi βj tk = 21 [αi ti , βj tj ]. Now take all αi , βj zero except e.g. α1 = β2 = ǫ → IC12k tk = [t1 , t2 ] etc., gives Lie algebra.
1 N
.
8 In fact, Lie algebra completely specifies group in non-small neighbourhood of identity. This means that for U (α)U (β) = U (γ), γ = γ(α, β) can be found from Lie algebra. We have shown this above in small neighbourhood of identity, i.e. to 2nd order in α, β, only. Check to 3rd order: Write X = αiti, Y = βiti, can verify that I 1 exp[IX] exp[IY ] = exp[I(X + Y ) − [X, Y ] + ([X, [Y, X]] + [Y, [X, Y ]]) + quadratic and higher]. 2 12 | {z } has the form Iγi ti , γi real
Lie algebra implies: 1. Cijk = −Cjik (antisymmetric in i, j).
Cijk can be chosen antisymmetric in i, j, k (see later).
2. Cijk real. Conjugate of Lie Algebra is
h
t†j , t†i
i
∗ † = −ICijk tk , which is negative of Lie Algebra because t†i = ti .
∗ ∗ Thus Cijk tk = Cijk tk , but tk linearly independent so Cijk = Cijk .
3. t2 = tj tj is invariant (commutes with all ti, so tranforming give eIαiti t2e−Iαk tk = t2). [t2 , ti ] = tj [tj , ti ] + [tj , ti ]tj = ICjik {tj , tk } = 0 by (anti)symmetry in (j, i) j, k. Examples: Rest mass, total angular momentum. tj tj doesn’t have to include all j, only subgroup.
2.2
9
(Matrix) Representations
Physically, matrix representation of any symmetry group G of nature formed by particles: (†)
General transformation of aσ : U (α)a†σ U †(α) = Dσσ′ (α)a†σ′ , with invariant vacuum: U (α)|0i = |0i . U (α)a†σ |0i is 1 particle state, so must be linear combination of a†σ′ |0i, i.e. U (α)a†σ |0i = Dσσ′ (α)a†σ′ |0i. Thus U (α)a†σ a†σ′ |0i = Dσσ′′ (α)Dσ′ σ′′′ (α)a†σ′′ a†σ′′′ |0i etc.
D(α) matrices furnish a representation of G , matrix generators (ti)σ′σ with same Lie algebra [ti, tj ]σσ′ = ICijk (tk )σσ′ . Since U (α)U (β) = U (γ), must have Dσ′′ σ′ (α)Dσ′ σ (β) = Dσ′′ σ (γ).
If Abelian limit of page 7 is obeyed, Dσ′σ (α) = eIαiti
σ′σ
.
Similarity transformation (ti)σσ′′′ → (t′i)σσ′′′ = Vσσ′ (ti)σ′σ′′ (V −1)σ′′σ′′′ also a representation. Particle states require V unitary. a†σ |0i → a′†σ |0i = Vσσ′ a†σ′ |0i, then othogonality h0|a′σ′ a′†σ |0i = δσ′ σ requires V † V = 1.
But in general, representations don’t have to be unitary.
In “reducible” cases, can similarity transform ti → V tiV −1 such that ti is block-diagonal matrix, each block furnishes a representation, e.g.: (ti)σ′σ =
(ti)jk 0 0 (ti)αβ
. σ′σ
Particles of one block don’t mix with those of other — can be treated as 2 separate “species”. Each block can have different t2, corresponds to different particle species.
Irreducible representation: Matrices (ti)σ′σ not block-diagonalizable by similarity transformation. In this sense, these particles are elementary. Size of matrix written as m(r) × m(r), where r labels representation. Corresponds to single species, single value of t2: t2 = C2(r)1 (consistent with [t2, ti] = 0). C2(r) is quadratic Casimir operator of representation r.
10
11 Fundamental representation of G: Generators written (ti)αβ . Matrices representing elements used to define group G (also called defining representation). Adjoint representation of G (r = A): Generators (tj )ik = ICijk , satisfy Lie algebra. Use Jacobi identity
[ti , [tj , tk ]] + [tj , [tk , ti ]] + [tk , [ti , tj ]] = 0.
From Lie algebra, [ti , [tj , tk ]] = I[ti , Cjkl tl ] = −Cjkl Cilm tm , so Jacobi identity is Cjkl Cilm + Ckil Cjlm + Cijl Cklm = 0 (after removing contraction with linearly independent tm ), or, from Cijk = −Cjik , −ICkjl IClim + ICkil ICljm − ICijl ICklm = 0, which from (tj )ik = ICijk reads [ti , tj ]km = ICijl (tl )km .
Conjugate representation has generators t∗i = −tTi (obey the same Lie algebra as ti). ∗
If −t∗i = U tiU † (U unitary), then e−Iαiti = U eIαiti U †, i.e. conjugate representation ≡ original representation, −→ representation is real. T
For invariant matrix g (“metric”), i.e. eIαiti geIαiti = g, G transformation leaves φT gψ invariant. φT gψ
→ =
φT
T
i ti e|Iα {z }
geIαi ti ψ
=
=U e−Iαi ti U †
† φT U e−Iαi ti U U eIαi ti U † gψ |{z} =1
T
=U eIαi ti U †
=
T
Iαi ti i ti φT U e−Iαi ti U † e|−Iα geIαi ti ψ {z } |e {z }
† φT eIαi ti U † eIαi ti U † gψ | {z } =1
=g
=
φT gψ.
12 Semi-simple Lie algebra: no ti that commutes with all other generators (no U(1) subgroup). Semi-simple group’s matrix generators must obey tr[ti] = 0. Make all tr[ti ] = 0 except one, tr[tK ], via t′i = Mij ti (this is just rotation of vector with components tr[ti ]). Determinant of U (α)U (β) = U (γ) is eIαi tr[ti ] eIβi tr[ti ] = eIγi tr[ti ] , i.e. (αi + βi − γi )tr[ti ] = 0. But only tr[tK ] 6= 0, so αK + βK − γK = 0. Must have CijK αi βj = 0 (recall γK ≃ αK + βK + CijK αi βj ), or CijK = CKij = 0 for all i, j, so from Lie group we have [tK , ti ] = 0 for all i.
13 Normalization of generators chosen as tr[titj ] = C(r)δij . Nij =tr[ti tj ] becomes Mik Nkl (M T )lj after basis transformation ti → Mij tj . Nij components of real symmetric matrix N , diagonalizable via M N M T when M real, orthogonal. Also N is positive definite matrix αT N α = αi tr[ti tj ]αj =tr[αi ti αj tj ] =tr[(αi ti )† αj tj ] ≥ 0 (because for any matrix A, tr[A† A] = A†βα Aαβ = A∗αβ Aαβ =
P
αβ
|Aαβ |2 ≥ 0).
After diagonalization, Nij = 0 for i 6= j and above implies Nii > 0 (no sum). Then multiply each ti by real number ci , changes Nii → c2i Nii > 0. Choose ci such that each Nii (no sum over i) all equal to positive C(r).
Representation dependence of quadratic Casimir operator: C2(r)m(r) = C(r)d(G) . Definition of quadratic Casimir operator gives tr[t2 ] = C2 (r)m(r). Normalization of generators tr[ti tj ] = C(r)δij =⇒ tr[t2 ] = C(r)d(G).
Example: 2 and 3 component repesentations of rotation group have different spins
1 2
and 1.
Antisymmetric structure constants: Cijk =
I − C(r) tr[[ti, tj ]tk ] .
From Lie algebra, tr[[ti , tj ]tl ] = ICijk tr[tk tl ] = ICijl C(r).
Structure constants obey CjkiClki = C(A)δjl . In adjoint representation, quadratic Casimir operator on page 10 is (t2 )jl = C2 (A)δjl = −Cjik Ckil . But C2 (A) = C(A): C2 (A)m(A) = C(A)d(G) from representation dependence of quadratic Casimir operator on page 13, and m(A) = d(G).
14
2.3 2.3.1
External symmetries Rotation group representations
(Spatial) rotation of vector v → Rv preserves v T v, so R orthogonal (RT R = 1). Rotation |θ| about θ: U (θ) = e−IJ ·θ . Lie algebra is [Ji, Jj ] = Iǫijk Jk (see later).
Or use J3 and raising/lowering operators J± = (J1 ± IJ2).
15
16 Irreducible spin j representations: (j)
(j)
(J3 )m′m = mδm′m and (J± )m′m = [(j ∓ m)(j ± m + 1)]δm′,m±1 , where m = −j, −j + 1, ..., j , spin j = 0, 12 , 1, ... , number of components n.o.c.= 2j + 1 and J 2 = j(j + 1) . Let |m, ji be J3 = m and J 2 = F (j) orthonormal eigenstates. J± changes m by ±1 because [J± , J3 ] = ∓J± , so J± |m, ji = C± (m, j)|m ± 1, ji. C∓ (m, j) =
p
F (j) − m2 ± m (states absorb complex phase):
|C∓ (m, j)|2 = hm, j|J± J∓ |m, ji (J±† = J∓ ) and J± J∓ = J 2 − J32 ± J3 . Let j be largest m value for given F (j) (m bounded because m2 = hm, j|J32 |m, ji = F (j) − hm, j|J12 + J22 |m, ji < F (j)). F (j) = j(j + 1) because J+ |j, ji = 0, so J− J+ |j, ji = (F (j) − j 2 − j)|j, ji = 0. Let −j ′ be smallest m: J− | − j ′ , ji = 0 =⇒ F (j) = j ′ (j ′ + 1), so j ′ = j (other possibility −j ′ = j + 1 > j). So m = −j, −j + 1, ..., j, i.e. n.o.c.= 2j + 1. Since n.o.c. is integer, j = 0, 12 , 1, ....
17 Spin decomposition of tensors: e.g. 2nd rank tensor Cij (n.o.c.=9), representations are j = 0, 1, 2: scalar (n.o.c.=1) + antisymmetric rank 2 tensor (n.o.c.=3) + symmetric traceless rank 2 (n.o.c.=5) components. Cij = 31 δij Ckk + 12 (Cij − Cji ) + 12 (Cij + Cji − 32 δij Ckk ). Component irreducible representations signified by 1
+
3
+
Counting n.o.c. shows they are equivalent respectively to the j = 0 (2j + 1 = 1), j = 1 (2j + 1 = 3) and j = 2 (2j + 1 = 5) representations: 1 3 δij Ckk
→ 13 δij Ckk is like scalar ≡ spin 0,
( 21 (Cjk − Ckj ))|i6=k,j = 21 ǫijk Cjk → Ril 12 ǫljk Cjk (because ǫR2 = Rǫ) is like vector ≡ spin 1, 1 2 (Cij
+ Cji − 32 δij Cii ) → Ril Rjm 12 (Clm + Cml − 32 δlm Ckk ) is like rank 2 tensor ≡ spin 2.
5.
18 Direct product: 2 particles, spins j1, j2, also in a representation: eIJ ·θ |m1, j1; m2, j2i = (eIJ
(j1 ,j2 ) ·θ
From eIJ ·θ |m1 , j1 ; m2 , j2 i = (eIJ
(j1 ) ·θ
1
)m′1 m1 (eIJ
(j2 ) ·θ
2
(j )
(j )
(j ,j )
)m′1m′2m1m2 |m′1, j1; m′2, j2i where Jm′1m2′ m
1 m2
= Jm′1m δm′2m2 + δm′1m1 Jm′2m . 1
1
2
2
)m′2 m2 |m′1 , j1 ; m′2 , j2 i, i.e. total rotation is rotation of each particle in turn.
Thus J3|m1, j1; m2, j2i = (m1 + m2)|m1, j1; m2, j2i, (j ,j2 )
so |m1, j1; m2, j2i is combination of J3 1
, J (j1,j2)2 eigenstates |2; m1 + m2, ji,
with j = m1 + m2, m1 + m2 + 1, ..., j1 + j2.
Triangle inequality: representation for (j1, j2) contains j = |j1 − j2|, |j1 − j2| + 1, j1 + j2 . Number of orthogonal eigenstates |m1 , j1 ; m2 , j2 i = Number of orthogonal eigenstates |2; m, ji, i.e. (2j1 + 1)(2j2 + 1) =
Pj1 +j2
j=jmin (2j
+ 1),
so jmin = |j1 − j2 |.
Example: Representation for 2 spin 1 particles (j1, j2) = (1, 1): From triangle inequality, this is ≡ sum of irreducible representations j = 0, 1, 2. Also from tensor representation on page 17: product of 2 vectors uivj (2nd rank tensor) is 3 × 3 = 1 + 3 + 5.
19 2.3.2
Poincar´ e and Lorentz groups
Poincar´e (inhomogeneous Lorentz) group formed by coordinate transformations xµ → x′µ = Λµν xν + aµ, preserving spacetime separation: gµν dxµdxν = gµν dx′µdx′ν . Implies: Transformation of metric tensor: gρσ = gµν ΛµρΛνσ or (Λ−1)µν = Λνµ . Identity: Λµν = δ µν , aµ = 0 .
Poincar´e group defined by U (Λ, a)U (Λ, a) = U (ΛΛ, Λa + a) . This is the double transformation x′′ = Λx′ + a = Λ(Λx + a) + a.
Poincar´ e group generators: J µν and P µ , appearing in U (1 + ω, ǫ) ≃ 1 + 12 Iωµν J µν − IǫµP µ . Obtained by going close to identity, Λµν = δ µν + ω µν and aµ = ǫµ .
Choose J µν = −J νµ . Allowed because ωρσ = −ωσρ : Transformation of metric tensor reads gρσ = gµν (δ µρ + ω µρ )(δ νσ + ω νσ ) ≃ gσρ + ωρσ + ωσρ .
Transformation properties of P µ, J µν : U (Λ, a)P µU †(Λ, a) = ΛρµP ρ
20
and U (Λ, a)J µν U †(Λ, a) = ΛρµΛσν (J ρσ − aρP σ + aσ P ρ) . Apply Poincar´e group to get U (Λ, a)U (1 + ω, ǫ) | {z } =U (Λ(1+ω),Λǫ+a)
U † (Λ, a) | {z }
=U (Λ−1 ,−Λ−1 a)
= U (1 + ΛωΛ−1 , Λ(ǫ − ωΛ−1 a)).
Expand both sides in ω, ǫ: U (Λ, a)(1 + 12 IωJ − IǫP )U † (Λ, a) = 1 + 12 IΛωΛ−1 J − IΛ(ǫ − ωΛ−1 a)P , equate coefficients of ω, ǫ.
So P µ transforms like 4-vector, Jij like angular momentum.
Poincar´ e algebra: I[J ρσ , J µν ] = −g σν J ρµ − g ρµJ σν + g σµJ ρν + g ρν J σµ → (homogeneous) Lorentz group, I[P µ, J ρσ ] = g µρP σ − g µσ P ρ , and [P µ, P ν ] = 0 . Obtained by taking Λµν = δ µν + ω µν , aµ = ǫµ in transformation properties of P µ , J µν , to first order in ω, ǫ gives P µ − 21 Iωρσ [P µ , J ρσ ] + Iǫν [P µ , P ν ] = P µ + 12 ωρσ g µσ P ρ − 21 ωρσ g µρ P σ , and J µν + 12 Iωρσ [J ρσ , J µν ] − Iǫρ [P ρ , J µν ] = J µν − g ρµ ǫρ P ν + g ρν ǫρ P µ + 21 ωρσ (g ρν J σµ − g σν J ρµ ) + 12 ωρσ (g σµ J ρν − g ρµ J σν ).
21 2.3.3
Relativistic quantum mechanical particles
Identify H = P 0, spatial momentum P i, angular momentum Ji = 21 ǫijk J jk (i.e. (J1, J2, J3) = (J 23, J 31, J 12)). [H, P ] = [H, J ] = 0 → P , J conserved. Rotation group [Ji, Jj ] = Iǫijk Jk is subgroup of Poincar´e group. µ
aµ Explicit form of Poincar´ e elements: U (Λ, a) = e|−IP {z } × translate aµ
(V : magnitude of 4-velocity’s spatial part),
eβ e|−IK·ˆ {z }
boost along eˆ by V =sinh β
×
IJ ·θ e|{z}
rotate |θ| about θ
where boost generator K = (J 10, J 20, J 30), obeys [Ji, Kj ] = Iǫijk Kk and [Ki, Kj ] = −Iǫijk Jk . K not conserved: [Ki, H] = IPi, because boost and time translation don’t commute.
Lorentz transformation of 4-vectors: (Ki)µν = I(δ0µδiν + δiµδ0ν ) and (Ji)µν = −Iǫ0iµν , so µ h iµ h iµ 2 −IK·ˆeβ µ ˆ e = 1 − IKieˆi sinh β − (Kieˆi)2 (cosh β − 1) = 1 + IJiθˆi sin θ − Jiθˆi (1 − cos θ) . and eIJ ·θθ ν ν
Follows directly from (Ki eˆi )3 = Ki eˆi and (Ji θˆi )3 = Ji θˆi .
ν
ν
Particles: Use P µ, P 2 = m2 eigenstates. Distinguish momentum p from list σ, i.e. a†σ → a†σ (p). (†)
Commutation relations for aσ (p): [a†σ (p), a†σ′ (p′)]∓ = [aσ (p), aσ′ (p′)]∓ = 0 and [aσ (p), a†σ′ (p′)]∓ = δσσ′ (2p0)δ (3)(p − p′) . As on page 3, but with different normalization (Lorentz invariant).
(†)
Application of general transformation of aσ on page 9 to Lorentz transformation complicated by mixing between σ and p. Simplify by finding 2 transformations, one that mixes σ and one that mixes p, separately.
22
Lorentz transformation for
(†) aσ (p):
U (Λ)a†σ (p)U †(Λ) =
Dσ′σ (W (Λ, p))a†σ′ (Λp) .
23
Same as general transformation on page 9, but because σ → {σ, p}, mixing of σ with p must be allowed.
Construction of W : Choose reference momentum k and transformation L to mix k but not σ (defines σ): Lµν (p)k ν = pµ and U (L(p))a†σ (k)|0i = a†σ (p)|0i . Then W (Λ, p) = L−1(Λp)ΛL(p) . W mixes σ but not k, i.e. belongs to little group of k: W µν k ν = k µ . General transformation is U (Λ)a†σ (p)|0i = U (Λ)U (L(p))a†σ (k)|0i, using definition of L. Multiplying by 1 = U (L(Λp))U (L−1 (Λp)) gives U (Λ)a†σ (p)|0i = U (L(Λp)) U (L−1 (Λp)) U (Λ)U (L(p))a†σ (k)|0i = U (L(Λp))U (W (Λ, p))a†σ (k)|0i. But W doesn’t change k, so U (Λ)a†σ (p)|0i = U (L(Λp))Dσ′ σ (W (Λ, p))a†σ′ (k)|0i. Then L(Λp) changes k to Λp but doesn’t change σ ′ .
(†)
Poincar´ e transformation for aσ (p): U (Λ, a)a†σ (p)U †(Λ, a) = e−IΛp·aDσ′σ (W (Λ, p))a†σ′ (Λp) . (†)
In Lorentz transformation for aσ (p), use U (Λ, a) = e−IP
µ
aµ
U (Λ) (from explit form of Poincar´e elements on page 21).
24 2.3.4
Quantum field theory
Lorentz invariant QM: H =
R
d3xH (x) , scalar field H (x) (i.e. U (Λ, a)H (x)U †(Λ, a) = H (Λx + a) ),
obeys cluster decomposition principle (two processes with large spatial separation evolve independently).
Causality: [H (x), H (y)] = 0 when (x − y)2 ≥ 0 . Required for Lorentz invariance of S-matrix. Intuitive reason: signal can’t propagate between 2 spacelike separated events.
(†)
In QM (general): H built from aσ . In QFT: H built from H (x) built from products of Fields ψl−c(x) =
R
+ D3p vlσ (x; p)ac† σ (p) and ψl (x) =
Lorentz invariant momentum space volume D3 p =
d3 p 2p0
R
D3p ulσ (x; p)aσ (p) , obeying
= d4 pδ(p2 + m2 )θ(p0 ) obeys D3 Λp = D3 p. (c)±
Poincar´ e transformation for fields: U (Λ, a)ψl
(c)±
(x)U †(Λ, a) = Dll′ (Λ−1)ψl′
† Take single particle species in representation labelled j, allow ac† σ (p) 6= aσ (p).
(Λx + a) .
25
x dependence of u, v: ul σ (x; p) = eIp·xul σ (p) , vl σ (x; p) = e−Ip·xvl σ (p) . (†)
Take Λ = 1 in Poincar´e transformation for fields and for aσ (p) on page 24 and 23, e.g. U (1, a)ψl+ (x)U † (1, a) = = ψl+ (x + a) =
R
R
D3 p ulσ (x; p)U (1, a)aσ (p)U † (1, a) =
R
D3 p eIp·a ulσ (x; p)aσ (p)
D3 p ulσ (x + a; p)aσ (p), equate coefficients of aσ (p) (underlined), gives ul σ (x; p)eIp·a = ulσ (x + a; p). ±(c)
Klein-Gordon equation: (∂ 2 − m2)ψl Act on e.g. ψl−c (x) =
R
(x) = 0 .
2 2 2 2 D3 p e−Ip·x vl σ (p)ac† σ (p) with (∂ − m ), use p = −m .
(j)
(j)
Transformation of u, v: Dll′ (Λ)ul′ σ (p) = Dσ′σ (W (Λ, p))ulσ′ (Λp) , Dll′ (Λ)vl′ σ (p) = Dσσ′ (W −1(Λ, p))vlσ′ (Λp) . (†)
E.g. consider v, use Poincar´e transformation for fields and for aσ (p) on page 24 and 23. LHS: U (Λ, a)ψl−c (x)U † (Λ, a) = RHS: Dll′ (Λ−1 )ψl−c ′ (Λx + a) =
R
R
† D3 p vlσ (x; p)U (Λ, a)ac† σ (p)U (Λ, a) =
D3 p Dll′ (Λ−1 )vlσ (x + Λa; p)ac† σ (p) =
R
R
(j)
D3 p e−IΛp·a vlσ (x; p)Dσ′ σ (W (Λ, p))ac† σ ′ (Λp), and
D3 Λp Dll′ (Λ−1 )vlσ (x + Λa; Λp)ac† σ (Λp) (p → Λp). (j)
−1 Use D3 Λp = D3 p, equate coefficient of ac† (Λ, p)). σ (Λp) (underlined), multiply by Dl′′ l (Λ)Dσ ′′ σ (W
p dependence of u, v: ulσ (p) = Dll′ (L(p))ul′ σ (k) . (p dependence of v is the same.) In transformation of u, v, take p = k so L(p) = 1, Λ = L(q) so Λp = L(q)k = q, then W (Λ, p) = L−1 (Λp)L(q) = 1.
26 2.3.5
Causal field theory
Since [ψl+(x), ψl−c ′ ]∓ 6= 0, causality on page 24 only gauranteed by taking H (x) to be functional of complete field ψl (x) = κψl+(x) + λψl−c(x) (so representations Dll′ (Λ−1) for ψl+(x) and ψl−c(x) the same), with
Causality: [ψl (x), ψl′ (y)]∓ = [ψl (x), ψl†′ (y)]∓ = 0 when (x − y)2 > 0 by suitable choice of κ, λ. Now h0|H|0i = ∞, i.e. consistency with gravity not gauranteed by QFT. Complete field: ψl (x) =
R
(p) . D3p κeIp·xulσ (p)aσ (p) + λe−Ip·xvlσ (p)ac† σ
27 2.3.6
Antiparticles
H (x) commutes with conserved additive Q: [Q, H (x)] = 0 . Imposed in order to satisfy [Q, H] = 0.
This is achieved as follows: Commutation of fields with conserved additive Q: [Q, ψl (x)] = −ql ψl (x) , and Field construction of H (x): H = (Mi, Lj label particle species).
P
L1 L2 M1 † M2 † M1 M2 ψlL11 ψlL22 . . . ψm . . . with q ψ + q + . . . − qm − qm − ... = 0 m l l 2 1 1 2 1 2
Antiparticles: For every particle species there is another species with opposite conserved quantum numbers . Commutation of fields with conserved additive Q implies [Q, ψl−c (x)] = −ql ψl−c (x) and [Q, ψl+ (x)] = −ql ψl+ (x), c c† c c but since [Q, ac† σ ] = qσ aσ and [Q, aσ ] = −qσ aσ (no sum) from page 4, ql = qσ and ql = −qσ , i.e. qσ = −qσ .
28 2.3.7
Spin in relativistic quantum mechanics
Lorentz group algebra simplified by choosing generators Ai = 21 (Ji − IKi) and Bi = 21 (Ji + IKi) , behaves like 2 independent rotations: [Ai, Aj ] = Iǫijk Ak , [Bi, Bj ] = Iǫijk Bk and [Ai, Bj ] = 0 , i.e. relativistic particle of type (A, B) (i.e. in eigenstate of A2 = A(A + 1), B 2 = B(B + 1)) ≡ 2 particles at rest, ordinary spins A, B (in representation sense). In terms of degrees of freedom, (A, B) = (2A+1) × (2B+1). Triangle inequality: ordinary spin J = A + B , so j = |A − B|, |A − B| + 1, ..., A + B . Derived on page 18.
Can have eigenstates of A2 = A(A + 1), B 2 = B(B + 1) and J 2 = j(j + 1) simultaneously, because [J 2, Ai] = [J 2, Bi] = 0 .
Use J 2 = (A + B)2 and A, B commutation relations.
Simultaneous eigenstate also with any K = I(A − B) or K 2 not possible.
Example: (A, B) =
( 21 , 12 )
is representation of 4-vector: (A3)µν , (B3)µν can have eigenvalues
± 21
29 only.
From triangle inequality on page 28, j = 0, 1. Also follows from tensor representation on page 17: 2 × 2 = 1 + 3. More generally, rank N tensor is i.e.
( 12 , 21 )N
=
N N 2, 2
( 12 , 12 )N
=
+lower spins, where
(N, 0) and (0, N ) are purely spin j = N .
P N2
A=0
N N 2, 2
P N2
B=0 (A, B),
≡ traceless symmetric rank N tensor, j = 0, ..., N .
30 2.3.8
Irreducible representation for fields (c)†
If particles created by aσ (p) have spin j, must take field with same spin j but any (A, B) consistent with triangle inequality on page 28: ψab(x) =
R
Ip·x −Ip·x c† D p κ e uab σ (p)aσ (p) + λ e vab σ (p)aσ (p) (l = ab, l′ = a′b′), 3
(A)
(B)
Lorentz transformation uses generators Aa′b′ab = Ja′a δb′b , Ba′b′ab = δa′aJb′b where a = −A, −A + 1, ..., A and b = −B, −B + 1, ..., B .
31 2.3.9
Massive particles
Choose k = (0, 0, 0, m) (momentum of particle at rest). Then W is an element of the spatial rotation group, i.e. rotation group apparatus of subsubsection 2.3.1 applies to relativity too. L(p) in terms of K: L(p) = exp[−I pˆ · Kβ] .
In this case, L(p) includes boost in p direction by 4-velocity (B)
(A)
(j)
(j)∗
(A)
|p| m
= sinh β.
(B)
Conditions on u, v: Jσ′σ uab σ′ (0) = Jaa′ ua′b σ (0) + Jbb′ uab′ σ (0) , −Jσ′σ vab σ′ (0) = Jaa′ va′b σ (0) + Jbb′ vab′ σ (0) . In transformation of u, v on page 25, take Λ = R and p = 0 (so p = k, Rp = p, L(p) = 1, W (R, p) = L−1 (Rp)RL(p) = L−1 (p)R = R), (j)
(j)
(j)∗
so e.g. Dll′ (R)vl′ σ (0) = Dσσ′ (R−1 )vlσ′ (0), and use Dσσ′ (R−1 ) = Dσ′ σ (R) because irreducible representations of R are unitary. (j)∗
Take l = ab etc., Daba′ b′ (R)va′ b′ σ (0) = Dσ′ σ (R)vabσ′ (0). Generators of D(j)∗ (R) and D(R) are −J (j)∗ and A + B.
u, v relation: vab σ (0) = (−1)j+σ uab (j)∗
−σ (0) ′
up to normalization.
(j)
Conditions on u, v with −Jσσ′ = (−1)σ−σ J−σ,−σ′ from page 16 gives vab σ (0) ∝ (−1)σ uab
−σ (0),
absorb proportionality constant into u, v.
The p dependence of u, v are the same, so vab σ (p) = (−1)j+σ uab
−σ (p) .
32 2.3.10
Massless particles
Take reference vector k = (0, 0, 1, 1). Little group transformation: W (θ, µ, ν) ≃ 1 + IθJ3 + IµM + IνN with M = J2 + K1, N = −J1 + K2 . W has 3 degrees of freedom: For (ti )µν k µ = 0, take ti = (J3 , M, N ), check with Lorentz transformation of 4-vectors on page 21.
Choice of states: (J3, M, N )|k, σi = (σ, 0, 0)|k, σi . Since [M, N ] = 0, try eigenstates for which M |k, m, ni = m|k, m, ni, N |k, m, ni = n|k, m, ni. Then m, n continuous degrees of freedom, unobserved: [J3 , M ] = IN , so M (1 − IθJ3 )|k, m, ni = (m − nθ)(1 − IθJ3 )|k, m, ni, i.e. (1 − IθJ3 )|k, m, ni is eigenvector of M , eigenvalue m − nθ. Similarly, [J3 , N ] = −IM , so (1 − IθJ3 )|k, m, ni is eigenvector of N , eigenvalue n + mθ. Avoid this problem by taking m = n = 0, so left with states J3 |k, σi = σ|k, σi.
ˆ σ is helicity, component of spin in direction of motion. Since J3 = J · k, Representation for massless particles: Dσ′σ (W ) = eIθσ δσ′σ . U (W )|k, σi = (1 + IθJ3 + IµM + IνN )|k, σi = (1 + Iθσ)|k, σi. For finite θ, U (W )|k, σi = eIθσ |k, σi.
33 p dependence of u: ul σ (p) = Dll′ (L(p))ul σ (k) . (p dependence of v is the same.)
As on page 25.
Little group transformation of u, v: ul σ (k)eIθ(W,k)σ = Dll′ (W )ul σ (k) , vl σ (k)e−Iθ(W,k)σ = Dll′ (W )vl σ (k) . Transformation of u, v on page 25 reads ul σ (Λp)eIθ(Λ,p)σ = Dll′ (Λ)ul σ (p). Take Λ = W , p = k.
Rotation of u, v: (J3)ll′ ul′ σ (k) = σul σ (k) , (J3)ll′ vl′ σ (k) = −σvl σ (k) . Take W to be rotation about 3-axis in little group transformation of u, v.
M , N transformation of u, v: Mll′ ul σ (k) = Nll′ ul σ (k) = 0 . Same for v. Take W = (1 + IµM + IνN ) in little group transformation of u, v.
u, v relation: vl σ (p) = u∗l σ (p) . Implied (up to proportionality constant) by rotation (note (J3 )ll′ is imaginary) and M , N transformation of u, v.
Allowed helicities for fields in given (A, B) representation: σ = ±(B − A) for particle/antiparticle . J = A + B, so rotation of u is σuab σ (0) = (a + b)uab σ (0). M , N transformation of u is Maba′ b′′ ua′ b′
σ
= Naba′ b′ ua′ b′
σ
= 0.
Using M = IA− − IB+ and N = −A− − B+ , where A± = A1 ± IA2 , B± = B1 ± IB2 are usual raising/lowering operators, (A− )aba′ b′ ua′ b′
σ
= (B+ )aba′ b′ ua′ b′
Similar for v, gives σ = A − B.
σ
= 0, i.e. must have a = −A, b = B or uab
σ
= 0. So σ = B − A.
34 2.3.11
Spin-statistics connection
Determine which of ∓ for given j is possible for causality on page 26 to hold. Demand more general condition h i R 3 † ∗ Ip·(x−y) ∗ −Ip·(x−y) ˜ e κe ∓ λλ = 0 for (x − y)2 > 0, [ψab(x), ψ˜a˜˜b(y)]∓ = D p πab,˜a˜b(p) κ˜ where ψab, ψ˜a˜˜b for same particle species and πab,˜a˜b(p) = uab σ (p)˜ u∗a˜˜b σ (p) = vab σ (p)˜ va˜∗˜b σ (p). (†)
In field on page 30, use commutation relations for aσ (p) on page 22. va˜∗˜b σ (p) holds for massive particles from u, v relation on page 31. uab σ (p)˜ u∗a˜˜b σ (p) = vab σ (p)˜ va˜∗˜b σ (p)]∗ in massless case from u, v relation on page 33). (uab σ (p)˜ u∗a˜˜b σ (p) = [vab σ (p)˜ ˜ ˜∗ . Relation between κ, λ of different massive fields: κ˜ κ∗ = ±(−1)2A+2B λλ
p Explicit calculation shows πab,˜a˜b (p) = Pab,˜a˜b (p) + 2 p2 + m2 Qab,˜a˜b (p), ˜
where (P, Q)(p) are polynomial in p, obey (P, Q)(−p) = (−1)2A+2B (P, −Q)(p). Take (x − y)2 > 0, use frame x0 = y 0 , write ∆+ (x) =
R
D3 peIp·x :
i i h h ˜ ˜ ∗ 2A+2B ∗ ∗ 2A+2B ∗ ˜ ˜ ˜ κ ± (−1) λλ Qab,˜a˜b (−I∇)δ (3) (x − y). [ψab , ψa˜˜b ]∓ = κ˜ κ ∓ (−1) λλ Pab,˜a˜b (−I∇)∆+ (x − y, 0) + κ˜ Commutator must vanish for x 6= y, so coefficient of P zero.
35
Relations between κ, λ of single field: |κ|2 = |λ|2 and ±(−1)2A+2B = 1 . ˜ B = B, ˜ relation between κ, λ of different fields reads |κ|2 = ±(−1)2A+2B |λ|2 . For A = A,
Spin-statistics: Bosons (fermions) have even (odd) 2j and vice versa . From triangle inequality on page 28, j − (A + B) is integer, so ±(−1)2j = 1, i.e. in [ψab , ψ˜a˜˜b ]∓ , must have − (+) for even (odd) 2j.
Relation between κ, λ of single massive field: λ = (−1)2AeIcκ , c is the same for all fields. ˜ 2: Divide relation between κ, λ of different massive fields on page 34 by |˜ κ|2 = |λ|
κ κ ˜
˜
˜
= ±(−1)2A+2B λλ˜ = (−1)2A+2A λλ˜ .
Absorb κ into field, eIc into ac† σ (p) (does not affect commutation relations on page 22). (−1)2A can’t be absorbed into ac† σ (p) since this is independent of A, nor absorbed into v since this is already chosen. So Massive irreducible field: ψab(x) = or more fully as ψab(x) =
R
R
(p) , D3p eIp·xuab σ (p)aσ (p) + (−1)2Ae−Ip·xvab σ (p)ac† σ
(j) D3p Daba′b′ (L(p)) eIp·xua′b′ σ (0)aσ (p) + (−1)2A+j+σ e−Ip·xua′b′
Use u, v relation on page 31.
c† (p) . (0)a −σ σ
2.4 2.4.1
36
External symmetries: fermions Spin
1 2
fields
2j is odd −→ particles are fermions. j =
1 2
representations include (A, B) = (A,B)
In each case, group element acts on 2 component spinor Xab 1 2 ,0
i.e. X ( ) ≡ XL = 0, 21
and X ( ) ≡ XR =
( 21 ,0)
( 12 ,0)
X 1 ,0 , X− 1 ,0
2
2
0, 21 0, 21
X
( )
0, 12 0,− 12
,X
( )
,
1 2, 0
and (A, B) = 0,
(“left-handed”)
(“right-handed”).
“Handedness”/chirality refers to eigenstates of helicity for massless particles (see later).
X is a field operator, but argument x suppressed.
1 2
,
Lorentz transformation of spinors: U (Λ)XL/R U †(Λ) = hL/R (Λ)XL/R
1 ,0 / 0, 1 1 1 2 ·θ −IK 2 ,0 / 0, 2 ·ˆ eβ
IJ ( 2
From explicit form of Lorentz group elements on page 21, hL/R (Λ) = e ( 21 ,0)
Lorentz group generators for spinors: Ji where σi are the Pauli σ matrices σ1 = which obey rotation group algebra (A)
Follows from Ji = Ji and
(0) Ji
( 12 )
= 0 and Ji
(B)
+ Ji
1
0 1 1 0
=
1 2 σi ,
, σ2 =
( 21 ,0)
Ki
=
0 −I I 0
37 (D acts on the 2 a, b components).
I 12 σi ,
)( )
(0, 21 )
Ji
and σ3 =
e
=
( )( )
1 2 σi ,
1 0 0 −1
(0, 12 )
Ki
, where
= −I 21 σi ,
,
1 1 σ , σ 2 i 2 j = Iǫijk 2 σk . (A)
and Ki = I(Ji
(B)
− Ji ) on page 28,
= 12 σi from irreducible representation for spin j on page 16. 1
1
Explicit form of hL/R : hL/R = eI 2 σiθi e∓ 2 σieˆiβ .
Note σ matrices are Hermitian.
Product of σ matrices: σiσj = δij + Iǫijk σk .
Follows by explicit calculation.
Direct calculation of hL/R : hL/R = (cos 2θ + Iσiθˆi sin 2θ )(cosh β2 ∓ σieˆi sinh β2 ) . From product of σ matrices, T 2 = 1 where T = σi eˆi (T = σi θˆi ). Then exT = cosh x + T sinh x (eIxT = cos x + IT sin x).
38
1 2 ,0
Write hL = h and Xa = (X1, X2) = XL = X ( ) , which transforms as 1. Xa′ = habXb . ˙
Can also defined spinor transforming with h∗: Use dotted indices for h∗, so 2. Xa′†˙ = ha∗˙ bXb˙† .
Then hR = h∗−1T .
From explicit form of hL/R on page 37. 1
Use upper indices for h−1, so X †a˙ = (X †1, X †2) = XR = X (0, 2 ) , ˙
transformation is 4. X ′†a˙ = h∗−1T
a˙
X b˙
˙
†b˙
Dotted indices because h∗ is used.
, where we define (hT )a b = hb a. ′a
−1T a
Conjugate of this turns dotted indices into undotted indices, so 3. X = h
b
Xb .
Conjugate of spinors: (X a)† = X †a˙ , (Xa)† = Xa†˙ . Definition of X †a˙ in terms of X a, Xa†˙ in terms of Xa. ˙
Check that if Xa transforms as transformation 1., Xa†˙ transforms as transformation 2.: Xa′†˙ = (Xa′ )† = (hab Xb )† = h∗a˙ b Xb˙† .
Spinor metric: ǫab =
0 1 −1 0
39 , ǫab = −ǫab. Note ǫabǫbc = δac. ǫ is unitary matrix.
Pseudo reality: ǫacσc dǫdb = −(σ T )ab . Follows by explicit calculation. Since σi∗ = σiT , shows rotation group representation by σ matrices is real (see page 11).
Xa†˙ is 0, 12 , i.e. right-handed, like X †a˙ .
˙
Pseudo reality implies h∗ , hR same up to unitary similarity transformation: ǫa˙ c˙ hc∗˙ d ǫd˙b˙ = (h∗−1T )a˙ b˙ . For unitary representations, follows because A = B † from definition of page 28, so conjugation makes
This means dotted indices are for right-handed ( 0, 12 ) fields. (Similarly, undotted indices are for left-handed (
1 2, 0
) fields.)
1 2, 0
→ 0, 21 .
Raising and lowering of spinor indices: X a = ǫabXb . This is definition of X a in terms of Xa.
40
Follows that Xa = ǫabX b . Same definition / behaviour for dotted indices. Check that if Xa transforms as transformation 1. on page 38, X a transforms as transformation 3.: X ′a = ǫab Xb′ = ǫab hbc Xc . From pseudo reality, ǫab hbc ǫcd = h−1T
a
, or ǫab hbc = h−1T d
Right-handed from left-handed fields: X
†a˙
a
ab
ǫbc , so X ′a = h−1T b
a
ǫbc Xc = h−1T b
†
= ǫ Xb .
So all fields can be expressed in terms of left-handed fields.
Scalar from 2 spinors: XY = X aYa = −YaX a = Y aXa = Y X . X a′ Ya′ = h−1T
a
X c hab Yb = h−1 c
a c
hab X c Yb . X, Y anticommute (spinor operators).
Hermitian conjugate of scalar: (XY )† = (X aYa)† = (Ya)†(X a)† = Ya˙†X †a˙ = Y †X † .
a
b
X b.
4-vector σ matrices σaµb˙ : σai b˙ = (σi)ab˙ , σa0b˙ = (σ0)ab˙ and σ0 = 4-vector σ matrices with raised indices: σ µ
ab ˙
1 0 0 1
˙
= ǫbcσcµd˙ǫa˙ d , so σ i
41 .
ab ˙
˙ = −(σi)ab , σ0
ab ˙
˙ = (σ0)ab .
Second result follows from definition by explicit calculation. ˙ ba
Inner product of 4-vector σ matrices: gρω σeωf˙σ ρ Outer product of 4-vector σ matrices: σaνb˙ σ ρ
˙ ba
˙
= −2δeaδf˙b .
= −2g νρ .
˙
X aσaµb˙ Y †b is vector . ˙
˙
˙
Need to show X ′a σaµb˙ Y ′†b = Λµν X a σaνb˙ Y †b . Since X ′a σaµb˙ Y ′†b = h−1T need to show h−1 gives Λµρ = − 21 σ ρ
c µ σ a cd˙
˙ ba
h−1
h∗−1T
c µ σ a cd˙
d˙
b˙
= Λµν σaνb˙ . Contracting with σ ρ
h∗−1T
d˙
b˙
a
˙ ba
X c σaµb˙ h∗−1T c
b˙
Y d˙
†d˙
,
and using outer product of 4-vector σ matrices
= − 12 tr σ ρ h−1 σ µ h∗−1T , which is equivalent because σ matrices linearly independent
(or multiply this by gρω σeωf˙ and use inner product of 4-vector σ matrices).
Verify last result using direct calculation of hL/R on page 37 and Lorentz transformation of 4-vectors on page 21.
42 Convenient to put left- and right-handed fields together as 4 component spinor: ψT =
1 2 ,0
1 ,0 2 − 21 ,0
(0, 12 )
( ) ( ) C1X 1 ,0 , C1X , C2Y0, 1 2
2
(0, 21 )
, C2Y0,− 1 2
Lorentz group generators: Ji =
1 2 σi
0
0 1 2 σi
, where C1, C2 scalar constants.
and Ki =
0 I 12 σi 0 −I 12 σi
.
Follows from Lorentz group generators for spinors on page 37.
This is the chiral (Weyl) representation, others representations from similarity transformation: Ji′ = V JiV −1, Ki′ = V KiV −1, X ′ = V X etc.
In our index notation, ψ =
Xa ˙ Y †b
. Preferable to express in terms of left-handed fields only: ψ =
Xa (ǫbcYc)†
.
43 2.4.2
Spin
1 2
in general representations
Any 4×4 matrix can be constructed from sums/products of gamma matrices (next page): Gamma matrices (chiral representation): γ 0 = −I
0 1 1 0
and γ i = −I
0 σi −σi 0
, or γ µ = −I
µ
0 σ σµ 0
.
Gamma matrix construction of 4×4 matrix same in representations related by similarity transformation γ ′µ = V γ µV −1 . Anticommutation relations for γ µ: {γ µ, γ ν } = 2g µν . This is representation independent. Check by explicit calculation in chiral representation.
Define γ5 =
1 0 0 −1
Then PL = 21 (1 + γ5) =
= −Iγ 0γ 1γ 2γ 3.
1 0 0 0
similarly PR = 12 (1 − γ5) =
Check last equality explicitly in chiral representation.
projects out left-handed spinor: PL
0 0 0 1
projects out right-handed spinor: PR
2 Note PL/R are projection operators: PL/R = PL/R , PL/R PR/L = 0.
=
Xa (ǫbcYc)†
Xa (ǫ Yc)† bc
Xa 0 =
,
0 (ǫbcYc)†
.
Any 4×4 matrix is linear combination of 1, γ µ, [γ µ, γ ν ], γ µγ5, γ5.
44
Because these are 16 non-zero linearly independent 4×4 matrices. Non-zero because their squares, calculated from anticommutation relations on page 43, are non-zero. Linearly independent because they are orthogonal if we define scalar product of any two to be trace of their matrix product: tr[1γ µ ] = 0, because tr[γ µ ] = 0 in chiral representation, therefore in any other representation. tr[1[γ µ , γ ν ]] = 0 by (anti)symmetry. From anticommutation relations, tr[1γ µ γ5 ] = −tr[γ5 γ µ ] = −tr[γ µ γ5 ] = 0 and also tr[1γ5 ] =tr[γ5 ] = 0 by commuting γ 0 from left to right. Next, tr[γ ρ [γ µ , γ ν ]] = 0: From anticommutation relations, γ52 = −γ 0 γ 1 γ 2 γ 3 γ 0 γ 1 γ 2 γ 3 = −γ 0 2 γ 1 2 γ 2 2 γ 3 2 = 1. So tr[γ ρ [γ µ , γ ν ]] =tr[γ52 γ ρ [γ µ , γ ν ]] = − tr[γ5 γ ρ [γ µ , γ ν ]γ5 ] =tr[γ5 γ ρ [γ µ , γ ν ]γ5 ]. To show tr[γ ρ γ µ γ5 ] = 0, first consider case ρ = µ. Then γ ρ γ µ = g ρρ , and result is ∝tr[γ5 ] = 0. If e.g. ρ = 1, µ = 2, tr[γ ρ γ µ γ5 ] = Itr[γ 0 γ 3 ] = 0. tr[γ µ γ5 ] = 0 already shown. tr[[γ µ , γ ν ]γ ρ γ5 ] = 0 from anticommutation relations. tr[[γ µ , γ ν ]γ5 ] = 0 because tr[γ ρ γ µ γ5 ] = 0. Finally tr[γ µ γ5 γ5 ] =tr[γ µ ] = 0.
So all spinorial observables expressible as representation independent sums/products of gamma matrices.
Lorentz group generators from γ µ: J µν =
− I4 [γ µ, γ ν ] .
45
Check by explicit calculation in chiral representation using Ji = 12 ǫijk Jjk and K = (J 10 , J 20 , J 30 ).
Similarity transformation on page 42 equivalent to similarity transformation on page 43.
Infinitesimal Lorentz transformation of γ µ: I[J µν , γ ρ] = g νργ µ − g µργ ν , Follows from anticommutation relations for γ µ on page 43 and Lorentz group generators from γ µ .
Lorentz transformation of γ µ: D(Λ)γ µD−1(Λ) = Λνµγ ν , i.e. γ µ transforms like a vector. Agrees with infinitesimal Lorentz transformation of γ µ , because for infinitesimal case, LHS is (1 + 12 Iωρσ J ρσ )γ µ (1 − 12 Iωωη J ωη ) = γ µ − 21 Iωρσ [γ µ , J ρσ ], and RHS is (δνµ + ωρσ g µσ )γ ρ = γ µ − 12 ωρσ (g µρ γ σ − g µσ γ ρ ).
General boost: γ µ(Λp)µ = D(Λ) γ µpµ D−1(Λ) . From Lorentz transformation of γ µ .
Reference boost:
γ µ pµ m
= −D(L(p))γ 0D−1(L(p)) . Equivalent of p = L(p)k on page 23, −γ 0m = γ µkµ.
In general boost, take p = k, Λ = L(q). Then γ µ qµ = D(L(q)) γ µ kµ D−1 (L(q)) = −D(L(q)) γ 0 m D−1 (L(q)).
Parity transformation matrix: β = Iγ 0 =
0 1 1 0
46 . Note β 2 = 1.
Pseudo-unitarity of Lorentz transformation: J µν† = βJ µν β , D† = βD−1β . βγ 0 β = γ 0 = −γ 0† and βγ i β = −γ i = −γ i† , or βγ µ β = −γ µ† . Then use Lorentz group generators from γ µ on page 45. Infinitesimal D† is 1 − 21 Iωmuν J µν† = β(1 − 12 Iωmuν J µν )β.
Adjoint spinor: X = X †β . Allows construction of scalars, vectors etc. from spinors: Covariant products: XY is scalar , Xγ µY is vector . First case: X ′ Y ′ = X ′† βY ′ = X † D† βDY . From pseudo-unitarity of Lorentz transformation, D† β = βD−1 β 2 = βD−1 , so X ′ Y ′ = X † βD−1 DY = X † βY . Second case: X ′ γ µ Y ′ = X † D† βγ µ DY = X † βD−1 γ µ DY as before. Lorentz transformation of γ µ on page 45 can be rewritten D−1 γ µ D = Λ−1
µ ν γ ν
= Λµν γ ν , so X ′ γ µ Y ′ = Λµν X † βγ ν Y .
Vanishing products: ψ R/LψL/R and ψ L/R γ µψL/R , where ψL/R = PL/R ψ (see page 43 for PL/R ). Using γ5† = γ5 in chiral representation and γ µ γ5 = −γ5 γ µ , ψ R ψL = ψ † PL βPL ψ = ψ † βPR PL ψ = 0. Similarly, ψ L γ µ ψL = ψ † PR βγ µ PL ψ = ψ † βγ µ PR PL ψ = 0.
47 2.4.3
The Dirac field
Group 4 possibilities for
Likewise, vσT =
(A,B) uab σ
together as 4 component spinor: uTσ =
( 21 ,0)
( 12 ,0)
(0, 12 )
2
2
(0, 12 )
u 1 ,0,σ , u− 1 ,0,σ , u0, 1 ,σ , u0,− 1 ,σ . 2
2
(0, 12 ) (0, 12 ) −v 1 ,0,σ , −v− 1 ,0,σ , v0, 1 ,σ , v0,− 1 ,σ . ( 21 ,0)
( 12 ,0)
2
2
2
2
First 2 v components multiplied by (−1)2A = −1 to remove it from massive irreducible field on page 35.
Dirac field: ψ(x) =
Xa (ǫbcYc)†
=
R
d3 p (2π)3
Ip·x −Ip·x c† e uσ (p)aσ (p) + e vσ (p)aσ (p) .
(Note D3p → d3p for convention.) Anticommutation relations for spin 12 : [aσ (p), a†σ′ (p′)]+ = (2π)3δσσ′ δ (3)(p − p′) (i.e. no 2p0 factor). p dependence of spin
1 2
u, v: uσ (p) =
q
m D(L(p))uσ (0) p0
This is just the p dependence of u, v on page 25.
and vσ (p) =
q
m D(L(p))vσ (0) . p0
Condition on spin
1 2
u, v:
(0, 21 ) 1 ∗ − 2 σi σ′σ v0b σ′ (0)
=
(0, 21 ) (0, 12 ) 1 1 ′ ′ 2 σi bb v0b′ σ (0) , 2 σi σ σ u0b σ ′ (0)
=
48
(0, 21 ) 1 ′ 2 σi bb u0b′ σ (0) .
From conditions on u, v on page 31.
Spin
1 2
u, v relation: v1
or 2 σ (0)
1
= −(−1) 2 +σ u1
or 2 −σ (0)
and v3
or 4 σ (0)
1
= (−1) 2 +σ u3
or 4 −σ (0) .
( 1 ,0) (0, 12 ) ( 1 ,0) (0, 1 ) From u, v relation on page 31, va02 σ (0) = (−1)σ ua02 −σ (0) and v0b σ2 (0) = (−1)σ u0b −σ (0). ( 12 ,0)
Then multiply va0
σ
(0) by (−1)2A = −1 as discussed on page 47.
Form of u, v: uTσ= 1 (0) = 2
√1 (1 2
Solution to condition on spin Use spin
1 2
0 1 0), uTσ=− 1 (0) = 2
1 2
√1 (0 2
2
√1 (0 2
2
√1 (0 2
2 2
2
2
0 1 0), uTσ=− 1 (0) = 2
2 2
1 2, 0
√1 (0 2
2
1 2
2
√1 (−1 2
0 1 0) .
2
and 0, 21 parts individually.
T 1 0 0), vσ= 1 (0) = 2
√1 (0 2
From allowed helicities for fields in given (A, B) representation on page 33, e.g. for allowed σ = 0 −
T 1 0 − 1), vσ=− 1 (0) =
(0, 12 ) (0, 12 ) (0, 12 ) (0, 1 ) and v = v = 0. Components for u constrained similarly. v is v0,−21 , 1 = −v0, 1 ,− 1 0, 1 , 1 0,− 1 ,− 1
u, v relation, and adjust normalizations of
Massless u, v: uTσ= 1 (0) =
T 1 0 1), vσ= 1 (0) =
for particle and σ =
1 2
Majorana particle = antiparticle:
− 0 for antiparticle, i.e. ψL (x) =
ac† σ (p)
=
a†σ (p) ,
so
T ψM
R
d3 p (2π)3
T 1 0 0), vσ=− 1 (0) = 2
1 2, 0
√1 (0 2
0 1 0) .
field such as neutrino, only
1 1 ,0 ,0 ( ( ) ) c† eIp·x u−21 (p)a− 12 (p) + e−Ip·x v 1 2 (p)a 1 (p) .
= Xa, X
2
†b˙
2
(i.e. Ya = Xa on page 47).
2
49 2.4.4
The Dirac equation
Representation independent definition of spin 12 u, v: (Iγ µpµ + m)uσ (p) = 0 and (−Iγ µpµ + m)vσ (p) = 0 . For u and v, reference boost gives
γµp −I m µ uσ (p)
last step from p dependence of spin
1 2
−1
= D(L(p))βD (L(p))uσ (p) =
q
m p0 D(L(p))βuσ (0),
u, v on page 47.
In the chiral representation, and therefore any other representation, βuσ (0) = uσ (0) and βvσ (0) = −vσ (0), so −I
γ µ pµ m uσ (p)
=
q
m p0 D(L(p))uσ (0)
= uσ (p), last step from p dependence of spin
±(c)
Dirac equation: (γ µ∂µ + m) ψl
1 2
u, v again, likewise −I
(x) = 0 .
Act on Dirac field on page 47 with (γ µ ∂µ + m), then use representation independent definition of spin
2
2
1 2
u, v.
Consistent with Klein-Gordon equation ∂ − m ψ ±(c)(x) = 0. From page 25. To check, act on Dirac equation from left with (γ ν ∂ν − m): 0 = (γ ν ∂ν − m) (γ µ ∂µ + m) ψ ±(c) = γ ν γ µ ∂ν ∂µ − m2 ψ ±(c) =
γ µ pµ m vσ (p)
1 ν µ 2 {γ , γ }∂ν ∂µ
− m2 ψ ±(c) = g µν ∂ν ∂µ − m2 ψ ±(c)
using anticommutation relations for γ µ on page 43.
= −vσ (p).
50 2.4.5
Dirac field equal time anticommutation relations
Projection operators from u, v: ul σ (p)ul′ σ (p) = Define Nll′ (p) = ul σ (p) ul′ σ (p) = so N (p) =
m p0 [D(L(p))uσ (0)]l
1 2p0
(−Iγ µpµ + m)ll′ , vl σ (p)v l′ σ (p) =
[u†σ (0)D† (L(p))β]l′ from p dependence of spin
1 2
1 2p0
(−Iγ µpµ − m)ll′ .
u, v on page 47.
m −1 p0 D(L(p))N (0)D (L(p)).
Explicit calculation from the form of u, v on page 48 gives N (0) = 21 (β + 1) which is true in any representation. So N (p) =
1 2p0 D(L(p))
Iγ 0 m + m D−1 (L(p)), then use reference boost on page 45.
Equal time anticommutation relations: [ψl (x, t), ψl†′ (y, t)]+ = δll′ δ (3)(x − y) . Define Rll′ = [ψl (x, t), ψl†′ (y, t)]+ =
R
d3 p Ip·(x−y) [ul σ (p)[u(p)β]l′ σ (2π)3 e
using Dirac field and anticommutation relations for spin From projection operators from u, v, R =
R
1 2
d3 p Ip·(x−y) (2π)3 2p0 e
+ vl σ (−p)[v(−p)β]l′ σ ],
on page 47. Then in“v” term, take p → −p.
Iγ 0 p0 − Iγ · p + m + Iγ 0 p0 + Iγ · p − m β = 1δ (3) (x − y).
2.5
51
External symmetries: bosons
Scalar boson field: ψ(x) =
R
d3 p 1
3
(2π) 2 (2p0 ) 2
Ip·x e a(p) + e−Ip·xac†(p) .
From general form of irreducible field on page 35, where u(p) = u(0) and v(p) = u(p), absorb overall u(0) into field.
Vector boson
field ( 12 , 12 ),
µ
spin 1: ψ (x) =
R
d3 p 3
1
(2π) 2 (2p0 ) 2
0 Ip·x µ c† −Ip·x µ vσ (p)aσ (p) , uσ (0) = vσ0 (0) = 0 . e uσ (p)aσ (p) + e
(j) (j)
Transformation of u, v on page 25 for Λ = R gives e.g. (Jk Jk )σ′ σ uµσ′ = (Jk Jk )µν uνσ . (j) (j)
But (Jk Jk )σ′ σ = j(j + 1)δσ′ σ and (Jk Jk )i j = 2δ i j , (Jk Jk )0µ = 0 from Lorentz transformation of 4-vectors on page 21, so j(j + 1)uiσ (0) = 2uiσ (0), j(j + 1)u0σ (0) = 0, i.e. j = 0 and uiσ (0) = 0 or j = 1 and u0σ (0) = 0.
Projection operator for vector boson
( 12 , 12 ),
spin 1:
uµσ (p)uν∗ σ (p)
Derivation similar to that for projection operators from u, v on page 50.
=
vσµ(p)vσν∗(p)
=
1 2p0
g
µν
+
pµ pν m2
.
52 Projection operator shows there is problem for m → 0. From allowed helicities for fields in given (A, B) representation on page 33, can’t construct ( 12 , 12 ) 4-vector field, where helicity σ = 0, from massless helicity σ = ±1 particle. But can construct 4-component field: Massless helicity ±1 field: Aµ(x) =
R
(p) , where σ = ±1 . D3p eIp·xuµσ (p)aσ (p) + e−Ip·xvσµ(p)ac† σ
Lorentz transformation of massless helicity ±1 polarization vector: e−Iθσ uσ (p) = Λ−1uσ (Λp) + ω(W, k)p . Simplest approach: take p dependence and rotation of u on page 33 to be true, gives uµσ (k) ∝ (1, σ, 0, 0). Then M , N transformation on page 33 cannot be true, in fact M uσ ∝ (0, 0, 1, 1) ∝ k and likewise for N . Then D(W )uσ (k) = eIθσ [uσ (k) + ω(W, k)k]. Since D(Λ) = Λ, multiplying this from the left by e−Iθσ Λ−1 D(ΛL(p)))D(W −1 ) = e−Iθσ Λ−1 L(Λp) gives result.
Lorentz transformation of massless helicity ±1 field: U (Λ)Aµ(x)U †(Λ) = ΛνµAν (Λx) + ∂ µα(x) . Use Lorentz transformation of uµ
σ
above in U (Λ)Aµ (x)U † (Λ) =
R
D3 p eIp·x uµσ (p)e−Iθ(Λ,p)σ aσ (Λp) + e−Ip·x vσµ (p)eIθ(Λ,p)σ ac† (Λp) . σ
As for Poincar´e transformation for fields on page 24, up to gauge transformation. This implies Fµν = ∂µAν − ∂ν Aµ is Lorentz covariant, as expected:
it is antisymmetric, i.e. is (1, 0) or (0, 1) if F µν = ± I2 ǫµνλρFλρ, so from σ = ±(A − B), can have σ = ±1.
53 Brehmstrahlung: Adding emission of massless helicity ±j boson with momentum q ≃ 0 to process with particles n with momenta pn modifies amplitude by factor ∝ uµ1µ2...µj σ (q) where gn is coupling of boson to fermion n and ηn = ±1 for outgoing / incoming particles. Lorentz invariance condition: qµ1
P
µ
µ
µj
gn pn1 pn2 ...pn η n n pn ·q
P
µ
µ
µj
gn pn1 pn2 ...pn η n n p·q
,
= 0.
Can show this for massless helicity ±1 boson: Lorentz transformation of polarization vector on page 52 implies amplitude not Lorentz invariant unless this is true.
Einstein’s principle of equivalence: Helicity ±2 bosons have identical coupling to all fermions . For soft emssion of graviton from process involving multiple fermions of momentum pn , Lorentz invariance condition reads
P
µ2 n ηn gn pn = 0. But momentum conservation is
P
µ2 n ηn p n
= 0, so gn same for all particles.
Constraint on particle spins: Massless particles must have helicity ≤ 2 and ≥ −2 . Lorentz invariance condition can be written
P
µ2 n ηn gn pn
µ
. . . pnj = 0.
For j > 2 this overconstrains 2 → 2 processes, since momentum conservation alone =⇒ it depends on scattering angle only.
2.6 2.6.1
54
The Lagrangian Formalism Generic quantum mechanics
Lagrangian formalism is natural framework for QM implementation of symmetry principles. Can be applied to canonical fields (e.g. Standard Model): Fields ψl (x, t) behave as canonical coordinates, i.e. with conjugate momenta pl (x, t) such that [ψl (x, t), pl′ (y, t)]∓ = Iδ 3(x − y)δll′ and [ψl (x, t), ψl′ (y, t)]∓ = [pl (x, t), pl′ (y, t)]∓ = 0 as usual in QM. In practice, find suitable pl (x, t) by explicit calculation of [ψl (x), ψl†′ (y)]∓. ˙ = Lagrangian formalism: Action A [ψ, ψ] ˙ = where Lagrangian L[ψ(t), ψ(t)]
R
R∞
˙
−∞ dtL[ψ(t), ψ(t)]
is stationary,
d3x pl (x, t)ψ˙ l (x, t) − H[ψ(t), p(t)] .
Coordinates obey field equations ψ˙ l =
δH δpl
when A is stationary.
55 2.6.2
Relativistic quantum mechanics
˙ If L[ψ(t), ψ(t)] =
R
d3xL (ψ(x), ∂µψ(x)),
Lagrangian density L (x) is scalar and A =
R
d4xL (x) is Lorentz invariant.
In practice, determine L from classical field theory, e.g. electrodynamics, then L (ψ(x), ∂µψ(x)) = pl (x)ψ˙ l (x) − H (ψ(x), p(x)) and pl from L : pl (x) = Stationary A requirement gives field equations ∂µ ∂(∂∂L = µψ ) l
∂L ∂ψl
∂L (x) ∂ ψ˙ l (x)
.
(Euler-Lagrange equations),
e.g. Klein-Gordon equation for free spin 0 field, Dirac equation for free spin
1 2
field etc.
(some definition of derivative with respect to operator field ψl must be given here). A must be real.
Let A depend on N real fields. Stationary real and imaginary parts of A → 2N field equations.
Noether’s theorem: Symmetries imply conservation: A invariant under ψl (x) → ψl (x) + IαFl [ψ; x] → conserved current J µ(x), ∂µJ µ = 0, for stationary A . If α made dependent on x, A no longer invariant. But change must be δA =
R
R d4 xJ µ ∂µ α(x) = − d4 x∂µ J µ α(x)
so that δA = 0 when α constant. Now take A stationary: δA = 0 even though α depends on x, so ∂µ J µ = 0.
Scalar boson (0, 0): Lscalar =
− 21 ∂µψ∂ µψ h
−
1 2 2 2m ψ .
56
i ˙ consistent with pl from L on page 55. ˙ Scalar boson field on page 51 implies ψ(x, t), ψ(y, t) = δ (3) (x − y), i.e. p = ψ, −
Field equations on page 55 give Klein-Gordon equation ∂ 2 − m2 ψ = 0 as required. Note ψ is a single operator.
Dirac fermion ( 21 , 0) + (0, 12 ): LDirac = −ψ (γ µ∂µ + m) ψ . Recall ψ is a column of 4 operators, and covariant quantities on page 46. Recall equal time anticommutation relations on page 50, [ψl (x, t), ψl†′ (x, t)]+ = δll′ δ (3) (x − y), so p = ψ † , consistent with pl from L on page 55. Field equations give Dirac equation (γ µ ∂µ + m) ψ = 0 as required.
Vector boson ( 12 , 21 ), spin 1: Lspin
1 vector
= − 41 Fµν F µν − 12 m2ψµψ µ, where Fµν = ∂µψν − ∂ν ψµ .
From vector boson field and projection operator for vector boson ( 21 , 12 ), spin 1, on page 51, h
i 0 ˙ ψi (x, t), ψj (y, t) + ∂j ψ (y, t) = δij δ (3) (x − y), i.e. conjugate momentum to ψi is pi = ψ˙ i + ∂i ψ 0 = F i0 , −
consistent with pl from L on page 55. ψ0 is auxilliary field because p0 = 0. Also ∂µ ψ µ = 0 and Klein-Gordon equation ∂ 2 − m2 ψ µ = 0, which is found from field equations on page 55.
2.7
57
Path-Integral Methods
Follows from Lagrangian formalism. Assume H is quadratic in the pl . Gives direct route from Lagrangian to calculations, all symmetries manifestly preserved along the way. Can work in simpler classical limit then return to QM later. Result is that bosons described by ordinary numbers, fermions by Grassmann variables.
LSZ reduction gives S-matrix from vacuum matrix elements of time ordered product of functions of fields, RQ IA [ψ] h0, out|T ψl (xA ),ψl (xB ),... |0, ini x,l dψl (x)ψlA (xA )ψlB (xB )...e A B RQ . given by path integral as = h0, out|0, ini dψ (x)eIA [ψ] x,l
l
Contribution mostly from field configurations for which A is minimal, i.e. fluctuation around classical result.
Noether’s theorem again (see page 55): A invariant under ψl (x) → ψl (x) = ψl (x) + IαFl [ψ; x]. So if α dependent on x, assuming measure
Q
x,l
RQ
x,l dψl (x) exp[IA ] →
RQ
R 4 µ dψ (x) exp I A − d x∂ J (x)α(x) l µ x,l
dψl (x) invariant. This is just change of variables, so h∂µJ µ(x)i = 0.
2.8
Internal symmetries
58
Consider unitary group representations.
Unitary U(N ): elements can be represented by N × N unitary matrices U (U †U = 1). Dimension d(U(N ))= N 2. 2N 2 degrees of freedom in complex N × N matrix, U † U = 1 is N 2 conditions or N 2 Hermitian N × N matrices: N diagonal reals, N 2 − N off-diagonal complexes but lower half conjugate to upper.
Special unitary SU(N ): same as U(N ) but U ’s have unit determinant (det(U ) = 1). Thus tr[ti] = 0, i.e. group is semi-simple. d(SU(N ))= N 2 − 1. Fundamental representation denoted N . Normalization of fundamental representation: tr[titj ] = 21 δij (i.e. C(N ) = 12 ).
U(1) (Abelian group): elements can be represented by phase eIqα . One generator: the real number q.
SU(2): Fundamental representation denoted 2, spin
1 2
representation of rotation group.
SU(2) is actually the universal covering group of rotation group. 3 generators ti =
σi 2,
[ti, tj ] = Iǫijk tk .
Adjoint representation denoted 3. C(3) = 2. ǫjki ǫlki = (d(SU(2)) − 1)δjl = 2δjl . σ∗
2 representation is real, 2 = 2 (i.e. − 2i = U σ2i U †, pseudoreal), and gαβ = ǫαβ and δαβ . SU(3): 8 generators
λi 2,
structure constants fijk .
Fundamental representation 3: λi are 3 × 3 Gell-Mann matrices. Adjoint representation 8. C(8) = 3. Group is complex, 3 6= 3. Example: 3 × 3 = 8 + 1, i.e. quark and antiquark can be combined to behave like gluon or colour singlet.
59
60 2.8.1
Abelian gauge invariance
Global gauge invariance: Consider complex fermion / boson field ψl (x), arbitrary spin. Each operator in Lfree is product of (∂µ)ψl with (∂µ)ψl†, invariant under U(1) transformation ψl → eIqα ψl (whence ∂µψl → eIqα ∂µψl ) if α independent of spacetime coords. q are U(1) generators, or charges.
Local gauge invariance: Find L invariant when α = α(x). Leads to renormalizable interacting theory. i h Iqα iqα ∂µψl + Iq(∂µα)ψl 6= eIqα ∂µψl , ∴ replace ∂µ by 4-vector “derivative” Dµ, In Lfree, ∂µψl → ∂µe ψl = e such that Dµ gauge transformation Dµψl → eIqα Dµψl . Simplest choice: Dµ − ∂µ is 4-component field: Covariant derivative: Dµ = ∂µ − IqAµ(x) . Gauge transformation: Aµ → Aµ + ∂µα whenever ψl → eIqα ψl . Write transformation as Aµ → A′µ (α). Require Dµ ψl → Dµ′ eiqα ψl = eiqα Dµ ψl , i.e. eIqα ∂µ ψl − Iq(∂µ α)ψl − IqA′µ ψl = eIqα [∂µ ψl − IqAµ ψl ], so A′µ = Aµ + ∂µ α.
61 Use Dν to find invariant (free) Lagrangian for Aµ, quadratic in (∂ν )Aµ: From Dµ gauge transformation, DµDν . . . ψl → eIqα DµDν . . . ψl . Products DµDν . . . contain spurious ∂ρs, but Fµν from Dµ: qFµν = I [Dµ, Dν ], where electromagnetic field strength Fµν = ∂µAν − ∂ν Aµ . [Dµ , Dν ] ψl = [∂µ , ∂ν ] +Iq([∂µ , Aν ] − [∂ν , Aµ ]) − q 2 [Aµ , Aν ] ψl . | {z } | {z } =0
=0
Fµν is gauge invariant.
′ Iqα ′ Fµν ψl → Fµν e ψl = eIqα Fµν ψl , i.e. Fµν = Fµν , or Fµν → Fµν . Also check explicitly from Fµν = ∂µ Aν − ∂ν Aµ .
Conversely, choose Aµ to be massless helicity ±1 field, whose Lorentz transformation on page 52 implies Lorentz invariant free Lagrangian for Aµ must be gauge invariant. Fµν in representation of U(1): Since Fµν → Fµν , Fµν transforms in adjoint representation of U(1). Example: QED Lagrangian for fermions: LDirac,
QED
= −ψ (γ µDµ + m) ψ − 41 Fµν F µν . | {z } LDirac, free +Iqψγ µ Aµ ψ
Interactions due to Iqψγ µ Aµ ψ. Most general Lagrangian locally gauge invariant under U(1) (ψ → eIqα ψ, Aµ → Aµ + ∂µ α), assuming P, T invariance and no mass dimension > 4 terms (Wilson: no contribution).
62 2.8.2
Non-Abelian gauge invariance
Global gauge invariance: Each term in Lfree proportional to (∂ µ)ψl γ (∂µ)ψl†′ γ , γ = 1, . . . , N . Then Lfree invariant under ψl
γ
→ Uγδ ψl δ , where U = exp[Iαiti] , αi spacetime independent.
So ψγ is in fundamental representation of group G =SU(N ) formed by matrices Uγδ , i = 1, . . . , d(G).
Local gauge invariance: Spacetime derivatives in Lagrangian appear as Covariant derivative: Dµ = ∂µ − IAµ(x) with Aµ = Aµiti, ti contain couplings, Aµi for i = 1, . . . , d(G) are (for) massless helicity ±1 gauge fields. To achieve Dµψ → U Dµψ, require Transformation of covariant derivative: Dµ → U DµU † , which requires Transformation of gauge fields: Aµ → U AµU † − I (∂µU ) U † .
63
i = Fµν = I[Dµ, Dν ] = ∂µAν − ∂ν Aµ − I[Aµ, Aν ] . Non-Abelian field strength: TiFµν
Fµν in adjoint representation: Fµν → U Fµν U † . In infinitesimal case, Fi ti
αβ
→ F i [(1 + Iαk tk )ti (1 − Iαk tk )]αβ = F i [ti + Iαk [tk , ti ]]αβ
= F i [ti + Iαk (ICkij )tj ]αβ = F i ti
k αβ + Iαk (tji )tj
αβ
= F i tj
Example: QCD Lagrangian for fermions:LDirac, i Fj More general result is − 41 gij Fµν
µν
k δ + Iα (t ) = F i tj ji k ji αβ QCD
= −ψ α (γ µDαβ
µ
αβ Uji
= Uij Fj ti
αβ ,
i.e. F i → Uij F j .
i + m) ψβ − 41 Fµν Fi
, but can always diagonalize and rescale so gij → δij .
µν
.
2.9
64
The Standard Model
Symmetry of vacuum is G=SU(3)colour×U(1)e.m. gauge group. SM: At today’s collider energies, some “hidden” (broken) symmetries become apparent: G=SU(3)colour×SU(2)weak
isospin ×U(1)weak hypercharge .
SM fermions and their SU(3)C ×SU(2)L ×U(1)Y representations, written as (SU(3)C rep.,SU(2)L rep.,U(1)Y hypercharge = generator / [coupling ≡ Y ]). The SU(3)C charges (3 for quarks, none for leptons) are not shown but, since SU(2)L is broken, particles differing only in T3 (component of weak isospin SU(2)L ) are shown explicitly, namely uL / νe (T3 = 1/2) + and dL / eL (T3 = −1/2). Recall ψL/R = 12 (1 ± γ5 )ψ. Note e.g. uL annihilates u− L and creates uR , and νe is left-handed. Table 2.9.1:
Names
Label
Representation under SU(3)C ×SU(2)L ×U(1)Y
Quarks
QL = (uL , dL )
(3, 2, 61 )
u†R
(3, 1, − 32 )
d†R Leptons
EL = (νe , eL ) e†R
Gauge fields are written as
(3, 1, 31 ) (1, 2, − 21 ) (1, 1, 1)
SU(3): gbµ = giµ gs λ2i
bµ = Aiµ g σi SU(2): A 2
bµ = Bµ g ′Y . U(1): B
We have only discussed the “1st generation” of fermions, in fact
(EL, e†R , QL, u†R , d†R )K ,
65 K = 1, 2, 3.
where (e1, e2, e3) = (e, µ, τ ), (νe1, νe2, νe3) = (νe, νµ, ντ ), (u1, u2, u3) = (u, c, t) and (d1, d2, d3) = (d, s, b). Allow mixing between particles of different generations with same transformation properties.
From Table 2.9.1, can construct the full Lagrangian by including all renormalizable invariant (1, 1, 0) terms. K
These are all possible terms of form ψ γ µDµψ K : K K µ K bµ + B bµ)]Q K + uK γ µ[∂µ − I(b b bµ)]dK . gµ + A Lquark = Q L γ µ[∂µ − I(b d g + B )]u + gµ + B µ µ R γ [∂µ − I(b L R R R K bµ + B bµ)]E K + eK γ µ[∂µ − I B bµ]eK . Llepton = E L γ µ[∂µ − I(A L R R K
K
More general ψ γ µDµRKM ψ M for some constant matrix R not allowed, gives terms ψ γ µ∂µψ M for K 6= M . i Lspin 1 = − 14 Fµν (b g )F i
µν
i b (b g ) − 41 Fµν (A)F i
Recall only real part of L to be taken.
µν
b − 1 Fµν (B)F b µν (B). b (A) 4
66 Higgs mechanism
2.9.1
Mass terms mψ L/R ψR/L are all (1, 2, ± 12 ) → violate gauge symmetry and thus renormalizability. Instead introduce Yukawa coupling λφH ψ L/R ψR/L which is (1, 1, 0), i.e. invariant (thus renormalizable), so φH is (1, 2 = 2, 12 ) scalar field, then hide (“break”) symmetry so that λh0|φH |0i = m. LHiggs = Lpure
Higgs
+ LHiggs−fermion .
bµ + B bµ) and SU(2)L components φT = (φH 1, φH 2) = (φ+ , φ0 ), (ǫφ† )T = (φ0†, −φ+†), Writing Dµ = ∂µ − I(A H H H H H H Lpure
Higgs
= − 21 (DµφH )†DµφH − V (φH ) , Higgs potential V (φH ) = K
K
m2H † 2 φH φH
+ λ4 (φ†H φH )2
K
KM † M M KM LHiggs−fermion = −GKM E L aφH aeM e R − Gu Q L a (ǫφH )a uR − Gd Q L a φH a dR .
All three terms are (1, 1, 0), i.e. invariant. Consider e.g. second term: From table 2.9.1, Y = − 61 − 12 + Write SU(2)L transformation of φH as φ′H
a
= Uab φH
a
= 0.
1
b
(U = eI 2 σi αi from page 59),
−1∗ ∗ from ǫac σcd ǫTdb = −σab . (Same transformation as φTH : φ′H so (ǫφ′H )†a = (ǫφH )†b Uba
Also QL′
2 3
a
−1∗ T .) = φH b Uba = φH b Uba
∗ = Uab QL b , so QL′† a = Uac QL† c , so QL′† a (ǫφ′H )†a = QL† a (ǫφH )†a . Note uR is an SU(2)L singlet.
67
m2H > 0: Stationary L (vacuum) occurs when all fields vanish. Spontaneous symmetry breaking (SSB): =⇒ |φ2H0| = v 2 =
|m2H | λ .
m2H
< 0: tree level vacuum obeys
From Higgs potential on page 66. σ Ivξi (x) 2i
Infinite number of choices for φ0H = h0|φH |0i. Take general φH = e
∂V (φH ) ∂φ φH =φH0
0 v + η(x)
=0
.
Vacuum taken as ξi = η = 0, no longer invariant under symmetry transformations.
= −m2W Wµ†W µ − 21 m2Z ZµZ µ , where mZ =
Lgauge
mass
Wµ =
√1 (A1 µ 2
From Lpure
Lgauge
dynamic
v 2
p
g 2 + g ′2 , mW = v2 g = mZ cos θW , cos θW = √
− IA2 µ) , Zµ = cos θW A3 µ − sin θW Bµ . Higgs .
Start with Lgauge
mass
= − 21 gAi
σi µ2
1
+ g ′ Bµ 2
0 v
2 = −1 2
g(A1 µ + IA2 µ ) −gA3 µ + g ′ Bµ
= − 21 |∂ µW ν − ∂ ν W µ|2 − 41 |∂ µZ ν − ∂ ν Z µ|2 − 41 |∂ µAν − ∂ ν Aµ|2 ,
where Aµ = sin θW A3 µ + cos θW Bµ must be massless photon. From Lspin 1 . Start with Lgauge
dynamic
= − 41 |∂ µ Aνi − ∂ ν Aµi |2 − 14 |∂ µ B ν − ∂ ν B µ |2 .
2 v 2 .
g g 2 +g ′2
,
68 In Lquark and Llepton, between left-handed fermions: bµ + B bµ)L = (A
e
1 2
+ Y Aµ +
g 2
′
cos θW − g Y sin θW Zµ 1 √g Wµ e − 2 + Y Aµ + 2
√g W † 2 µ − g2 cos θW
′
− g Y sin θW Zµ
bµ + B bµ)R = eY Aµ + (−g ′ sin θW Y )Zµ , where e = g sin θW = g ′ cos θW . between right-handed: (A
So charge/e is Q = T3 + Y , i.e. charge of uL, dL, νe, eL is 23 , − 31 , 0, −1 and of uR , dR , eR is 23 , − 31 , −1.
!
,
Lfermion
mass
KM M KM M eR − uK = −eK L me L mu uR −
K M dL mKM d dR
69 , where mKM = GKM ψ ψ v.
From Lquark/lepton .
KM M Can always transform uK′ R = AuR uR , likewise for uL , dL , dR , νe , eL , eR . µ K′ A matrices must be unitary so that kinetic terms retain their previous forms, uK′ R γ ∂µ uR etc.
Choose A matrices such that new mass matrices m′u = AuL muA†uR etc. diagonal, entries mK′ u : Lfermion
mass
Then LW
=
P
K′
K′ K K′ K′ K′ K′ K′ K′ K −eL me eR − uL mu uR − dL md dR .
K
−fermion
K′
K µ K µ K′′ µ † KN N ′ ∝ dL γ µWµuK uL + eK′ L γ Wµ νe L + eL γ Wµ νe = dL γ Wµ (V )
(proportionality constant is − √Ig2 ), where CKM matrix V = AuL A−1 dL . Analogous leptonic matrix absorbed into νeK′′ = AeL A−1 νe
KN
νeN ′.
K (In contrast to uK L , any combination of νe is mass eigenstate because mass matrix is zero.)
70 2.9.2
Some remaining features K
Neutrino mass by adding to LHiggs−fermion a term −Gνe E a (ǫφH )†aνeR → −ν emνe νeR , where νeR is (1, 1, 0). Expect mνe ∼ v to be similar order of magnitude to quark and charged lepton masses. T MR νeR , can only come from higher scale symmetry breaking, Also allowed SM invariant term − 21 νeR
so MR ≫ v ∼ mνe , i.e. no right handed neutrinos at low energy. Gives Lneutrino
mass
T = − 21 (ν e νeR )
i.e. seesaw mechanism: mass
m2νe MR
0 m νe m νe M R
ν Te νeR
′T ≃ − 21 (ν ′e νeR )
m2 − Mνe R
0
0 MR
of (almost) left handed νe′ (i.e. mνe suppressed by
!
m νe MR ).
ν ′T e ′ νeR
,
71 Invariance with respect to parity P , charge conjugation C and time reversal T transformations. CP T conserved, but CP -violation due to phases in CKM matrix. CP and P violating terms
θ κλρσ i i F ǫ F 2 ρσ κλ 64π
allowed, but are total derivatives and therefore non-perturbative.
Current observation suggest θ consistent with zero (no CP violation in QCD). Cancelled by (harmless) anomaly (subsubsection 2.9.4) of global symmetry ψf → eIγ5αf ψf when but this introduces unobserved CP violating phase e−Iθ on quark masses.
P
αf = − 21 θ,
Peccei-Quinn mechanism: where (ǫφH )† in LHiggs−fermion on page 66 is replaced with second Higgs, which transforms differently to first Higgs and can soak up this phase at least at some GUT scale.
72 2.9.3
Grand unification SU(2)L
Suppose SM unifies to single group G at scale MX , then tU(1)Y (diagonal), ti
SU(3)C
, tj
are generators of G.
Tracelessness requires sum of Y values (=elements of tU(1)Y ) to vanish, which is the case from Table 2.9.1. SU(2)L 2
Normalization of generators as on page 13, so tr[tU(1)Y 2] =tr[ti
SU(3)c 2
] =tr[tj
], so
gs2(MX ) = g 2(MX ) = 35 g ′2(MX ) (after dividing by 2×no. generations). Implies sin2 θW (MX ) =
3 8
from page 67.
Within couplings’ exp. errors, unification occurs (provided N = 1 SUSY is included) at MX = 2 × 1016 GeV. To find simplest unification with no new particles, note SM particles are chiral, require complex representations: In general, define all particles fL to be left-handed, then antiparticles fR = fL† are right-handed. Then if fL in representation R of some group G, fR is in representation R. If fL, fR equivalent (have same transformation properties), then R = R, i.e. pseudoreal representation. SM particles fL = (EL, e†R , QL, u†R , d†R ) require complex representation because fR = fL† inequivalent, e.g. 3 6= 3. Pseudoreal representation possible if particle content enlarged to fL → FL so that FL, FR = FL† equivalent. e.g. in SO(10), can fit 15 particles of each generation into real 16 representation, requires adding 1 νeR .
73 SU(5) is simplest unification. Since SU(3)×SU(2)×U(1) ⊂ SU(5), all internal symmetries accounted for by fermions ψα with α = 1, ..., 5. 0 0 0 0 0 0 0 0 0 0 0 0 λi 0 0 2 SU(3)C SU(2)L SU(2) SU(3) , i = 1, ..., 8, ti = g 0 0 0 0 0 = ti Choose ti = gs , i = 1, ..., 3. 0 0 = ti 0 0 0 σi 0 0 0 0 0 0 0 0 2 0 0 0 0 0
U(1) generator must commute with generators above 1 3 0 0 0 1 0 3 U(1) ′ t = 2g 0 0 31 0 0 0 0 0 0
and be traceless. Tentatively take 0 0 0 0 0 0 = 2tU(1)Y . − 12 0 0 − 21
Then fermions form 5 (fundamental) and 10 representations of SU(5): 1 3 2 1 1 d 0 uR −uR uL dL R 2 d 0 u1R u2L d2L R 3 3 3 , 0 u d L L . d R e 0 R eL —“ 0 −νeL
(Note: all particles left-handed, 1,2,3 superscripts are colour indices, 10 matrix is antisymmetric).
In SO(10), 1 generation fits into 16 = 1 + 5 + 10, and 1 is identified with right-handed neutrino.
74 2.9.4
Anomalies
Gauge anomalies modify symmetry relations (Ward identities), spoils renormalizability and maybe unitarity. Anomaly occurs because A =
R
d4xL respects symmetry, but not measure
Relevant example: Let A be invariant under ψα = Uk
αβ ψβ ,
RQ
x,l
dψl (x) = d[ψ]d[ψ]d[A].
where Uk = exp[Iγ5αtk ] is chiral symmetry (global) and each ψα is a Dirac field. Problem: although A is invariant, measure is not: Resulting change in path-integral µν ρσ 1 is “as if” L changes by αJk [A], where Jk = − 16π 2 ǫµνρσ Fi Fj tr[{ti , tj }tk ].
Noether’s theorem on page 57:
R
d[ψ]d[ψ]d[A] exp[IA ] →
R
d[ψ]d[ψ]d[A] exp[I A +
R R
d[ψ]d[ψ]d[A] exp[IA ]
4
d xα(x) [Jk −
µν ρσ 1 i.e. conservation violation: h∂µJkµ(x)iA = − 16π 2 ǫµνρσ Fi Fj tr[{ti , tj }tk ] (hiA means no A integration).
µ ∂µJk (x)] ],
Anomalous (non-classical) triangle diagrams between Jkµ, Fiµν and Fjρσ modify Ward identities. Physical theories must be anomaly free (i.e. tr[{ti, tj }tk ] cancel), e.g. real representations: tr[{ti, tj }tk ] = 0. tr[{t∗i , t∗j }t∗k ] =tr[{(−U ti U † ), (−U tj U † )}(−U tk U † )] = −tr[{ti , tj }tk ]. But tr[{t∗i , t∗j }t∗k ] =tr[{tTi , tTj }tTk ] =tr[{ti , tj }tk ].
SM is anomaly free.
SM in 10 + 5 of SU(5), in real representation 16 of SO(10) (thus tr[{ti , tj }tk ]5 = −tr[{ti , tj }tk ]10 ).
3 3.1
Supersymmetry: development
75
Why SUSY?
Attractive features of SUSY:
1. Eliminates fine tuning in Higgs mass. 2. Gauge coupling unification. 3. Radiative electroweak symmetry breaking: SUSY =⇒ Higgs potential on page 66 with µ2 < 0. 4. Excess of matter over anti-matter (large CP violation, not in SM) possible from SUSY breaking terms. 5. Cold dark matter may be stable neutral lightest SUSY particle (LSP) = gravitino / lightest neutralino. 6. Gravity may be described by local SUSY = supergravity.
76 SM is accurately verified but incomplete — e.g. does not + cannot contain gravity, so must break down at / before energies around Planck scale MP = (8πG)−1/2 = 2.4 × 1018 GeV. In fact, SM cannot hold without modification much above 1 TeV, otherwise we have √ Gauge hierarchy problem: Since v = |h0|φH |0i| = 246GeV and λ = O(1), |mH | = | λv| = O(100) GeV . If ΛUV > O(1)TeV, fine tuning between ∆m2H from quantum loop corrections (Fig. 3.1) and tree level (bare) m2H : ∆m2H =
λφ 2 Λ 8π 2 UV
− |{z} 3
colour“3′′
λ
|κt |2 2 Λ 8π 2 UV
+
... |{z}
.
smaller terms
(Largest from − 4φ |φH |4 and top quark (κt ≃ 1 because mt ≃ v).) No similar problem for fermion and gauge boson masses, but these masses affected by m2H . Avoid fine tuning by taking ΛUV ∼ 1 TeV, i.e. modify SM above this scale.
φ
ψ
φH
φH
(a) Fermion field ψ, Lagrangian term −κψ φH ψψ, giving 1-loop contribution to Higgs mass of
|κ |2 − 8πψ2 Λ2UV .
(b) Boson field φ, Lagrangian term λφ |φH |2 |φ|2 , giving 1-loop contribution λ to Higgs mass of 8πφ2 Λ2UV .
Figure 3.1: Fermion and boson contributions to Higgs mass parameter m2H .
77 One solution: Higgs is composite of new fermions bound by new strong force at ΛUV ≃ 1 TeV → difficult. Alternatively, forbid bare m2H |φH |2 term by some new symmetry δφH = ǫ × something. Various choices for “something” bosonic (leads to “little Higgs” models, extra dimensions). For a standard symmetry, “something” would be I [Qa, φH ], i.e. φH → eIǫQa φH e−IǫQa . Try “fermionic” generator Qa, which must be a
1 2, 0
spinor (so ǫ a spinor of Grassmann variables).
Relation with momentum: {Qa , Q†b˙ } is
1 2, 0
× 0, 12 =
{Qa, Q†b˙ }
1 1 2, 2
=
78
2σaµb˙ Pµ .
from triangle inequality. Only candidate is P µ (see Coleman-Mandula theorem later).
Lorentz invariance requires combination σaµb˙ Pµ , factor 2 comes from suitable normalization of Qa . 6 0 because for any state |Xi, Note {Qa , Q†b˙ } = 2 2 hX|{Qa , (Qa )† }|Xi = hX|Qa (Qa )† |Xi + hX| (Qa )† Qa |Xi = (Qa )† |Xi + Qa |Xi ≥ 0. If equality holds for all |Xi, Qa = 0.
At least one of Pµ non-zero on every state, so Qa affects every state, not just Higgs,
i.e. every particle has a superpartner with opposite statistics and spin difference of 1/2, together called a supermultiplet. This fermion-boson symmetry is supersymmetry. For every fermion field (component) ψf with −κf φH ψ f ψf , introduce boson field φf with −λf |φH |2|φf |2. From Fig. 3.1, contribution of this supermultiplet (component) to
∆m2H
is
∆m2H
|κf |2 2 λf 2 = 2 ΛUV − 2 ΛUV . |8π {z } | 8π{z } boson
fermion
Just requiring fermion-boson symmetry guarantees λf = |κf |2, and ∆m2H = 0 (+ finite terms) to all orders.
3.2
Haag-Lopuszanski-Sohnius theorem
79
Reconsider symmetries: So far assumed generators are bosonic. Now generalize to include fermionic ones.
Generalize additive observable on page 4 to Q = Qσσ′ a†σ aσ′ . (If Qσσ′ are components of Hermitian matrix, unitary transformation of particle states gives back original result.) Since Q is bosonic, a†σ , a†σ′ both bosons or both fermions, i.e. Qσσ′ = 0 if a†σ bosonic, a†σ′ fermionic, or vice versa. SUSY: Allow for Q’s containing fermionic parts to also be generators of symmetries that commute with S-matrix. For convenience, distinguish between fermionic and bosonic parts of any Q. Fermionic Q = Qσρa†σ aρ + Rσρa†ρaσ , where σ sums over bosonic particles, ρ over fermionic particles. Such a generator converts bosons into fermions and vice versa. E.g. action of Q on 1 fermion + 1 boson state using (anti) commutation relations on page 3 gives Qa†ρ′ a†σ′ |0i = Qσρ′ a†σ a†σ′ |0i + Rσ′ ρ a†ρ a†ρ′ |0i.
But as a symmetry implies there are fermions and bosons with similar properties.
80 Identify symmetry generators ti also with fermionic Q.
Graded parameters αi, βj obey αiβj = (−1)ηiηj βj αi, where grading ηi = 0(1) for complex (Grassmann) αi. Graded generator tj obeys αitj = (−1)ηiηj tj αi, where ηi = 0(1) for bosonic (fermionic) generator ti. For transformations O → eIαiti Oe−Iαiti to preserve grading of any operator O, αi has same grading as ti. Graded Lie algebra: [(−1)ηiηj titj − tj ti] = ICijk tk . Repeating steps in derivation of Lie algebra on page 7 gives 1 2 ICijk αi βj tk
= 21 [αi ti βj tj − βj tj αi ti ] = 21 [(−1)ηi ηj αi βj ti tj − αi βj tj ti ].
Fermionic generators Qi: U (Λ)QiU †(Λ) = Cij (Λ)Qj , so Qi furnishes representation of Lorentz group. (A,B)
Choose Qi = Qab
(A,B)
in (A, B) representation: [A, Qab
(A)
(A,B)
] = −Jaa′ Qa′b
(A,B)
and [B, Qab
(B)
(A,B)
] = −Jbb′ Qab′
.
Anticommutators of fermionic generators can be used to build bosonic generators of various (A, B). (A,B)
Coleman-Mandula theorem puts limits on allowed bosonic generators, and hence allowed (A, B) for the Qab
.
81 Coleman-Mandula theorem: Only bosonic generators are of internal + Poincar´e group symmetries. Simple argument: Additional conserved additive rank ≥ 1 tensors constrain scattering amplitude too much. Only 1 4-vector, P µ: Consider 2→2 scattering, c.m. frame. Conservation of momentum and angular momentum → amplitude depends on scattering angle θ. Second conserved additive 4-vector Rµ gives additional constraints unless Rµ ∝ P µ. Only 1 2nd rank tensor, J µν : Assume rank 2 conserved additive tensor Σµν . Additive property means [Σµν , a†σ (p)] = Cµν σ (p)a†σ (p). Then Cµν (p, σ) = ασ (m2)pµpν + βσ (m2)gµν . Lorentz transformation of RHS is U (Λ)[Σµν , a†σ (p)]U † (Λ) = [U (Λ)Σµν U † (Λ), U (Λ)a†σ (p)U † (Λ)] = [Λρµ Λδν Σρδ , Dσ′ σ (W (Λ, p))a†σ′ (Λp)] = Dσ′ σ (W (Λ, p))Λρµ Λδν Cρδ (Λp, σ)a†σ′ (Λp), and of LHS is Cµν (p, σ)Dσ′ σ (W (Λ, p))a†σ′ (Λp), so Λµρ Λνδ Cρδ (p, σ) = Cµν (Λp, σ). Only candidates are pµ pν and gµν . 2 ′µ ′ν ′ν In 2→2 scattering, conservation of Σµν implies ασ1 (m21)pµ1 pν1 + ασ2 (m22)pµ2 pν2 = ασ1 (m21)p′µ 1 p1 + ασ2 (m2 )p2 p2 . ′µ µ ′µ µ ′µ 2 2 P µ conservation: pµ1 +pµ2 = p′µ +p =⇒ p = p , i.e. no scattering (allowed p = p 1 2 1,2 1,2 1,2 2,1 if ασ1 (m1 ) = ασ2 (m2 )).
No higher rank tensors: Generalize last argument to higher rank tensors.
Allowed representations for fermionic generators (C,D)
(C,D)†
(A,B) Qab :
1 2
A + B = , i.e.
1 2, 0
(C+D,C+D)
{QC,−D , QC,−D } = XC+D,−C−D : Firstly, Q(C,D)† is of type (A, B) = (D, C) because B † = A.
or 0,
1 2
82 , and j = 12 .
(C,D)† e(D,C) , we find b = −C because [B− , Q e(D,C) ] = −[A+ , Q(C,D) ]† = 0 using B− = A†+ . Similarly a = D. Writing QC,−D = Q C,−D a,b a,b (C,D) (C,D)† (C,D) e (D,C) Now {QC,−D , QC,−D } = {QC,−D , Q D,−C }, must have A3 = C + D and B3 = −C − D, i.e. A, B ≥ C + D.
But since A must be ≤ (C + D) (from triangle inequality), it must be = C + D. Similarly for B.
Since X (C+D,C+D) is bosonic, CM theorem means it must be P µ ( 12 , 21 ), or internal symmetry generator ((0, 0)).
Latter implies C = D = 0, not possible by spin-statistics connection. Final result is relation with momentum on page 78.
Take Qa to be
1 2, 0
spinor, i.e. [A, Qa] =
− 21 σabQb ,
[B, Qa] = 0
(Q†a
will be 0, ). 1 2
Qa not ruled out by reasoning of CM theorem, because no similar conservation law:
Take |ii, |ji to have definite particle number. hj|ii = 6 0 =⇒ even difference in fermion numbers, so hj|Qa|ii = 0. Can have multiple generators Qar , r = 1, ..., N −→ simple SUSY is N = 1, extended SUSY is N ≥ 2. Summary: CM theorem: only bosonic generators are (0, 0) (internal symmetry), HLS theorem: only fermionic generators are
1 2, 0
and 0, 12 (Qar ).
1 1 2, 2
(P µ), and (1, 0) and (0, 1) (J µν ).
Relation with momentum for any N :
{Qar , Q†bs ˙ }
=
83
2δrsσaµb˙ Pµ .
µ From allowed representations for fermionic generators on page 82, {Qar , Q†bs ˙ } = 2Nrs σab˙ Pµ . Nrs is Hermitian, µ † † ∗ µ † because {Qar , Q†bs ˙ } = 2Nsr σba˙ Pµ . So N diagonalized by unitary matrix W . ˙ } = 2Nrs σba˙ Pµ , but {Qbs , Qar ˙ } = {Qbs , Qar µ Writing Q′ar′ = Wr′ r Qar gives {Q′ar , Q′†bs ˙ } = 2nr δrs σab˙ Pµ (no sum over r on RHS), where nr are eigenvalues of Nrs .
√ Writing Qar = Q′ar / nr gives result if nr > 0 (otherwise we have a factor -1): ′ † Taking Q′†bs ˙ = (Qar ) and operating from right and left with |X(p)i and hX(p)| gives
on LHS: hX(p)|{Q′ar , (Q′ar )† }|X(p)i = | (Q′ar )† }|X(p)i|2 + |Q′ar }|X(p)i|2 > 0, and on RHS: 2nr (p0 ± p3 ), where ± for a = 1, 2. If p0 ≥ ∓p3 , then nr > 0 as required.
SUSY implies h0|H|0i = 0 for supersymmetric vacuum (Qar |0i = Q†ar ˙ |0i = 0). Commutation with momentum: [Qa, P µ] = 0 . [Qa , P µ ] is µ
1 2, 0
So [Qb , P ] =
×
1 1 2, 2
kσbµa˙ Q†a˙
= 1, 21 + 0, 12 . No 1, 12 generator, but Q†a˙ is 0, 21 .
h i † µ and therefore Qa˙ , P = −k ∗ Qb σbµa˙ , or, using ǫ matrix, Q†a˙ , P µ = k ∗ σ µ ˙
˙
ab ˙
Qb .
Jacobi identity: 0 = [[Qa , P µ ], P ν ] + [[P µ , P ν ], Qa ] + [[P ν , Qa ], P µ ] = kσaµb˙ [Q†b , P ν ] − kσaνb˙ [Q†b , P µ ] = |k|2 [σ µ , σ ν ]ab Qb . Since [σ µ , σ ν ]ab 6= 0 for all µ, ν, must have |k|2 = k = 0.
84 Anticommuting generators: {Qar , Qbs} = ǫabZrs , with (0, 0) generators Zrs = −Zsr . {Qar , Qbs } is (1, 0) + (0, 0). From CM theorem, only (1, 0) generator is J µν . But {Qar , Qbs } commutes with P µ from commutation with momentum on page 83, while (linear combinations of) J µν doesn’t. So only possibility is (0, 0) generators, which in general commute with P µ from CM theorem. Lorentz invariance requires ǫab (= −ǫba ), but whole expression must be symmetric under ar ↔ bs so Zrs = −Zsr .
Antisymmetry of Zrs → vanish for N = 1. Zrs are central charges due to following commutation relations: Commutation with central charges: [Zrs, Qat] = [Zrs, Q†at ˙ ] = 0. † † † Jacobi identity 0 = [{Qar , Qbs }, Qct ˙ , Qar }, Qbs ]. ˙ }, Qar ] + [{Qct ˙ ] + [{Qbs , Qct
2nd, 3rd terms vanish from commutation with momentum on page 83. Thus [Zrs , Q†at ˙ ] = 0. † † Generalized Jacobi identity 0 = −[Zrs , {Qat , Q†bu ˙ , Zrs ]}. ˙ , [Zrs , Qat ]} − {Qat , [Qbu ˙ }] + {Qbu
1st, 3rd terms vanish because Zrs commutes with P µ and Q†bu ˙ . µ Commutator in 2nd term must be [Zrs , Qat ] = Mrstv Qav , so 0 = {Q†bu ˙ , [Zrs , Qat ]} = 2Mrstu σab˙ Pµ , i.e. Mrstu = 0 so [Zrs , Qat ] = 0.
† Commuting central charges: [Zrs, Ztu] = [Zrs, Ztu ] = 0.
[Zrs , Ztu ] = [{Q1r , Q2s }, Ztu ] = 0, etc.
R-symmetry: Zrs = 0 gives U(N) symmetry Qar → VrsQas (N = 1 case: Qa → eIφQa, always true).
85 Consider (anti)commutation relations of all internal symmetry generators ti with SUSY algebra.
CM theorem: [ti, P µ] = [ti, J µν ] = 0 .
Since ti is (0, 0), must have [ti, Q 1 r ] = −(ai)rsQ 1 s . Matrices ai represent G . 2
2
Jacobi identity 0 = [[ti , tj ], Q 21 r ] + [[Q 21 r , ti ], tj ] + [[tj , Q 21 r ], ti ] = ICijk [tk , Q 21 r ] + (ai )rt [Q 21 t , tj ] − (aj )rt [Q 21 t , ti ] = −ICijk (ak )rs Q 12 s + (ai )rt (aj )ts Q 12 s − (aj )rt (ai )ts Q 12 s , i.e. [ai , aj ]rs = ICijk (ak )rs .
Simple SUSY and internal symmetry generators commute .
If N =1, numbers ai cannot represent G unless ai = 0.
3.3
86
Supermultiplets
Supermultiplets: Particles that mix under SUSY transformations, furnish representation of SUSY. Irreducible representations of SUSY: doesn’t contain 2+ supermultiplets separately mixing under SUSY. Action of Qar or Q†ar ˙ converts one particle into another of the same irreducible supermulitplet (superpartners). µ µ 2 Since [Qar , P µ] = [Q†ar ˙ , P ] = 0, all particles in supermultiplet have same P (and hence same mass P ).
Equal number of bosonic and fermionic degrees of freedom in supermultiplet: nB = nF . P
Convention:
X
over supermultiplet states,
P
all X
† over complete basis. Choose given r for Qar ≡ Qa , Q†ar ˙ ≡ Qa˙ below.
Supermultiplet’s states |X i have same pµ , and (−1)2s |X i = ±1|X i for spin s bosonic/fermionic |X i, so Pµ′ ≡
P
2σaµb˙ Pµ′ =
2s X hX |(−1) Pµ |X i
= pµ (nB − nF ). We show Pµ′ = 0: From relation with momentum for any N on page 83,
P
† 2s X hX |(−1) Qa Qb˙ |X i +
P
2s † X hX |(−1) Qb˙ Qa |X i =
Since Qa |X i, |X i in same supermultiplet, limit 2σaµb˙ Pµ′ = hX |(−1)2s Qa Q†b˙ |X i +
P
all
P
all Y
→
P
† 2s X hX |(−1) Qa Qb˙ |X i +
P
Y . Conversely, extend
2s † X,Y hY |Qa |X ihX |(−1) Qb˙ |Y i =
But (−1)2s Q†b˙ |Y i = −Q†b˙ (−1)2s |Y i, so 2σaµb˙ Pµ′ =
P
P
P
2s † X,all Y hX |(−1) Qb˙ |Y ihY |Qa |X i.
P
X
→
† 2s X hX |(−1) Qa Qb˙ |X i +
† 2s X hX |(−1) Qa Qb˙ |X i + −
P
P
all X
(so
P
P
all X |X ihX |
2s † Y hY |Qa (−1) Qb˙ |Y i.
† 2s Y hY |(−1) Qa Qb˙ |Y i
= 0.
= 1).
Massless supermultiplets: In frame where p = p = 0, p = p , 1
Q− 1 r , Q†
˙ − 21 r
2
2
3
0
{Q 1 r , Q†1˙ 2
2s
87 {Q 1 r , Q 1˙ } = 0 2 −2s . † {Q− 1 r , Q 1˙ } = 0 †
0
} = 4p δrs
{Q− 1 r , Q†1˙ } = 0 2
2s
2
−2s
annihilate supermultiplet states , so Zrs annihilate supermultiplet states . †
For any |Xi in supermultiplet, 0 = hX|{Q− 12 r , Q
− 21˙ r
† }|Xi = | Q− 12 r |Xi|2 + |Q− 21 r |Xi|2 (no sum over r).
All supermultiplet states reached by acting on maximum helicity state |λmaxi with the Q 1 r . 2
Q†1˙ give no new states: Consider |Xi not containing Q 12 r . Then Q†1˙ |Xi = 0 (Q†1˙ commutes across to act directly on |λmax i). 2r
2r
2r
Then Q†1˙ Q 12 r |Xi = {Q 21 r , Q†1˙ }|Xi = 4E|Xi (no sum over r), i.e. Q†1˙ just removes Q 12 r . 2r
2r
2r
Q†1˙ |Xi (or Q 1 r |Xi) has helicity greater (or less) than |Xi by 21 . 2r
2
Range of helicities in supermultiplet:
N! n!(N −n)!
Recall [J3 , Q†1˙ ] = 21 Q†1˙ . 2r
2r
helicity λmax − n2 states , λmin = λmax − N2 .
Obtain supermultiplet states |Xi by acting on maximum helicity state |λmax i with any n of Q 21 1 , Q 12 2 , ..., Q 21 N . Order doesn’t matter since Q 12 a anticommute, each generator cannot appear more than once since Q21 r = 0. 2
Constraint on particle spins on page 53 implies λmin ≥ −2 and λmax ≤ 2, i.e. N ≤ 8 .
88 SM particles probably belong to ∼ massless supermultiplets. Superpartners (masses ∼ M ) of SM particles (∼ m) not seen, i.e. M ≫ m, so SUSY is broken, at energy < mSUSY . SUSY restored at mSUSY ∴ mSUSY ≫ M − m ∼ M (superpartner has same mass) ∴ supermultiplets ∼ massless.
Most likely scenario is simple SUSY: quark, lepton (spin 21 ) superpartners are scalars (0): squarks, sleptons. Higgs (0), gauge bosons (1), graviton (2) superpartners fermionic: Higgsino + gauginos ( 21 ), gravitino ( 32 ). Alternatives: In simple SUSY, SM gauge bosons cannot be superpartners of SM fermions. SM fermions, gauge bosons in different representations (recall simple SUSY and SM SU(3)×SU(2) commute from page 85).
Quarks, leptons cannot be in same supermultiplet as any beyond-SM vector (gauge) bosons. Gauge bosons are in adjoint representation of a group. If e.g. helicity + 12 fermion, +1 gauge boson in same supermultiplet, fermion in adjoint = real representation. But SM is chiral, i.e. helicity + 12 fermions belong to complex representations.
Superpartners of gauge bosons must be helicity ± 21 , not ± 23 , fermions. Helicity ± 32 particle couples only to Qa (principle of equivalence on page 53: ±2 only couples to P µ ).
Extended SUSY probably cannot be realised in nature. In extended SUSY, helicity ± 12 fermions either in same supermultiplet as gauge bosons (N ≥ 3), or each other (N = 2).
89 Massive supermultiplets: For particles with masses M ≫ mSUSY , e.g. heavy gauge bosons in SU(5).
In frame where p = p = p = 0, p0 = M , 1
so Q 1 r , Q† 2
aA(a,r) =
− 21˙ r
2
3
{Q 1 r , Q†1˙ } 2 s
{Q 1 r , Q
= 2M δrs
2
2
{Q− 1 r , Q†1˙ } 2 s
=0
2
†
†
{Q− 1 r , Q
− 12˙ s
− 12˙ s
2
}=0
} = 2M δrs
(or Q− 1 r , Q†1˙ ) lower (or raise) spin 3 component by 12 .
√ 1 Qar 2M
2
,
2r
are fermionic annihilation / creation operators: {aA, a†B } = δAB , {aA, aB } = {a†A, a†B } = 0.
Define Clifford “vacuum” |Ωi: aA|Ωi = 0. |Ωi has given spin j, range of spin 3 is −j ≤ σ ≤ j. Supermultiplet is all states a†A1 . . . a†An |Ωi, spins ranging from Max(j − N2 , 0), . . ., j + N2 . Simple SUSY: States with spin j ± 21 , and 2 sets of states with spin j. If j = 0, there are two bosonic spin 0 states and two fermionic states of spin
1 2
(i.e. spin 3 is ± 21 ).
90 3.3.1
Field supermultiplets (the left-chiral supermultiplet)
Simplest case: N = 1, scalar φ(x) obeying
[Q†b˙ , φ(x)]
= 0 : Write [Qa, φ(x)] = −Iζa(x) , which is a
1 2, 0
field.
{Q†a˙ , ζb(x)} = 2σbµa˙ ∂µφ(x) . Thus [Q, φ] ∼ ζ and {Q†, ζ} ∼ φ, i.e. φ and ζ are each others’ superpartners. {Q†a˙ , −Iζb } = {Q†a˙ , [Qb , φ]} = [{Q†a˙ , Qb }, φ] = 2σbµa˙ [Pµ , φ]. Poincar´e transformation for fields on page 24: [Pµ , φ] = I∂µ φ.
{Qa, ζb(x)} = 2IǫabF (x) . {Qa , −Iζb } = {Qa , [Qb , φ]} = −{Qb , [Qa , φ]} = −{Qb , Iζa } ∝ ǫab . F is (0, 0) in
[Qa, F (x)] = 0 . F has no superpartner, it is auxiliary field.
1 2, 0
×
1 2, 0
= (0, 0) + (1, 0).
2Iǫab [Qc , F ] = [Qc , {Qa , ζb }] = −[Qa , {Qc , ζb }] = −2Iǫcb [Qa , F ]. For a = c 6= b, 2Iǫcb [Qc , F ] = −2Iǫcb [Qc , F ], so [Qc , F ] = 0. ˙ [Q†a˙ , F (x)] = −σ µ ab ∂µζb(x) .
2Iǫab [Q†c˙ , F ] = [Q†c˙ , {Qa , ζb }] = [{Q†c˙ , Qa }, ζb ] − [Qa , {Q†c˙ , ζb }] = [2σaµc˙ Pµ , ζb ] − [Qa , 2σbµc˙ ∂µ φ] = 2Iσaµc˙ ∂µ ζb − 2Iσbµc˙ ∂µ ζa . Then ǫda ǫab [Q†c˙ , F ] = δ db [Q†c˙ , F ] = ǫda σaµc˙ ∂µ ζb − ǫda σbµc˙ ∂µ ζa = ǫda σaµc˙ ∂µ ζb − σbµc˙ ∂µ ζ d . Sum d = b: δ dd [Qc†˙ , F ] = 2[Qc†˙ , F ] = −2σbµc˙ ∂µ ζ b . Then [Q†a˙ , F ] = ǫa˙ c˙ [Q†c˙ , F ] = −ǫa˙ c˙ σbµc˙ ǫbd ζd = −σ µ
ad ˙
ζd , using 4-vector σ matrices with raised indices on page 41.
Conjugation of everything above gives right-chiral supermulitplet.
To simplify algebra, use 4-component Majorana spinors, ψ =
Majorana conjugation: ψ = ψ T γ5E where E =
Xa ˙ X †b
ab
91 , with notation of page 42.
ǫ 0 0 −ǫa˙ b˙ (= ǫab)
.
From ψ = ψ † β = (X a , Xb˙† ).
(Anti)commutation relations: {Q, Q} = −2Iγ µPµ and [Q, P µ] = 0 , where Majorana Q =
Qa ˙ Q†b = (ǫbcQc)†
Directly from relations on page 83.
Infinitesimal transformation of operator O: δO = [IαQ, O] , where Grassmann spinor α =
αa ˙ α †b
˙
, so αQ = αaQa + αb†˙ Q†b . Implies δO † = [IαQ, O †] .
(δO)† = [IαQ, O]† = [IαQ, O † ] = δ(O † ), because, as for scalar from 2 spinors and Hermitian conjugate of scalar on page 40, ˙
(αQ)† = (αa Qa + αb†˙ Q†b )† = (Q†a˙ α†a˙ + Qb αb ) = (−α†a˙ Q†a˙ − αb Qb ) = (αa†˙ Q†a˙ + αb Qb ) = αQ.
Product rule for δ: δ(AB) = (δA)B + AδB , where A, B can be fermionic / bosonic. [IαQ, AB] = I(αQAB − ABαQ) = I(αQAB − AαQB + AαQB − ABαQ) = [IαQ, A]B + A[IαQ, B].
If Abelian limit on page 7 obeyed, finite transformation is O → eIαQOe−IαQ.
.
92 (Anti)commutation relations on page 90 can be simplified with 4-component spinor notation and definitions φ =
√1 (A 2
+ IB) , ψ =
√1 2
ζa ˙ ζ †b
and F =
√1 (F 2
− IG) :
1. δA = αψ , 2. δB = −Iαγ5ψ . [Q, φ] =
[Qa , φ] = −Iζa ˙ [Q , φ] = ǫba˙ [Q†a˙ , φ] = 0 †b˙
[Qa , φ ] = −[Q†a˙ , φ]† = 0 ˙ ˙ [Q†b , φ† ] = −ǫba˙ [Qa , φ]† = −Iζ†b˙ †
and [Q, φ† ] =
!
,
so [Q, A] = [Q, √12 (φ + φ† )] = −Iψ, and [Q, B] = [Q, √I2 (φ† − φ)] = −γ5 ψ from definition of γ5 on page 43..
3. δF = αγ µ∂µψ , 4. δG = −Iαγ5γ µ∂µψ . [Q, F ] =
[Q, F † ] =
[Qa , F ] = 0 ˙ †b˙ [Q , F ] = −σ µ ba ∂µ ζa †
†b˙
and, using σ µ †
˙ µ bc∗
˙ bc∗
∂µ ζc†˙ †
˙ = σ µ cb ,
µ cb ˙
∂µ ζc†˙ †
[Qa , F ] = −ǫab [Q , F ] = ǫab σ = ǫab σ = ǫab σ ˙ [Q†b , F ] = −ǫba [Qa , F ] = 0
[Q, F ] = [Q, √12 (F + F † )] = − √I2
[Q, G] = [Q, √I2 (F − F † )] =
√1 2
˙ −Iσaµb˙ ∂µ ζ †b ˙ −Iσ µ bd ∂µ ζd ˙ Iσaµb˙ ∂µ ζ †b ˙ −Iσ µ bd ∂µ ζd
!
!
= −Iγ µ ∂µ ψ and
= −γ5 γ µ ∂µ ψ.
µ cb ˙
ǫc˙d˙∂µ ζ
†d˙
=
˙ −σaµd˙∂µ ζ †d
!
, so
93
5. δψ = ∂µ(A + Iγ5B)γ µα + (F − Iγ5G)α . ˙
[IαQ, ψ] =
=
√I 2
√
=I 2
αc {Qc , ζa } + αd†˙{Q†d , ζa } ˙ ˙ ˙ αc {Qc , ζ †b } + αd†˙{Q†d , ζ †b }
√I 2
!
˙
=
˙ α (2Iǫca F ) + αd†˙ǫdc˙ (2σaµc˙ ∂µ φ) ˙ ˙ ˙˙ ǫca αa (2σcµd˙∂µ φ† )ǫbd − αd†˙ǫdc˙ ǫba˙ (2Iǫc˙a˙ F † ) c
−αa (IF ) − α†c˙ (σaµc˙ ∂µ φ) ˙ ˙ −αa (σ µ ba ∂µ φ† ) − α†b (IF † )
=
√I 2
αc {Qc , ζa } + αd†˙ǫdc˙ {Qc†˙ , ζa } ˙ ˙ ˙˙ αc {Q†c˙ , ζd }† ǫbd + αd†˙ǫdc˙ ǫba˙ {Qc , ζa }†
!
!
αa (F − IG) − Iσaµc˙ α†c˙ ∂µ (A + IB) ˙ ˙ α†b (F + IG) − Iσ µ ba αa ∂µ (A − IB)
.
For L = − 21 ∂µA∂ µA − 21 ∂µB∂ µB − 12 ψγ µ∂µψ + 12 (F 2 + G2) + m(F A + GB − 12 ψψ) 2
2
+g[F (A + B ) + 2GAB − ψ(A + Iγ5B)ψ] , above transformations 1 — 5 leave action A = For example, for m = g = 0, δL = −∂µ δA∂ µ A − ∂µ δB∂ µ B − (δψ)γ µ ∂µ ψ + F δF + GδG.
R
d4xL invariant .
We have replaced − 12 ψγ µ ∂µ δψ → − 12 (δψ)γ µ ∂µ ψ, since difference with analogous term in hermitian conjugate (h.c.) of L (must be added to make L real) is total derivative ∂µ f which doesn’t contribute to A : (Xγ µ ∂µ Z)† = (X † βγ µ ∂µ Z)† = ∂µ Z † γ µ† β † X = −∂µ Z † βγ µ X = −∂µ Zγ µ X = Zγ µ ∂µ X − ∂µ (Zγ µ X).
No derivatives of F , G (or F ) appear, so they are auxiliary fields, i.e. can be expressed in terms of the other supermultiplet fields by solving equations of motion
∂L ∂F
=
∂L ∂G
= 0.
3.4
94
Superfields and Superspace
As P µ generates translations in spacetime x on field φl (x) via [P µ, φl (x)] = I∂ µφl (x), find formalism where
Q generates translations in superspace (x, θ = θa, θ Definition of superfield: δS = αQS .
†b˙
T
) on superfield S(x, θ) via [IαQ, S(x, θ)] = αQS(x, θ) .
From infinitesimal transformation of operator O on page 91.
µ ∂µ , where α, β run over the 4 spinor indices. Condition on Q: {Qα , Q β } = 2γαβ µ µ From (anti)commutation relations on page 91, [{Qα , Qβ }, S] = −2Iγαβ [Pµ , S], which from above reads {Qα , Q β }S = 2γαβ ∂µ S.
µ µ ∂ + γ µθ∂µ (explicitly, Qα = − ∂θ∂ (γ5E)βα + γαβ θβ ∂µ = (γ5E)αβ ∂θ∂ + γαβ θβ ∂µ). Definition of Q: Q = − ∂θ β
For a (and b) Grassmann,
∂ ∂a
β
is left derivative: (anti)commute a left then remove it, e.g. 2
∂ ∂a ba
∂ = − ∂a ab = −b.
T
This choice satisfies condition on Q: We use (γ5 E) = −1, (γ5 E) = −γ5 E (direct calculation), and from Majorana conjugation on page 91. Now Q δ = Qα† βαδ = −
∂ ∂θ† β
∂ ∂θ δ
∂ ∂θ
α
=
∂ ∂θ β
(γ5 E)βα
µ∗ † (γ5 E)βα βαδ + γαβ θβ βαδ ∂µ . But γ µ∗ = −βγ µT β,
µ µ∗ † . Thus, using {γ5 E, β} = 0, Q δ = − and β 2 = 1 so that γαβ θβ βαδ = −θα γαδ
So Q δ =
∂ ∂θ α (γ5 E)αβ (γ5 E)βδ
µ µ − θα (γ5 Eγ µ )αδ ∂µ . Then {Qα , Q β } = δσβ (γ5 E)βα (γ5 Eγ µ )σδ ∂µ + γαδ ∂µ = 2γαδ ∂µ .
µ − θα (γ5 E)αβ γβδ ∂µ .
95 Superfields from superfields I: S = S1 + S2 , S = S1S2 , etc. are superfields when S1, S2 are superfields. 1st case clearly obeys SUSY transformation using Q above. 2nd case: From product rule for δ on page 91, δS1 S2 = (δS1 )S2 + S1 δS2 = (αQS1 )S2 + S1 αQS2 = αQ(S1 S2 ).
Superfields from superfields II: S ′ = Dβ S , S ′ = D β S , ∂ − γ µθ∂µ obeys {Dβ , Qγ } = {D β , Qγ } = 0 . where superderivative D = − ∂θ
First case: δS ′ = δDβ S = [IαQ, Dβ S] = Dβ [IαQ, S] since αQ is a commuting object. Then δS ′ = Dβ αγ Qγ S = −αγ Dβ Qγ S since αγ , Dβ anticommute. Then δS ′ = αγ Qγ Dβ S = αQS ′ .
Summary: Function of superfields and their superderivatives is a superfield.
R quantum number assignments for superspace: θL/R has R quantum number R = ±1 . Recall definition of R-symmetry on page 84: Qa → eIRL φ Qa and Q†a → eIRR φ Q†a , where R = RL/R = ±1. Assignments for superspace follow from definition of superfield (i.e. [Q, S(x, θ)] = −IQS(x, θ)) and definition of Q on page 94. Note θR ∼ θL∗ .
96
From now on, write e.g. Aµγ µ = A. /
General form of superfield: S(x, θ) = C(x) − I[θγ5]ω(x) − θγ5θ M (x) − 21 [θθ]N (x) + I2 [θγ5γµθ]V µ(x) I 2
−I[(θγ5θ)θ] λ(x) +
4 (Pseudo)scalar fields: C, M , N , D,
1 / ω(x) 2∂
2 Spinor fields: ω, λ,
−
1 2 4 [θγ5 θ]
D(x) +
1 Vector field: V µ
1 2 2 ∂ C(x)
,
(i.e. 8 bosonic and 8 fermionic degrees of freedom). This is most general Taylor series in θT = (θ1 , θ2 , θ3 , θ4 ), or θ = θT γ5 E = (−θ2 , θ1 , θ4 , −θ3 ), in manifestly Lorentz invariant form: 2nd term −Iθγ5 ω(x) most general quantity linear in θα . Next 3 bilinears in θα is expansion for any bilinear B = B12 θ1 θ2 + B13 θ1 θ3 + B14 θ1 θ4 + B23 θ2 θ3 + B24 θ2 θ4 + B34 θ3 θ4 : θγ5 ( I1 γ0 or I1 γ3 )θ = 2θ2 θ3 ± 2θ4 θ1 , θγ5 ( I1 γ1 or I1 γ2 )θ = 2θ2 θ4 ± 2θ3 θ1 (gamma matrices on page 43), θ(1 or γ5 )θ = 2θ1 θ2 ± 2θ4 θ3 . 6th term has all 4 possible cubics in θα : (θγ5 θ)θ = 2(θ4 θ3 θ2 , −θ4 θ3 θ1 , θ1 θ2 θ4 , −θ1 θ2 θ3 ). Last term ∝ [θγ5 θ]2 = 8θ1 θ2 θ3 θ4 . Any higher products of θα vanish, since θα2 = 0 (no sum). E.g. θ1 θ2 θ3 θ4 θ3 = −θ1 θ2 θ32 θ4 = 0.
Superfield transformation: δS = Iαγ5ω + θ(−/ ∂ C + Iγ5M + N − Iγ5V/ )α +
I 2 [θγ5 θ]α(λ
97 + ∂/ ω)
− I2 [θθ]αγ5(λ + ∂/ ω) + I2 [θγ5γ µθ]αγµλ + I2 [θγ5γ ν θ]α∂ν ω + 21 [(θγ5θ)θ] I ∂/ M − γ5∂/ N − I∂µV/ γ µ + γ5(D + 21 ∂ 2C) α − I4 [θγ5θ]2αγ5(/ ∂ λ + 21 ∂ 2ω) .
From definition of superfield and of Q, on page 94, i.e. δS = α Note for M = 1, γ5 γµ and γ5 ,
∂ (θM θ) ∂θ
∂ − ∂θ
µ
+ γ θ∂µ S.
= 2M θ. See also Weinberg III (hardback), page 63.
Component field transformations: δC = I(αγ5ω) , δω = (−Iγ5∂/ C − M + Iγ5N + V/ )α , δM = −α(λ + ∂/ ω) , δN = Iαγ5(λ + ∂/ ω) , δVµ = α(γµλ + ∂µω) , / γ µ] + Iγ5D)α and δD = Iαγ5∂/ λ . δλ = ( 21 [∂µV, Compare superfield transformation above with general form of superfield on page 96.
D-component of general superfield (coefficient of [θγ5θ]2) is candidate for SUSY Lagrangian. If L ∝ [S]D ∝ D + 21 ∂ 2 C, then δL ∝ δD + 12 ∂ 2 δC = ∂µ (Iαγ5 γ µ λ + 21 ∂ µ δC), i.e. a derivative, so action A =
R
d4 xL obeys δA = 0.
98 3.4.1
Chiral superfield
Definition of chiral superfield X(x, θ): λ = D = 0 , Vµ = ∂µB . This is superfield because these conditions are preserved by SUSY transformations: δD = Iαγ5 ∂/ λ = 0, / γ µ ] + Iγ5 D)α = 12 [∂µ V, / γ µ ]α. So require [∂µ V, / γ µ ] = 0. and δλ = ( 12 [∂µ V, But [∂µ V, / γ µ ] = ∂µ Vν [γ ν , γ µ ] = (∂µ Vν − ∂ν Vµ ){γ ν , γ µ } = 2g νµ (∂µ Vν − ∂ν Vµ ), so ∂µ Vν − ∂ν Vµ = 0 which requires Vµ = ∂µ B. Finally, δV µ = α(γµ λ + ∂µ ω) = ∂µ αω, i.e δB = αω, i.e. Vµ = ∂µ B condition is preserved.
Chiral superfield decomposition: X(x, θ) = Φ±(x, θ) = φ±(x) −
√
√1 [Φ+ (x, θ) 2
+ Φ−(x, θ)] with left / right-chiral superfields
2 θψL/R (x) + [θPL/R θ]F±(x) ± 21 [θγ5γµθ]∂ µφ±(x) ∓ √12 (θγ5θ)θ/ ∂ ψL/R (x) − 81 [θγ5θ]2∂ 2φ±(x) ,
with ψL/R = PL/R ψ , φ± =
√1 (A 2
± IB) , F± =
√1 (F 2
T
∓ IG) . Recall ψ =
√1 2
ζa, ζ
†b˙
from page 92.
Write C = A, ω = −Iγ5 ψ, M = G, N = −F , Z = B for later convenience in general form of superfield on page 96, so X = A − θψ + 21 θθF − I2 θγ5 θG + I2 θγ5 γµ θ∂ µ B + 21 (θγ5 θ)θγ5 ∂/ ψ − 81 [θγ5 θ]2 ∂ 2 A.
Left / right transformation: δψL/R =
√
2(∂µφ±γ µPR/L + F±PL/R )α , δF± =
√
√ 99 2α/ ∂ ψL/R , δφ± = 2αψL/R .
So left-chiral supermultiplet here = that on page 90 (ζa = ψL, φ = φ+, F = F+), right-chiral by conjugation.
Compact form: Φ±(x, θ) = φ±(x±) ∓
√
T T 2θL/R E ψL/R (x±) ± θL/R EθL/R F±(x±) ,
where θL/R = PL/R θ , xµ± = xµ ± θRT Eγ µθL . T This is left / right-chiral superfield on page 98, by Taylor expansion in θR Eγ µ θL :
Make use of θT = −θγ5 E (Majorana conjugation on page 91) and E 2 = (γ5 E)2 = −1. Firstly, rewrite as Φ± (x, θ) = φ± (x± ) −
√
2θPL/R ψ(x± ) + θPL/R θF± (x± ),
T EψL/R = −θγ5 E 21 (1 ± γ5 )E 21 (1 ± γ5 )ψ = −θ 21 (−γ5 ∓ 1) 21 (1 ± γ5 )ψ = ±θPL/R ψ. Similarly, xµ± = xµ ± 12 θγ5 γ µ θ, so because θL/R
1st term in underlined equation above: φ± (x± ) = φ± (x) ± 12 (θγ5 γµ θ)∂ µ φ± (x) − 12 ( 12 θγ5 γ µ θ)( 12 θγ5 γ ν θ)∂µ ∂ν φ± (x) = φ± (x) ± 12 (θγ5 γµ θ)∂ µ φ± (x) − 21 ( 12 θγ5 θ)2 12 {γ µ , γ ν }∂µ ∂ν φ± (x), then use 21 {γ µ , γ ν } = g µν . √ √ 2nd term: − 2θPL/R ψ(x± ) = − 2θPL/R ψ(x) ∓
√1 (θγ5 γ µ θ)θ∂µ PL/R ψ(x) 2
√ = − 2θPL/R ψ(x) ∓
√1 (θγ5 θ)θγ µ ∂µ PL/R ψ(x). 2
T T 3rd term: θPL/R θF± (x± ) = θPL/R θF± (x) ± 21 θγ5 γ µ θθPL/R θ∂µ F± (x) = θPL/R θF± (x) + [(θR Eγ µ θL )(θL/R EθL/R )∂µ F± (x)],
term in square brackets vanishes due to 3 occurences of “2-component” θL/R .
100 Compact form on page 99 is most general function of x± and θL/R . Conjugate of right-chiral superfield is left-chiral superfield.
Superspace “direction” of chiral superfield: D∓Φ± = 0 , where D∓ = PR/LD . Can be used to define left / right-chiral superfield. ∂ Follows from D∓ xµ± = 0, using D∓ = ∓ǫαβ ∂θR/L
β
− (γ µ θL/R )α ∂x∂ µ .
Chiral superfield from superfield: For general superfield S, D±α D±β S (i.e. D±T ED±S) is left / right-chiral. D±γ (D±α D±β S) = 0 because the D±α anticommute and there are only 2 of them.
F -component of chiral superfield (coefficient of θPLθ) is candidate for SUSY Lagrangian. If L ∝ [Φ± ]F = F± , then δL ∝ α/ ∂ ψL/R , i.e. a derivative, so action A =
R
d4 xL obeys δA = 0.
101 3.4.2
Supersymmetric Actions
From now on, only work with left-chiral superfields, write Φ+ = Φ, φ+ = φ, F+ = F .
Supersymmetric action from chiral superfields: A = where f is left-chiral, K is real superfield.
R
d4x[f ]F +
R
d4x[f ]∗F +
R
d4x 12 [K]D ,
Superpotential f is polynomial in left-chiral superfields: Recall function of left-chiral superfields (but not complex conjugates thereof) is left-chiral superfield. Superderivatives (and therefore derivatives) of left-chiral superfield is not left-chiral. Left-chiral superfield of form D+α D+β S (see chiral superfield from superfield on page 100) allowed in f , then can write f = D+T ED+ h, since D+ annihilates all left-chiral superfields after differentiating with product rule. T But D+T ED+ h ∝ coeff. of θR EθR (neglecting spacetime derivatives with don’t contribute to A from now on). T EθR ) = (θγ5 θ)2 in h, so Then [D+T ED+ h]F ∝ coeff. of (θLT EθL )(θR
i.e. D+α D+β S terms not necessary in f , can be included in [K]D .
R
d4 x[D+T ED+ h]F ∝
Kahler potential K depends on left / right-chiral superfields but is more general.
R
d4 x[h]D ,
102 R quantum number assignments for potentials f and K: Rf = 2 , RK = 0 . First case: Want
R
d4 x[f ]F to have RF = 0. From chiral superfield decomposition on page 98, term in left-chiral superfield Φ
containing F component is θPL θF = ±θLT EθL F . Since RθLT EθL = 2 and want RF = 0, must have RΦ = 2. Second case: Want
R
d4 x[K]D to have RD = 0. From general form of superfield on page 96,
term containing D component is ∝ [θγ5 θ]2 (D + 21 ∂ 2 C). But R[θγ5 θ]2 = 0, so for RD = 0 must have RS = 0.
For renormalizable theory of chiral superfields:
Superpotential f is at most cubic in left-chiral superfields Φn. Operators O in L must have mass dimension dM (O) ≤ 4 for renormalizability. From definition of D, dM (θ) = − 12 .
So dM ([S]F ) = dM (S) + 1, dM ([S]D ) = dM (S) + 2, so operators (represented by S here) in f (K) have mass dimensionality ≤ 3(2 Since dM (Φ± ) = dM (φ± ) = 1, f can only be cubic polynomial in Φ±n .
General form of Kahler potential: K(Φ, Φ∗) =
P
∗ mn gmn Φm Φn ,
gmn Hermitian.
As noted above, operators in K have mass dimensionality ≤ 2. Φm Φn terms (or conjugates thereof) don’t appear, since they are left-chiral so have no D term.
Kahler potential part of L :
1 ∗ 2 [K(Φ, Φ )]D
= −∂µφ∗n∂ µφn + Fn∗Fn −
1 µ 2 ψ nL γ ∂µ ψnL
+
103 .
1 µ 2 (∂µ ψ nR )γ ψnR
Direct calculation gives 21 [K(Φ, Φ∗ )]D = gmn −∂µ φ∗m ∂ µ φn + Fm∗ Fn − 21 ψ mL γ µ ∂µ ψnL + 21 (∂µ ψ mR )γ µ ψnR .
′ Take Φm = Nmn Φn , gmn = (N † gN )mn . Since gmn Hermitian, choose unitary N to diagonalize it.
µ ′ ′ Diagonal terms positive to get positive coefficient for −∂µ φ′∗ n ∂ φn , absorb them into φn then drop primes.
2
∂f (φ) ∂ f (φ) ∗ ψ ψ + F Superpotential part of L : [f (Φ)]F = − 21 ∂φ . Similar for f . mL n nR ∂φn n ∂φm
From direct calculation.
L from chiral superfields: L = −∂µφ∗n∂ µφn − 21 ψ nLγ µ∂µψnL + 12 (∂µψ nR )γ µψnR ∂ 2 f (φ) − 21 ∂φ ψ ψmL n ∂φm nR
where Fn = −
∂f (φ) ∂φn
∗
−
1 2
∂ 2 f (φ) ∂φn ∂φm
∗
(ψ nR ψmL)∗ − V (φ) ,
P ∂f (φ) 2 and scalar field potential V (φ) = n ∂φn .
Sum Kahler potential and superpotential parts (and conjugate) of L above, use field equations to get Fn = −
∂f (φ) ∂φn
∗
.
∂f (φ) Minimized V (φ0) ≥ 0, equality (tree level) only when ∂φn
104 φ=φ0
= 0 (note φn0 = h0|φn|0i).
Tree-level expansion of V : V (φ) = V (φ0) + (M †M )mn∆φ†m∆φn ,
where ∆φn = φn − φn0 , bosonic mass matrix Mmn = From Taylor’s theorem and
P
n Mmn
P ∂f 2 P ∂f V = n ∂φn ≃ n ∂φn φ=φ0 +
2
∗ ∂f ∂φn
∂ f ∂φm ∂φn
∂ 2f ∂φm ∂φn φ=φ0
.
= 0 (see page 105), φ=φ0
2 † † ∆φ m = V (φ0 ) + ∆φm (M M )mn ∆φn . φ=φ0
Matrix M can be diagonalised because it is complex symmetric.
From L from chiral superfields on page 103, fermionic mass term is − 21 Mmnψ nR ψmL, so fermions and bosons have same mass, as required in SUSY. Free Lagrangian density: L0 =
P n
− ∂µ∆φ∗n∂ µ∆φn − m2n∆φ∗n∆φn
− 21 ψ nLγ µ∂µψnL + 12 (∂µψ nR )γ µψnR − 12 mnψ nR ψnL − 12 mn(ψ nR ψnL)∗ .
∗ (ψ nR ψmL )∗ L0 = −∂µ ∆φ∗n ∂ µ ∆φn − (M † M )mn ∆φ∗m ∆φn − 12 ψ nL γ µ ∂µ ψnL + 21 (∂µ ψ nR )γ µ ψnR − 12 Mmn ψ nR ψmL − 21 Mmn
from L from chiral superfields on page 103 and tree-level expansion of V above, then diagonalize M .
3.5
105
Spontaneous supersymmetry breaking
∂f (φ) Broken SUSY vacuum condition: h0|F |0i = 6 0 or ∂φn
φ=φ0
6= 0 or V (φ0) > 0 .
In left / right transformation on page 99, want to make one of h0|δΨl |0i = 6 0, where Ψl = ψnL , φn or Fn . Cannot make h0|ψnL |0i = 6 0 since vacuum Lorentz invariant, nor h0|∂µ φn |0i = 6 0 since h0|φn |0i constant. Only possibility is h0|Fn |0i = 6 0 (whence h0|δψnL |0i =
√ 2αh0|Fn |0i),
which from L from chiral superfields on page 103 is equivalent to last 2 statements.
∂f (φ) Spontaneous SUSY breaking requires f such that ∂φn
φ=φ0
Spontaneous SUSY breaking gives rise to massless spin
1 2
∂V ∂φn
φ=φ0
P ∂f 2 P ∂ 2 f ∂f ∗ = 0. But V = n ∂φn , so 2 n ∂φm ∂φn ∂φn
= 0 has no solution.
goldstino. = 0, i.e.
φ=φ0
P
n Mmn
∗ ∂f ∂φn
= 0. φ=φ0
So M has at least one zero eigenvalue, so from free Lagrangian density on page 104, since M eigenvalues are the mn , there is at least one linear combination of ψnL with zero mass.
In local SUSY, goldstino absorbed into longitudinal component of gravitino, gives it a mass.
106 3.5.1
O’Raifeartaigh Models
Theories in which left-chiral fields Xn, Yi have R = 0, 2. Most general superpotential is f (X, Y ) =
P
i Yi fi (X) .
From R quantum number assignments for potentials f and K on page 102, i.e. Rf = 2 (X ∗ , Y ∗ not allowed since f left-chiral).
SUSY broken when no. fields X < no. fields Y . Write scalar components of X, Y as x, y. Condition
∂f (x,y) ∂yi
= 0 implies fi (x) = 0, i.e. more conditions than fields Xi ,
only possible to satisfy by careful choice of the fi (x).
Scalar field potential: V =
2 P P ∂fi (x) 2 i yi ∂xn . i |fi (x)| + n
P
From general superpotential above and definition of scalar field potential in L from chiral superfields on page 103.
Simplest (renormalizable) model: Fields X, Y1, Y2. Then choice f1(X) = X − a, f2(X) = X 2 is general, for which V = |x|4 + |x − a|2 + |y1 + 2xy2|2. Renormalizability allows fi to be quadratic only. Then take linear combinations of Yi and shift and rescale X.
Vacuum matrix elements: x0 = y10 = 0, y20 arbitrary , or h0|Fy1 |0i = h0|Fy2 |0i = 0 and h0|Fx|0i = a . (†)
(†)
(†) Solve Fw† = − ∂V ∂w = 0 where w = x , y1 , y2 . Recall w0 = h0|w|0i.
3.6
107
Supersymmetric gauge theories
Gauge transformation of left-chiral supermultiplet: (φ, ψL, F )n(x) → exp[ItAΛA(x)]
nm
(φ, ψL, F )m(x) .
Fields in same supermultiplet have same transformation properties under internal symmetry transformation U : (†)
(†)
(†)
Since [Q, aB/F ]∓ ∼ aF/B , where aB/F are annihilation (creation) operators for bosonic and fermionic superpartners, (†)
(†)
(†)
general transformation of aσ on page 9 is same for aB , aF because [U, Q] = 0 from page 85. So from complete field on page 26, transformation same for all fields in supermultiplet.
Gauge transformation of left-chiral superfields: Φ(x, θ) → exp[ItAΛA(x+)]Φ(x, θ) (column vector Φ). Implied by gauge transformation of left-chiral supermultiplet above and compact form on page 99.
Gauge transformation of right-chiral superfields: Φ†(x, θ) → Φ†(x, θ) exp[−ItAΛA(x−)] . From conjugate of gauge transformation of left-chiral superfields above. Note ΛA (x) is real function of x and ΛA∗ (x+ ) = ΛA (x− ).
Global gauge invariance =⇒ local for
R
4
d x[f (Φ)]F but not for
R
d4x[K(Φ, Φ†)]D .
No derivatives / conjugates of Φ in f , so argument y of ΛA (y) irrelevant. But K also contains Φ† , so e.g. Φ†n Φn not invariant under gauge transformation of left and right-chiral superfield above because ΛA (x+ ) 6= ΛA (x− ).
108 Define gauge connection Γ(x, θ), with transformation property Gauge transformation of gauge connection: Γ(x, θ) → exp[ItAΛA(x−)]Γ(x, θ) exp[−ItAΛA(x+)] , then “new” right-chiral superfield Φ†(x, θ)Γ(x, θ) has transformation property Gauge transformation of new right-chiral superfield: Φ†(x, θ)Γ(x, θ) → Φ†(x, θ)Γ(x, θ) exp[−ItAΛA(x+)] . Global gauge invariance =⇒ local for
R
d4x[K(Φ, Φ†Γ)]D . Also implies invariance under
Extended gauge transformations: Φ(x, θ) → exp[ItAΩA(x, θ)]Φ(x, θ) , Γ(x, θ) → exp[ItAΩA†(x, θ)]Γ(x, θ) exp[−ItAΩA(x, θ)] with any left-chiral superfields ΩA(x, θ). We are extending ΛA (x+ ) to ΩA (x, θ) = ΛA (x+ ) −
√
2θLT E ψΛL (x+ ) + θLT EθL FΛ+ (x+ ) in compact form on page 99.
Φ(x, θ) is left-chiral (depends on x+ , θL only), so cannot introduce x− or θR in ΩA (x, θ).
Hermitian connection: Γ(x, θ) = Γ†(x, θ) . Take Γ → 12 (Γ + Γ† ) or
1 2I (Γ
− Γ† ). Ordinary gauge transformation of Γ above intact, since Γ, Γ† have same transformation.
Connection from gauge superfields: Γ(x, θ) = exp[−2tAV A(x, θ)] , with real gauge superfields VA. Form preserved by gauge transformation of gauge connection above, new V A independent of tA representation.
Extended gauge transformation of gauge superfields: V A(x, θ) → V A(x, θ) +
I A 2 [Ω (x, θ)
109 − ΩA†(x, θ)] + . . . ,
where “. . .” = commutators of generators, which vanish for zero coupling and in U(1) (Abelian). From extended gauge transformations and definition of gauge superfields above.
Gauge superfields in terms of components: V A(x, θ) = C A(x) − Iθγ5ω A(x) − I2 θγ5θM A(x) − 12 θθN A(x) + I2 θγ5γ µθVµA(x) − Iθγ5θθ[λA(x) + 21 ∂/ ω A(x)] − 14 (θγ5θ)2(DA(x) + 12 ∂ 2C A(x)) . From general form of superfield on page 96.
Transformation superfields in terms of components: ΩA(x, θ) = W A(x) −
√
2θPLwA(x) + W A(x)θPLθ
+ 21 θγ5γµθ∂ µW A(x) − √12 θγ5θθ/ ∂ PLwA(x) − 81 (θγ5θ)2∂ 2W A(x) . From chiral superfield decomposition on page 98, taking left-chiral part (Φ+ there).
Transformation of supermultiplet fields: C A(x) → C A(x) − ImW A(x) + . . . , ω A(x) → ω A(x) + √12 wA(x) + . . . , VµA(x) → VµA(x) + ∂µReW A(x) + . . . , M A(x) → M A(x) − ReW A(x) + . . . , N A(x) → N A(x) + ImW A(x) + . . . , λA(x) → λA(x) + . . . , DA(x) → DA(x) + . . . . From all 3 above results. Again, “. . .” = commutators of generators, which vanish for zero coupling and in U(1) (Abelian).
Wess-Zumino gauge: C A, ω A, M A, N A = 0 → V A(x, θ) =
I µ A 2 θγ5 γ θVµ (x)
− Iθγ5θθλA(x) −
110
1 2 A 4 (θγ5 θ) D (x) .
√ Transformation of supermultiplet fields on page 109: take ImW A (x) = C A (x), wA (x) = − 2ω A (x), W A (x) = M A (x) − IN A (x). Sufficient for Abelian case because “. . .” = 0. For non-Abelian case, add nth order (in gauge couplings) terms to ImW A (x), wA (x), W A (x) to cancel nth order terms from commutators of m ≤ n − 1th order terms.
Wess-Zumino gauge is not supersymmetric. Ensuring δC A , δω A , δM A , δN A = 0 in component field transformations on page 97, requires also having VµA , λA = 0, which requires δVµA , δλA = 0. δVµA = 0 satisfied, but δλA = 0 requires DA = 0, i.e. V A = 0.
Gauge invariant L for chiral supermultiplet:
1 † 2 [Φ ΓΦ]D
= − 21 (Dµφ)†Dµφ − 21 ψ Lγ µDµψL + 21 F †F
√
+I 2ψ LtAλAφ − 21 DAφ†tAφ + h.c. , where covariant derivative Dµ = ∂µ − ItAVµA (gauge transformation for gauge superfield derived next on page 111). In Wess-Zumino gauge, Γ(x, θ) = 1 − Iθγ5 γµ θtA VµA (x) − 21 θγ5 γ µ θθγ5 γ ν θtA tB VµA (x)VνB (x) + 2Iθγ5 θtA θλA (x) + 21 (θγ5 θ)2 tA DA (x). Then use left-chiral superfield in chiral superfield decomposition on page 98. Note D term is coefficient of − 41 (θγ5 θ)2 minus 21 ∂ 2 × first “φ” term.
111 Gauge transformation of gauge supermultiplet fields: δgaugeVµA(x) = CABC VµB (x)ΛC (x) + ∂µΛA(x) (which is infinitesimal version of transformation of gauge fields on page 62), and δgaugeλA(x) = CABC λB (x)ΛC (x) , δgaugeDA(x) = CABC DB (x)ΛC (x) (i.e. λA, DA in adjoint representation). Firstly ΛA (x+ ) = ΛA (x) + 12 (θγ5 γµ θ)∂ µ ΛA (x) − 18 (θγ5 θ)2 ∂ 2 ΛA (x). Then gauge transformation of gauge connection on page 108 (using connection from gauge superfields) reads exp[−2tA V A′ (x, θ)] = exp[ItA ΛA† (x+ )] exp[−2tA V A (x, θ)] exp[−ItA ΛA (x+ )]. Write as exp[−2tA V A′ (x, θ)] = ea eX eb where X = −2tA V A (x, θ) = −2tA [ I2 θγ5 γµ θVµA (x) − Iθγ5 θθλA (x) − 41 (θγ5 θ)2 DA (x)] , and small quantities b + a = 2tA ImΛA (x+ ) = −Iθγ5 γµ θtA ∂ µ ΛA (x) , b − a = −2ItA ReΛA (x+ ) = −2ItA [ΛA (x) − 81 (θγ5 θ)2 ∂ 2 ΛA (x)] . To first order in a, b, −2tA V A′ (x, θ) = X + 12 [X, b − a] + b + a + . . . and “. . .” means higher order in a, b, as well as terms of O(X n )O(b − a) with integer n ≥ 2 which vanish: since (1, γ5 ) = PL ± PR , 3 X ∼ θL θR (× further θ factors), while b + a ∼ θL θR , so O(X 2 )O(b − a) ∼ θL3 θR (× . . .) = 0.
So V A′ (x, θ) = V A (x, θ) + CABC V B (x, θ)ΛC (x) + I2 θγ5 γµ θ∂ µ ΛA (x), compare superfield expansion in Wess-Zumino gauge on page 110.
Wess-Zumino gauge is gauge invariant (i.e. C A, ω A, M A, N A remain zero) under ordinary gauge transformations. Note θ dependence of V A′ (x, θ) same as V A (x, θ), i.e. C A′ , ω A′ , M A′ , N A′ = 0.
112 3.6.1
Gauge-invariant Lagrangians
Construct gauge-invariant Lagrangian from A Gauge-covariant spinor superfield: 2tAWLα (x, θ) = D−T ED− exp[2tAV A(x, θ)]D+α exp[−2tAV A(x, θ)]
which is left-chiral because D−WL = 0 (Dα−Dβ−Dγ− = 0), i.e. A A (x, θ) → exp[ItAΩA(x, θ)]2tAWLα (x, θ) exp[−ItAΩA(x, θ)] . Extended gauge transformation of WLA: 2tAWLα
From extended gauge transformations and connection from gauge superfields on page 108, A 2tA WLα → D−T ED− exp[ItA ΩA ] exp[2tA V A ] exp[−ItA ΩA† ] D+α exp[ItA ΩA† ] exp[−2tA V A ] exp[−ItA ΩA ].
Use product rule for the D± here. Since ΩA is left-chiral, D− ΩA = D+ ΩA† = 0. A → exp[ItA ΩA ]D−T ED− exp[2tA V A ] exp[−ItA ΩA† ] exp[ItA ΩA† ]D+α exp[−2tA V A ] exp[−ItA ΩA ] Then 2tA WLα
= exp[ItA ΩA ]D−T ED− exp[2tA V A ]D+α exp[−2tA V A ]exp[−ItA ΩA ]. Not finished, because D+α exp[−ItA ΩA ] 6= 0 (however, D−T ED− exp[−ItA ΩA ] does vanish). But D−T ED− D+α = D+α D−T ED− − 4 [PL ∂/ D− ]α , so D−T ED− D+α exp[−ItA ΩA ] = 0. A So 2tA WLα
→ exp[ItA ΩA ] D−T ED− exp[2tA V A ]D+α exp[−2tA V A ] exp[−ItA ΩA ],
where all derivatives in quantity between
and
evaluated before multiplying on left / right with exp[±ItA ΩA ].
Form of WLA: WLA(x, θ) = λA L (x+ ) +
1 µ ν A 2 γ γ θL Fµν (x+ )
113
A + θLT EθLDλ / A R (x+ ) − IθL D (x+ ) .
Note exp[−2tA V A (x, θ)] = 1 − Iθγ5 γµ θtA VµA (x) − 12 θγ5 γ µ θθγ5 γ ν θtA tB VµA (x)VνB (x) + 2Iθγ5 θtA θλA (x) + 21 (θγ5 θ)2 tA DA (x). After performing all D± in gauge-covariant spinor superfield on page 112, choose gauge where VA (X) = 0 at given x = X, 1 µ ν A A T A gives WLA (X, θ) = λA / λA L (x+ ) + 2 γ γ θL (∂µ Vν (X+ ) − ∂ν Vµ (X+ )) + θL EθL ∂ R (X+ ) − IθL D (X+ ).
Then convert to gauge covariant form consistent with this, A i.e. ∂µ VνA − ∂ν VµA → Fµν (the non-Abelian field strength of page 63) and ∂/ → D. /
A
A Aµν F − 21 λ (Dλ) Gauge supermultiplet Lagrangian: Lgauge = − 21 Re([WLAT EWLA]F ) = − 41 Fµν / A + 21 DADA .
From form of WLA above.
Additional pieces: Additional non perturbative part: Lθ = where g is coupling appearing in tA.
g2θ − 16π 2 Im
WLAT EWLA
=
g2θ − 16π 2
A
A
/ 5λ − Iλ Dγ
1 Aµν Aρσ ǫ F F µνρσ 4
Fayet-Iliopolis term: LFI = ξD , where ξ is arbitrary constant, D is for Abelian supermultiplet.. Corresponding action is supersymmetric, because δD = Iαγ5 ∂/ λ is derivative.
,
114 Explicit check that action from Lgauge is supersymmetric. In VA (X) = 0 gauge, component field transformations on page 97 at x = X: δVµA = αγµ λA , δDA = Iαγ5 ∂/ λA , and A ν A [γ , γ µ ] + Iγ5 DA )α where Fµν = ∂µ VνA − ∂ν VµA + CABC VµB VνC is non-Abelian field strength of page 63. δλA = ( 14 Fµν A Aµν A A Aµν Then using δ(Fµν F ) = 2Fµν δF Aµν etc., δ(Fµν F ) = 2F Aµν α(γν ∂µ − γµ ∂ν )λA , δ(DA DA ) = 2IDA αγ5 ∂/ λA A
A
A µ ν [γ , γ ] + Iγ5 D]/ ∂ λA + δ(CABC λ V/ B λC ) up to derivatives and δ(λ ∂/ λA ) = 2α[ 41 Fµν A
(we will show δ(CABC λ V/ B λC ) = 0 later). For this last expression, use [γ µ , γ ν ]γ ρ = −2g µρ γ ν + 2g νρ γ µ − 2Iǫµνρσ γσ γ 5 (expand in 16 matrices on page 44, Lorentz and space inversion invariance limits this to these three terms. 1st coefficient by taking µνρ = 121: [γ 1 , γ 2 ]γ 1 = −2g 11 γ 2 , correct because from anticommutation relations for γ µ on page 43, γ 1 γ 2 = −γ 2 γ 1 and γ 1 2 = 2. Similarly for µνρ = 211 to get 2nd coefficient. 3rd coefficient from µνρ = 123: [γ 1 , γ 2 ]γ 3 = −2Iǫ1230 γ0 γ5 , LHS is 2γ 1 γ 2 γ 3 = −2γ0 γ 0 γ 1 γ 2 γ 3 = −2Iγ0 γ5 ). A
A αγσ γ5 ∂ρ λA + 2IDA αγ5 ∂/ λA . 2nd term replaceable, after integration by parts, So δ(λ ∂/ λA ) = −F Aµν α(γν ∂µ − γµ ∂ν )λA − Iǫµνρσ Fµν A A Aµν by Iǫµνρσ (∂ρ Fµν )αγσ γ5 λA , which vanishes since e.g. ǫµνρσ ∂ρ ∂µ VνA = 0, 1st and 3rd terms cancel with δ(Fµν F ) and δ(DA DA ). A
A
A
Thus we are left with δ(CABC λ V/ B λC ) = CABC λ (δ V/ B )λC = CABC λ γµ λC αγ µ λB = 0, where antisymmetry of CABC used in 1st step. Last step requires explicit calculation.
115 Lagrangian for chiral superfield interacting with gauge fields: L = −(Dµφ)†n(Dµφ)n −
1 µ 2 ψ nL γ (Dµ ψL )n
+
1 µ 2 Dµ ψ nL γ ψnL
−
1 ∂ 2f T 2 ∂φn ∂φm ψnL EψmL
−
1 2
∂ 2f ∂φn ∂φm
∗
T ψnL EψmL
√ √ A A g2θ Aµν Aρσ A Aµν −V (φ) + I 2ψ nL(tA)nm λAφm − I 2φ†nλ (tA)nm ψmL − 41 Fµν / A + 64π F − 21 λ (Dλ) F , 2 ǫµνρσ F where potential V (φ) =
∂f (φ) ∂φn
∂f (φ) ∂φn
∗
+
1 2
A
ξ +
φ∗n(tA)nm φm
A
ξ +
φ∗k (tA)kl φl
∗
.
Sum of gauge invariant L for chiral supermultiplet on page 110, superpotential part of L on page 103 and 3 Lagrangians on page 113 gives L =
1 2
† Φ exp −2tA V A Φ D + 2Re[f (Φ)]F − 21 Re WLAT EWLA F − ξ A DA −
g2 θ AT A 16π 2 Im(WL EWL ),
(using Majorana conjugation on page 91, e.g. ψ R = ψ † PR β = ψPL = ψ T Eγ5 PL = ψ T PL Eγ5 = ψLT Eγ5 ) √ 2 ∂f (φ) f T L = −(Dµ φ)†n (Dµ φ)n − 12 ψ n γ µ (Dµ ψ)n + Fn† Fn − Re ∂φ∂n ∂φ ψ Eψ + 2Re F − 2 2Im(tA )nm ψ nL λA φm m n n ∂φn m √ A A Aµν F − 21 λ (Dλ) / A+ +2 2Im(tA )mn ψ nR λA φ†m − φ†n (tA )nm φm DA − ξ A DA + 21 DA DA − 14 Fµν Then use field equations for auxiliary fields: Fn = −
∂f (φ) ∂φn
∗
g2 θ Aµν Aρσ F . 64π 2 ǫµνρσ F
and DA = ξ A + φ∗n (tA )nm φm .
or explicitly
116 Perturbative non-renormalization theorem for F term: For general SUSY gauge theory with coupling g absorbed into gauge superfields, † AT 1 A A L = Φ exp −2tAV Φ D + 2Re [f (Φ)]F + 2g2 Re WL EWL F . 1 2
For supersymmetric, gauge-invariant cut-off λ,
AT 1 † A Lλ = Aλ(Φ, Φ , V, D, ...) D + 2Re [f (Φ)]F + 2g2 Re WL EWL F , 1 2
λ
where one-loop effective gauge coupling gλ−2 = const − 2b ln λ.
Implies coefficients in terms of superpotential unchanged by renormalization, i.e. these coefficients receive no radiative corrections.
In MSSM, Higgs mass term appears in superpotential, so no hierarchy problem to all orders.
117 A Lorentz invariance: h0|(ψnL , λA , Fµν )|0i = 0, so h0|L |0i = −h0|V |0i.
Tree-level vacuum is minimum of V (φ). Unbroken SUSY vacuum: Fn0 = −
h
i
∂f (φ) ∂φn φ=φ 0
= 0 , D0A = ξ A + φ∗n0(tA)nm φm0 = 0 ⇐⇒ V (φ) = 0.
Can write V = Fn∗ Fn + DA DA > 0, so V = 0 (if allowed) is a minimum. In this case, Fn0 = D0A = 0. From left / right transformation on page 99, δψL = δF = δφ = 0. Argument holds in reverse. Note: no overconstraining on N φ components: Fn0 = D0A = 0 is N conditions, not N + D, where D is dimensionality of group, i.e. A = 1, . . . , D, because only N − D conditions needed to satisfy Fn0 = on page 108,
∂f (Φ) ∂ΩA
B
=0=
∂f (Φ) ∂([eItB Ω ]nm Φm ) ∂Φn ∂ΩA ΩC =0
=
∂f (φ) ∂φ
= 0: f (Φ) invariant under extended gauge transformations φ=φ0
∂f (Φ) ∂Φn (ItA Φ)n ,
i.e.
∂f (φ) ∂φn (tA φ)n
= 0, which is already D conditions.
Existence of any supersymmetric field configuration =⇒ unbroken SUSY vacuum. From above, SUSY field configuration has V = 0, which is absolute minimum so lower than V for non SUSY field configuration.
ξ A = 0: To check vacuum is unbroken SUSY, enough to check that f (φ) has no φ∗ , so in invariant under φ → eIΛ
A
tA
φ with ΛA complex. If
∂f (φ) ∂φn ∂f (φ) ∂φn
= 0 can be satisfied. e e true for φΛ = eIΛA tA φ. = 0 true for φ,
Choose ΛA such that φΛ† φΛ minimum (which exists because it is real and positive), i.e. i.e. DA = 0, so unbroken SUSY vacuum condition obeyed.
∂ Λ† Λ ∂ΛA (φ φ )
Λ ∝ φΛ∗ n (tA )nm φm = 0,
118 3.6.2
Spontaneous supersymmetry breaking in gauge theories
So break SUSY by 1. Making
∂f ∂φ
= 0 impossible (already considered on page 105)
or 2. Fayet-Iliopoulos term ξD. Simple example: 2 left-chiral Φ± with U(1) quantum numbers ±e, so spinor components are left-handed parts of electron / positron. Then only possibility is f (Φ) = mΦ+Φ−, so from V defined on page 115, V (φ+, φ−) = m2|φ+|2 + m2|φ−|2 + (ξ + e2|φ+|2 − e2|φ−|2)2, which cannot vanish for ξ 6= 0, so SUSY broken. Note U(1) symmetry intact for |ξ|
1032 years. K M N Relevant processes are uR dR → (˜ s∗R or ˜b∗R ) → eL uL and uL → uL . (No uR dR → d˜∗R : coupling is λKM N D D U K
with λKM N antisymmetric in KM since D D
M
antisymmetric in colour indices to get colour singlet coupling.)
121 Rule out some / all baryon lepton number violating terms on page 120 by symmetry, e.g.: 1.) Rule out all by baryon number (B) and lepton number (L) assignments BU,U = BD,D = ± 31 , LU,U = LD,D = 0, BN,N = BE,E = 0, LN,N = LE,E = ±1, BθL = BθR = LθL/R = 0. Or rule out some by requiring only conservation of linear combination, e.g. L, B, B − L etc. K
M
N
2.) Rule out [D D U ]F by LN = LE = 0, LU = LD = LU = LD = −1, LE = −2. “Conventional” L for quarks, leptons by taking LθL/R = ±1. L for squarks, sleptons is then unconventional. 3.) Rule out all by R parity, a discrete global symmetry, = 1 for SM particles and = −1 for their superpartners, i.e. ΠR = (−1)2s(−1)3(B−L) . Lightest SUSY particle (LSP) (lightest particle with ΠR = −1) completely stable. Cannot decay into other ΠR = −1 particles (heavier), or into ΠR = 1 particles only (violates R-parity conservation).
Colliders: sparticles produced in pairs.
Non-zero vacuum expectation values for scalar components of N K → charged lepton, charge − 3e quark masses via 1st, 2nd terms in baryon / lepton number violating terms on page 120, but charge
2e 3
quarks remain massless.
122 Spontaneous breakdown of SU(2)×U(1) for massive quarks, leptons, W ±, Z by Higgs superfields in table 4.1.2: Names
Table 4.1.2: Required Higgs superfields in the MSSM. Label Representation under SU(3)C ×SU(2)L ×U(1)Y
Higgs(ino) H1 = (H10 , H1− )
(1, 2, − 21 )
H2 = (H2+ , H20 )
(1, 2, 21 )
Higgs-chiral superfield couplings in superpotential [f (Φ)]F : (K, M = generation) M
M
M
K − K − E U K 0 K 0 K + K 0 hD KM [(D H1 − U H1 )D ]F , hKM [(E H1 − N H1 )E ]F , hKM [(D H2 − U H2 )U ]F . K
† M E.g. last term is just Qa (ǫH2 )a U , leads to −GKM u Q L a (ǫφH )a uR term in LHiggs−fermion on page 66.
Note second Higgs H2 needed for last term, because we need a (1, 2, 21 ) left-chiral superfield. (H1† is (1, 2, 21 ), but is right-chiral.)
6 0 gives mass to u-type quarks. 6 0 gives mass to d-type quarks and charged leptons, h0|φH 0 |0i = h0|φH 0 |0i = 2
1
2
∂ f (φ) ψ nR ψmL . In superpotential part of L on page 103, fermions get mass from first term − 21 ∂φ n ∂φm
So from e.g. first term of Higgs-chiral superfield couplings in superpotential [f (Φ)]F above, mass term for dK from
∂ 2 f (φ) K K ˜K∗ dR dL ∂ d˜K L ∂ dR
K
= φH10 dR dK L.
For more Higgs superfields, number of H1 and H2 type superfields must be equal. Higgsinos produce SU(2)-SU(2)-U(1) anomalies: For H1 , anomaly ∝ for H2 , anomaly ∝
P
t23 y =
2 1 g 2
P
t23 y =
2 1 g 2
1 ′ 2g
+ − 12 g
2
1 ′ 2g
= 12 g 2 g ′ ,
2 − 12 g ′ + − 12 g − 12 g ′ = − 12 g 2 g ′ . No anomalies from gauginos, in adjoint representation.
123 Most general renormalizable Lagrangian for a gauge theory with R parity or B − L conserved consists of 1. sum of [Φ∗ exp(−V )Φ]D terms for quark, lepton and Higgs chiral superfields, 2. sum of [ǫαβ Wα Wβ ]F for gauge superfields, 3. sum of Higgs-chiral superfield couplings in superpotential [f (Φ)]F on page 122, and + − T 0 0 4. µ term: Lµ = µ H1 ǫH2 F = µ H2 H1 − H2 H1 F .
µ has no radiative corrections due to perturbative non-renormalization theorem for F term on page 116.
Gauge hierarchy problem on page 76 explicitly solved as follows: Unbroken SUSY: 1-loop correction to Higgs mass from any particle cancelled by that particle’s superpartner. For broken SUSY, replace ΛUV with ∼ mass of particle, so 1-loop correction to Higgs mass from top is 2
|κt |2 ∆m2s , 8π 2
where ∆m2s is mass splitting between top and stop.
No fine-tuning if this is . 1 TeV, so since |κt| ∼ 1, stop mass is
0, ∼ power of some M ) → δm2H ∼ M 2 ln ΛU V , OK provided M ∼ mSUSY . 10TeV. So SUSY breaking terms must be superrenormalizable, called soft terms.
127 Soft SUSY breaking R parity / B − L conserving SM invariant terms: (Sum over SU(2), SU(3) indices and generations K, M ) 1. LSR ⊃ 2. LSR ⊃
P
S
P
X
M S2 −MKM φK† S φS , where S = Q, U , D, L, E superfields and φS their component scalars ,
λX mX λX , where X = gluino, wino, bino ,
D K T M D D K T ∗ M 3. Trilinear terms: LSR ⊃ −AD KM hKM (φQ ) ǫφH1 φD − CKM hKM (φQ ) φH2 φD E K T M E E K T ∗ M −AE KM hKM (φL ) ǫφH1 φE − CKM hKM (φL ) φH2 φE U U K T ∗ M T M −AUKM hUKM (φK Q ) ǫφH2 φU − CKM hKM (φQ ) φH1 φU
defined in Higgs-chiral superfield couplings in superpotential [f (Φ)]F on page 122, where hD,E,U KM 4. LSR ⊃ − 21 BµφTH2 ǫφH1 − 12 m2H1 φ†H1 φH1 − 12 m2H2 φ†H2 φH2 where µ defined in µ term on page 123. Recall Hermitian conjugate is added to L . So LSR ⊃ −Re
BµφTH2 ǫφH1
− m2H1 φ†H1 φH1 − m2H2 φ†H2 φH2 .
Choose H1, H2 superfields’ phases so Bµ real, positive: LSR ⊃ −BµRe φTH2 ǫφH1 −m2H1 φ†H1 φH1 −m2H2 φ†H2 φH2
To respect approximate symmetries, choose ASKM , B ∼ 1:
128
ASKM for chiral symmetry: Reflected by small Yukawa coupling of light quarks. B for Peccei-Quinn symmetry: Reflected by small µ term on page 123. S CKM terms involve scalar components of left- and right-chiral superfields,
→ quadratic divergences =⇒ fine-tuning and hierarchy problems. In fact these divergences from tadpole graphs which disappear into vacuum, cannot occur since no SM invariant scalars.
Note SM superpartners acquire mass even if no electroweak symmetry breaking (i.e. if SM particles massless).
S S ASKM , B, CKM are arbitrary and complex → > 100 parameters even without CKM terms.
But expect soft terms to arise from some underlying principle.
129 SUSY breaking at tree-level (see below and next page) ruled out: Predicts squark mass(es) too small, would have effect in accurately measured e+e− → hadrons. This is good, otherwise get fine-tuning: tree-level SUSY breaking would require mass parameter M in Lagrangian. Would affect all SM masses, so M must coincidentally be ∼ electroweak symmetry breaking scale v = 246 GeV. No SUSY breaking at tree-level implies no SUSY breaking at all orders.
SUSY breaking at tree-level for 3 generations ruled out. Tree-level mass sum rule on page 119 holds for each set of colour and charge values. Gives e.g. 2(m2d + m2s + m2b ) ≃ 2(5 GeV)2 = So each squark mass is
10, get mh > mZ .
√ 3 2m4t GF 2π 2 sin2 β
ln
2 Mst . m2t
138 Condition which leads to electroweak symmetry breaking: 4(m2H1 + |µ|2)(m2H2 + |µ|2) ≤ (Bµ)2 . From parameter relations with v1 and v2 on page 135, 4(m2H1 + |µ|2 )(m2H2 + |µ|2 ) = m4A sin2 2β − m2Z (m2Z + 2m2A ) cos2 2β. But Bµ = m2A sin 2β, so 4(m2H1 + |µ|2 )(m2H2 + |µ|2 ) = (Bµ)2 − m2Z (m2Z + 2m2A ) cos2 2β, so inequality follows from cos2 β ≥ 0. These inequalities mean that second derivative matrix of V has negative eigenvalue at SU(2) respecting point φH1 = φH2 = 0, i.e. V unstable (not minimum) there: Quadratic part of scalar Higgs potential on page 134 can be written as four terms ˜T
φ
˜ Mφ2 φ,
where φ˜ is separately the imainary and real parts of (φH1− , φH2+ ) and (φH10 , φH20 ), and
Mφ2
=
± 21 Bµ m2H1 + |µ|2 ± 12 Bµ m2H2 + |µ|2
So mass eigenvalues m2 obey (m2H1 + |µ|2 − m2 )(m2H2 + |µ|2 − m2 ) − 14 (Bµ)2 = 0, 2
2
solutions are 2m = 2|µ| +
m2H1
+
m2H2
±
q
(2|µ|2 + m2H1 + m2H2 )2 + (Bµ)2 − 4(m2H1 + |µ|2 )(m2H2 + |µ|2 ).
Inequality above implies one of these solutions is negative.
Example: For tan β = ∞ (β = π2 ), parameter relations with v1 and v2 on page 135 imply m2H1 + |µ|2 > 0 and m2H2 + |µ|2 < 0, so electroweak symmetry broken. Alternatively, radiative corrections give
d 2 d ln µr mH2
= x hU33 + . . . (hU33 is top quark Yukawa coupling), x > 0.
(Similarly for stop masses, smaller x.) So although m2H2 > 0 at µr = MX , may have m2H2 < (≪)0 at µr = v.
.
4.5
Sparticle mass eigenstates
139
Consider gauginos and higgsinos. Particles with different SU(2)×U(1) but same U(1)e.m. transformation properties can mix after electroweak symmetry breaking.
Neutralinos: 4 neutral fermionic mass eigenstates χ e0i , i = 1(lightest), ..., 4, mixtures of bino, neutral wino, neutral higgsinos.
Bilinear terms in these fields appearing in Lagrangian can be written − 21 (λ0 )T Mχe0 λ0 , where λ0 = (λbino , λneutralwino , λH10 , λH20 ), Mχe0
mbino 0 −cβ sW mZ sβ sW mZ 0 mwino cβ cW mZ −sβ sW mZ (sβ = sin β etc.). µ dependent part from µ term on page 123. = −cβ sW mZ cβ cW mZ 0 −µ sβ sW mZ −sβ cW mZ −µ 0
mZ dependent part from 1. for Φ = Hi and V = SU(2)×U(1) fields after electroweak symmetry breaking on page 123. mbino , mwino dependent terms from SUSY breaking terms for gauginos from 2. on page 127. Mχe0 symmetric, diagonalized by unitary matrix.
χ e01 can be LSP, and therefore candidate for cold dark matter.
Charginos: 4 charged fermionic mass eigenstates
χ e± i ,
140 i = 1, 2, mixtures of charged winos and charged higgsinos.
Bilinear terms in these fields appearing in Lagrangian can be written − 21 (λ+ )T Mχec λ− , where λ+ = (λ+ charged
wino , λH2+ ),
Define λc = (λ+ , λ− )T , Mχec =
λ− = (λ− charged
0 Mχe†c Mχec 0
wino , λH1− ),
Mχec =
√ m I 2m s wino W β √ . µ I 2mW cβ
, so contribution to Lagrangian including hermitian conjugate
is − 12 (λc )T Mχec λc . Squared mass eigenvalues can be obtained from diagonalization of Mχe2c = gives
m2χe+ 1,2
=
m2χe− 1,2
=
1 2
m2wino
+
2m2W
Q/U 2
!
,
p 4 2 2 2 2 2 2 2 + |µ| ± (mwino − |µ| ) + 4mW cos 2β + 4mW (mwino + |µ| − 2mwino Re(µ sin 2β)) .
Slepton, squark mass matrix: m2squark = where m2L/Rsquark,KM = MKM
Mχe†c Mχec 0 0 Mχec Mχe†c
2
2 m2Lsquark mTLRsquark m2LRsquark m2Rsquark
,
+ (m2quark,K + m2Z (T3 − Q sin2 θW ) cos 2β)δKM (no K sum),
S2 MKM from 1. on page 127, mquark from unbroken SUSY condition msquark,KK = mquark,K (no K sum),
and term proportional to m2Z from Higgs-chiral superfield couplings in superpotential [f (Φ)]F on page 122.
m2LRsquark,KM
=
(AUKM
− δKM mquarkµ tan β)mquark .
2 First term from 3. on page 127, second term from ∂φ∂f 0 . H1
Further reading
Stephen P. Martin, hep-ph/9709356 Very topical (not much basics) — helps you understand what people are talking about
Ian J. R. Aitchison, hep-ph/0505105 Complete derivations up to and including the MSSM
Michael E. Peskin, 0801.1928 [hep-ph] Recent, readable
Steven Weinberg, The Quantum Theory Of Fields III Very complete derivations, quite formal but clear
5
141
Supergravity
Construct coordinates xµ over whole of curved spacetime with metric tensor gµν (x), and locally inertial coordinate system at each point with “flat” metric tensor ηab = diag(1, 1, 1, −1). Vierbein eaµ(x): Defined by gµν (x) = ηabeaµ(x)ebν (x) .
Curved space transformation of vierbein: e′aµ(x′) =
∂xν a ∂x′µ e µ (x) .
Lorentz transformation of vierbein: eaµ(x) → Λab(x)ebµ(x) . Infinitesimal curved space transformation: xµ → xµ + ξ µ(x) . Infinitesimal Lorentz transformation: Λab(x) = δ ab + ω ab(x) . Work in weak field limit, i.e. eaµ(x) ≃ δµa , then a → µ etc.
√ 142 Weak field representation: eµν (x) = ηµν + 2κφµν , i.e. gµν (x) = ηµν + 2κ(φµν + φνµ) , where κ = 8πG .
Particle content of supergravity: spin-2 graviton hµν = φµν + φνµ and spin- 32 gravitino ψµ.
Graviton transformation: φµν (x) → φµν (x) +
1 2κ
h
∂ξµ (x) − ∂x ν
i + ωµν (x) .
Follows from performing infinitessimal curved space and Lorentz transformations on page 141.
Gravitino transformation: ψµ(x) → ψµ(x) + ∂µψ(x) . Required for low energy interactions, similar to requirement of invariance under Aµ (x) → Aµ (x) + ∂ µ α(x) found on page 52.
Goal: Put φµν , ψµ into one superfield, ξµ, ωµν , ψ into another.
143 Graviton, gravitino as functions of supermultiplet components of Metric superfield: Hµ(x, θ) =
CµH (x)
−
I[θγ5]ωµH (x)
−I[(θγ5θ)θ]
H − θγ5θ Mµ (x) − 21 [θθ]NµH (x) + I2 [θγ5γν θ]VµH ν (x)
λH µ (x)
I 2
+
1 / ωµH (x) 2∂
−
1 2 4 [θγ5 θ]
DµH (x)
+
1 2 H 2 ∂ Cµ (x)
This is general form of superfield on page 96, but with extra spacetime index µ.
,
Specifically, H Graviton, gravitino components of metric superfield Hµ: φµν (x) = Vµν (x) − 31 ηµν V Hλ λ (x) , 1 2 ψµ (x)
1 1 ρ H ρ H = λH µ − 3 γµ γ λρ (x) − 3 γµ ∂ ωρ (x) .
Graviton, gravitino transformations on page 142 equivalent to Transformation of metric superfield: Hµ(x, θ) → Hµ(x, θ) + ∆µ(x, θ) , where Transformation superfield: ∆µ(x, θ) =
Cµ∆(x)
−
I[θγ5]ωµ∆(x)
∆ − θγ5θ Mµ (x) − 21 [θθ]Nµ∆(x) + I2 [θγ5γν θ]Vµ∆ ν (x) I 2
−I[(θγ5θ)θ]
λ∆ µ (x)
+
1 / ωµ∆(x) 2∂
−
1 2 4 [θγ5 θ]
Dµ∆(x)
+
1 2 ∆ 2 ∂ Cµ (x)
See later for dependence of these fields on ψ, ξµ, ωµν of graviton, gravitino transformations on page 142.
.
Gravity-matter coupling: Aint = 2κ
R
144
d4x [HµΘµ]D , where
Supercurrent from left-chiral superfields: Θµ =
I 12
† 4Φn∂µΦn − 4Φn∂µΦ†n + (DΦ†n)γµ(DΦn) .
Supercurrent is a superfield, contains conserved current and energy momentum tensor: Supercurrent conservation law: γ µDΘµ = D 23 Im
h
(Φ) M ∂f∂M
i
.
˜ i, where λ ˜ i dimensionless. M here defined as follows: Each coupling constant λi in f is written λi = M dM (λi)λ SUSY current from supercurrent components: S µ = −2ω Θµ + 2γ µγ ν ωνΘ . SUSY current conservation law: ∂µS µ = 0 .
Follows from supercurrent conservation law above.
Θ Θ − 12 Vνµ + ηµν V Θλ Energy-momentum tensor from supercurrent components: Tµν = − 21 Vµν λ , obeys
Energy-momentum tensor conservation law: ∂µT µν = 0 ,
Follows from supercurrent conservation law above.
Relation between energy-momentum tensor and momentum:
R
d3xT 0ν = P ν .
Follows from SUSY transformation of ωµΘ , take time component and integrate over x: I But
R
R
R d3 xS 0 , Q = 2γν d3 xT 0ν .
d3 xS 0 = Q (as in bosonic generator case), then use relation with momentum for any N on page 83.
Further useful results: R µ = 2C Θµ , MµΘ = ∂µM , NµΘ = ∂µN .
145 Constraint on transformation superfield ∆µ: ∆µ = DγµΞ , where Ξ obeys (DD)(DΞ) = 0 . Ensures Aint in gravity-matter coupling on page 144 is invariant under transformation of metric superfield on page 143, follows from supercurrent conservation law on page 144.
Recal transformation parameters ξµ, ωµν , ψ in graviton, gravitino transformations on page 142. Transformation parameter components of transformation superfield: ∆ Vµν
+
∆ Vνµ
=
1 − 2κ
h
∂ξµ ∂xν
+
∂ξν ∂xµ
−
∂ξ λ 2ηµν ∂x λ
i
1 1 1 ρ ∆ ρ ∆ , λ∆ µ − 3 γµ γ λρ − 3 γµ ∂ ωρ = 2 ∂µ ψ .
From constraint on transformation superfield ∆µ above, and graviton, gravitino transformations on page 142. ∆ = D∆σ + ∂ σ ∂ ρCρ∆ , Further constraints on components of transformation superfield: − 12 ǫνµκσ ∂κVνµ
∂Mµ∆ = ∂Nµ∆ = 0 .
Again, from constraint on transformation superfield ∆µ above. H Auxiliary fields: bσ = DHσ + 12 ǫνµκσ ∂κVνµ + ∂ σ ∂ ρCρH , s = ∂ µMµH , p = ∂ µNµH , invariant.
From further constraints on components of transformation superfield above. H H Choose CµH = Vµν = φµν − φνµ = ωνH = 0 . Then hµν = 2φµν . − Vνµ ∆ ∆ Can be done by suitable choice of Cµ∆ , Vµν − Vνµ , ων∆ in transformation of metric superfield on page 143.
146 To summarize, components of superfields are: Θ ∋ T κσ , S σ , Rσ , M , N .
Hµ ∋ hκσ , ψσ , bσ , s, p. R
σ
Gravity-matter coupling in terms of components: Aint = κ d4x[T κσ hκσ + 21 S ψσ + Rσ bσ − 2M s − 2N p] . For dynamic part of gravitational action, use:
Einstein superfield Eµ: CµE = bµ , ωµE = 23 Lµ − 21 γµγ ν Lν , MµE = ∂µs , NµE = ∂µp , ν E E / ωµE , DµE = ∂µ∂ ν bν − ∂ 2bµ , where Lν = Iǫνµκργ5γµ∂κψρ , = − 23 Eµν + 21 ηµν E ρρ + 21 ǫνµσρ∂ σ bρ , λE Vµν µ = ∂µ γ ων − ∂
and linearized Einstein tensor Eµν = =
1 2κ
1 2
∂µ∂ν hλλ
2
λ
λ
+ ∂ hµν − ∂µ∂ hλν − ∂ν ∂ hλµ −
Rµν − 12 g µν R .
ηµν ∂ 2hλλ
λ ρ
+ ηµν ∂ ∂ hλρ
Dynamic part of gravitational Lagrangian: LE = 34 [EµH µ]D = Eµν hµν − 21 ψ µLµ − 43 (s2 + p2 − bµbµ) . Now put together matter Lagrangian LM , and LE and integrand of Aint above and eliminate auxiliary fields: σ Lagrangian of nature: L = LM + Eµν hµν − 21 ψ µLµ + κ[T κσ hκσ + 21 S ψσ ] + 43 κ2 M 2 + N 2 − 14 RµR µ . Everything except LM of order κ2.
Vacuum energy density: ρVAC = −LVAC = −LM
VAC
−
3 2 2 4 κ (M
+ N 2).
Only s and p can acquire vacuum expectation values.
Solution to Einstein field equations (ρVAC uniform) for ρVAC ≷ 0 is de Sitter / anti de Sitter space: spacetime embedded in 5-D space with x25 ± ηµν xµxν = R2 and ds2 = ηµν xµxν ± dx25 for ρVAC ≷ 0. ρVAC < 0 corresponds to O(3,2), which includes N = 1 SUSY, but ρVAC > 0 corresponds to O(4,1), which excludes unbroken N = 1 SUSY.
In unbroken SUSY, LM
VAC
= 0 so from vacuum energy density above, ρVAC < 0.
ρVAC < 0 is unstable. However, anti de Sitter space cannot form since positive energy S1 > |ρVAC| is needed for its surface tension.
147
148 Local supersymmetry: α in definition of superfield on page 94 becomes dependent on x. Only change is to matter action: δ
R
4
R
µ
d xLM = − d4xS (x)∂µα(x) (usual definition of current).
R σ Cancelled by term κ d4x 21 S ψσ in gravity-matter coupling in terms of components on page 146
if SUSY transformation of gravitino modified to: δψµ → δψµ + κ2 ∂µα.
But ψµ → ψµ + κ2 ∂µα is of same form as gravitino transformation on page 142, i.e. leaves term − 12 ψ µLµ in dynamic part of gravitational Lagrangian LE on page 146 unchanged. So supersymmetric gravity is locally supersymmetric.
