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68-505: Originally the fourth color was labelled " l ilac" to alliterate with " l epton". Pati–Salam is an alternative to the Georgi–Glashow SU(5) unification also proposed in 1974. Both can be embedded within an SO(10) unification model . The Pati–Salam model states that the gauge group is either SU(4) × SU(2) L × SU(2) R or (SU(4) × SU(2) L × SU(2) R )/ Z 2 and the fermions form three families, each consisting of

136-703: A C 2 ⊕ C 3 {\displaystyle \mathbb {C} ^{2}\oplus \mathbb {C} ^{3}} splitting restricts SU(5) to S(U(2)×U(3)) , yielding matrices of the form with kernel { ( α , α − 3 I d 2 , α 2 I d 3 ) | α ∈ C , α 6 = 1 } ≅ Z 6 {\displaystyle \{(\alpha ,\alpha ^{-3}\mathrm {Id} _{2},\alpha ^{2}\mathrm {Id} _{3})|\alpha \in \mathbb {C} ,\alpha ^{6}=1\}\cong \mathbb {Z} _{6}} , hence isomorphic to

204-518: A Z 2 {\displaystyle \mathbb {Z} _{2}} matter parity to the chiral superfields with the matter fields having odd parity and the Higgs having even parity to protect the electroweak Higgs from quadratic radiative mass corrections (the hierarchy problem ). In the non-supersymmetric version the action is invariant under a similar Z 2 {\displaystyle \mathbb {Z} _{2}} symmetry because

272-897: A Φ + 3 b Φ 2 = λ 1   , {\displaystyle \ 2a\Phi +3b\Phi ^{2}=\lambda \mathbf {1} \ ,} where λ is a Lagrange multiplier. Up to an SU(5) (unitary) transformation, The three cases are called case I, II, and III and they break the gauge symmetry into   S U ( 5 ) ,   [ S U ( 4 ) × U ( 1 ) ] / Z 4   {\displaystyle \ SU(5),\ \left[SU(4)\times U(1)\right]/\mathbb {Z} _{4}\ } and   [ S U ( 3 ) × S U ( 2 ) × U ( 1 ) ] / Z 6 {\displaystyle \ \left[SU(3)\times SU(2)\times U(1)\right]/\mathbb {Z} _{6}} respectively (the stabilizer of

340-400: A leaves x unchanged is equivalent to saying that a and x commute: Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring). An automorphism of a group G is inner if and only if it extends to every group containing G . By associating the element a ∈ G with

408-455: A left and right inverse , namely φ g − 1 . {\displaystyle \varphi _{g^{-1}}.} Thus, φ g {\displaystyle \varphi _{g}} is both an monomorphism and epimorphism , and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation. When discussing right conjugation,

476-415: A left-handed neutrino , C 0 ⊗ C ⊗ C {\displaystyle \mathbb {C} _{0}\otimes \mathbb {C} \otimes \mathbb {C} } . For the first exterior power ⋀ 1 C 5 ≅ C 5 {\displaystyle {\textstyle \bigwedge }^{1}\mathbb {C} ^{5}\cong \mathbb {C} ^{5}} ,

544-580: A prediction of the Georgi–Glashow model. The SM gauge fields can be embedded explicitly as well. For that we recall a gauge field transforms as an adjoint, and thus can be written as A μ a T a {\displaystyle A_{\mu }^{a}T^{a}} with T a {\displaystyle T^{a}} the S U ( 5 ) {\displaystyle SU(5)} generators. Now, if we restrict ourselves to generators with non-zero entries only in

612-428: A right action of G on itself. A common example is as follows: Describe a homomorphism Φ {\displaystyle \Phi } for which the image, Im ( Φ ) {\displaystyle {\text{Im}}(\Phi )} , is a normal subgroup of inner automorphisms of a group G {\displaystyle G} ; alternatively, describe a natural homomorphism of which

680-646: A triplet in SU(3), a singlet in SU(2), and under the Y = − ⁠ 1 / 3 ⁠ representation of U(1) (as α = α ); this matches a right-handed down quark , C − 1 3 ⊗ C ⊗ C 3 {\displaystyle \mathbb {C} _{-{\frac {1}{3}}}\otimes \mathbb {C} \otimes \mathbb {C} ^{3}} . The second power ⋀ 2 C 5 {\displaystyle {\textstyle \bigwedge }^{2}\mathbb {C} ^{5}}

748-401: A unique non-trivial class of non-inner automorphisms, and when n = 2 , the symmetric group, despite having no non-inner automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete. If the inner automorphism group of a perfect group G is simple, then G is called quasisimple . An automorphism of a Lie algebra 𝔊 is called an inner automorphism if it

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816-401: Is abelian . The group Inn( G ) is cyclic only when it is trivial. At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called complete . This is the case for all of the symmetric groups on n elements when n is not 2 or 6. When n = 6 , the symmetric group has

884-429: Is a normal subgroup of the full automorphism group Aut( G ) of G . The outer automorphism group , Out( G ) is the quotient group The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out( G ) , but different non-inner automorphisms may yield the same element of Out( G ) . Saying that conjugation of x by

952-432: Is a finite non-abelian p -group , then G has an automorphism of p -power order which is not inner. It is an open problem whether every non-abelian p -group G has an automorphism of order p . The latter question has positive answer whenever G has one of the following conditions: The inner automorphism group of a group G , Inn( G ) , is trivial (i.e., consists only of the identity element ) if and only if G

1020-436: Is a homomorphism, (2) φ g {\displaystyle \varphi _{g}} is also a bijection, (3) Φ {\displaystyle \Phi } is a homomorphism. The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn( G ) . Inn( G )

1088-394: Is a linear combination of the following terms: The first column is an Abbreviation of the second column (neglecting proper normalization factors), where capital indices are SU(5) indices, and i and j are the generation indices. The last two rows presupposes the multiplicity of   N c   {\displaystyle \ \mathrm {N} ^{\mathsf {c}}\ }

1156-527: Is a ring, and g is a unit ), then the function is called (right) conjugation by g (see also conjugacy class ). This function is an endomorphism of G : for all x 1 , x 2 ∈ G , {\displaystyle x_{1},x_{2}\in G,} where the second equality is given by the insertion of the identity between x 1 {\displaystyle x_{1}} and x 2 . {\displaystyle x_{2}.} Furthermore, it has

1224-517: Is achieved in the Georgi–Glashow model via a fundamental 5 {\displaystyle \mathbf {5} } which contains the SM Higgs, with H + {\displaystyle H^{+}} and H 0 {\displaystyle H^{0}} the charged and neutral components of the SM Higgs, respectively. Note that the T i {\displaystyle T_{i}} are not SM particles and are thus

1292-467: Is allowed by the new vector bosons introduced from the adjoint representation of SU(5) which also contains the gauge bosons of the Standard Model forces. Since these new gauge bosons are in (3,2) −5/6 bifundamental representations , they violated baryon and lepton number. As a result, the new operators should cause protons to decay at a rate inversely proportional to their masses. This process

1360-956: Is always a natural homomorphism Φ : G → Aut ( G ) {\displaystyle \Phi :G\to {\text{Aut}}(G)} , which associates to every g ∈ G {\displaystyle g\in G} an (inner) automorphism φ g {\displaystyle \varphi _{g}} in Aut ( G ) {\displaystyle {\text{Aut}}(G)} . Put identically, Φ : g ↦ φ g {\displaystyle \Phi :g\mapsto \varphi _{g}} . Let φ g ( x ) := g x g − 1 {\displaystyle \varphi _{g}(x):=gxg^{-1}} as defined above. This requires demonstrating that (1) φ g {\displaystyle \varphi _{g}}

1428-558: Is called dimension 6 proton decay and is an issue for the model, since the proton is experimentally determined to have a lifetime greater than the age of the universe. This means that an SU(5) model is severely constrained by this process. As well as these new gauge bosons, in SU(5) models, the Higgs field is usually embedded in a 5 representation of the GUT group. The caveat of this is that since

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1496-512: Is embedded within an even larger GUT group). As mentioned above, both the Pati–Salam and Georgi–Glashow SU(5) unification models can be embedded in a SO(10) unification . The difference between the two models then lies in the way that the SO(10) symmetry is broken, generating different particles that may or may not be important at low scales and accessible by current experiments. If we look at

1564-844: Is indeed equal to S U ( 3 ) × S U ( 2 ) × U ( 1 ) {\displaystyle SU(3)\times SU(2)\times U(1)} by noting that [ ⟨ 24 H ⟩ , G μ ] = [ ⟨ 24 H ⟩ , W μ ] = [ ⟨ 24 H ⟩ , B μ ] = 0 {\displaystyle [\langle \mathbf {24} _{H}\rangle ,G_{\mu }]=[\langle \mathbf {24} _{H}\rangle ,W_{\mu }]=[\langle \mathbf {24} _{H}\rangle ,B_{\mu }]=0} . Computation of similar commutators further shows that all other S U ( 5 ) {\displaystyle SU(5)} gauge fields acquire masses. To be precise,

1632-681: Is not zero (i.e. that a sterile neutrino exists). The coupling   H u   10 i   10 j   {\displaystyle \ \mathrm {H} _{\mathsf {u}}\ \mathbf {10} _{i}\ \mathbf {10} _{j}\ } has coefficients which are symmetric in i and j . The coupling   N i c   N j c   {\displaystyle \ \mathrm {N} _{i}^{\mathsf {c}}\ \mathrm {N} _{j}^{\mathsf {c}}\ } has coefficients which are symmetric in i and j . The number of sterile neutrino generations need not be three, unless

1700-452: Is obtained via the formula ⋀ 2 ( V ⊕ W ) = ⋀ 2 V 2 ⊕ ( V ⊗ W ) ⊕ ⋀ 2 W 2 {\displaystyle {\textstyle \bigwedge }^{2}(V\oplus W)={\textstyle \bigwedge }^{2}V^{2}\oplus (V\otimes W)\oplus {\textstyle \bigwedge }^{2}W^{2}} . As SU(5) preserves

1768-446: Is of the form Ad g , where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is 𝔊 . The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra. If G is the group of units of a ring , A , then an inner automorphism on G can be extended to

1836-753: Is the Higgs fields 5 H and   5 ¯ H   {\displaystyle \ {\overline {\mathbf {5} }}_{\mathrm {H} }\ } which are interesting. The two relevant superpotential terms here are   5 H   5 ¯ H   {\displaystyle \ 5_{\mathrm {H} }\ {\bar {5}}_{\mathrm {H} }\ } and   ⟨ 24 ⟩ 5 H   5 ¯ H   . {\displaystyle \ \langle 24\rangle 5_{\mathrm {H} }\ {\bar {5}}_{\mathrm {H} }~.} Unless there happens to be some fine tuning , we would expect both

1904-562: The U ( 1 ) {\displaystyle U(1)} hypercharge (up to some normalization N {\displaystyle N} .) Using the embedding, we can explicitly check that the fermionic fields transform as they should. This explicit embedding can be found in Ref. or in the original paper by Georgi and Glashow. SU(5) breaking occurs when a scalar field (Which we will denote as 24 H {\displaystyle \mathbf {24} _{H}} ), analogous to

1972-417: The Higgs field and transforming in the adjoint of SU(5), acquires a vacuum expectation value (vev) proportional to the weak hypercharge generator When this occurs, SU(5) is spontaneously broken to the subgroup of SU(5) commuting with the group generated by Y . Using the embedding from the previous section, we can explicitly check that S U ( 5 ) {\displaystyle SU(5)}

2040-449: The SU(4) × SU(2) L × SU(2) R gauge symmetry As complex representations: A generic invariant renormalizable superpotential is a (complex) SU(4) × SU(2) L × SU(2) R and U(1) R invariant cubic polynomial in the superfields. It is a linear combination of the following terms: i {\displaystyle i} and j {\displaystyle j} are

2108-550: The Standard Model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5) . The unified group SU(5) is then thought to be spontaneously broken into the Standard Model subgroup below a very high energy scale called the grand unification scale . Since the Georgi–Glashow model combines leptons and quarks into single irreducible representations , there exist interactions which do not conserve baryon number, although they still conserve

Pati–Salam model - Misplaced Pages Continue

2176-444: The Standard Model 's true gauge group S U ( 3 ) × S U ( 2 ) × U ( 1 ) / Z 6 {\displaystyle SU(3)\times SU(2)\times U(1)/\mathbb {Z} _{6}} . For the zeroth power ⋀ 0 C 5 {\displaystyle {\textstyle \bigwedge }^{0}\mathbb {C} ^{5}} , this acts trivially to match

2244-697: The Y = ⁠ 1 / 2 ⁠ representation of U(1) (as weak hypercharge is conventionally normalized as α = α ); this matches a right-handed anti- lepton , C 1 2 ⊗ C 2 ∗ ⊗ C {\displaystyle \mathbb {C} _{\frac {1}{2}}\otimes \mathbb {C} ^{2*}\otimes \mathbb {C} } (as C 2 ≅ C 2 ∗ {\displaystyle \mathbb {C} ^{2}\cong \mathbb {C} ^{2*}} in SU(2)). The C 3 {\displaystyle \mathbb {C} ^{3}} transforms as

2312-423: The conjugation action of a fixed element, called the conjugating element . They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group . If G is a group and g is an element of G (alternatively, if G

2380-399: The representations ( 4 , 2 , 1 ) and ( 4 , 1 , 2 ) . This needs some explanation. The center of SU(4) × SU(2) L × SU(2) R is Z 4 × Z 2L × Z 2R . The Z 2 in the quotient refers to the two element subgroup generated by the element of the center corresponding to the two element of Z 4 and the 1 elements of Z 2L and Z 2R . This includes

2448-404: The representations things like ( 4 , 1 , 2 ) and ( 6 , 1 , 1 ) is purely a physicist's convention(source?), not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, it is standard among GUT theorists. The weak hypercharge , Y, is the sum of the two matrices: It is possible to extend

2516-470: The weak interaction . In this extended model, ( 4 , 2 , 1 ) ⊕ ( 4 , 1 , 2 ) is an irrep and so is ( 4 , 1 , 2 ) ⊕ ( 4 , 2 , 1 ) . This is the simplest extension of the minimal left-right model unifying QCD with B−L . Since the homotopy group this model predicts monopoles . See 't Hooft–Polyakov monopole . This model was invented by Jogesh Pati and Abdus Salam . This model doesn't predict gauge mediated proton decay (unless it

2584-439: The (unextended) Pati–Salam group which is the composition of an involutive outer automorphism of SU(4) which isn't an inner automorphism with interchanging the left and right copies of SU(2) . This explains the name left and right and is one of the main motivations for originally studying this model. This extra " left-right symmetry " restores the concept of parity which had been shown not to hold at low energy scales for

2652-460: The Georgi–Glashow model. The fermion sector is then composed of an anti fundamental 5 ¯ {\displaystyle {\overline {\mathbf {5} }}} and an antisymmetric 10 {\displaystyle \mathbf {10} } . In terms of SM degrees of freedoms, this can be written as and with d i {\displaystyle d_{i}} and u i {\displaystyle u_{i}}

2720-434: The Higgs field is an SU(2) doublet, the remaining part, an SU(3) triplet, must be some new field - usually called D or T. This new scalar would be able to generate proton decay as well and, assuming the most basic Higgs vacuum alignment, would be massless so allowing the process at very high rates. While not an issue in the Georgi–Glashow model, a supersymmeterised SU(5) model would have additional proton decay operators due to

2788-551: The Pati–Salam group so that it has two connected components . The relevant group is now the semidirect product ( [ S U ( 4 ) × S U ( 2 ) L × S U ( 2 ) R ] / Z 2 ) ⋊ Z 2 {\displaystyle \left([SU(4)\times SU(2)_{L}\times SU(2)_{R}]/\mathbf {Z} _{2}\right)\rtimes \mathbf {Z} _{2}} . The last Z 2 also needs explaining. It corresponds to an automorphism of

Pati–Salam model - Misplaced Pages Continue

2856-538: The SU(5) is embedded in a higher unification scheme such as SO(10) . The vacua correspond to the mutual zeros of the F and D terms. Let's first look at the case where the VEVs of all the chiral fields are zero except for Φ . The F zeros corresponds to finding the stationary points of W subject to the traceless constraint   T r [ Φ ] = 0   . {\displaystyle \ Tr[\Phi ]=0~.} So,   2

2924-502: The Standard Model's group action preserves the splitting C 5 ≅ C 2 ⊕ C 3 {\displaystyle \mathbb {C} ^{5}\cong \mathbb {C} ^{2}\oplus \mathbb {C} ^{3}} . The C 2 {\displaystyle \mathbb {C} ^{2}} transforms trivially in SU(3) , as a doublet in SU(2) , and under

2992-588: The Standard Model's representation F ⊕ F* of one generation of fermions and antifermions lies within ∧ C 5 {\displaystyle \wedge \mathbb {C} ^{5}} . Similar motivations apply to the Pati–Salam model, and to SO(10) , E6, and other supergroups of SU(5). Owing to its relatively simple gauge group S U ( 5 ) {\displaystyle SU(5)} , GUTs can be written in terms of vectors and matrices which allows for an intuitive understanding of

3060-520: The TeV scale). The gauge algebra 24 decomposes as This 24 is a real representation, so the last two terms need explanation. Both ( 3 , 2 ) − 5 6 {\displaystyle (3,2)_{-{\frac {5}{6}}}} and   ( 3 ¯ , 2 ) 5 6   {\displaystyle \ ({\bar {3}},2)_{\frac {5}{6}}\ } are complex representations. However,

3128-530: The VEV). In other words, there are at least three different superselection sections, which is typical for supersymmetric theories. Only case III makes any phenomenological sense and so, we will focus on this case from now onwards. It can be verified that this solution together with zero VEVs for all the other chiral multiplets is a zero of the F-terms and D-terms . The matter parity remains unbroken (right up to

3196-536: The adjoint Higgs to be absorbed. The other real half acquires a mass coming from the D-terms . And the other three components of the adjoint Higgs,   ( 8 , 1 ) 0 , ( 1 , 3 ) 0   {\displaystyle \ (8,1)_{0},(1,3)_{0}\ } and   ( 1 , 1 ) 0   {\displaystyle \ (1,1)_{0}\ } acquire GUT scale masses coming from self pairings of

3264-473: The canonical volume form of C 5 {\displaystyle \mathbb {C} ^{5}} , Hodge duals give the upper three powers by ⋀ p C 5 ≅ ( ⋀ 5 − p C 5 ) ∗ {\displaystyle {\textstyle \bigwedge }^{p}\mathbb {C} ^{5}\cong ({\textstyle \bigwedge }^{5-p}\mathbb {C} ^{5})^{*}} . Thus

3332-445: The color charge in the SU(4) C group, while the other part of the weak hypercharge is in the SU(2) R . When those two groups break then the two parts together eventually unify into the usual weak hypercharge U(1) Y . The N = 1 superspace extension of 3 + 1 Minkowski spacetime N=1 SUSY over 3 + 1 Minkowski spacetime with R-symmetry (SU(4) × SU(2) L × SU(2) R )/ Z 2 U(1) A Those associated with

3400-675: The direct sum of both representation decomposes into two irreducible real representations and we only take half of the direct sum, i.e. one of the two real irreducible copies. The first three components are left unbroken. The adjoint Higgs also has a similar decomposition, except that it is complex. The Higgs mechanism causes one real HALF of the   ( 3 , 2 ) − 5 6   {\displaystyle \ (3,2)_{-{\frac {5}{6}}}\ } and   ( 3 ¯ , 2 ) 5 6   {\displaystyle \ ({\bar {3}},2)_{\frac {5}{6}}\ } of

3468-529: The evidence for neutrino oscillations , unless a way is found to introduce an infinitesimal Majorana coupling for the left-handed neutrinos. Since the homotopy group is this model predicts 't Hooft–Polyakov monopoles . Because the electromagnetic charge Q is a linear combination of some SU(2) generator with ⁠ Y / 2 ⁠ , these monopoles also have quantized magnetic charges Y , where by magnetic , here we mean magnetic electromagnetic charges. The minimal supersymmetric SU(5) model assigns

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3536-667: The expression g − 1 x g {\displaystyle g^{-1}xg} is often denoted exponentially by x g . {\displaystyle x^{g}.} This notation is used because composition of conjugations satisfies the identity: ( x g 1 ) g 2 = x g 1 g 2 {\displaystyle \left(x^{g_{1}}\right)^{g_{2}}=x^{g_{1}g_{2}}} for all g 1 , g 2 ∈ G . {\displaystyle g_{1},g_{2}\in G.} This shows that right conjugation gives

3604-455: The generation indices. We can extend this model to include left-right symmetry . For that, we need the additional chiral multiplets ( 4 , 2 , 1 ) H and ( 4 , 2 , 1 ) H . Georgi%E2%80%93Glashow model In particle physics , the Georgi–Glashow model is a particular Grand Unified Theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model,

3672-474: The individual models, the most important difference is in the origin of the weak hypercharge . In the SU(5) model by itself there is no left-right symmetry (although there could be one in a larger unification in which the model is embedded), and the weak hypercharge is treated separately from the color charge. In the Pati–Salam model, part of the weak hypercharge (often called U(1) B-L ) starts being unified with

3740-528: The inner automorphism f ( x ) = x in Inn( G ) as above, one obtains an isomorphism between the quotient group G / Z( G ) (where Z( G ) is the center of G ) and the inner automorphism group: This is a consequence of the first isomorphism theorem , because Z( G ) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing). A result of Wolfgang Gaschütz says that if G

3808-432: The kernel of Φ {\displaystyle \Phi } is the center of G {\displaystyle G} (all g ∈ G {\displaystyle g\in G} for which conjugating by them returns the trivial automorphism), in other words, Ker ( Φ ) = Z ( G ) {\displaystyle {\text{Ker}}(\Phi )={\text{Z}}(G)} . There

3876-423: The left and right-handed electron, respectively. In addition to the fermions, we need to break S U ( 3 ) × S U L ( 2 ) × U Y ( 1 ) → S U ( 3 ) × U E M ( 1 ) {\displaystyle SU(3)\times SU_{L}(2)\times U_{Y}(1)\rightarrow SU(3)\times U_{EM}(1)} ; this

3944-404: The left-handed up and down type quark, d i c {\displaystyle d_{i}^{c}} and u i c {\displaystyle u_{i}^{c}} their righthanded counterparts, ν {\displaystyle \nu } the neutrino, e {\displaystyle e} and e R {\displaystyle e_{R}}

4012-399: The matter fields are all fermionic and thus must appear in the action in pairs, while the Higgs fields are bosonic . As complex representations: A generic invariant renormalizable superpotential is a (complex) S U ( 5 ) × Z 2 {\displaystyle SU(5)\times \mathbb {Z} _{2}} invariant cubic polynomial in the superfields. It

4080-673: The model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes, particularly SO(10) in basic and SUSY variants. (For a more elementary introduction to how the representation theory of Lie algebras are related to particle physics, see the article Particle physics and representation theory .) Also, this model suffers from the doublet–triplet splitting problem . SU(5) acts on C 5 {\displaystyle \mathbb {C} ^{5}} and hence on its exterior algebra ∧ C 5 {\displaystyle \wedge \mathbb {C} ^{5}} . Choosing

4148-409: The quantum number B – L associated with the symmetry of the common representation. This yields a mechanism for proton decay , and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. Nevertheless, the elegance of

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4216-481: The right-handed neutrino. See neutrino oscillations . There is also a ( 4 , 1 , 2 ) and/or a ( 4 , 1 , 2 ) scalar field called the Higgs field which acquires a non-zero VEV . This results in a spontaneous symmetry breaking from SU(4) × SU(2) L × SU(2) R to (SU(3) × SU(2) × U(1) Y )/ Z 3 or from (SU(4) × SU(2) L × SU(2) R )/ Z 2 to (SU(3) × SU(2) × U(1) Y )/ Z 6 and also, See restricted representation . Of course, calling

4284-404: The superpartners of the Standard Model fermions. The lack of detection of proton decay (in any form) brings into question the veracity of SU(5) GUTs of all types; however, while the models are highly constrained by this result, they are not in general ruled out. Inner automorphism In abstract algebra an inner automorphism is an automorphism of a group , ring , or algebra given by

4352-554: The superpotential,   a Φ 2 + b < Φ > Φ 2   . {\displaystyle \ a\Phi ^{2}+b<\Phi >\Phi ^{2}~.} The sterile neutrinos, if any exist, would also acquire a GUT scale Majorana mass coming from the superpotential coupling ν  . Because of matter parity, the matter representations   5 ¯   {\displaystyle \ {\overline {\mathbf {5} }}\ } and 10 remain chiral. It

4420-412: The triplet terms and the doublet terms to pair up, leaving us with no light electroweak doublets. This is in complete disagreement with phenomenology. See doublet-triplet splitting problem for more details. Unification of the Standard Model via an SU(5) group has significant phenomenological implications. Most notable of these is proton decay which is present in SU(5) with and without supersymmetry. This

4488-586: The unbroken subgroup is actually Under this unbroken subgroup, the adjoint 24 transforms as to yield the gauge bosons of the Standard Model plus the new X and Y bosons . See restricted representation . The Standard Model's quarks and leptons fit neatly into representations of SU(5). Specifically, the left-handed fermions combine into 3 generations of   5 ¯ ⊕ 10 ⊕ 1   . {\displaystyle \ {\overline {\mathbf {5} }}\oplus \mathbf {10} \oplus \mathbf {1} ~.} Under

4556-485: The unbroken subgroup these transform as to yield precisely the left-handed fermionic content of the Standard Model where every generation d , u , e , and ν correspond to anti- down-type quark , anti- up-type quark , anti- down-type lepton , and anti- up-type lepton , respectively. Also, q and ℓ {\displaystyle \ell } correspond to quark and lepton. Fermions transforming as 1 under SU(5) are now thought to be necessary because of

4624-426: The upper 3 × 3 {\displaystyle 3\times 3} block, in the lower 2 × 2 {\displaystyle 2\times 2} block, or on the diagonal, we can identify with the S U ( 3 ) {\displaystyle SU(3)} colour gauge fields, with the weak S U ( 2 ) {\displaystyle SU(2)} fields, and with

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