SUSY gauge theories SUSY QCD Consider a SUSY SU ( N ) with F flavors - PowerPoint PPT Presentation

SUSY gauge theories

SUSY QCD Consider a SUSY SU ( N ) with F “flavors” of “quarks” and squarks Q i = ( φ i , Q i , F i ) , i = 1 , . . . , F , where φ is the squark and Q is the quark. Q i = ( φ i , Q i , F i ) , in the antifundamental representation. Note the the bar ( ) is part of the name not a conjugation, the conjugate fields are ∗ † † Q † i , Q † i = ( φ ∗ i , F ∗ i , F ∗ i ) , Q i = ( φ i , Q i ) .

SUSY QCD matter content is: SU ( N ) SU ( F ) SU ( F ) U (1) B U (1) R F − N Q 1 1 F F − N Q − 1 1 F W = 0

R -charge [ R, Q α ] = − Q α . chiral supermultiplet: R ψ = R φ − 1 , normalize the R -charge by Rλ a = λ a , R -charge of the gluino is 1, and the R -charge of the gluon is 0.

Group Theory: Bird Tracks Identify the group generator with a vertex as in Fig. ?? . a m ) (T = r n n m Figure 1: Bird-track notation for the group generator T a .

Bird Tracks quadratic Casimir C 2 ( r ) and the index T ( r ) of the representation r , ( T a r ) m l ( T a r ) l n = C 2 ( r ) δ m n , m = T ( r ) δ ab , ( T a r ) m n ( T b r ) n are given diagrammaticly as m C (r) = δ 2 n n m ab T(r) = δ b a Figure 2:

Bird Tracks Contracting the external legs: In the first diagram setting m equal = Figure 3: to n and summing over n yields a factor of d ( r ). In the second diagram setting a equal to b and summing yields a factor d ( Ad ). d ( r ) C 2 ( r ) = d ( Ad ) T ( r ) .

Casimirs N 2 − 1 d ( ) = N, d ( Ad ) = = 1 T ( ) 2 , T ( Ad ) = N so C 2 ( ) = N 2 − 1 2 N , C 2 ( Ad ) = N .

Sum over Generators For the fundamental representation : n = 1 p − 1 ( T a ) l p ( T a ) m 2 ( δ l n δ m N δ l p δ m n ) . 1 1 = − 2Ν 2 Figure 4: We can reduce the sums over multiple generators to an essentially topo- logical exercise

Anomalies Since we can define an R -charge by taking arbitrary linear combina- tions of the U (1) R and U (1) B charges we can choose Q i and Q i to have the same R -charge. For a U (1) not be to broken by instanton effects the SU ( N ) 2 U (1) R anomaly diagram vanishes Figure 5: fermion contributes its R -charge times T ( r ). Sum over gluino, quarks: 1 · T ( Ad ) + ( R − 1) T ( ) 2 F = 0 , so R = F − N F

Renormalization group √ 2 g , λ = g 2 . For SUSY to be a consistent tree-level SUSY: Y = quantum symmetry these relations must be preserved under RG running. the β function for the gauge coupling at one-loop is � 11 ≡ − g 3 b dµ = − g 3 µ dg 3 T (Ad) − 2 3 T ( F ) − 1 � β g = 3 T ( S ) 16 π 2 , 16 π 2 For SUSY QCD: b = (3 N − F )

Renormalization group the β function for the Yukawa coupling is : � � 2 ( F ) Y j + Y j Y 2 ( F ) (4 π ) 2 β j Y † 1 + 2 Y k Y j † Y k = Y 2 + Y k Tr Y k † Y j − 3 g 2 m { C m 2 ( F ) , Y j } , where Y 2 ( F ) ≡ Y j † Y j 2 ( F ) Y j represents the scalar loop corrections to the fermion legs Y † 2 Y k Y j † Y k contains the 1PI vertex corrections Y k Tr Y k † Y j represents fermion loop corrections to the scalar leg C m 2 ( F ) is the quadratic Casimir of the fermion fields in the m th gauge group, and represents gauge loop corrections to the fermion legs

SUSY QCD RG For SUSY QCD the Yukawa coupling of quark i with color index m , gluino a , and antisquark j with color index n is given by √ Y jn m δ j 2 g ( T a ) n im,a = i . φ = Y (Q) 2 Q Q λ φ = Y ( ) λ 2 Q λ λ λ = Tr Y Y φ φ Q Figure 6: Feynman diagrams and associated bird-track diagrams. Y 2 ( Q ) = 2 g 2 C 2 ( ) , Y 2 ( λ ) = 2 g 2 2 F T ( )

SUSY QCD RG no scalar corrections corresponding to Y k Y † j Y k . As for the fermion loop correction it always has a quark (antisquark) and gluino for the internal lines so we have Y k Tr Y k † Y j = Y kq im,a ( Y kq fp,b ) † Y jn m δ j fp,b = 2 g 2 C 2 ( )( T a ) n i , gauge loop corrections are 2 ( F ) , Y j } = ( C 2 ( ) + C 2 ( Ad )) Y j . { C m all the terms in β j Y proportional to C 2 ( ) cancel: √ (4 π ) 2 β j 2 g 3 ( C 2 ( ) + F + 2 C 2 ( ) − 3 C 2 ( ) − 3 N ) = √ Y 2 g 3 (3 N − F ) = − √ 2(4 π ) 2 β g = so the relation between the Yukawa and gauge couplings is preserved under RG running

SUSY QCD Quartic RG SUSY also requires the D -term quartic coupling λ = g 2 . The auxiliary D a field is given by in ( T a ) m ∗ D a = g ( φ ∗ in ( T a ) m n φ mi − φ n φ mi ) and the D -term potential is V = 1 2 D a D a The β function for a quartic scalar coupling at one-loop is (4 π ) 2 β λ = Λ (2) − 4 H + 3 A + Λ Y − 3Λ S , Λ (2) corresponds to the 1PI contribution from the quartic interactions H corresponds to the fermion box graphs A to the two gauge boson exchange graphs Λ Y to the Yukawa leg corrections Λ S corresponds to the gauge leg corrections

SUSY QCD Quartic RG (a) (b) (d) (c) (e) + Figure 7:

SUSY QCD Quartic RG 1 1 1 1 − 4 Ν − 4 Ν + 4 Ν 2 = 4 2 2 Ν − 2 Ν − 1 = + 4 Ν 2 2 Ν Figure 8: The bird-track diagram for the sum over four generators quickly reduces to the sum over two generators and a product of identity matrices.

SUSY QCD Quartic RG in ( T a ) m ∗ jq ( T a ) p ∗ ( φ ∗ in ( T a ) m mi )( φ ∗ jq ( T a ) p n φ mi − φ n φ q φ pj − φ q φ pj ) , (with flavor indices i � = j , the case i = j is left as an exercise) we have 2 F + N − 6 1 Λ (2) � � ( T a ) m n ( T a ) p � � δ m n δ p = q + 1 − q , N 2 N N − 2 1 � � ( T a ) m n ( T a ) p � � δ m n δ p − 4 H = − 8 q − 4 1 − q , N 2 N N − 4 1 � � ( T a ) m n ( T a ) p � � δ m n δ p 3 A = 3 q + 3 1 − q , N 2 N N − 1 Λ Y � � ( T a ) m n ( T a ) p = 4 q , N N − 1 − 3Λ S � � ( T a ) m n ( T a ) p = − 6 q . N individual diagrams that renormalize the gauge invariant, SUSY break- ing, operator ( φ ∗ mi φ mi )( φ ∗ pj φ pj ) but the full β function for this operator vanishes and the D -term β function satisfies β g 2 T a T a , β λ = β g 2 = 2 gβ g . So SUSY is not anomalous at one-loop, and the β functions preserve the relations between couplings at all scales.

SUSY RG Figure 9: The couplings remain equal as we run below the SUSY thresh- old M , but split apart below the non-SUSY threshold m . If we had added dimension 4 SUSY breaking terms to the theory then the couplings would have run differently at all scales

one-loop squark mass Figure 10: The squark loop correction to the squark mass. d 4 k − ig 2 ( T a ) l i n ( T a ) m � Σ squark (0) = l (2 π ) 4 k 2 � Λ 2 dk 2 . − ig 2 16 π 2 C 2 ( ) δ m = n 0

one-loop squark mass Figure 11: The quark–gluino loop correction to the squark mass. √ d 4 k 2 g ) 2 ( T a ) l n ( T a ) m � (2 π ) 4 Tr ik · σ ik · σ Σ quark − gluino (0) = ( − i l ( − 1) k 2 k 2 (2 π ) 4 2 k 2 d 4 k − 2 g 2 C 2 ( ) δ m � = n k 4 � Λ 2 4 ig 2 dk 2 . 16 π 2 C 2 ( ) δ m = n 0

one-loop squark mass (b) (a) Figure 12: (a) The squark–gluon loop and (b) the gluon loop. kµkν ( g µν +( ξ − 1) ) d 4 k ( ig ) 2 ( T a ) l i n ( T a ) m � k 2 k µ ( − i ) k ν k 2 Σ quark − gluino (0) = l (2 π ) 4 k 2 � Λ 2 ξig 2 dk 2 , 16 π 2 C 2 ( ) δ m = n 0 kµkν ( g µν +( ξ − 1) ) d 4 k 1 2 ig 2 � ( T a ) l n , ( T b ) m � δ ab g µν � i k 2 Σ gluon (0) = k 2 ( − i ) l (2 π ) 4 k 2 � Λ 2 − (3+ ξ ) ig 2 dk 2 . C 2 ( ) δ m = n 16 π 2 0

one-loop squark mass Adding all the terms together we have � Λ 2 dk 2 = 0 . ig 2 16 π 2 C 2 ( ) δ m Σ(0) = ( − 1 + 4 + ξ − (3 + ξ )) n 0 The quadratic divergence in the squark mass cancels! In fact for a massless squark all the mass corrections cancel. This means that in a SUSY theory with a Higgs the Higgs mass is protected from quadratic divergences from gauge interactions as well as from Yukawa interactions

Flat directions F < N D a = g ( φ ∗ in ( T a ) m in ( T a ) m ∗ n φ mi − φ n φ mi ) and the scalar potential is: V = 1 2 D a D a define d n m ≡ � φ ∗ in φ mi � n in φ ∗ d m = � φ mi � maximal rank F . In a SUSY vacua: n D a = T am ( d n m − d m ) = 0 n Since T a is a complete basis for traceless matrices, we must therefore have that the difference of the two matrices is proportional to the identity matrix: n d n m − d m = αI

Flat directions F < N d n m can be diagonalized by an SU ( N ) gauge transformation U † d U In this diagonal basis there will be at least N − F zero eigenvalues v 2   1 v 2 2     ...      v 2  d =   F   0     ...     0 n where v 2 i ≥ 0. In this basis d m must also be diagonal, and it must also have N − F zero eigenvalues. This tells us that α = 0, and hence that n m = d n d m

Flat directions F < N n d n m and d m are invariant under SU ( F ) × SU ( F ) transformations since φ mi V i φ mi → j , V ∗ j d n i � φ ∗ in �� φ mi � V i → j , m � φ ∗ jn φ mj � = d n → m . Thus, up to a flavor transformation, we can write   v 1 ...       v F ∗ � = � φ � =   � φ .   0 . . . 0    . .  . .   . .   0 . . . 0 D -term potential has flat directions, as we change the VEVs, we move between different vacua with different particle spectra, generically SU ( N − F ) gauge symmetry