BRANE INDUCED SUSY BREAKING AND DE SITTER SUPERGRAVITY
Recent burst of interest in the study of spontaneous breaking of local supersymmetry and the emergence of positive cosmological constant. Explore the Volkov-Akulov model for SUSY breaking and its application in cosmology and inflation.
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Dmitri Sorokin, INFN Padova Section based on arXiv:1511.03024 with I. Bandos, L. Martucci and M. Tonin BRANE INDUCED SUSY BREAKING AND DE SITTER SUPERGRAVITY VII Round Table Italy-Russia, Dubna, 25-27 November 2015
MOTIVATION Recent (2014-15) burst of interest in the study of spontaneous breaking of local supersymmetry and the emergence of positive cosmological constant due to Volkov- Akulov goldstini in D=4 sugra. In applications to cosmology and inflation (S. Ferrara talk). One of the aims: provide a solid 4D effective field theory ground for cosmological models constructed in the framework of 10D string theory (e.g. KKLT-like models involving anti- D3-branes to generate dS vacua) Most of the constructions of D=4 sugra with spontaneously broken susy use constrained (nilpotent) superfield description of the Volkov-Akulov goldstini (scalar partners are composed of goldstino bilinears) Antoniadis, Dudas, Ferrara, Kehagias, Farakos, Kallosh, Linde, Porrati, Sagnotti, dall Agata, Zwirner; Bergshoeff, Freedman, Kallosh, Van Proeyen; Hasegawa,Yamada;Kuzenko; Antoniadis, Markou; Earlier related studies of local susy breaking & super-Braut-Englert-Higgs effect in sugra: Volkov & Soroka 73, Deser & Zumino 77, , Lindstrom & Rocek 79, Samuel & Wess 83, Ivanov & Kapustnikov 84 - 90, Our goal: to generate susy breaking by coupling supergravity to the original VA model which is the most straightforwardly related to the 3-brane constructions 2
ORIGINAL VOLKOV-AKULOV MODEL 72 AS A 3-BRANE (Hughes & Polchinski 86, Kallosh 98) Goldstino as the manifestation of susy breaking = + + ( ) ' ( ) ( ) nonlinear terms x x x = = = M m ( , , ) 3 , 2 , 1 , 0 , 1,2 z x m Volkov-Akulov superspace = = = Pauli m m m m , , , matrices x i i SUSY transform. = i d + id m 0 m m m d SUSY invariant VA 1-form: M Consider a 4d worldvolume (of a 3-brane) placed in superspace ) 4 , 4 ( spinor field appears on the brane worldvolume 1 1 ( ) ( ), ( ) ( ) x f x x f x ( , ) = = non-linear susy transform. 1 1 m m m ( ) ( ) ( ) ( ) x f x x f if x xm , ( ) x m m susy breaking parameter 3-brane tesnion m x 3
VOLKOV-AKULOV ACTION volume of 4d surface in superspace 2 = mnpq = = + + f 2 4 2 m n p q m m det x ( ( )) ( . .) ... S E E E E f d E x f i c c 0 0 0 0 0 VA n m ! 4 M M 4 + 4 = = + = 2 2 m m m m n m n m m n m ( ) [ ( )] ( ( )) E dx if d d dx if dx E x 0 0 n n n Constrained superfield realization of the VA goldstino (Ivanov, Kapustnikov 77; Rocek 78 , ,Samuel, Wess 83; Casalbuoni et al. 89 , , Komargodski, Seiberg 09 ,..) Example: = = 2 = + + 2 0 S s ( , ) ( ) ( ) ( ) S x s x x F x 1 chiral superfield constrained by: L L L L 2 F ( ) = + + = + + + + 8 4 4 m S S . . ( ) S d z S f d z d c c d x i f F F F F NS L m F F From general theory of non-linear realizations it follows (Ivanov & Kapustnikov 77) that the VA model is universal: all models of spontaneous susy breaking involving goldstino should be related to the VA model by a non-linear transformation of whose general form was obtained only quite ricently Kuzenko & Tyler 10 -11 (x ) 4
COUPLING VA GOLDSTINO TO N=1, D=4 SUGRA, SUPER-BEHIGGS EFFECT AND DE SITTER VACUA Volkov -Soroka model 73 Deser, Zumino 77, , Lindstrom, Rocek 79; Ferrara et al; Samuel, Wess 83; Ivanov, Kapustnikov 84 - 90, 2014-15 Antoniadis, Dudas, Ferrara, Kehagias, Farakos, Kallosh, Linde, Porrati, Sagnotti, dall Agata, Zwirner: Bergshoeff, Freedman, Kallosh, Van Proeyen; Hasegawa,Yamada;Kuzenko; Antoniadis, Markou; , Schillo, van der Woerd, Wrase Recent constructions are mainly based on the superconformal approach to supergravity (see Freedman and Van Proeyen Book for a review) Alternatively, one can use Poincar superspace and superfield forlmalism (e.g. Samuel, Wess 83) Sugra coupled to scalar (z), vector V(z), ? (z), and goldstino S(z) superfields ( ) 2 V ( , , , ) K e S S E = + + + 8 6 2 A M tr W W Ber ( , ) g( SYM kin.term , ) S d z E e d z W S S S 3 m 3 NS L 2 2 4 2 Kahler potential chiral measure Lagrange multiplier superpotential = M m = = ( , , ) A M A A A B A z x ( ), E dz E z T dE E M B - sugra constraints ab = = ( + a a a b a a b , ) ( ) ( ) T iE E T iE E G z iE E R z E E T 1 ab a 2 5
COUPLING VA GOLDSTINO TO N=1, D=4 SUGRA, SUPER-BEHIGGS EFFECT AND DE SITTER VACUA component pure supergravity + VA goldstino to the second order in (upon integrating out the auxiliary fields) (Bergshoeff at. all; Hasegawa,Yamada 15) AdS sugra action (Townsend 77) ( ( ( ) = + l m + a + 4 2 2 2 lmn ab ( , ) 6 2 S d x e e m m f 1 + R n b 2 2 ) if + + m + 4 a a 2 . d x e c c i a a 2 ) + b + + O + 2 4 4 8 abc f . ( ,..., ) d x e c c 3 a c 2 2 = 0 In the unitary gauge the remaining first line in the action describes a massive gravitino field coupled to gravity with the cosmological constant 2 3 m = 2 f 2 6
VOLKOV-AKULOV BRANE COUPLED TO SUPERGRAVITY For finding a more direct relation of 4D effective theories with spontaneously broken local susy to 10D stringy constructions which use anti-D3-branes to induce a de Sitter vacuum, it may be useful to couple to supergravity the original Volkov-Akulov 3-brane model, without resorting to the constrained superfields The action has a suggestive geometric form of the sum of three different volumes in N=1, D=4 curved superspace E = + + det 8 6 2 4 A M a i M Ber ( ( )) S d z E d z f d z 3 m VA + SG L 2 2 4 2 pullback on 3-brane worldvolume = + i + i a i M m a m a a E ( ( )) ( ) ( , , ) ( ) ( , , ) ( ) ( , , ) z x E x E x E x i = = 1 m i m i ( ) , ( ) ( ) x f x in the static gauge for worldvolume diffeomorphisms: a i zM E ( ( )) x All the couplings of VA goldstino to sugra fields is encoded in . In the WZ gauge we obtain: = 2 ( + a + + + + + [ ] a M m a m a c a c b a c a ( ) ( ) ( ) . .) 2 ( ( )) ( ( )) E z dx e x i x i c c e G e G x e R R x b ( ) . . c c + + 3 4 ab abcd ( ) ( ) ( ) O O d T T i i b cd 6 2 ~ ~ m n ) [ ) = ( ] ( c 2 2 T e e i G i G i R [ ] [ ] [ ] ab a b m n a b c a b a b 7
VOLKOV-AKULOV BRANE COUPLED TO SUPERGRAVITY Integrating out the auxiliary fields R(x) and ??(x) from the action we get ( R ) = + l m + a + 4 2 2 2 lmn ab ( , ) 6 2 S d x e e m m f 1 + 3 SG br n b 2 2 ( ( ) + if + + m + 4 a a 2 . d x e c c i a a 2 ) differ from constrained superfield action, but should be related by non-linear field redefinitions a la Kuzenko & Tyler + b + + O + 2 4 4 8 abc f . ( ,..., ) d x e c c 3 a c 2 2 8
VOLKOV-SOROKA MODEL OF THE SUPER-HIGGS EFFECT o in 1973 Volkov & Soroka gauged non-linearly realized super-Poincar group Introduced gauge fields associated with local super-Poincar generators local super-Poincar transform: G 1 G+ G 1dG(x) ( 2 ) ( ) ( ) ( i x a x e + = = a m a m a ( ) ( ) ( ) e x dx = e x a x = + a + b a a a b a ) ( ) e l x m ( ) ( ) ( ) x dx x x ( ) ( ) ( ) ( ) x x l x m = ab m ab m ab ( ) ( ) ( ) x dx x l x = ab ab ( ) ( ) ( ) - does not transform under susy x l x Constructed translation and susy invariant objects with the use of Stueckelberg-like fields: = + + a + a a a a ( ) ( ) x ( ) ( ) 2 ( i . ) E x e = = x X X x c c generalization of VA one-form + ( ) ( ) ( ) ( ) x x = + a a a a ( ) ( ), ( ) ( ) ( ) x x x a x i Volkov-Soroka action: = de Sitter sugra action ( ) = + + + + 4 2 lmn ab det ( ) ( ) . . c c S d x E R c m 1 VS l m n a b 2 2 - is independent field constant c c can be absorbed by c = 9
GENUINE N=1, D=4 SUPERGRAVITY Freedman, van Nieuwenhuizen & Ferrara 76 Deser & Zumino 76 ( ) = + ) + 4 lmn det ( ) ( . . c c S d x e R 1 SG l m n 2 2 SYSY transformations: = a a a ( ) 2 ( ) e x i = ( ) ( ) x x ab = n + n + [ m ] abnp a b cnp ( ) . . c c x i ie m m p c p The constructed models of local susy breaking demonstrate that coupling of N=1 D=4 supergravity to Volkov-Akulov goldstino in one way or another results (in the unitary gauge) in the Volkov-Soroka action + = ( | ) S S S =0 SG VA VS 10
CONCLUSION We have reviewed models of matter coupled N=1, D=4 supergravity with spontaneously broken supersymmetry triggered by the presence of VA goldstini. Spontaneous SUSY breaking generates a positive contribution to the vacuum energy thus allowing for the appearance of de Sitter vacua in the theory. This makes these supergravity models useful and relevant for cosmological and inflationary model building The realization of the local susy breaking with the use o the space-filling Volkov-Akulov 3-brane coupled to gravity and matter supermultiplets may by useful for finding a more direct connection of these 4d effective field theories to string theory constructions involving anti-D3-branes, like the KKLT model and its refinings ( ) 2 V ( , ) K e E = + + 8 6 tr W W Ber ( ) g( ) S d z E e d z W 3 m 3 + VA + matter SG L 2 2 4 2 F det + 2 4 a i M ( , , ( )) ( ( )) f d V z 11
