Dimensional Reduction, Regularisation

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“Loops and Legs in Quantum FieldApril Theory", 15-20, 11th 2012 DESY Workshop on ElementaryWernigerode, Germany Particle Physics Tim Jones Dimensional Reduction (and all that) Liverpool University E-mail: PoS(LL2012)011 di- and (1.2) (1.1) (1.4) SQCD . Tim Jones DRED DR case appears 4. It is useful < D ) as a variation of 0 is often termed e σ DRED cannot be used in supersym- ( W ]. A pedagogical introduction . As a consequence, for exam- d MS σ 4 W non-supersymmetric 0 ],[ c σ 3 DREG DREG c W b σ Λ b ) i j W a D x W Λ ( − ade β µν c σ f ˜ 4 abc g . ψ Λ W ) b , = + σ abc g f 0 νρ νρ ) αβ f , ˜ ˆ j 0 (1.3) ε g D g 4 ) W ) + ) 2 µν i j x a 2 0 ˆ 0 ( a g subtraction method g g , i R ======, abc i i ( Λ i p W x -scalars. Consequently the interactions = 4, and renormalisation is effected by subtracting these ρ ρ ρ g f ig ∂ ( ( = ( ε µν µν µν ν ν ν − ˜ ˆ ˜ ˜ ˆ g g g g g g µν , the UV divergences of the four-dimensional theory are and -dimensional indices respectively, and = = = ≡ ≡ ≡ µν DREG g D indices whose only non-vanishing components are in the µν µν µν µν µν µν g D a a µ µ µ α σ i ˜ ˆ ˆ g g g g g g a x p σ W W W ··· ], to maintain the equality of Bose-Fermi degrees of freedom DREG W δ δ δψ ν 2 , ψ [ a µ R amounts to defining σ is used. Which is not to say that regularisation by dimensional reduction ψγ DREG g , DRED ambiguities to be discussed below) see [ 4. Just as in DREG = version of the familiar are 4-dimensional and DRED j transform as scalars, called , ] introduced i a σ 1 DRED W ]. Essentially and 5 (modulo µ A (Dirac) fermion represents 4 degrees of freedom as long as we define the Dirac matrix So we define The dimensionally reduced form of the gauge transformations: Siegel [ -dimensional subspace; in particular, ˆ show that trace to satisfy Tr1 manifested by the occurrence of poles in poles. The where to define hatted quantities with D where Dimensional Reduction (and all that) 1. Introduction characteristic of supersymmetry, an equality notple, preserved by the relationship between theis quark-quark-gluon not and preserved quark-squark-gluino when couplings in metric theories; but it lacks convenience.DREG For a formal discussion ofto the Siegel’s proposal, equivalence and of a first discussionin of its Ref. application [ to the mensional regularisation PoS(LL2012)011 . ) N ( DRED SU Tim Jones in are equiv- ]. abcd 7 K DREG ], [ 6 and . DRED bce cde bde f f f ade ace abe , d d d ) ) 3 6 = = = K K in supersymmetric theories, where su- 9 7 8 + + K K K 2 5 K K ( ( -scalar. . 1 1 2 2 except bde ) cde bde ε 2 d for example: a basis for tensors d d = = H ) 2 4 3 ace abe ade N + H H d d d ( 1 = = = H SU , , ( -scalar to the gauge boson, which is of course protected 4 5 6 1 4 1 3 ε K K K K K 2 2 1 1 = 4 bc cd bd = = H δ δ δ 3 1 + ab ac ad H H 3 -scalars is itself gauge invariant. Since it is not forbidden, such a δ δ δ -scalar, and both its Yukawa coupling and the quartic interaction ε H ε = = = 3 1 2 with gauge group K K K ponderous in non-susy theories. Nevertheless, for the QCD since then ) mass 3 DRED ( SU -scalar interactions remain in step with the corresponding gauge interactions, and its ε -scalars a natural basis is ε renormalise differently from the gaugegenerated. coupling. New quartic group theory structures are Un-supersymmetric theories: Again a mass for the Softly-broken supersymmetric theories: Radiative corrections generate a mass for the Supersymmetric Theories: The mass remains zero. -scalar quartic couplings ε This makes Moreover, a Let us consider We have three classes of theories which behave differently under renormalisation using • • • are both gauge invariant by themselves. This means that in general they will not renormalise like reducible in So for alent. For some loop calculations where the above interactions arise, see Refs. [ mass will therefore be generated by loop corrections Dimensional Reduction (and all that) the corresponding gauge interactions; and in thestructures case will of be the quartic generated coupling, by different radiative group corrections, theory and require subtraction. persymmetry transformations connect the 1.2 from developing a mass by gauge invariance. 1.1 Evanescent Couplings and Masses is given by PoS(LL2012)011 ) 4 σ > ) no ] by (1.7) (1.8) (1.6) (1.5) 1.14 (1.13) (1.14) (1.12) D ) even 11 ), ( 1.12 Tim Jones 1.10 1.6 ) is not true 1.5 ) and Eq. ( 1.6 4, it would seem one can ··· ··· ] it was proposed that the + + < , in which Eqs. ( 9 S D 0 (1.9) σδ σδ 4 ), one can use the basic relation ˆ g g Q = -parameter can then be absorbed ργ ργ c 1.3 ˆ . D g g 0 < ) does not hold and so Eq. ( να να µνρσ ˆ = αβγδ g g ˆ ε σ ε αβγδ µρ ) ˆ ε ˆ 4 space, µβ µβ g σ 6 1.11 < ˆ σδ ) ˆ g g γ ˆ = g ε 00 (1.10) (1.11) + ρ c ˆ − − is problematic. Of course Eq. ( γ αβγδ ργ D = = ν ˆ ˆ ε g + 3 ˆ γ σδ σδ

ˆ 1 µ g g 5 5 for 4 νβ 4 − (effectively by analytical continuation from ˆ ˆ γ γ γ µνρσ g 2 ργ ργ 5 , , ˆ µνρσ ε quasi D ˆ = ( g g γ D , µ µ ˆ ε µα ˆ γ γ 0 ˆ g )( νβ νβ νρ = ˆ 4 ˆ Tr g g ε DRED = = ) ν − 4 µα µα µ 5 ˆ ˆ g g ε D γ − µνρσ in this manner, and the problem does not arise. , )( µνρσ = = A σ D . 1 ˆ ( γ ε + derivation of the anomaly. In Ref. [ µνρσ αβγδ αβγδ ˆ D ε 4 space, from a ˆ ( DRED ε ε 4 the definition of = 4, there is no problem: Eq. ( < DREG D and µνρσ µνρσ > D ˆ ε ε D ] drew attention to the following issue. Given 8 DREG ) simply be used also in 4, then < as 1.13 is the usual 4-dimensional tensor. Then, using Eq. ( ambiguities ] the relation if D µνρσ 10 4). This proposal was successful in that the authors then found that the Adler-Bardeen αβγδ 4 so one cannot define ˆ ε ε < 5 Siegel himself[ Once again, for To avoid all ambiguities we must avoid assuming relations like Eq. ( From > γ DRED D D to show that and hence by consideration of that This indicates that for giving an unambiguous can be used instead without ambiguity; the dependence on the into redefinitions of the renormalised metric and torsion. Stöckinger has formalised this[ distinguishing “normal” are not true. By starting with our theory defined in such a space one avoids the problem. longer follows. In that case you can impose relation Eq. ( though they are truemodels[ in “normal" four-dimensional space. For example, in two dimensional if to theorem held, in other words thattwo the loops two-loop in corrections both to the axial anomaly summed to zero at 1.4 we have, and hence that Dimensional Reduction (and all that) 1.3 define where PoS(LL2012)011 j i ) a . (2.1) (2.6) (2.3) (2.2) (2.5) (2.4) R k a φ j R φ Tim Jones i 2 theories φ = ( j = i jk i ) Y N 6 1 R calculated using ( γ C = , in that there exists , . W aa and δ ]

2 g ] 1 at three and four loops DRED ) . 2 = β ) R

r . = ( ] R , 2 # ( and C that the redefinition exists; in N ) [ C R ) [ bcd 1 ( Tr f R − )] Tr ( C ) Q for [ R 2 6 Q C ( acd DRED knp g ]. 6 π f Tr Y 2 g Q m PC 15 6 16 [ knp 6 p = + DRED ( NSVZ g scheme of the the chiral supermultiplet, ) Y g non-trivial 2 − 4 γ β ], [ Tr 4 m ab R g 4 X 3 p ( ) δ X g 14 ) − ) of the gauge group, which has structure con- Tr C − G 1 ] l 3 R 1 ( + 3 )] a G 2 n − ( NSVZ X − ( ) ) r 3 C X R R r C 4 1 2 2 ( R R 2 l X C 5 2 ( ( n 2 6 , C + -functions. Exploiting the fact that − − ) − 2 P C − 1 b ) + 1 1 β PC P 2 X [ [ R 1 Q X klm klm 2 a π X " scheme in that it preserves (under renormalisation) cou- Tr Tr − Y Y R results at three loops. However, there is an analytic re- 3 2 4 4 2 2 ( 16 g g g g g ( 3 g 1 π : g Tr 1 g g − − 1 = = = = − r 16 β ] which differs from both − r = DRED = and the cubic part of the superpotential is 1 2 3 4 r NSVZ = 13 = j X X X X = g ab i = ) ],[ δ g δ 1 supersymmetric gauge theories: ) R ( which connects them. It is 12 R = NSVZ δβ NSVZ g ) ( 3 C ) 3 β ) 2 DRED N ( T g ) Y 2 g connection , 3 , β ( 2 π ) in g g ( β g G 0 − 16 ( β g ( C jkl 3 -function 1 theory by (comparatively) simple calculations. Subsequently, special cases of DRED is anomalous dimension in the Y → β scheme is like the − ikl g = ↔ ) , Y N R g 2 1 NSVZ ( γ T . = NSVZ NSVZ NSVZ j i = abc begin to deviate from the P f Q In the abelian case (for simplicity) There is a third scheme [ The The coupling constant redefinition linking the two schemes is uniquely determined up to an and generates just the right shift in Here and Here and where DRED definition of the abelian case for example,terms the (with redefinition different consists tensor of structure) a in single the term, but it affects four distinct which transforms according to thestants representation pling constant relations that are a consequence of supersymmetry; but are finite beyond one loopfor it a was general possible to determine these results were confirmed by the Karlsruhe group [ overall constant: 2.1 The Dimensional Reduction (and all that) 2. The a particular form for where PoS(LL2012)011 ) and (3.1) (3.2) (3.3) 2 ) DRED Tim Jones ··· or quantity in + d as arbitrary as F c DREG F not b physical F f a N F f 2 i 3 c Tr N 4 N ( i 2 2 t t 2 2 2 c − µ µ m m N , c 7 8 N )) . g 3 3ln 3ln i ( − 3 3 c calculation (using ζ c π , − − α 5 3 N N 5 4 12 g + 21 66 + 2 ) ) QCD 1 c − − h µ µ N f 42 3 ) c ( ( 6 N µ N π π ( − i ]: − 3 3 f f 2 9 -functions beyond one loop is , DRED DREG 132 3 3 16 20 N N 3 6 − h β − α α 2 i i c g ) DRED 2 t c c . c 2 3 N ) + 2 N ( 9 N 3 + + m c N f 3 ζ g 4 1 1 N − − N 36 3 4 c 2 c c c − ) = ) ) 2 N c − + N N N 3 µ µ µ 3 N f 4 ( ( ( − 21 44 N 102 h h h h DREG t DREG DRED t t = = − = = ( + + , thus rendering the theory supersymmetric. : m m m ) ) ) ) a 2 3 1 4 ( ( ( ( F g g g g = = β β β β 2 3 4 2 -function ) ) ) DRED π pole pole t t 2 2 2 β π π π m m 16 found in the corresponding 4-loop 16 16 16 2 ( ( ( ) : ··· + example. In d DREG R c DRED SQCD R QCD b R a This has been calculated through four loops[ Thus to establish the relation between the schemes we must calculate a Note that the higher order group theory invariants of the form The lesson of this result is that the form of A R is replaced by the adjoint, a Tr from which we can deduce that ( do not appear here; andR indeed they cancel in those calculations when the fermion representation both schemes. 4. The whereas in Dimensional Reduction (and all that) generally believed; the set of renormalisation schemesdoes which not is permit spanned arbitrary by assignments parameter of redefinitions coefficients of all terms. 3. Physical quantities and the schemes PoS(LL2012)011 - β (5.2) (5.1) (5.3) ]. 19 Tim Jones γ extend to the QCD are given by: . g ∂ ) not b ∂ G ]: X j ( 1 . The same results hold in the i , -functions to all orders. C − 22 γ + ∗ ) β 2 )). O ··· ··· π -parameters. + cc = + DRED MM 2.1 A 2 i j l 16 2 ) , i + ( ˜ − l m mass terms ) m 2 jk ) ) j ) g Y Γ 2 )] ∂ ··· Y g ) ( Y i µ ∂ φ R ( , ( l i i ( G ( l ˜ + g Y Γ ( C ( 2 Γ C , 2 i mass terms of genuine particles under renor- 2 C -functions for the soft parameters correspond- ˜ 2 scheme can be extended to all orders [ m B − 2 ∗ i 0 − m β g + [ − g l i D Y β ) − l 2 ∑ ∂ 7 ) φφ ∂ Tr j jk g 1 ∂ , and the + 1 is a bit more tricky [ + h DRED b h ∂ h 2 i i i | − 2 2 DRED O ( ( 2 l l 2 i l r − ˜ m γ γ 2 ) g m m ]. I will return to this issue in the discussion of soft i ) 2 m 2 2 ) ) | ] that the 3 is free of such structures to all orders (manifestly so for g Y π 2 linkage described above does ∂ = = = Y 17 , = g , ∂ M m g 0 . The | 21 16 g i j 2 ( ( M b 2 2 ( i jk i h 2 β jk β m A ], [ C β ( + − l Mg SQCD interaction NSVZ DRED ∗ 20 Y | = = 3 = mass term 2 i 2 2 i φ then being the anomalous dimension in that scheme. ↔ = ∗ = ˜ m for m m OO γ β β 2 i jk O -function φφ ˜ Y , the β NSVZ DREG = DRED M X g 2 β ↔ m , and β ∗ -scalar mass mixes with the ) is independent of ˜ ε , we can prove [ 0 2 i denotes terms involving gaugino masses and DRED lmn m -functions Y the ]: β β ··· 18 = ( DRED + and soft breaking -function for the is the chiral supermultiplet anomalous dimension in β is known through three loops, but there is no all orders form for it. j i lmn 0 DRED in the absence of chiral superfields, of course, see Eq. ( γ Y scheme, with of course By an analytic redefinition of the form The These new terms in QCD cannot be removed by analytic coupling constant redefinitions; it It is possible that Using Here In DRED -functions. NSVZ DRED g we can make where the NSVZ X Thus the underlying supersymmetric theory determines these soft 5.2 The soft scalar mass where where follows that the function ansatz of Ryttovβ and Sannino[ 5. malisation [ ing to the gaugino mass Dimensional Reduction (and all that) 5.1 The soft β PoS(LL2012)011 and (6.5) (6.2) (6.3) (6.4) (6.1) DREG value in Tim Jones DRED practical   ) G ( ) C 2 1 ) − ) ) 1 − Q 1 ) R ) are simultaneously valid. − ( − 2 ) ) T 2 π f 2 6.2 scheme. Moreover, it is unclear to π N π 16 ) 2 ( 16 R 16 2 RS ( ( ]. + is universally adopted for calculations ( g 2 -function: 2 1 ) T 1 g 23 f g β ) , G ) ( N 1 G G ( m − C ) and Eq. ( ( g γ ) DRED 2 g C β 2 2 3 C QCD 2 2 6.1 − π − employed in the softly-broken case. For a recent − 1 − 8 0 16 ( O schemes, and relate the results to each other and to 1 Q ( 1 ) beyond one loop. / ( 2 ( 2 ) ) -function for the gaugino mass; a soft breaking term. / g 2 = 6.3 2 ) G ) β G ( π ( π 2 G DRED M G ( C DREG ( π C β 16 3 16 2 scheme, see Ref. [ C C g g 2 16 2 scheme by the formula 3 6 g and − MS − 6 − attempts to extend this advantage to encompass supersymmetry 1 1 susy) they equate this to − = =   was a remarkable achievement, and of crucial NSVZ = = 2 DRED 3 π m N deduction Eq. ( g DRED γ is precisely the NSVZ g NSVZ m 16 γ β m QCD DREG RS γ = RS g β equal to the ] presents the following ansatz for the anzatz for scheme; but I don’t know how to calculate in the 17 is the fermion mass anomalous dimension. In the special case of a single fermion adjoint not RS m (1979) 193. γ and also the matching to the 84 The introduction of But in this case the So: I can calculate in the Ref. [ NSVZ [1] W. Siegel, “Supersymmetric Dimensional Regularization via Dimensional Reduction,” Phys. Lett. B beyond one loop in supersymmetry; with making loop calculations inwith non-abelian other gauge regulators theories such as much explicitpreserves simpler gauge UV invariance; to cut-offs. organise An important thanas part they of well. are this is In the spitedescribed fact of above, that the the practical difficulties advantages with mean defining that the method inexample relevant a to rigorous LHC manner physics, with thatDREG a we careful discussion have of the relationship between which is me why there should exist a scheme in which Eq. ( 7. Summary References Consequently it is given in the the to deduce that then i.e. Dimensional Reduction (and all that) 6. The where multiplet (corresponding to PoS(LL2012)011 (2008) (1980) (1994) Tim Jones 78 94 333 (1980) 479. 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Yamada, “Decoupling I. of Jack the and epsilon D. scalar R. T. Jones, “The Gaugino Beta function,” Phys. Lett. B [17] T. A. Ryttov and F. Sannino, “Supersymmetry inspired QCD beta function,” Phys. Rev. D [16] I. Jack, D. R. T. Jones and A. Pickering, “The Connection between DRED and NSVZ,” Phys. Lett. B [13] V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, “Instanton[14] Effects in R. Harlander, P. Kant, L. Mihaila and M. Steinhauser, “Dimensional Reduction applied[15] to QCD R. at V. Harlander, L. Mihaila and M. Steinhauser, “The SUSY-QCD beta function to three loops,” Eur. [11] D. Stockinger, “Regularization by dimensional reduction: consistency, quantum action principle, and [12] D. R. T. Jones, “More On The Axial Anomaly In Supersymmetric Yang-mills Theory,” Phys. Lett. B Dimensional Reduction (and all that) [10] R. W. Allen and D. R. T. Jones, “The N=1 Supersymmetric Sigma Model With Torsion: The Two PoS(LL2012)011 Tim Jones 432 10 (1998) 73 [hep-ph/9712542]. 426 Phys. Lett. B (1998) 114 [hep-ph/9803405]. Higgs boson production in gluon fusion,” arXiv:1208.1588 [hep-ph]. Dimensional Reduction (and all that) [21] I. Jack, D. R. T. Jones and A. Pickering, “Renormalization invariance and the soft Beta functions,” [23] A. Pak, M. Steinhauser and N. Zerf, “Supersymmetric next-to-next-to-leading order corrections to [22] I. Jack, D. R. T. Jones and A. Pickering, “The soft scalar mass beta function,” Phys. Lett. B